Fluid model for charged particle transport in

UNIVERSIDADE TÉCNICA DE LISBOA INSTITUTO SUPERIOR TÉCNICO

0.15

0.10

0.05

3.2

0.0

2.4

1.6

1.6

3.2 0.8

4.8 6.4

0.0

Fluid model for charged particle transport in capacitively coupled radio-frequency discharges Aurel Salaba¸s (Mestre)

Dissertação para obtenção do Grau de Doutor em Física

Orientador: Doutor Luís Paulo da Mota Capitão Lemos Alves Co-Orientador: Doutor Carlos Renato de Almeida Matos Ferreira Júri: Presidente: Reitor da Universidade Técnica de Lisboa Vogais: Doutor Carlos Renato de Almeida Matos Ferreira Doutor Gérard Gousset Doutor Jacques Jolly Doutor Luís Paulo da Mota Capitão Lemos Alves Doutor João Pedro Saraiva Bizarro

Outubro de 2003

To the memory of acupuncturist Ioan Ladea.

i M ODELO

FLUIDO PARA O TRANSPORTE DE PARTÍCULAS CARREGADAS EM DESCARGAS CAPACITIVAS DE RÁDIO - FREQUÊNCIA

Nome: Aurel Salaba¸s Doutoramento em Física Orientador: Professor Luís Paulo da Mota Capitão Lemos Alves Provas concluídas em: Resumo Neste trabalho desenvolve-se um modelo fluido auto-consistente, que descreve o transporte das partículas carregadas em descargas capacitivas de rádio-frequência. O modelo resolve as equações de continuidade e de transporte do momento para electrões e iões, as equações de transporte da energia média electrónica e a equação de Poisson para o potencial de plasma. Este sistema de equações fluido adopta como condição de fecho a aproximação da energia média local, que introduz uma dependência espacio-temporal nos parâmetros electrónicos através do perfil da energia média. A formulação coerente adoptada no presente trabalho melhora os modelos tipo fluido deste domínio. Em particular, as equações de transporte dos electrões são directamente obtidas calculando os momentos dum desenvolvimento a dois termos da equação de Boltzmann electrónica, o qual é também utilizado na aproximação da energia média local. O mesmo quadro de aproximações é também utilizado na dedução das condições fronteira para o fluxo electrónico. A equação iónica de transporte do momento escreve-se incluindo o seu termo não linear de inércia, através duma generalização do conceito de campo eléctrico efectivo. Actualiza-se também a cinética colisional da fase gasosa relativamente a trabalhos anteriores. O modelo aplica-se a uma configuração cilíndrica de eléctrodos pararelos, produzindo descargas em hélio, hidrogénio e misturas de silano-hidrogénio, a uma frequência fixa (13.56 MHz), para uma larga gama de valores de pressão e tensões aplicadas. O modelo é testado em relação a simulações anteriores e a medidas experimentais. Os resultados do modelo permitem em geral melhorar as previsões dos valores de parâmetros eléctricos da descarga.

Palavras-chave: Modelização de plasmas, Modelo fluido, Transporte de partículas carregadas, Descarga de rádio-frequência, Hidrogénio, Silano.

ii F LUID

MODEL FOR CHARGED PARTICLE TRANSPORT IN CAPACITIVELY COUPLED RADIO - FREQUENCY DISCHARGES

Abstract In the present work, a two-dimensional self-consistent fluid model is developed to describe the charged particle transport in capacitively coupled radio-frequency discharges. The model solves the continuity and momentum transfer equations for electrons and ions, coupled with the transport equations for the electron mean energy and Poissont’s equation for the plasma potential. The set of fluid equations is closed using the local mean energy approximation, which introduces a space-time dependence for the electron parameters via the mean energy profile. The coherent formulation adopted here improves earlier fluid models. In particular, the electron transport equations are directly obtained from the moments of the two-term development of the electron Boltzmann equation, which is also used in the local mean energy approximation. The same approximation framework is also employed in deducing the electron flux boundary conditions. The ion momentum transfer equation is written here including its non-linear inertia term, by generalizing the concept of effective electric field. The gas phase kinetics is also updated with respect to previous works. The model is applied to a cylindrical parallel plate configuration, producing discharges in helium, hydrogen and silane-hydrogen mixtures, at fixed frequency (13.56 MHz), for a wide range of pressures and applied voltages. The model validation is made against previous calculations and experimental measurements. The reported results generally improve the predicted values of the discharge electrical parameters.

Key words: Plasma modeling, Fluid model, Charged particle transport, Radio-frequency discharge, Hydrogen, Silane.

iii

Acknowledgments A thesis is seldom the work of a single person in isolation and, normally, it is the result of a research activity involving a bunch of people. I am pleased now to acknowledge those who have, in various way, contributed. I express my sincere thanks to Prof. C.M. Ferreira and Prof. G. Musa for inspiring the fascination of theoretical and experimental physics in myself. True to the form of great teachers they are the desired example for any student. Thanks goes to my supervisor, Prof. L.L. Alves, who first introduced me to plasma modeling. I have profited greatly from many discussions and I am grateful for his tremendous involvement in each stage of this work. Many thanks are due for the arduous task of reading the manuscript. I would like to thank Prof. G. Gousset for enlightening discussions and for the constant encouragement he has given to me. Special thanks goes to Vasco. His deep and romantic fascination with science has shown me that research can be so entertaining. His everyday company and friendship, together with that of the mathematicians group, is one of the best memories of these years. Thanks to you Cedric for discussions and remarks. It has been a pleasure to receive your help. Aikido lessons and entertaining company in Bairro Alto evenings are well appreciated. I am pleased to thank to all the other members of EG group, Prof. J. Loureiro, Prof. E. Tatarova, Prof. M. Pinheiro, M. Dias, J. Henriques and I. Dionisio for the excellent and friendly working atmosphere at Centro de Física de Plasmas. Thanks to Ilda, and Anabela for helping me time and again. I take the opportunity to thank Rosa. She put a roof over my head when I arrived in Lisbon and I felt at home ever since. Thanks to Joca for his exceptional friendship. Aurora and Patxi, I thank you too. Special thanks goes to Loric˘a. Understanding and supportive you contributed significantly to my “bien-être” during the last years. Be now ready for the Alps! Thanks to my family and to my friends for their tender encouragement. I am grateful to the Portuguese MCT for the fellowship PRAXIS XXI, BD/15716/98. In addition, the Gulbenkian foundation has contributed to the financial support of conferences abroad. Aurel Salaba¸s, Lisbon, December 2002

Contents Resumo

i

Abstract

ii

Acknowledgement

iii

List of tables

viii

List of figures

xi

1 Introduction

1

1.1

General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Etching and deposition assisted by low temperature plasmas . . . . . . . . . .

2

1.3

A brief history on a-Si:H . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3.1

Amorphous materials . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3.2

a-Si:H researches . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.3.3

a-Si:H device technology . . . . . . . . . . . . . . . . . . . . . . . . .

6

Radio frequency discharges . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.4.1

The plasma and the sheath . . . . . . . . . . . . . . . . . . . . . . . .

9

1.4.2

Self-bias voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.4.3

Coupled electrical power . . . . . . . . . . . . . . . . . . . . . . . . .

13

1.4.4

α and γ regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1.4.5

γ’- powder regime . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

Discharge modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

1.5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

1.5.2

Statistical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

1.5.3

Fluid models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

1.5.4

Hybrid models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

Overview thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

1.4

1.5

1.6

iv

C ONTENTS

v

2 Theoretical considerations

23

2.1

General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.2

From Boltzmann equation to fluid equations . . . . . . . . . . . . . . . . . . .

26

2.2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

2.2.2

Derivation of moment equations from general the Boltzmann equation .

28

2.2.3

Derivation of electron moment equations from the two-term Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

Validity of drift-diffusion approximation . . . . . . . . . . . . . . . . .

36

Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.3.2

Electron transport equations . . . . . . . . . . . . . . . . . . . . . . .

39

2.3.3

Ion transport equation . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.3.4

Poisson’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

2.3.5

Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

Closure conditions for the moment equation system . . . . . . . . . . . . . . .

45

2.4.1

Kinetic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

2.4.2

Electron transport parameters . . . . . . . . . . . . . . . . . . . . . .

49

2.4.3

Ions transport parameters . . . . . . . . . . . . . . . . . . . . . . . . .

55

The discharge electrical parameters . . . . . . . . . . . . . . . . . . . . . . . .

56

2.5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

2.5.2

Self-bias voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

2.5.3

Coupled electrical power . . . . . . . . . . . . . . . . . . . . . . . . .

60

2.2.4 2.3

2.4

2.5

3 Numerics 3.1

61

General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

3.1.1

Computational mesh . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

3.1.2

Finite-difference method . . . . . . . . . . . . . . . . . . . . . . . . .

65

Advancing equations in time . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

3.2.1

Explicit, implicit and semi-implicit numerical schemes . . . . . . . . .

68

3.2.2

The Courant-Friedrichs-Lewy restriction . . . . . . . . . . . . . . . .

69

3.2.3

Crank-Nicholson algorithm . . . . . . . . . . . . . . . . . . . . . . .

71

Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

3.3.1

General aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

3.3.2

The Scharfetter-Gummel discretization scheme . . . . . . . . . . . . .

73

3.4

The solution to transport equations . . . . . . . . . . . . . . . . . . . . . . . .

75

3.5

Numerical treatment of Poisson’s equation . . . . . . . . . . . . . . . . . . . .

77

3.2

3.3

C ONTENTS

vi

3.6

Effective electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

3.7

The convergence criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

4 Simulation results for He discharges

86

4.1

General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

4.2

Electric field and charged-particle distributions . . . . . . . . . . . . . . . . .

88

4.3

Electron mean energy and ionization rate distribution . . . . . . . . . . . . . .

92

4.4

Discharge currents and particle fluxes . . . . . . . . . . . . . . . . . . . . . .

95

4.5

The effect of gas pressure and applied voltage . . . . . . . . . . . . . . . . . .

97

4.6

The influence of the ion inertia term and the effective electric field . . . . . . . 103

4.7

Comparison with experimental results . . . . . . . . . . . . . . . . . . . . . . 105

5 Simulation results for H2 and SiH4 -H2 discharges

108

5.1

General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.2

Simulations in pure hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2.1

General characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.2.2

Positive charge distribution . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2.3

Ionization and dissociation rates . . . . . . . . . . . . . . . . . . . . . 116

5.2.4

The electric field distribution. Field inversion and field reversal . . . . . 121

5.2.5

Comparison with experimental results. The influence of the ion inertia term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.3

5.2.6

Effects of reactor dimensions . . . . . . . . . . . . . . . . . . . . . . . 127

5.2.7

Similarity law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Simulations in SiH4 -H2 mixtures . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.3.1

Typical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.3.2

Effects of SiH4 dilution . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6 Conclusions

164

6.1

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.2

Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

A Appendix

169

A.1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

A.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

A.3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

References

173

C ONTENTS Index

vii 191

List of Tables 2.1

Mechanisms for charged particle production in helium.The label EBE means that the rate coefficient is calculated from the corresponding electron collision cross-section, using the EEDF obtained from the solution to the homogeneous EBE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

2.2

Mechanisms for charged particle production and destruction in hydrogen. . . .

47

2.3

Mechanisms for charged particle production and destruction in silane-hydrogen gas mixtures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

48

Tabulation of electron transport parameters and rate coefficients. Line after line, the results on this table were obtained by running a homogeneous Boltzmann equation solver for a specific (E/N )Boltz value. . . . . . . . . . . . . . . . . .

51

2.5

Low field mobilities for hydrogen ions in H2 . . . . . . . . . . . . . . . . . . .

56

4.1

Set of parameters used for simulations in helium. . . . . . . . . . . . . . . . .

87

4.2

Work conditions for helium discharge simulations. . . . . . . . . . . . . . . . 103

4.3

Electrical parameters, Vdc (in V), Weff (in W) and maximum time-average electron density ne (in 109 cm−3 ), for helium discharges operating in conditions of Table 4.2. Simulation results (A)-(C) were obtained assuming different ion transport conditions: (A), constant ion mobility and no effective electric field; (B), field-dependent mobility and effective electric field calculated without inertia term; (C), field-dependent mobility and effective electric field calculated with inertia term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.4

Comparison of computed and measured values of electrical parameters for helium discharges working conditions (C2b-He) and (C3a-He) (see Table 4.2). Simulation results (A)-(C) were obtained in this work, assuming the same ion transport conditions as in Table 4.3. Simulation results (C’) were obtained for the same conditions as (C), by updating Vdc at each time step. Simulation results (D) were obtained in a previous work using an earlier model version [Leroy 1996]. There is at least a 40% uncertainty in measured values of Vdc . . . . . . . 106

viii

L IST

OF TABLES

5.1

ix

Set of parameters used for simulations in pure hydrogen and silane-hydrogen mixtures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.2

Work conditions for pure hydrogen discharge simulations. . . . . . . . . . . . 111

5.3

Plasma and electric parameters for discharge simulations in helium and hydrogen at p = 1 Torr and Vrf = 217 V. . . . . . . . . . . . . . . . . . . . . . . . . 112

5.4

Calculated and measured values of electrical parameters for hydrogen discharges working in conditions (C1-H2 ) and (C2-H2 ) (see Table 5.2). Simulation results (A)-(D) were obtained in this work assuming different ion transport conditions: (A), constant ion mobility and no effective electric field; (B), fielddependent ion mobilities and effective electric field calculated without inertia term; (C), field-dependent ion mobilities and effective electric field calculated with inertia term; (C’), same conditions as (C), but updating the self-bias voltage after each time step; (D), same conditions as (C), but considering only the H+ 2 ion. Simulation results (E) were obtained in a previous work using an earlier model version [Leroy 1996]. There is at least a 40% uncertainty in measured values of Vdc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.5

Calculated values of the coupled electric power Weff , the space-average, rms

reduced electric field E rms /N and the maximum electron density value nmax , e

for ccrf hydrogen discharges operating at different applied voltages V rf and gas pressures p. In these simulations, the inter-electrode distance was kept constant at d = 3.2 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.6

Calculated values of the coupled electric power Weff , the space-average, rms

reduced electric field E rms /N and the maximum electron density value nmax , e for ccrf hydrogen discharges operating at inter-electrode distances d and gas

pressures p. In these simulations, the rf applied voltage was kept constant at Vrf = 350 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.7

Work conditions for discharge simulations in silane-hydrogen mixtures. . . . . 144

L IST

OF TABLES

5.8

x

Calculated and measured values of electrical parameters for silane-hydrogen discharges operating in conditions C1-C3 (see Table 5.7). Simulations results (A)-(D) were obtained in this work, assuming different ion transport conditions: + − (A), three types of ions (H+ 2 , SiH3 and SiH3 ), with constant mobilities and no

effective electric field; (B), same ions as in (A), with field-dependent mobilities and effective electric field calculated with inertia term; (C), six types of ions + + + − (H+ , H+ 2 , H3 , SiH2 , SiH3 and SiH3 ), with field-dependent mobilities and ef-

fective electric field calculated with inertia term; (D), same ions as in (C), with field-dependent mobilities and effective electric field calculated without inertia term. Simulation results (E) were obtained in previous works using an earlier model version [Leroy 1996; Leroy et al. 1998]. There is at least a 40% uncertainty in measured values of Vdc . . . . . . . . . . . . . . . . . . . . . . . . . . 154

List of Figures 1.1

Schematic cross section of a single junction p-i-n superstrate device, [Catalano 1991]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.2

Schematic representation of a capacitively coupled radio frequency discharge. .

8

1.3

Time-averaged electric field (1) and plasma potential (2), as a function of the axial distance z between the rf electrode (left) and the grounded electrode (right), for a typical ccrf discharge. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4

10

Typical electric-field (column a) and plasma-potential (column b) axial distributions between the rf electrode (left) and the grounded electrode (right) at different times during a rf period T . The electric field is in (Vcm2 ) and the plasma potential in (V). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.5

11

Steady-state voltages in a typical ccrf discharge: rf generator potential (1), rf electrode potential (2), potential difference between the plasma and the rf electrode (3). In this figure Vdc =-63.7 V and the applied rf voltage Vrf =217 V. . . .

1.6

13

Ignition potential of a ccrf discharge as a function of pressure. The discharge is in hydrogen at frequency f = 3 MHz and inter-electrode distance d = 2.64 cm. The left-hand branch up to the jump is the γ discharge regime and the righthand branch (solid curve) is the α discharge regime. Dashed curve plots the transition from α to γ regime. Source: [Levitskii 1957; Raizer 1991]. . . . . .

15

2.1

Schematic representation of the simulation domain and its boundaries. . . . . .

43

2.2

Particle (A) and energy (B) reduced electron diffusion coefficient in helium (a), in 7% SiH4 − 93% H2 mixture (b) and in hydrogen (c). Blue and red lines are obtained from Boltzmann and Maxwellian calculations, respectively. . . . . . .

2.3

53

Particle (A) and energy (B) reduced electron mobility coefficient in helium (a), in 7% SiH4 − 93% H2 mixture (b) and in hydrogen (c). Blue and red lines are

2.4

obtained from Boltzmann and Maxwellian calculations, respectively. . . . . . .

54

Schematic diagram of a ccrf reactor (a) and echivalent electric scheme (b). . . .

57

xi

L IST

xii

OF FIGURES

3.1

Schematic representation of the computational mesh. The vertical boundaries are determined by the reactor axis (at r0 = 0) and the lateral grid (at rnr = R). The horizontal boundaries are defined by the rf electrode (at z0 = 0) and the grounded electrode (at znz = d). . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

63

The staggered mesh. Scalar variables are calculated at mesh-node locations (marked with filled circles), while vector components are defined at halfdistance between nodes (empty squares and empty circles). . . . . . . . . . . .

65

3.3

The general computational flow chart of the model. . . . . . . . . . . . . . . .

67

3.4

The Courant-Friedrichs-Lewy condition imposes a restriction upon the integration time-step of an explicit scheme. When condition v∆t < ∆x is satisfied the algorithm is stable (a). Otherwise, the algorithm becomes unstable (b). . . . . .

3.5

69

Generic form of the matrix equation. The matrix A is tridiagonal, with the solid lines indicating nonzero coefficients, while the blank areas correspond to zero coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6

The matrix equation corresponding to Poisson’s equation.

76

The four off-

diagonals of the pentadiagonal matrix A are not contiguous and the blank areas indicate zero coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7

81

Discretization map for the effective-electric-field equation. The nine radial and axial half-points marked in red are used in the discretization of its inertia term. .

82

4.1

Schematic configuration of the rf PECVD reactor under study. . . . . . . . . .

87

4.2

Time evolution of the convergence error, for the following time-average parameters: electron density (1), ion density (2), electron mean energy (3), plasma potential (4), and self-bias voltage (5). The curves were obtained for a helium discharge operating at p = 1 Torr and Vrf = 217 V. . . . . . . . . . . . . . . .

4.3

88

Time-average, 2d steady-state profile of axial Ez /N (a) and radial E r /N (b) reduced electric field components, for a helium discharge operating in conditions of Fig. 4.2. Time evolution (during one rf cycle) of axial E z /N (at r = 0) (c) and radial E r /N (at z = d/2) (d) reduced electric field components, for the same conditions as before. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4

89

Time-average, 2d steady-state profile of the plasma potential V p (a), for a helium discharge operating in conditions of Fig. 4.2. Time evolution (during one rf cycle) of plasma potential (at r = 0) (b), for the same conditions as before. .

90

L IST

xiii

OF FIGURES

4.5

Time-average, 2d steady-state profiles of the electron density ne (a) and ion density np (b), for a helium discharge operating in conditions of Fig. 4.2. For one rf cycle, the corresponding time variations at r = 0 are plotted in (c) and (d), respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6

91

Time-average, 2d steady-state profile of the electron mean energy ε (a), for a helium discharge operating in conditions of Fig. 4.2. For one rf cycle, the spacetime variation of the mean energy density ne ε (in eVcm−3 ) at r = 0 is plotted in (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.7

93

Spatial contour plot of the time-average, steady-state, helium ionization rate (in cm−3 s−1 ) (a), for a discharge operating in conditions of Fig. 4.2. For one rf cycle, the space-time variation of helium ionization rate at r = 0 is plotted in (b). 94

4.8

(a) Time variation, during one rf cycle, of steady-state voltages for a helium discharge operating in conditions of Fig. 4.2: the rf applied potential (1); the potential at driven electrode (2); the potential difference between the plasma and the driven electrode (3). (b) Time variation, during one rf cycle, of the rf applied potential (1) and the discharge current (2), for a helium discharge operating in conditions of Fig. 4.2. . . . . . . . . . . . . . . . . . . . . . . . .

4.9

96

(a) Time variation, during one rf cycle, of discharge currents at the rf electrode (z = 0) for a helium discharge operating in conditions of Fig. 4.2: electron current (1); ion current (2); displacement current (3); total current (as a sum of the previous currents) (4). (b) Time variation, during one rf cycle, of discharge current components, calculated at z = 0 (curves 1-3) and z = d/2 (curves 1’3’), for a helium discharge operating in conditions of Fig. 4.2. The labels 1-3 (and the corresponding prime labels) are for the same current components as in (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

−2 −1

4.10 Space-time contour plots at r = 0 of the axial electron flux (in cm s ) (a), the axial energy density flux (in eVcm−2 s−1 ) (b), the axial ion flux (in cm−2 s−1 ) (c), during one rf cycle, and for a helium discharge operating in conditions of Fig. 4.2. The fluxes are positive when transport is oriented towards the grounded electrode and negative when transport is oriented towards the driven electrode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

4.11 Spatial contour plot of the time-average, steady state, electron density (in cm−3 ), for helium discharges operating at Vrf = 217 V and the following pressures: p = 0.5 Torr (a), p = 1.0 Torr (b), and p = 3.0 Torr (c). . . . . . . . . . 100

L IST

OF FIGURES

xiv

4.12 Axial profile (at r = 0) of the time-average axial reduced electric field Ez /N , for heliumm discharges operating at Vrf = 350 V and the following pressures: p = 0.5 Torr (1), p = 1.0 Torr (2), and p = 3.0 Torr (3). . . . . . . . . . . . . . 101 4.13 The self-bias voltage Vdc , as a function of the applied rf voltage (at p = 0.5 Torr) (a) and gas pressure (at Vrf = 217 V) (b), for helium discharges. Coupled electrical power Weff , as a function of the applied rf voltage (at p = 0.5 Torr) (c) and gas pressure (at Vrf = 217 V) (d), for a helium discharge. . . . . . . . . 102 4.14 Axial component of different reduced electric fields affecting the ion transport, at r = 0 and t = 8T /20, for a helium discharge operating in conditions of Fig. 4.2. The curves correspond to: Ez /N , calculated assuming a constant eff ion mobility and no effective electric field (1); EHe + ,z /N , calculated for a field-

dependent mobility, with and without the ion inertia term (2 and 3, respectively); Ez /N , for the same conditions as curves 2 and 3 (4 and 5, respectively). . . . . 105 5.1

(a) Typical time evolution of convergence errors (obtained for a hydrogen discharge operating at p = 0.5 Torr and Vrf = 217 V), for the following timeaverage parameters: electron density (1), H2+ , H3+ and H + ion densities (2-4), electron mean energy (5), plasma potential (6), and self-bias voltage (7). The time evolution of the self-bias voltage Vdc and the coupled electrical power Weff are shown in (b) and (c), respectively. . . . . . . . . . . . . . . . . . . . . . . 110

5.2

Axial profiles (at r = 0) of the time-average, steady-state, electron density ne , for a ccrf discharge operating at p = 1 Torr and Vrf = 217 V in helium (red curve) and hydrogen (black). . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.3

Spatial contour-plots of the time-average, steady-state, ionization rate (in cm3 s−1 ) of helium (a) and hydrogen (b), for a ccrf discharge operating in conditions of Figure 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.4

Time evolution of the electron conduction current (a), the ion conduction current (b), and the total discharge current (c), for hydrogen (black lines) and helium (red lines) discharges operating at p = 1 Torr and Vrf = 217 V. In (c), the time evolution, during one rf period, of the corresponding rf electrode potential is also plotted (dashed lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.5

Spatial contour plot of the time-average, steady-state, densities (in cm −3 ) of the + + following hydrogen ions: H+ 2 (a), H3 (b) and H (c). The results were obtained

for a hydrogen discharge operating at p = 0.5 Torr and Vrf = 200 V. . . . . . . 117

L IST

OF FIGURES

5.6

xv

Spatial contour plot of the time-average, steady-state, hydrogen ionization rate (in cm−3 s−1 ) (a), for a discharge operating in conditions of Fig. 5.5. For one rf cycle, the space-time variation of hydrogen ionization rate at r = 0 is plotted in (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.7

Space-time averaged hydrogen dissociation rate, as a function of gas pressure and for different applied voltages. . . . . . . . . . . . . . . . . . . . . . . . . 120

5.8

(a) Axial profile (at r = 0) of the axial reduced electric field Ez /N , at different times along the rf period. The results are for a hydrogen discharge operating at p = 0.5 Torr and Vrf = 200 V. (b) Axial profile components (at r = 0) of the axial reduced electric field Ez /N , at t = T /20. The results are for a hydrogen discharge operating at p = 0.5 Torr and (1) Vrf = 200 V; (2) Vrf = 400 V. . . . 122

5.9

(a) Time variation, during one rf cycle and for simulations (A) and (B), of the total current It (curves 1 and 3, respectively) and the total potential at the rf electrode Ut (curves 2 and 4, respectively), for a helium discharge operating at p = 0.5 Torr and Vrf = 217 V. (b) As in (a), but for a hydrogen discharge operating at same pressure and applied voltage conditions. . . . . . . . . . . . 125

5.10 Axial profile (at r = 0) of the time-average, steady-state plasma potential V p (a) and axial reduced electric field Ez /N (b), for hydrogen discharges operating in conditions of Fig. 5.5, at different values of the rf electrode radius R. . . . . 128 5.11 Spatial contour plot of the time-average, steady-state electron density (in cm −3 ), for hydrogen discharges operating in conditions of Fig. 5.5 and for the following values of the electrode radius: R = 3.2 cm (a), R = 6.4 cm (b) and R = 12.8 cm (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.12 Axial profile (at r = 0) of the time-average, steady-state electron density ne (a) and electron mean energy ε (b), for hydrogen discharges operating in conditions of Fig. 5.5 and for different values of the rf electrode radius. . . . . . . . . . . 131 5.13 Axial profile (at r = 0) of the time-average, steady-state plasma potential V p (a) and axial reduced electric field Ez /N (b), for hydrogen discharges operating in conditions of Fig. 5.5, at different values of the inter-electrode distance d. . . 133 5.14 Axial profile (at r = 0) of the time-average, steady-state electron density ne (a), electron mean energy ε (b), and energy density ne ε (c), for hydrogen discharges operating in conditions of Fig. 5.5, at different values of inter-electrode distance d.134 5.15 Spatial contour plot of the time-average, steady-state electron density (in cm −3 ), for hydrogen discharges operating in conditions of Fig. 5.5 and for the following values of inter-electrode distance: d = 2.4 cm (a), d = 3.2 cm (b) and d = 6.4 cm (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

L IST

OF FIGURES

xvi

5.16 Plot of the space-average, rms reduced electric field E rms /N , as a function of N d, for different values of the rf applied voltage Vrf (at d = 3.2 cm) (a), and

for different values of inter-electrode distance d (at Vrf = 350 V) (b). . . . . . . 139 5.17 Effective electrical power coupled to the plasma Weff , as a function of N d, for different values of the rf applied voltage Vrf (at d = 3.2 cm) (a), and for different values of inter-electrode distance d (at Vrf = 350 V) (b). . . . . . . . . . . . . 140

1 5.18 Similarity curve of E ? /N = ( E rms /N )(Weff )− 2 vs. N d, for hydrogen dis-

charges operating at dref = 3.2 cm for different Vrf values. . . . . . . . . . . . 142

5.19 Similarity curve of E ? /N vs. N Λ [Λ = d (dref /lref ) (d/l)−1 ], for hydrogen discharges operating at various rf applied voltages and inter-electrode distances. 143 5.20 Typical time evolution of the convergence error (obtained for a silane-hydrogen discharge operating in condition C3 ), for the following time-average parame+ − + + ters: electron density (1), SiH+ 3 , SiH2 and SiH3 ion densities (2-4), H3 , H2

and H+ ion densities (5-7), electron mean energy (8), plasma potential (9), and self-bias voltage (10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.21 Axial profile (at r = 0) of the time-average, steady-state negative nn (1) and positive np (2) charged particle densities (a), electron mean energy ε (b), plasma potential V p (c), and axial reduced electric field Ez /N (d), for silane-hydrogen discharges operating in condition C3. . . . . . . . . . . . . . . . . . . . . . . 146 5.22 Axial profile (at r = 0) of time-average, steady-state ion densities for a silanehydrogen discharge operating in condition C3 . Silane ions (a): SiH + 3 (solid − + − curve); SiH+ 2 (dashed); SiH3 (dotted). The SiH2 and SiH3 ion densities are + + multiplied by 10. Hydrogen ions (b): H+ 3 (solid curve); H2 (dashed); H (dot+ ted). The H+ 2 and H ion densities are multiplied by 3. . . . . . . . . . . . . . 147

5.23 Radial profile (at z ' 0) of time-average, steady-state ion densities for a silanehydrogen discharge operating in condition C3. Silane ions (a): SiH + 3 (solid + curve); SiH+ 2 (dashed). The SiH2 ion density is multiplied by 10. Hydrogen + + + + ions (b): H+ 3 (solid curve); H2 (dashed); H (dotted). The H2 and H ion

densities are multiplied by 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.24 Spatial contour plot of the time-average, steady-state, silane dissociation rate by electron impact (in cm−3 s−1 ), for silane-hydrogen discharges operating in conditions C1 (a), C2 (b) and C3 (c). . . . . . . . . . . . . . . . . . . . . . . . 150

L IST

xvii

OF FIGURES

5.25 Axial component of different reduced electric fields, at r = 0 and t = 8T /20, for a silane-hydrogen discharge operating in condition C3 (a). The curves coreff eff respond to: Ez /N (black), ESiH + /N (red) and E + /N (blue), calculated for ,z H ,z 3

3

a field-dependent mobility with and without the ion inertia term (dot and straight lines, respectively). A zoom of the near rf electrode region is given in (b). . . . 151 5.26 Axial profile (at r = 0) of different time-average, steady-state reduced electric fields, for a silane-hydrogen discharge operating in condition C3. The curves eff correspond to: Ez /N (solid) and ESiH + /N (dotted). . . . . . . . . . . . . . . 152 ,z 3

5.27 Space-time contour plots at r = 0 of different reduced electric fields (in Vcm2 ), during one rf cycle, and for a silane-hydrogen discharge operating in condition eff C3. Ez /N (a); EHeff+ ,z /N (b); and ESiH + /N (c). . . . . . . . . . . . . . . . . . 153 ,z 3

3

5.28 Time-average, 2d steady-state profiles of the electron density ne (a) and the electron mean energy ε (b), for a silane-hydrogen discharge (5% SiH 4 ) operating at p = 0.3 Torr and Vrf = 400 V. . . . . . . . . . . . . . . . . . . . . . . . 157 5.29 Spatial contour plot of the time-average, steady-state electron density (in cm −3 ), for silane-hydrogen discharges operating in conditions of Fig. 5.28, at different SiH4 relative concentrations: 5% (a), 10% (b), 30% (c), and 50% (d). . . . . . 158 5.30 Axial profiles (at r = 0) of the time-average, steady-state electron density ne (a), electron mean energy ε (b) and plasma potential V p (c), for silane-hydrogen discharges operating in conditions of Fig. 5.28, at various SiH 4 dilutions. . . . . 160 5.31 Self-bias voltage Vdc (a) and coupled electrical power Weff (b) as a function of silane relative concentration, for silane-hydrogen discharges operating in conditions of Fig. 5.28. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.32 Space-time-averaged dissociation rates of SiH4 (squares) and H2 (circles), as a function of silane relative concentration, for silane-hydrogen discharges operating in conditions of Fig. 5.28. . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.33 Time evolution of the convergence error for the time-average SiH − 3 density at different silane dilutions. The curves were obtained for silane-hydrogen discharges operating in conditions of Fig. 5.28. . . . . . . . . . . . . . . . . . . . 163

Chapter 1 Introduction 1.1 General introduction Low temperature plasmas (LTP) are extensively used for deposition or etching. Although the overwhelming majority of the gas in these discharges is neutral and in the electronic ground state, the plasma environment as a whole plays a major role in these processes. To obtain the desired chemical active species and an ion bombardment at the substrate, different plasma reactors have been developed. However, to tune the experimental conditions to the pointed aim is a complex task, often made in a semi-empirical way. Nowadays, plasma modeling has achieved a notable experience and the complexity degree of numerical codes is high enough to ensure a real contribution to the optimization of deposition/etching plasma reactors. To form and sustain these discharges, an electrical source is required to produce the ionization of the background gas. The electric field directly acts over charged particles only, influencing electrons and ions in totally different manners. For example, the injected energy is transferred mainly to electrons, being used in the ionization, recombination and dissociation of the background gas. The basic properties of the plasma are controlled by the charged particle dynamics; processes like excitation/relaxation and ionization/recombination being either directly or indirectly affected by the charged particle transport. For this kind of gas discharges the electron density and temperature are about 10 8 -1012 cm-3 and 5 eV respectively1 , the ionization degree is about 10-6 -10-3 , corresponding to weakly ionized gases, the pressure ranges between some mTorr and a few Torr, and the applied voltages are in the interval 50-500 V. The frequency range of the excitation electric field is used to distinguish between direct current (dc), radio frequency (rf) (105 -108 Hz) or microwave (109 -1011 Hz) discharges. Taking advantage of the ability to create and control the flow of charged particle in plasmas, 1

the ions and the neutrals have relatively low temperatures: typically 25-50 meV.

1

C HAPTER 1. I NTRODUCTION

2

a large number of technologies have been developed. In the next section, we strictly mention the basic ideas of some experimental techniques using LTP, a number of which is adopted in industrial applications.

1.2 Etching and deposition assisted by low temperature plasmas Integrated Circuits (IC) and Very Large Scale Integration (VLSI), based on sequential deposition of semiconductor and dielectric layers, have stimulated etching and deposition methods. Etching techniques such as Reactive Ion Etching (RIE) and Reactive Ion Beam Etching (RIBE), Chemically Assisted Ion Beam Etching (CAIBE) and Electron Cyclotron Resonance (ECR), are commonly used to meet the high-resolution and anisotropic 2 etching requirements of VLSI. The etching effect is promoted using extremely reactive plasmas obtained by LTP discharges in gases containing fluorine and chlorine (e.g. the etching of silicon and silicon oxides in a discharge of CF4 to form volatile SiF4 ). RIE combines two features: a special geometric design of the reactor chamber and low pressure (5-100 mTorr) conditions to achieve intense ion bombardments [Coburn 2000]. An enforced control of ion energy is obtained using RIBE. The reactive ions are produced in a separate chamber and a grid system accelerates them in the target direction. To maintain the beam focused, pressures less than 2 mTorr are suitable. Generally, to achieve good etch rates and acceptable etch-induced damages, the ion beam energies are below 1000 eV. The main difference between RIBE and CAIBE is the fact that CAIBE does not use reactive plasmas to generate the ion beam. The reactive gas is injected near the substrate which ensures a slightly greater degree of control of the side-wall profile [Vawter 2000]. The localization of the power absorption in microwave etching ECR is determined by a magnetic field that allows a resonant energy absorption by electrons. Compact and dense plasmas can thus be produced. Typical operating pressures are 0.1-some mTorr, the input frequency is f = 2.45 GHz and the magnetic field is 0.0875 T [Stevens 2000]. A large variety of plasma interactions have been used in deposition techniques of thin films (< 1 µm thickness). We can formally consider two sort of processes leading to film formation: 1. physical processes such as evaporation and sputtering, 2. chemical processes for which a chemical reaction is a preliminary condition for the film formation. 2

chemical etching leads to an isotropic etching of the material.


3

Sputtering processes involve knocking an atom out of the surface of a target to generate a topographic pattern of the surface or merely to clean it. By condensation on the substrate surface, the sputtered atom can be used. The former process, called sputtering deposition, is one of the main application of sputtering and an useful tool in coating technologies. A huge number of materials can be evaporated. Vacuum (10 -4 -10-5 Torr) evaporation techniques are widely used for the preparation of both non-metallic and metallic thin films. There are some important advantages: • the material evaporation temperature is considerably low under vacuum condition, • the formation of oxides and film pollution by impurities are substantial reduced, • in the absence of collisions, a straight line propagation of the material from the source to the substrate occurs, thereby, ensuring sharply and finely pattern on the substrate. Among the large variety of deposition methods there are two techniques considered as technologically mature, with certain advantages from the point of view of mass production: Chemical Vapors Deposition (CVD) and Plasma Enhanced Chemical Vapor Deposition (PECVD). The decrease of individual dimensions in IC devices helps to promote CVD at the expense of other traditional techniques, like evaporation or sputtering deposition. In a typical CVD process, reactant gases are used to form solid insulating or conducting layers on a wafer surface. The gas mixture enters the reaction chamber and is heated (thermally or radiativelly) as it approaches the deposition surface. Chemical reactions involving the source gases or reactive intermediate species occur at the deposition surface, forming the deposited layer. Volatile byproducts, frequently hazardous (flammable, corrosive), are transported out of the reaction chamber. In recent years, hot-wire CVD (in which the gas dissociation is obtained through a hot filament at about 2100 K) starts to be of common use. Advanced materials (e.g. metals, alloys, amorphous and poly/micro-crystalline semiconductors, ceramics and polymeric materials) in the areas of IC, opto-electronic devices and sensors or protective coatings are currently obtained using CVD. PECVD method uses the energetic, “hot” electrons which are present in a plasma (obtained by rf or microwave discharges) to create the chemical active species (the film precursors). An extremely important advantage of this techniques with respect to CVD relates to the fact that substrate temperature remains low, typically 500 K for amorphous hydrogenated silicon deposited. Devices using surface junctions have a maximum admitted treatment temperature of 1000 - 1200 K, but this value drops to 600 K for devices including metalizations. After deposition of the first metal layer on the wafer, CVD can no longer be used because it will melt the metal. Subsequent depositions must consequently be performed at low temperature, using PECVD.


4

PECVD has been used to deposit a variety of materials out of the gas phase: amorphous hydrogenated silicon (a-Si:H), micro-crystalline silicon (µc-Si:H), nanocrystalline silicon (ncSi:H) amorphous germanium, amorphous carbon and alloys of these materials. Large area substrates can be covered using this technique, which is extremely attractive in photovoltaics and flat screen displays. Amorphous hydrogenated silicon has received a considerable attention as an important candidate for large area photovoltaic power generation.

1.3 A brief history on a-Si:H 1.3.1 Amorphous materials The main feature that distinguish amorphous from crystalline materials is the disorder of the atomic structure. The amorphous material exhibits a short range order, but lacks the long range one. After a few inter-atomic spaces the correlation between atoms pair is lost as a consequence of a slight disorder in bond’s length and angles. Without the constraint of the crystals, the network organization is much more flexible. Increased temperature change the vibrational, translational and rotational movements of atoms, which can diffuse toward nucleation sites. The colossal interest of controlling this sort of structures comes from the information encoding based upon structural changes. Some materials allow both amorphous and crystalline structures with different local properties (e.g. reflectivity, conductivity) and high-density memories can be based on the ability to “read” and “write” in such materials. The interest in amorphous semiconductors began with these kind of researches in the 60’s and 70’s. The disorder, the defects and the possibility of alternative configurations influence the electronic properties of these materials in several ways: band tails, scattering, electronic states in the band gap, electronically induced metastable states [Street 1991]. In such networks there are no “correct” positions for atoms. Therefore it is impossible to say if an alternative structure is a defect or not, the coordination (each atom has a characteristic number of bonds to its immediate neighbors) being the only specific feature of random networks. In an ideal continuous random network of a-Si, all the silicon is four-fold coordinated. When the coordination of any silicon atom is less then ideal, the missing bonds are referred to as dangling. The three-fold coordinated silicon dangling bond is one example. A dangling bond is energetically very unfavorable, but it is difficult to avoid in films deposited from vapor phase and, in practice, the material voids contain always some dangling bonds [Ovshinski and Madan 1991]. The important point here is the fact that electronic properties of semiconductors are imposed by these deviant electronic configurations. However, these defects insert a large density of states in the band gap, thus controlling the conductivity. That is the main reason why elemental


5

amorphous silicon is a technologically useless3 material [Ovshinski 1991].

1.3.2 a-Si:H researches Hydrogenated amorphous silicon films have been reported in the works of [Sterling and Swann 1965] and [Chittick et al. 1969], where the passivation of dangling bonds by hydrogen was observed. Further researches [LeComber and Spear 1970] and [Spear et al. 1974] have shown that hydrogenated amorphous silicon has an improved photoconductivity and carrier with a fairly high mobility. Further, it is possible to manipulate the carriers numbers in doped films with boron or phosphorus by adding diborane (B2 H6 ) or phosphine (PH3 ) to the plasma gas, as reported in [Spear and LeComber 1975]. The interest in hydrogenated amorphous silicon owes its origin to the first practical device, a solar cell, which was built with a-Si:H [Carlson and Wronski 1976]. Since the 80’s, a-Si:H researches follow two major directions, not always in close connection: 1. studies of film growth, micro-structure and properties; 2. investigations of plasma sources leading to film deposition. Most of the studies were carried out in rf deposition systems using silane-hydrogen gas mixtures. The first kind of studies was responsible for several works devoted to film characterization [Matsuda et al. 1980, 1981; Vanier et al. 1984], growth and layer formation [Matsuda 1983; Tsai et al. 1986, 1987; Tsai 1988; Matsuda 1999; von Keudell and Abelson 1999; Cicala et al. 2001]. The second one was concerned with gas phase investigations using optical emission spectroscopy (OES) [Matsuda et al. 1983; Kampas and Kushner 1986], mass spectrometry [Henis et al. 1972; Allen et al. 1977; Perrin et al. 1984] and laser induced fluorescence (LIF) [Schmitt 1983]. We should also refer the studies concerning the ionization cross sections [Chatham et al. 1984], the rate constants and transport coefficients for silane plasma chemistry [Perrin et al. 1996], or the description of incoming fluxes at the substrate [Gallagher 1988; Vepˇrec et al. 1989; Hamers 1998]. At the interface between these two directions we can find the phenomenological analysis of surface reactions reported in [Perrin and Broekhuizen 1987; Perrin et al. 1998]. The results of this analysis are frequently used in modeling by introducing a surface reaction probability β (for SiH3 on a-Si:H β = 0.23 and for H on a-Si:H 0.4 (pd) crt the α regime cannot be achieved at all [Yatsenko 1981, 1982; Raizer 1991]. Observing ccrf discharges at moderate pressures, [Yatsenko 1980] noted a change in the luminosity accompanied by a sharp increased of the current discharge, suggesting [Yatsenko 1981] that the transition from α to γ regime is a consequence of the space-charge breakdown, involving electron secondary emission at the electrodes under ion bombardment. In contrast with the α bulk ionization that maintains the low current regime, the γ surface processes are


15

V (V)

700

500

γ α

300

100 −1

10

10

0

10

1

p (Torr) Figure 1.6: Ignition potential of a ccrf discharge as a function of pressure. The discharge is in hydrogen at frequency f = 3 MHz and inter-electrode distance d = 2.64 cm. The left-hand branch up to the jump is the γ discharge regime and the right-hand branch (solid curve) is the α discharge regime. Dashed curve plots the transition from α to γ regime. Source: [Levitskii 1957; Raizer 1991].


16

responsible for the abrupt increase in the electrodes sheath conductance and consequently for the on-set of high conductivity γ regime. [Godyak and Khanneh 1986] have shown that the α to γ transition is accompanied by an increase in the plasma density and a decrease in the electron temperature. Time-resolved and time-averaged light emission studies of α, γ and α to γ transition in rf discharges have also been presented by [Vidaud et al. 1988]. The α regime is preferred to the γ regime in many industrial applications due to the low power loss near the electrodes (low sheath conductance).

1.4.5 γ’- powder regime When a high power density is used for rf discharges in silane, the polymerization [Howling et al. 1993; Merad 1998] of SiH4 can occur. The corresponding dusty plasma regime is known as γ’ regime. The simplest polymerization reactions involve ion-molecule and ion-ion interactions Six Hy− + SiH4 ⇒ (Six+1 Hy− )∗ + (H, H2 Six Hy− + Sip Hq+ ⇒ Six0 Hy0 + (Si, H, H2

products) products)

(1.4) (1.5)

To avoid powder formation is a major concern in PECVD, since the falling of particles into the growing a-Si:H can decrease the efficiency of solar cells by 1% [Kausche and Plättner 1992]. The same process occurring in plasma-assisted deposition of thin films is responsible for the rejection of semiconductor devices. In consequence, several theoretical and experimental works have focused on the identity of the precursors responsible for powder generation. [Choi and Kushner 1993] proposed that negative ions, trapped in electropositive plasmas, could allow cluster formation. Once critically large clusters are formed, particle growth rapidly follows. Mass spectrometry experiments in silane plasmas for amorphous silicon deposition [Howling et al. 1993] indicate the presence of negative ions containing up to sixteen atoms of silicon, while positive ions have no more than six atoms! [Reents and Mandich 1994] present a possible scenario for positive ions to became initiators of particle formation in silane plasmas under the presence of moisture. The presence of water in the PECVD reactor studied by [Perrin et al. 1994] was below 0.1%, but, the comparison of the induction time (the period between the beginning of cluster growth and the moment at which it influence the discharge operation) with the diffusion time of radicals and positive ions, support the anions involvement in powder formation. In the case of silane plasmas used for deposition purposes, there is now a general agreement that negative ions play the major role in powder development. Being trapped in the plasma, the powder presence is not bad as long as it is not incorporated in the growing film. Moreover,


17

the γ’ powder regime is responsible for an increase in electron temperature [Boeuf et al. 1994], thus enhancing the radical production and the deposition rate.

1.5 Discharge modeling 1.5.1 Introduction Discharge modeling is nowadays considered as a practical tool for the design and optimization of experimental installations used in plasma-assisted processes. Physical simulations can help in reactor design, providing informations about its optimum dimensions, the location of pump ports and inlet manifolds, the type of power source or the location of pressure sensors and wafers. Quick answers concerning the effects of pressure, power, gas dilution and gas flow on the general performance of the system may be obtained using low-dimensional discharge models. For a given reactor, more accurate multi-dimensional plasma models are used to predict etch/deposition rates, sheath voltages, spatial distribution of radicals, etc. These aspects are covered by sophisticated models including the description of charged particle transport, sheath dynamics, gas phase and gas-surface chemistry. At the moment, there is no commercial software package dealing with all relevant physical and chemical processes occurring in a discharge plasma and at a wafer surface, providing a certain number of informations as a function of the reactor operation conditions, in a reasonable amount of time. System complexity and computer time restrictions, justify the development of models orientated to targeted informations, often involving a large set of approximations. The interest in first-order solutions obtained by using low-dimensional discharge models (0d, 1d) cannot be neglected. These approaches allow a quick estimation of fundamental plasma characteristics (e.g. plasma density, electron temperature), providing first-order comparison with experimental measurements. Sensitivity studies based on such models can help to test and tune the data base adopted, in view of further utilizations in multi-dimensional models. The complicated gas phase and gas-surface chemistry can be also reduced to a small set of relevant reactions, more suitable to be managed in terms of computational time. However, in rf discharges used for etching/deposition, there is a significant number of parameters such as the self-bias voltage, the power distribution within the discharge, the etching and deposition uniformity, that can only be adequately treated using multi-dimensional simulations (2d, 3d). Moreover, the actual existing trend to load 300 mm (instead of 200 mm) wafers in semiconductor reactors supposes a chamber scale-up and accentuates the importance of multi-dimensional models. The development of a discharge model is centered in one basic idea: the coupling between the applied electric field and the plasma. A self-consistent description must account for the net


18

creation and transport of charged particles within the electric field resulting from the external applied voltage and the internal space-charge distribution. Depending on the approach used to simulate plasma discharges, we can distinguish between three classes of models: • statistical (or particle) models • fluid (or continuum) models • hybrid models. We point out briefly the main force ideas behind each one of these classes.

1.5.2 Statistical models Statistical models define representative particles for each of the species in the reactor (e.g. electrons and ions) and follow their movements across a grid in phase space. This is done by integrating the usual (1d, 2d or 3d) equation of motion for each particle of mass m, charge q and velocity v, subject to an electric field E and a magnetic field B dv = q(E + v × B) . (1.6) dt Charge and current densities, assigned to the grid points by the weighting method [Birdsall m

1991], are used to solve the field equations and re-calculate the forces acting on particles when time is incremented. Present day works, combine Particle In Cell (PIC) with Monte Carlo Collisions (MCC) techniques [Birdsall 1991]. Based on probability functions and statistics, these methods determine the type of collisions (e.g. collisions types are determined by comparing random numbers with collision probabilities), reactions and trajectories in a self-consistent way. A typical computing sequence can be find in [Birdsall et al. 1998]. The space and time variation of the particle distribution functions is an output of these simulation models, which ensure an extremely accurate description of charged-particle transport. Their results are often used as benchmark for comparison with other models. Moreover, physical processes which are hard to include in macroscopic models (such as collisionless or stochastic heating [Popov and Godyak 1985; Goedde et al. 1988; Turner 1993]) can be included by PIC-MCC models [Vahedi et al. 1995; Rauf and Kushner 1997]. In terms of convergence time, the actual computational burden of statistical models for single processor machines is very significant, being 10-100 higher than for fluid models. However, PIC-MCC schemes can be made more attractive using speed-up procedures combined with multi-parallel processors.


19

1.5.3 Fluid models A discharge model where the charged particle transport is described using mean values for some plasma parameters is called a fluid model. In principle, the electron and ion kinetics in a weakly ionized gas can be described using the corresponding Boltzmann equations. The Boltzmann equation is a continuity equation in the phase-space (r, w), which determines the particle distribution function f (r, w, t), at each position (r, w) and time t, under the effect of external forces and collisional processes with gas molecules. The solution to the multi-dimensional, time-dependent Boltzmann equation constitutes an extremely hard numerical task, which provides a kinetic description with such detail often unnecessary for applications. Moreover, the self-consistent solution to the charged particle Boltzmann equations should involve the coupling with Poisson’s equation for the self-consistent space-charge field, which complicates further this problem. As an alternative solution one can assume that the main plasma characteristics are well described in terms of some mean properties: density, mean velocity, mean energy, etc. In this way, the particle Boltzmann equations can be replaced by some characteristic average equations (termed the Boltzmann equation moments [Delcroix and Bers 1994, Chap.9]), which correspond to an infinite series of hydrodynamic equations. The development of these fluid models involves two main hypothesis 1. The first hypothesis concerns the truncation of the infinite series of hydrodynamic equations generated from Boltzmannt’s equation, assuming that two or three moments for electrons (mass, momentum and energy balance equations) and two moments for ions (mass and momentum equations) are usually enough to obtain a reasonable description. 2. The second hypothesis concerns the closure of the selected set of moment equations, and requires supplementary assumptions. The problem is twofold. Firstly, the truncation of the equations hierarchy implies a truncation also of the calculated parameters. In LTP models it is usual to adopt the so-called small anisotropy approximation which corresponds to assume that particle thermal velocities dominate over their drift velocities, thus introducing an additional relationship on plasma fluxes. Secondly, the system of moments equations contains integrals over the unknown particle distribution function, which can be physically interpreted as the particle transport parameters and rate coefficients. Hence, to calculate these integrals it is necessary to make some assumptions about the particle distribution function, which is particularly delicate for electrons. In rf plasmas, electrons are not in local thermodynamical equilibrium due to the inefficient energy exchange between light electrons and heavy neutrals, which means that the EDF deviates from its


20

Maxwellian form. Nowadays there is a general agreement that fluid models must include the continuity and the momentum transfer equations for electrons and ions, the electron mean energy transport equation and Poisson’s equation, and that the problem is to be solved self-consistently either in 1d [Boeuf 1987] or 2d [Nienhuis 1998] geometries. However, concerning closure conditions for the problem, we can distinguish between two different approximations for the EDF calculation • the local field approximation (LFA) • the local mean energy approximation (LEA). The LFA assumes that both the electron transport parameters (eTP) and the electron rate coefficients (eRC), calculated at a given position r and time, are an exclusive function of the local reduced electric field E(r, t)/N , where E is the magnitude of the electric field and N is the gas density. In other words, the EDF at (r, t) is assumed to be the same that would exist in an uniform reduced electric field equal to E(r, t)/N , and this equilibrium with the local electric field implies that the energy gained by electrons from the electric field is locally compensated by their energy losses due to collisions. The original idea behind this approximation [Richards et al. 1987] was to use for eTP data from dc Townsend experiments together with calculated eRC for different E(r, t)/N values. The eRC were obtained by integration of the corresponding electron cross-section over a local EDF, calculated from the solution to the homogeneous (space and time independent) electron Boltzmann equation (EBE). Earlier discharge models such as [Graves and Jensen 1986; Graves 1987; Richards et al. 1987; Boeuf 1987, 1988; Gogolides et al. 1989; Passchier and Goedheer 1993b] used this approximation to ensure the closure of the equations system. However, the assumption normally fails in the sheath regions (where electrons are not in equilibrium with the electric field) leading to an over/increase of the eTP near the discharge walls. In general, the problem was avoided imposing constant eTP within the sheath region as a practical solution. The LEA considers the eTP and eRC at (r, t) as an exclusive function of the electron mean energy calculated from the energy balance equation in the fluid model. In practical terms, the electron coefficients are obtained from a local EDF (calculated by solving the homogeneous EBE), which is now considered as a function of the (local) electron mean energy. Although inexact, this approach is able to remove the non-physical variations of the eTP within the sheath regions, obtained when the LFA is adopted. Often, the transport equations in fluid models are further simplified by adopting the driftdiffusion approximation (DDA), which writes the charged particle fluxes as a sum of a drift plus a diffusion term, ignoring all other terms [Graves and Jensen 1986].


21

The outstanding advantage of fluid models, that up to now cannot be challenged by dimensional equivalent statistical models, is their rapid convergence. Run times of a few hours provide reliable results and thus simulations using fluid codes are by far the most common method employed in the analysis of LTP for materials processing. However, fluid models cannot provide the species distribution functions but only averages over the distributions. These lack of details (e.g. the energy distribution of ions impinging into the wafer is unavailable) is often compensate by hybrid codes. Subject to a set of assumptions, that are reflected both in the final accuracy and validity, the fluid models have to be confronted with experimental measurements and the results of correspondent statistical models have to be seen as a benchmark.

1.5.4 Hybrid models The term hybrid model is used to designate either simulations that consider two group of electrons as in [Boeuf and Pitchford 1991; Fiala et al. 1994] or, alternatively, complex and modular codes like the ones presented in [Ventzek et al. 1994; Li et al. 1998]. Simulations involving two groups of electrons consider two relatively independents electron populations: the first one includes the vast majority of the bulk electrons, and the second one is formed by the small fraction of energetic electrons heated by the applied electric field. To avoid the generalized treatment of the entire electron ensemble by time consuming methods, it is possible to describe them using fluid equation, whereas a MC method is adopted to control the transport of fast electrons. This hybrid approach combines a good description of the highly anisotropic sheath regions with reasonable computational times. It is also possible to develop hybrid numerical codes by combining different calculation modules describing different processes occurring either in gas phase or gas-surface interactions [Sommerer and Kushner 1992]. The modular approach is a powerful method in plasmareactor modeling, as it tries to achieve a proper balance between computational time, precision and a comprehensive physical description.

1.6 Overview thesis The thesis is organized in the following way. In Chapter 2 we review the basic theoretical ideas and the main difficulties of fluid models for charged particle transport. The global equation system adopted by this work is presented in this chapter and the peculiar aspects introduced by our approach are stressed. Chapter 3 summarizes the computational algorithms adopted to solve the equation system. The main numerical restrictions are also indicated.


22

The calculation results for ccrf discharges operating in pure helium are presented in Chapter 4. We provide comparisons of discharge electrical parameters with earlier calculations and experimental measurements. Chapter 5 reports simulation results for discharges sustained in pure hydrogen and silanehydrogen mixtures. The influence of reactor geometry and silane dilution over the discharge parameters is discussed. Comparisons with experimental measurements and other calculations are also available. The conclusions are presented in the Chapter 6.

Chapter 2 Theoretical considerations 2.1 General introduction Equilibrium plasmas are well described using statistical mechanics. The dynamical state of a system is represented by a point in the 6N -dimensional phase space, defined by the vector (r, w), where r represents the particle position and v the velocity. The main assumption behind statistical mechanics is that the macroscopically observed variables of the system are in relation, among all conceivable combinations of positions and velocities, to the most probable dynamical states. The latter can be generally found by solving Liouville’s equation, which states that the volume of a given set of points in the phase-space is constant throughout the motion of these points. A general solution to Liouville’s equation supposes the precise knowledge of all particle motions, i.e., the maximum of information that one can expect to have from the system [Liboff 1969]. However, it is often unnecessary to have such a detailed description of the system and, in many cases, it is sufficient to have information on the system’s particle distribution function f (r, w, t), defined as dNP , (2.1) dr dw where dNP is the number of particles inside the six-dimensional phase-space volume dr dw, f (r, w, t) ≡

located at time t around position (r, w). In this way, the 6N -dimensional phase-space, is replaced by a 6-dimensional phase-space µ for a single particle, which constitutes the framework of the so-called kinetic theory. The density of particles at position r and time t, n(r, t), can be obtained from the particle distribution function by using dNP dn = = f (r, w, t) dw ; dr

n(r, t) =

Z

w

f (r, w, t) dw .

(2.2)

The equation that describes the space-time evolution of f (r, w, t) can be derived from integration of Liouville’s equation, by introducing the two-particle distribution function f 12 to characterize particle binary interactions; further, the equation for the space-time evolution of

23

C HAPTER 2. T HEORETICAL

24

CONSIDERATIONS

f12 includes particle triplet interactions by means of function f123 and so on. The procedure yields the Born-Bogoliubov-Green-Kirkwood-Yvon (BBGKY) system of equations, which is completely equivalent to Liouville’s equation ([Delcroix and Bers 1994, chap.8]). The first equation of this system reads ∂f X + w · ∇f + · ∇w f + ∂t m

Z

X 12 · ∇w f12 dr 2 dw 2 = 0 m

,

(2.3)

where ∇ and ∇w represent the gradients in position and velocity space, respectively; X and X 12 represent the external and binary interaction forces acting over the particle; m is the particle mass; and dr 2 and dw2 are elemental integration volumes for all others particles. Equation 2.3 is a continuity equation in phase-space µ, stating that the time evolution of the particle distribution function is controlled by diffusion phenomena occurring in configuration space, the term w · ∇f , by the action of external forces affecting particle velocities, the term X · ∇w f , and by m R X 12 collisional effects described by the collision operator · ∇w f12 dr 2 dw 2 . m The solution to the BBGKY equations system requires a closure condition. In weakly ion-

ized gases, where binary encounters are dominant, this condition comes from writing the collision operator as ˆ )= J(f

Z

(f 0 f20 − f f2 ) g σ(χ, ϕ) dΩ dw2

,

(2.4)

yielding the so-called Boltzmann equation ∂f X ˆ ) + w · ∇f + · ∇w f = J(f ∂t m

.

(2.5)

In equation 2.4, f ≡ f (r, w, t), f2 ≡ f (r, w 2 , t), f 0 ≡ f (r, w , t), f20 ≡ f (r, w 2 , t) are the particle distribution functions before and after the collision event, respectively 1, g is the relative velocity of the particles involved in the collision, σ(χ, ϕ) is the differential collision cross section (χ and ϕ are the deflection and the azimuthal angles, respectively), and dΩ represents the elemental collision solid angle. The collision operator Jˆ has the following important properties: • local attributes: Eq. 2.4 uses the particle distribution function values in the collision point, ignoring their variation inside the particle interaction sphere. • conservation properties: Eq. 2.4 ensures the conservation laws of matter, momentum and energy, associated with the collision invariants 1, mw, mw 2 /2. • non-linear characteristics: the non-linearity of Eq. 2.4 is responsible for an irreversible character in the evolution of f (Boltzmann’s H theorem), with strong physical and philosophical implications [Liboff 1969]. 1

f2 and f20 can refer or not to the same kind of particles as f and f 0 .


25

CONSIDERATIONS

In stationary (∂f /∂t ≡ 0), homogeneous (∇f ≡ 0) conditions and in the absence of external forces (X/m ≡ 0), the Boltzmann equation reduces to ˆ )=0 J(f

.

(2.6)

In the case of systems formed by a single kind of particles the solution to Eq. 2.6 is the so2 called Maxwellian distribution function f (w) = Ce−A(w −w0 ) , where w0 is the referenceframe velocity. Under the above-mentioned conditions, it is possible to identify the temperature of the system with its kinetic temperature (associated with the thermal movement of particles), and the system is considered to be in a thermodynamic equilibrium state. In these equilibrium conditions, the system can be described by using common thermodynamic variables such as pressure, temperature and density. In a plasma environment with several constituents, the knowledge of the distribution functions corresponding to each population is determinant for an accurate evaluation of the macroscopic properties characterizing the whole ensemble. In general, the different particle distribution functions deviate from their Maxwellian form, and the plasma is considered to be in non-equilibrium conditions. As the various particle sub-systems of a weakly ionized gas (electrons, ions, neutrals) have different influence on plasma properties, the analysis of these particle populations is essential for quantitative evaluations in gas-discharge modeling. In this case, the description of the system requires the resolution of Boltzmannt’s equation 2.5 for each plasma component, with important distinction regarding their behavior. Electrons, being lighter than other species, receive most of the applied field energy during the time interval between collisions. As the energy transfer between electrons and neutrals is extremely inefficient (due to their different masses), and electron-ion collisions are relatively rare events in weakly ionized plasmas, the electron energy will remain high when compared to the neutral or ion energies. In the presence of external forces and spatial gradients, the particle motion acquires a drift component and their distribution function becomes anisotropic. Once again, this effect is particularly important for electrons with much smaller masses (hence much higher mobilities and diffusion coefficients) than ions or neutrals. Finally, electrons play a fundamental role in discharge maintenance: they are responsible for the fragmentation, dissociation and ionization of the background gas; they also control most of the discharge energy deposition process. These arguments altogether justify how important a correct estimation of the electron distribution function is in discharge modeling. As already mentioned, this non-equilibrium EDF is to be calculated from the solution to the electron Boltzmann equation. The very effective energy transfer between ions and neutrals, combined with the fact that the


26

CONSIDERATIONS

ion mean energy is generally much smaller than the electron mean energy, justifies the use of a near-Maxwellian ion distribution function (IDF) at gas temperature. The latter assumption is certainly correct under moderate electric field conditions, for which it is reasonable to consider that the ions are in equilibrium with the neutrals. However, in discharge regions where the electric field values are higher, the ion energy can increase far above the thermal energy of the neutrals, and the IDF should present non-equilibrium features. This is usually the case in the near-substrate region of plasma reactors. Although deposition is almost entirely due to neutral species, an improvement in film quality can be achieved through moderate ion bombardment.

2.2 From Boltzmann equation to fluid equations 2.2.1 Introduction The most natural approach to describe a plasma is the so-called hydrodynamic approach, in which the plasma is pictured as a single fluid [Holt and Haskell 1965]. This fluid is characterized by macroscopic quantities close to experiment, such as the mass density ρ m , the mean mass ∨

velocity v m and the total pressure tensor pt defined as ρm ≡

X

nα mα

(2.7)

α

vm ≡ ∨

pt ≡

P

hw α i α nα mα

n m α Pα α

X

nα mα h(w α − v α )(wα − v α )i

(2.8) .

(2.9)

α

In these equations, the symbol h i designates the mean value over a particle distribution function, and the subscript α is associated to the different plasma species (electrons, ions and neutrals), each of which has density nα , mass mα and total velocity wα = v thα +v α , where v thα is its thermal (random) component (which averages zero, hv thα i = 0) and v α is its oriented component due to the presence of electric fields and/or spatial gradients (naturally, hwα i = hv α i ≡ v α ). In this description, where the main physical quantities are referred to the mean mass velocity and then summed over all species to characterize the entire fluid, the information about individual components is “lost” being diluted in this macroscopic view. To describe charged particles transport properties and other phenomena that depend explicitly on the relative motions of the various plasma constituents, single-species equations are useful. These equations are written in terms of single-species global quantities, which are calculated as averages over the different distribution functions as follows (we omit the subscript α for simplicity, stressing however that the following expressions refer to a single kind of plasma species)


27

CONSIDERATIONS

• the particle density n(r, t) =

Z

w

f (r, w, t) dw

.

(2.10)

• the total mean velocity Z 1 v(r, t) = hwi = w f (r, w, t) dw n(r, t) w

.

(2.11)

• the mean energy ε(r, t) =

mw 2 2

Z mw 2 1 f (r, w, t) dw = n(r, t) w 2

.

(2.12)

According to the observations above, the particle mean energy can be written as the sum of a 2 random and an oriented-motion energies ε = εvth + εv , where εvth = 21 m hvth i and εv = 21 mv 2 .

In equilibrium conditions, corresponding to an isotropic distribution function, ε v = 0 and εvth = 3 k T 2 B

with kB the Boltzmann constant and T the kinetic temperature, associated to the thermal

random motion. ∨

• the kinetic pressure tensor p

pij = n(r, t)m vthi vthj = m

Z

w

(w − v)i (w − v)j f (r, w, t) dw

(2.13)

∨

which is related to the momentum flux density tensor P (Pij = nm hwi wj i) according to Pij = nmvi vj + pij . The tensor element Pij represents the mean rate of flow of the momentum i component through an unit area whose normal is along j direction. ∨

For an isotropic distribution function, the diagonal elements of p read

2 D 2 E 2 1 2 ∨ vthi = vthj = vthk = 3 hvth i. In this case, the tensor p turns into a scalar verify-

2 ing p = 31 nm hvth i = nkB T . The scalar pressure can be used to write the general expression of

2 the pressure tensor as pij = pδij + πij , where πij ≡ nm vthi vthj − 31 vth is the viscous stress

tensor and δij is the Kronecker symbol. In equilibrium conditions, it is usually assumed that the scalar pressure satisfies the equation of state for ideal gases, p = nkB T • the heat flow vector q (the thermal energy flux vector) Z n(r, t)m 2 m v vt = (w − v)(w − v)2 f (r, w, t) dw q= 2 2 w

(2.14)

The heat flow represents the mean rate at which the energy flows through a surface along the i direction, in a frame moving with velocity v. The thermalization of a given plasma species implies that there is no net thermal energy flux of this species through any surface within the


28

CONSIDERATIONS

plasma. The heat flow vector is connected to the energy flux vector Q by the relation Q = ∨

q + n(εv + εvth )v+ p v. The macroscopic quantities defined by Eqs. 2.10-2.14 can be calculated knowing the particle distribution function f (r, w, t), which is obtained from the solution to the corresponding particle Boltzmann equation. In non-equilibrium conditions, typical for real discharge plasmas, the resolution of the kinetic Boltzmann equation is a very complex and time-consuming numerical task. In fact, this is one of the reasons why we have chosen to adopt a fluid description of the plasma, based on the hydrodynamic approach, involving the conservation equation of matter, momentum and energy. As mentioned before, the other reason for this choice is the proximity to experiment which helps in model-validity checks. However, a problem remains. The hydrodynamic fluid equations are written in terms of the macroscopic quantities defined before by Eqs. 2.10-2.14, and the latter can only be calculated after solving the corresponding particle Boltzmann equation to get f (r, w, t). In order to break this loop we shall adopt the following procedure. Firstly, the macroscopic fluid equations will be deduced by calculating the so-called moments of the microscopic Boltzmann equation. Secondly, the particle distribution function will be evaluated by solving an approximate form (numerically simpler) of the Boltzmann equation. Thirdly, the macroscopic quantities involved in the fluid equations are obtained from the calculated distribution function by using an appropriated closure approximation to the problem. In the next section, the fluid equations for the conservation of matter, momentum and energy are deduced from the Boltzmann equation moments, which constitutes the first step of the above-described procedure. The remainder two steps will be discussed in later sections.

2.2.2 Derivation of moment equations from general the Boltzmann equation The time average variation of any function A(r, w, t) can be obtain by multiplying the Boltzmann equation 2.5 by A(r, w, t) and integrating over the entire velocity space. The resulting equation can be written as [Delcroix and Bers 1994] ∂(nA) ∂A X −n + ∇ · nwA − nw · ∇A − n ∇w A = C(A) ∂t ∂t m

,

(2.15)

where the right-hand-side (RHS) C(A) represents the average collision operator and the average value A is given by Z 1 A ≡ hAi ≡ A(r, w, t)f (r, w, t) dw n(r, t) w

.

(2.16)


29

CONSIDERATIONS

Equation 2.15 leads to the Boltzmann equation moments by identifying the function A with the velocity powers w 0 , w 1 , w 2 . Thus, the zero-order moment is for A ≡ w 0 and gives the particle conservation law; the first-order moment is for A ≡ mw1 and yields the momentum conservation equation; the second-order moment is for A = mw 2 /2 and it leads to the energy conservation equation. The procedure can continue for higher orders of velocity powers to generate the infinite set of the Boltzmann equation moments [Holt and Haskell 1965]. In this chained equation set the zeroth moment equation involves the particle flux n(r, t) and v(r, t), the 1st equation introduces the pressure tensor, the 2nd one the heat flow and so on; because the equation set must be truncated at some level, we are always left with more unknowns than equations. The fundamental conservation laws, obtained from Eq. 2.15, are • the particle conservation equation (A = 1) ∂n δn + ∇ · (nv) = ∂t δt

.

(2.17)

The term δn/δt concerns the creation and destruction of particles in kinetic processes like ionization and recombination, and it can be written as δn/δt ≡ nν, where ν/N is the particle net creation rate coefficient. • the momentum conservation equation (A = mw) m∂(nv) δnmv ∨ + m∇ · (nvv) = nX − ∇p − ∇· π + . (2.18) ∂t δt The terms on the left-hand-side (LHS) of Eq. 2.18 are associated to the hydrodynamical acceleration, while the terms on its RHS represent the sum of forces acting upon the particles per unit volume: the external forces nX, the forces due to the particle pressure −∇p, the stress related ∨

to viscosity −∇· π, and the friction forces resulting from particle collisions δ(nmv)/δt. The latter term is zero for collisions between particles of the same kind because, in this case, it only occurs an internal distribution of energy within the particle population. In weakly ionized gases, the collisional events involving charged particles are dominated by their impact upon neutrals (atoms or molecules). In this case, we can write δmnv ' −mnvνca δt

,

where νca represents the charged-particle/neutral mean collision frequency. Equation 2.18 can be written in a different form, by developing its non-linear inertial term ∇ · (nvv), and by using Eq. 2.17 together with the identity n(δv/δt) = δ(nv)/δt − v(δn/δt), to yield

δv ∂v ∨ + (v · ∇)v = nX − ∇p − ∇· π +nm mn ∂t δt

.

(2.19)


30

CONSIDERATIONS

For electrons in a weakly ionized plasma, Eq. 2.19 can be written in a simplified form after a certain number of approximations. The collision term for dominant electron-neutral encounters reads [Golant and Sakharov 1980] e−a me ne (δv e /δt) ' −me−a ' −me ne v e ν ce−a r ne (v e − v a )νc

,

(2.20)

where v a is the oriented velocity of neutrals (with mass ma ), me−a = me ma /(me + ma ) ∼ me r is the reduced mass of colliding particles, and ν e−a is the total electron-neutral momentum c transfer mean collision frequency. In principle, Eq. 2.20 is only valid if ν ce−a presents a weak dependence on the electron velocity. • the energy conservation equation (A = 21 mw 2 ) ∂(nε) δ(nε) = −∇ · Q + nv · X + ∂t δt

.

(2.21)

Equation 2.21 shows that the time variation of the total kinetic energy of a given plasma species is due to the energy transport due to particle motion ∇ · Q, the particle heating under the effect of external forces nv · X, and the energy transfer in collision processes δ(nε)/δt. The latter term is zero for collisions between particles of the same kind because, in this case, it only occurs an internal distribution of energy within the particle population. The overwhelming difference between the oriented and the thermal electron velocities v e vthe , makes it possible to neglect in Eq. 2.19 the non-linear term (v e ∇)v e , when compared to the pressure term ∇p, since ve2 L 2 1 vth ∼ me e 3 L

me (v e ∇)v e ∼ me ∇pe k B Te ∼ ne L

(2.22) ,

(2.23)

where L represents a characteristic gradient length in the discharge, and we have assumed equilibrium conditions. Moreover, if the the characteristic variation time of the electron parameters exceeds the inter-collisional time τce−a = 1/ν e−a c , Eq. 2.18 can be written in its stationary form as the term proportional to the time derivative can be neglected when compared to the collisional term

∂ve ν e−a c ve ∂t

.

(2.24)

∨

Finally, we saw in Section 2.2.1 that the viscous tensor π measures the deviation of the distribution function from spherical symmetry. In small-anisotropy conditions the scalar pressure term is dominant and the effects of viscous stress on the electron transport can be neglected [Golant and Sakharov 1980, p.196].


31

CONSIDERATIONS

Under the above conditions, the momentum equation, Eq. 2.18, for a weakly ionized gas, subject to an electric field E, reduces to ve =

−eE 1 ∇(kB Te ne ) e−a − me ν c me ν e−a ne c

,

(2.25)

where e is the electron charge. Introducing the electron mobility µ e and free diffusion De coefficients defined as e me ν e−a c k B Te ≡ me ν e−a c

µe ≡ De

(2.26) ,

(2.27)

Eq. 2.25 yields the electron flux equation Γe ≡ ne v e = −µe ne E − ∇(De ne )

,

(2.28)

or, assuming small temperature gradients, Γe ' −µe ne E − De ∇ne

.

(2.29)

Note that the transport parameters defined by Eqs. 2.26-2.27 verify Einstein’s relation De k B Te = µe e

.

(2.30)

According to Eq. 2.29, the electron flux is given as the sum of a drift term proportional to the electric field −µe ne E and a diffusion term controlled by the electron density gradient −D e ∇ne . This form of the electron flux, referred to as drift-diffusion flux is frequently used in fluid discharge models [Boeuf and Pitchford 1995a; Nienhuis et al. 1997]. The validity of the driftdiffusion approximation DDA for the present model will be discussed in Section 2.2.4 For electrons in a weakly ionized plasma, Eq. 2.21 can be written under the same approximation framework that was applied to the momentum equation 2.28. Thus, using the expression of the energy flux vector Q (cf. Section 2.2.1), and taking peij ∼ pe δij = (2/3) ne εδij , ε ∼ εvth 2 we can write ∂(ne ε) 2 δ(ne ε) + ∇ · (q + ne εv + ne εv) = −Γe E + ∂t 3 δt

,

(2.31)

where Γe ≡ ne v e is the electron flux given by Eq. 2.28. The electron heat-flow vector q can be expressed as

2

Hereafter, we will take ε ≡ εe .

2 q = − KTe ∇ε 3

,

(2.32)


32

CONSIDERATIONS

where KTe is the electron thermal conductivity coefficient defined by KT = −(5/2)ne De

,

(2.33)

as can be obtained from the third moment of the EBE [Golant and Sakharov 1980, p. 188]. The second term on the LHS of Eq. 2.31 can be regarded as the divergence of the electron energy flux Γε , defined as (cf. Eq. 2.29, 2.32 and 2.33) 2 5 5 5 Γε ≡ q e + ne v e (ε + ε) = − De ne ∇ε − µe εne E − De ε∇ne 3 3 3 3

,

(2.34)

or Γε = −µε ne εE − Dε ∇(ne ε)

,

(2.35)

where

5 µε ≡ µe (2.36) 3 5 (2.37) Dε ≡ De 3 are respectively, the electron mobility and free diffusion coefficient for energy transport. The electron mean energy balance equation 2.31, with the electron energy flux defined from Eqs. 2.35-2.37 can be written as ∂ (ne ε) δ(ne ε) + ∇ · Γε = −Γe E + ∂t δt

.

(2.38)

This equation is frequently used in fluid discharge models [Boeuf and Pitchford 1995a; Nienhuis et al. 1997].

2.2.3 Derivation of electron moment equations from the two-term Boltzmann equation In Section 2.2.1 it was shown that the knowledge of the particle distribution function f (r, w, t) was necessary in order to calculate the macroscopic quantities involved in the different hydrodynamic fluid equations. It is by now clear that this problem concerns mainly the electrons, since they are the non-equilibrium population of the plasma. In fact, in a weakly ionized gas, the ions and the neutrals can be considered as being in equilibrium at gas temperature Tg ; the former follow a Maxwellian distribution function at this temperature Tg and the latter satisfy the equation of state p = N kB Tg (hereafter, p and N will represent the total neutral gas pressure and density, respectively). We are thus left with the problem of calculating the non-equilibrium EDF by solving the corresponding general EBE, which constitutes a very complex and time-consuming numerical task. For this reason, the EDF is often obtained as the solution to an approximative form of the


33

CONSIDERATIONS

EBE, which turns out to be more adequate for numerical treatment. The procedure starts by assuming that the anisotropies affecting the electron population (due to the presence of electric fields and/or density gradients) are small, which justifies the writing of the EDF as a two-term expansion in velocity space. f (r, w, t) = f 0 (r, w, t) + f 1 (r, w, t) · eanisotropy

.

(2.39)

Here, f 0 represents the isotropic part of the full EDF, f 1 represents its first anisotropy, and eanisotropy is the unit vector along the total anisotropy direction. Note that both f 0 and f 1 are dependent on the absolute value of velocity only, and that they verify the normalization condition

Z

f (r, w, t) dw =

Z

∞

f (r, w, t)4πw 2 dw = ne (r, t)

.

(2.40)

0

When expansion 2.39 is introduced into the Boltzmann equation 2.5, the latter is decomposed in a set of two equations (termed the classical two-term approximation) ∂f 0 w 1 ∂ 1 1 2 eE ˆ 0) + ∇·f − w ·f = J(f 2 ∂t 3 3w ∂w me eE 1 ∂f 0 1 1 0 − ∇f , f = N σc (w) me w ∂w

(2.41) (2.42)

where σc (w) is the total electron-neutral momentum transfer cross-section. Equation 2.41 describes the isotropic properties of the plasma electrons . Its direct integration, over all velocities, gives the particle conservation equation 2.17; after multiplication by 1 m w2 , 2 e

its integration gives the energy conservation equation 2.38, with the energy flux defined

by Γε = µε ne εE − ∇(Dε ne ε) .

(2.43)

Equation 2.42 describes the anisotropic properties of the plasma electrons, and it was obtained assuming that the characteristic evolution time of first anisotropy is much higher than the time interval between collisions, i.e., ∂f 1 νce−a (v)f 1 ∂t

,

where νce−a (v) = N σc (v)v is the total electron-neutral momentum transfer collision frequency. In principle, in rf discharges f 0 and f 1 are time dependent. However, the isotropic part of the EDF is expected to be time independent at sufficiently low pressures. We consider that the evolution time of f 1 is even biger than that one of f 0 , f 1 being almost constant ∂f 1 /∂t ∼ 0. After multiplication by me w, the integration of Eq. 2.42 over all velocities yields the momentum conservation equation 2.28.


34

CONSIDERATIONS

The electron net creation rate coefficient νe /N , involved in the writing of the RHS of Eq. 2.17, is calculated as the sum of various gain/loss rate coefficients νj /N , each of which given by νj /N =

2 me

1/2 Z

∞

F (u, r, t) u σj (u) du

.

(2.44)

0

Here, u = me w 2 /2e is the electron kinetic energy (in eV), σj (u) is the collision cross-section for the corresponding gain/loss mechanism, and F (u, r, t) is an electron energy distribution function (EEDF) defined as F (u, r, t) u1/2 du ≡ f 0 (r, w, t)4πw 2 dw/ne (r, t) thus verifying the normalization condition (cf. Eq. 2.40) Z ∞ F (u, r, t) u1/2 du = 1

,

.

(2.45)

(2.46)

0

The electron transport parameters (eTP) involved in the writing of Eqs. 2.28 and 2.35 are given by [Alves et al. 1997] µe N = De N = µε N = Dε N =

1 Z 1 2e 2 ∞ u ∂F (u, r, t) − du 3 me σc (u) ∂u 0 1 Z 1 2e 2 ∞ u F (u, r, t) du 3 me σc (u) 0 1 R ∞ u2 ∂F (u, ,t) 1 2e 2 0 σc (u) ∂u du − R∞ 3 3 me u 2 F (u, r, t) du 0 21 R ∞ u2 F (u, r, t) du 1 2e 0 σc (u) , R∞ 3 3 me u 2 F (u, r, t) du

(2.47) (2.48) (2.49) (2.50)

0

where the denominator of Eqs. 2.49-2.50 represents the electron mean energy Z ∞ 3 ε= u 2 F (u, r, t) du . 0

Finally, the last term on the RHS of Eq. 2.38, representing the electron energy transferred in collisional (elastic and inelastic) processes is given by [Alves et al. 1997] δ(nε) δ(nε) δ(nε) = |el + |inel (2.51) δt δt δt 1 Z ∞ δ(nε) 2me 2e 2 kB Tg ∂F (u, r, t) 2 |el = N ne (r, t) du σel (u) u F (u, r, t) + δt ma me e ∂u 0 (2.52) X νj (r, t) δ(nε) |inel = N ne (r, t) Vj , (2.53) δt N j


35

CONSIDERATIONS

where ma is the mass of the neutral atom or molecule, σel (u) is the electron-neutral elastic collision cross-section, and Vj is the excitation energy of process j. The high mobility of electrons justifies a space-time calculation of their transport parameters, by using some consistent physical model. To solve this problem, some authors [Sommerer and Kushner 1992] have calculated the space-time profiles of the eTP by using a multidimensional, time-dependent MC code coupled to the fluid equations of the model. In order to avoid this time-consuming hybrid approach, the eTP 2.44 and 2.47-2.53 were defined here in terms of an EEDF that separates the contribution of the electron density (cf. Eq. 2.45), keeping however a space and time dependent profile. Even with this approximation we need to calculate F (u, r, t), which can require a considerable computational effort. In order to circumvent these difficulties, recent fluid models [Nienhuis et al. 1997] have adopted the so-called local mean energy approximation (LEA) by which the space-time dependence of the EEDF is introduced through the local electron mean energy value, i.e., F (u, r, t) ≡ F (u, ε(r, t)). This closure condition can be used to calculate the profile of the eTP in the following way. First, the different eTP are tabulated as a function of the electron mean energy by solving the stationary, space-independent EBE (written under the classical two-term approximation), for various electric field values. Second, the space-time profiles of the eTP are obtained from this table by using, at time t and for each position r, the values of the local mean energy profile ε(r, t), as obtained from the solution to the electron mean energy equation within the fluid model. Note that if the eTP are calculated using Eqs. 2.44 and 2.47-2.53, obtained from the twoterm approximation form of the EBE Eqs. 2.41-2.42, then, for coherence, one should adopt the electron flux Eqs. 2.28 and 2.43 instead of Eqs. 2.29 and 2.35, the latter obtained by applying the small anisotropy approximation to the general hydrodynamic momentum equation. The difference between these flux equations concern not only the presence of thermal gradients (associated to the spatial variation of De and Dε ), but also the use of non-equilibrium variation expressions for the particle and energy mobility and diffusion coefficients. Note that Eqs. 2.262.27 and Eqs. 2.36-2.37 can be obtained from Eqs. 2.47-2.50 by (i) assuming a constant total electron-neutral momentum transfer collision frequency νce−a = ν e−a c ; (ii) assuming an equilibrium Maxwellian EEDF at temperature kB Te /e = (2/3)ε. In this case, the eTP 2.49-2.50 become 5 e 5 µe = 3 3 me νce−a 5 5 kB T = De = 3 3 me νce−a

µε =

using (i)

(2.54)

Dε

using (i) and (ii) .

(2.55)

Most rf discharge models adopt the equilibrium version of the particle and energy flux equations 2.29 and 2.35, which happens to favour some numerical properties of the model like stabil-


36

CONSIDERATIONS

ity and convergence speed. In this thesis, we have adopted a set of electron transport equations that was deduced under the same approximation framework used to calculate the EEDF and the corresponding eTP: the two-term expansion of the EBE. Additionally, a Maxwellian distribution function was used outside the validity range of the small anisotropy approximation, i.e. when the expansion 2.39 fails.

2.2.4 Validity of drift-diffusion approximation The drift-diffusion approximation DDA is at the basis of many physical models and it corresponds to neglect the non-linear terms and the time derivative term in the momentum transport equation, as it was shown in Section 2.2.3. This approximation is justified in the collisional regime when the oriented energy is negligible compared to the thermal energy and when the variations of the electric field are small within a collision time. However, processing plasma reactors are frequently operated at low pressure, low frequency conditions; under these circumstances, it is necessary to investigate if the charged particle fluxes can be well described by DDA. In the drift-diffusion approximation the transport of charged particles is seen as the combined effect of a drift flux, caused by an electric field, and a diffusion flux, caused by density gradients (cf. Eq. 2.63 and Eq. 2.72). At fixed 13.56 MHz rf frequency (as used in the present simulations), the range of pressures for which the DDA formulation can be used depends mainly on the following two conditions. • The characteristic time between collisions with neutrals, τcβ−a = 1/νcβ−a (β = e, i), is much smaller than the rf oscillation period, Trf , τcβ−a Trf →

1 νcβ−a

13.56 × 106 1 =⇒ N . f (νcβ−a /N )

(2.56)

This condition corresponds to assume that the charged particles react almost instantaneously to the variations of the electric field. Condition 2.56 is satisfied for electrons (µe N ' 1022 V−1 s−1 cm−1 ) if p & 2 mTorr at Tg = 300 K. However, for ions (µHe+ N ' 3.3 × 1020 V−1 s−1 cm−1 [Chanin and Biondi 1957]; µH+2 ,H2 N ' 3.1 × 1020 V−1 s−1 cm−1 [Bretagne et al. 1994]; µSiH±3 ,SiH4 N ' 1019 V−1 s−1 cm−1 [McDaniel and Mason 1973; Perrin et al. 1996], cf. Section 2.4.3), condition 2.56 is verified at Tg = 300 K for pressures above 0.5 Torr only. For this reason, it is possible to use the stationary momentum transfer equation 2.63 for electrons, while it is necessary to keep the time derivative term in the momentum transfer equations 2.72-2.76 for the ions. • The collisional mean-free-path, λβ−a (β = e, i), is much smaller than the characteristic c gradient lengths, L, in the discharge


37

CONSIDERATIONS

vβ τcβ−a ' λβ−a L =⇒ N c

vβ β−a (νc /N )L

.

(2.57)

This condition corresponds to the so-called hydrodynamic condition, which limits the charged particle motion to a collision dominated transport regime. In this case, the different terms in the charged particle momentum transfer equation 2.63 and equation 2.72 should have the same order of magnitude and so, by using also equation 2.57, nβ vβ νcβ−a '

∇pβ nβ k B Tβ ' mβ mβ L

=⇒ =⇒

nβ vβ νcβ−a '

e nβ E mβ

=⇒

k B Tβ 1 mβ νcβ−a L λβ−a mβ vβ2 ' kB Tβ c kB Tβ L β−a 2 eEλc ' mβ vβ kB Tβ . vβ '

(2.58) (2.59)

These results confirm the validity of the small anisotropy approximation used in deducing the charged particle transport equations: the oriented energy gained by a particle, from the electric field between two consecutive collisions, is much smaller than its thermal energy. The above results can also be used to rewrite Eq. 2.57 as r p r kB Tβ /mβ k B Tβ m β µ β N N β−a = e e L (νc /N )L

.

(2.60)

Condition 2.60 is satisfied for both, electrons and ions, if p & 70 mTorr at Tg = 300 K and L ' 0.1 cm. The simulations carried out in this work are for ccrf discharges (∼3 cm interelectrode distance), operating at frequency 13.56 MHz and pressures above ∼ 70 mTorr (for silane-hydrogen mixtures) or 250 mTorr (for pure helium or hydrogen). Under these conditions, equations 2.56 and 2.60 are satisfied for electrons, and the DDA form of Eq. 2.63 can be used in the model. It should be noted that, in general, fluid models based on the DDA produce reliable results even for pressures bellow the validity limits of Eqs. 2.56 and 2.60 (see, e.g. [Stewart et al. 1994; Surendra 1995])

2.3 Model equations 2.3.1 Introduction In order to describe the charged particle transport in ccrf discharges, this thesis adopts a fluid representation based on the solution to the particle transport equations, coupled with Poisson’s equation and the electron mean energy equation. This fluid description incorporates a certain number of improved features already used by other authors:


38

CONSIDERATIONS

• i) The electron description, which initially used the continuity and momentum equations only [Ward 1958, 1962], now includes the third moment of the EBE accounting for the electron mean-energy transport [Graves and Jensen 1986; Richards et al. 1987]; • ii) an effective electric field replaces the standard electric field in the ion momentum equation [Richards et al. 1987; Nienhuis et al. 1997], which is written in a more complete form; • iii) The closure condition for the equations system is based on the local energy approximation [Nienhuis et al. 1997] rather than the local electric field approximation [Boeuf 1987, 1988]. The fluid formulation adopted here puts forward some physical aspects formerly set just to help the numerical resolution. As seen in Section 2.2.3, previous works adopted the driftdiffusion form of the electron and energy flux equations 2.29 and 2.35, which were obtained from the corresponding general hydrodynamical equations by dropping non-linear terms and by further considering a constant electron-neutral momentum transfer collision frequency, small temperature gradients and a Maxwellian EEDF at temperature k B Te 2 = ε e 3

.

(2.61)

The immediate consequences of such an approach are the following: i) the thermal diffusion term [cf. Golant and Sakharov 1980, pg. 183], associated to temperature gradients is removed from the flux equations. As stated by [Boeuf and Pitchford 1995a], the preservation of this term in the energy equation is responsible for a number of numerical difficulties; ii) the transport parameters 2.26-2.27 and 2.36-2.37 are written under a Maxwellian hypothesis, which puts away the non-equilibrium information contained in Eqs. 2.47-2.50. In order to correct these aspects, the electron flux equations adopted here were obtained from the two-term approximation form of the non-equilibrium EBE. For coherence, the eTP were calculated from Eqs. 2.47-2.50, which are deduced in the same approximation framework. Moreover, the ion momentum transfer equation is written in its complete form, by taking into account the ion inertia term ∇·(nv i v i ). This strong non-linear term was already considered by [Meyyappan and Govindan 1993; Nitschke and Graves 1994] in order to better describe the sheath regions where ion velocities may become considerable. In our case, the inclusion of the ion inertia term is done by decomposing the ion momentum transfer equation into (i) an ion drift-diffusion flux equation, having the same form as the electron flux equation, but involving an effective electric field ; (ii) an evolution equation for the effective electric field [Richards et al. 1987; Lymberopoulos and Economou 1993a; Young and Wu 1993a]. The ion effective electric field is more than a mere mathematical ability that allows the inclusion of time derivative


39

CONSIDERATIONS

and inertia terms in the ion momentum transfer equation. It has a physical deep meaning, representing an electric field with which the ion species are in equilibrium. In the following, we present the system of fluid equations and boundary conditions used in the model. All equations are written as a function of reduced parameters, calculated with respect to the total gas density N (r, z) = p(r, z)/kB Tg (r, z). In the case of a gas mixture P N (r, z) = α Nα (r, z), where Nα (r, z) = pα (r, z)/kB Tg (r, z) is the density of gas component

α in the mixture (pα representing its partial pressure). The equations are written in cylindrical

geometry (r, ϕ, z), assuming an azimuthal symmetry due to the rf reactor configuration (cf. Chapter 5).

2.3.2 Electron transport equations The fluid model describes the electron transport by using the continuity, the momentum transfer and the mean energy equation obtained from the zero-, first- and second-order moments of EBE (written under the two-term approximation). These equations are written as follows: • the continuity equation (cf. Eqs. 2.17 and 2.28) ∂ne 1 ∂(rΓer ) ∂Γez =− − + Se ∂t r ∂r ∂z

,

(2.62)

where Se (r, z, t) = ne (r, z, t) νe (r, z, t) represents the net creation rate of electrons due to kinetic processes, with νe /N the net creation rate coefficient. In the case of a gas mixture, the P latter is defined as νe /N = α (νeα /Nα )(Nα /N ), where νeα /Nα is the net electron creation rate coefficient in a gas component α. • the flux equation (cf. Eq. 2.28) Γeq = −(µe N )ne

Eq 1 ∂ [(De N )ne ] − N N ∂q

,

(2.63)

where (µe N )(r, z, t) and (De N )(r, z, t) are the reduced electron mobility and diffusion coefficient, respectively (cf. Eqs. 2.47-2.48) and (Eq /N )(r, z, t) is the q = r, z component of the reduced rf electric field. Note that most discharge models write the diffusion term as −

(De N ) ∂ne N ∂q

,

thus neglecting the thermal diffusion term, associated with the spatial variation of the electron temperature. • the mean-energy balance equation (cf. Eq. 2.38 and 2.43) 1 ∂(rΓεr ) ∂Γεz ∂ (ne ε) =− − − Γe · E − Sε , ∂t r ∂r ∂z

(2.64)


CONSIDERATIONS

40

where −Sε (r, z, t) is the electron energy dissipation rate in collisions (elastic and inelastic), which can be calculated by using [Alves et al. 1997] (cf. Eqs. 2.51-2.53) X Sel Sε Sinelα Nα α + = ne (r, z, t) N Nα Nα N α r Z ∞ 2me 2e Selα = ne (r, z, t) σelα (u)u2 (F (r, z, t)+ Nα mα me 0 kB Tg ∂F (r, z, t) du e ∂u X νx Sinelα = ne (r, z, t) Vxα α . Nα Nα x

(2.65)

(2.66) (2.67)

For each gas component α, mα is the mass of a neutral atom or molecule; σelα (u) is the electron-

neutral elastic collision cross-section; Vxα is the excitation energy of process x; and νxα /Nα is the electron-neutral inelastic rate coefficient, defined as (σxα (u) is the corresponding crosssection)

r Z ν xα 2e ∞ σxα (u)F (r, z, t)u du . (2.68) = Nα me 0 The sum in Eq. 2.67 is made for the electron collisions here considered: electronic (including

ionization and dissociation), vibrational and rotational excitations from ground state only. In Eq. 2.64, the quantities Γεq (r, z, t) are the q components of the electron energy flux which, in the DDA, are given by Γεq = −(µε N ) εne

Eq 1 ∂ [(Dε N )εne ] − N N ∂q

(2.69)

where (µε N )(r, z, t) and (Dε N )(r, z, t) are, respectively, the reduced mobility and diffusion coefficient for energy transport εTP (cf. Eqs. 2.49-2.50). As in the electron flux equation 2.63, the variation of the diffusion coefficient for energy transport is included in Eq. 2.69. The solution to the energy balance equation gives the space-time profile of the electron mean energy which, in the framework of the LEA, is used to feed the model with non-local transport features.

2.3.3 Ion transport equation The ions cannot respond to the rapid oscillations of the rf electric field due to their high inertia, and thus they receive very little energy from the electric field. Furthermore, the energy transfer between electrons and ions (with very different masses) is quite inefficient. These facts justify that the ion mean energy is much smaller than the electron mean energy, being similar to the neutral thermal energy. Thus, the fluid model describes the ion transport by considering only the first two moments of the ion Boltzmann equation, corresponding to the continuity and momentum transfer equations. These equations are written as follows:


41

CONSIDERATIONS

• the continuity equation (cf. Eq. 2.17) ∂ni 1 ∂(rΓir ) ∂Γiz =− − + Si , (2.70) ∂t r ∂r ∂z where Γiq (r, z, t) ≡ ni (r, z, t)viq (r, z, t) are the q components of the ion flux (viq are the corresponding components of the ion drift velocity), and Si (r, z, t) = ni (r, z, t)νi (r, z, t) represents the net creation rate of ions due to kinetic processes, with νi /N the net creation rate coefficient. Equation 2.70 is written for each ion species i = p, n, where p, n accounts for positive or negative ions. • the momentum transfer equation (cf. Eq. 2.18) ∂(ni v i ) e 1 = −∇r,z (ni v i v i ) ± ni E − νci−n ni v i − ∇r,z pi , (± for i = p, n) ∂t mi mi

(2.71)

where νci−n /N is the ion-neutral momentum transfer rate coefficient. Equation 2.71 has been obtained by dropping the contribution of the small viscous tensor, while keeping the time derivative ∂(ni v i )/∂t and the inertia ∇r,z · (ni v i v i ) terms. The contribution of the latter is more significant in the sheaths (where the ion velocities and density gradients are specially important) than in the plasma bulk. Equation 2.71 can be decomposed by developing its non-linear inertia term ∇ r,z · (ni v i v i ) and by using Eq. 2.70 to yield, for small relative ion temperature gradients (∇ r,z Ti )/Ti (∇r,z ni )/ni ), Γiq = ±(µi N )ni

Eieff q N

−

Di N ∂ni N ∂q

∂viq ∂ ∂ e = − v ir + v iz (Eq − Eieff ) v iq − ν i v iq ± q ∂t ∂r ∂z mi

(2.72) (± for i = p, n)

(2.73)

where, for each ion species i, µi N = e/[mi (νci−n /N )] and Di N = (µi N ) (kB Ti /e) are the (r, z, t) is the q comporeduced mobility and diffusion coefficients, respectively, whereas Eieff q nent of the so-called effective electric field [Richards et al. 1987]. With the introduction of Eieff , the ion momentum transfer equation 2.72 assumes the same drift-diffusion form as the q corresponding electron equation 2.63, which is especially adequate for a numerical treatment (cf. Chapter 3). As pointed out by [Gogolides and Sawin 1992], an alternative way to treat this problem is to use an effective ion mobility instead of an effective electric field. The left-hand-side of Eq. 2.73 can be evaluated from Eq. 2.72 by assuming low-varying ion mobilities and diffusion coefficients with time, yielding ∂(Eieff /N ) Di N ∂ ∂viq q ' ±(µi N ) − ∂t ∂t N ∂t

1 ∂ni niq ∂q

(2.74)


42

CONSIDERATIONS

or, by further neglecting the time variations of the relative ion density gradient, ∂(Eieff /N ) ∂viq q ' ±(µi N ) . (2.75) ∂t ∂t From Eq. 2.75, we conclude that Eieff defines an electric field with which the ion species i is q in equilibrium. This physical interpretation for the effective electric field justifies its use when the ion boundary conditions (cf. Section 2.3.5) or the field dependence of the ion mobilities (cf. Section 2.4.3) are discussed. The equation that describes the space-time evolution of (Eieff /N )(r, z, t) is finally given by q replacing Eq. 2.75 into the left-hand-side of Eq. 2.73, yielding ∂(Eieff /N ) ∂ νi 1 ∂ q v ir v iq − =− + v iz vi ± νci−n ∂t µi N ∂r ∂z µi N q

eff E q E iq − N N

!

.

(2.76)

Eq. 2.76 has highly non-linear characteristics, which can seriously affect the good convergence properties of the global equation system. Although this problem can be controlled by assuming an adequate dependence of ion mobilities with the effective electric field (cf. Section 2.4.3), most rf discharge models prefer to adopt simplified versions of Eq. 2.76, either by just taking Eieff ' Eq q as in [Boeuf 1987, 1988; Lymberopoulos and Economou 1993b; Dalvie et al. 1993; Boeuf and Pitchford 1995a], or by neglecting the first two terms on the RHS of Eq. 2.76 ) /∂t ' ± νci−n (Eq − Eieff ∂Eieff q q as considered by [Richards et al. 1987; Gogolides and Sawin 1992; Passchier and Goedheer 1993a,b; Young and Wu 1993b; Lymberopoulos and Economou 1993a, 1994, 1995; Goedheer et al. 1995; Nienhuis et al. 1997].

2.3.4 Poisson’s equation The self-consistent treatment of the charged particle transport in ccrf discharges is ensured by coupling the above set of equations with Poisson’s equation. The electric field in Eqs. 2.63-2.69 and 2.71 is strongly dependent on both the external applied voltage and the internal chargedparticle separation. The electric field writes E(r, z, t) = −∇r,z V (r, z, t)

,

(2.77)

where V (r, z, t) is the corresponding potential within the discharge calculated by using Poisson’s equation (assuming single charged ions) 1 ∂ ∂V ∂2V e r + =− 2 r ∂r ∂r ∂z ε0 where ε0 is the vacuum permittivity.

X p

np −

X n

nn − n e

!

,

(2.78)


43

CONSIDERATIONS

2.3.5 Boundary conditions The above physical model, describing the charged-particle transport within a plasma device, is to be solved over a specified integration domain. In order to fully define this mathematical problem, a set of adequate boundary conditions is required. Note that the correct simulation of the device is strongly dependent on the boundary conditions adopted. Figure 2.1 gives a schematic representation of the two-dimensional (2D) integration domain D(r, z), corresponding to the central region of the real cylindrical ccrf plasma reactor. The

simulation domain

D

Grounded electrode

r

B

Grounded grid

Driven electrode 0 A

C

z Figure 2.1: Schematic representation of the simulation domain and its boundaries.

polygon ABCD limits the model’s domain and defines two kinds of boundaries: • the physical boundaries AB, BC and DC (so-called external boundaries) representing the rf electrode, the grounded grid and the grounded electrode, respectively. • the internal segment AD (a so-called internal boundary) representing the symmetry axis of the reactor. The transport Eqs. 2.62-2.70 and 2.72, and Poisson’s Eq. 2.78 constitute a system of six (three scalar + three vectorial) first-order and one (scalar) second-order differential equations that require eight boundary conditions to be solved. At the reactor axis (corresponding to the r = 0 radial position), a set of von Neumann symmetry boundary conditions is imposed. This corresponds to assume a zero radial derivative for the charged particle density, mean energy and potential. ∂nβ = 0 (β = e, i) ∂r ∂(ne ε) = 0 ∂r ∂V = 0 . ∂r

(2.79) (2.80) (2.81)


CONSIDERATIONS

44

At all other boundaries, we have chosen to adopt physically meaningful flux boundary conditions, instead of either just imposing zero densities or zero density gradients at the discharge boundaries [Oh et al. 1990; Makabe et al. 1992; Nienhuis et al. 1997]. Consequently, at each physical boundary (electrode, wall), the perpendicular components of the different particle and energy fluxes are set to verify the Dirichlet boundary conditions [Alves et al. 1997; Punset 1998; Hagelaar et al. 2000] 1 ne hvi 2 1 ne hvui = 2 Epeff⊥ 1 = np vthp + γp np (µp N ) 4 N = 0 ,

(Γe )⊥ =

(2.82)

(Γε )⊥

(2.83)

(Γp )⊥ (Γn )⊥

(2.84) (2.85)

where hvi and hvui represent the average values of v and vu over the EEDF, vthp = (8kB Tp /πmp )1/2 is the thermal velocity for the positive ion species p, and   1 if Epeff⊥ is directed towards the wall γp =  0 if Epeff⊥ is in opposite direction to the wall .

Conditions 2.82-2.83 were obtained for totally absorbing physical boundaries, by imposing that all backward perpendicular fluxes are equal to zero. In the case of the electrodes, this

assumption corresponds to a perfect conductor condition. For coherence, conditions 2.82-2.83 where obtained under the small anisotropy approximation [Alves et al. 1997], also used to write the two-term development of the EBE and to deduce the electron transport equations 2.622.69. In the framework of this approximation, the electron diffusion and the drift fluxes almost compensate each other, yielding a small global flux that mixes the effects of these two kinds of motion. For positive ions, however, it is possible to separate between the thermal and the drift motion, as shown in condition 2.84. The latter defines a forward thermal ion flux equal to 41 np vthp for an isotropic ion distribution function, and a forward drift ion flux equal to γ p np µp Epeff⊥ , that depends on the direction of the perpendicular electric field. Note that the ion drift appearing in the boundary condition 2.84 is considered to occur under the influence of the effective electric field with which the ions are in equilibrium. Condition 2.85 expresses the confinement of negative ions in the discharge bulk, due to the influence of the sheath electric field. For ccrf discharges operating in the α regime the secondary electron emission is not necessary for discharge maintenance. This is the case of the simulation presented in this work, and therefore the contribution of this process to boundary conditions is neglected.


45

CONSIDERATIONS

At each physical boundary (electrode, walls) the applied potential must satisfy the Dirichlet boundary conditions V =

Vdc + Vrf cos(ωt) 0

at driven electrode at grounded electrode and walls ,

(2.86)

where ω = 2πf (f is the rf oscillation frequency), Vrf is the maximum tension applied to the driven electrode, and Vdc is an offset potential termed self-bias voltage, that develops in the case of an asymmetric configuration reactor (cf. Section 1.4.2). In the case of a large-surface electrodes, the potential variation along them must be included in the electrical boundary conditions. We assume here that the electrode surfaces are equipotential and so that no current flows along the electrodes and grid wall . To conclude, when the integration path is limited by two physical boundaries (wall-to-wall path; AB to CD) , there are four boundary conditions 2.82-2.86 to be satisfied at each wall. Alternatively, when the integration follows an axis-to-wall path (AD to BC), the system must satisfy conditions 2.79-2.81 at the reactor axis and 2.82-2.86 at the wall, again a total of eight boundary conditions.

2.4 Closure conditions for the moment equation system The fluid description of the charged particle transport in a discharge plasma involves the solution to a truncated set of moment equations, obtained from the integration of the corresponding kinetic Boltzmann equation in velocity space. As discussed in Sections 2.1 and 2.2, the truncation of this chained equation set leads to an open problem, where the number of unknowns exceeds the number of equations. The solution to this problem requires the introduction of adequate closure conditions, which are strongly dependent on the particle species. In the case of ions, for which only the first two moment equations are considered (i.e., the continuity and the momentum-transfer equations, cf. Section 2.3), it is reasonable to assume equilibrium conditions, thus adopting a maxwellian IDF at gas temperature (cf. Section 2.1). This closure condition for ions implies that the reduced ion diffusion coefficient can be calculated from its equilibrium expression Di N = µ i N

k B Tg e

,

(2.87)

where the reduced ion mobility can be directly obtained from experiment. In the case of electrons, for which the first three moment equations are considered (i.e., the continuity, the momentum-transfer and the energy-balance equations, Section 2.3), we can no longer assume equilibrium conditions, meaning that the EDF is to be obtained from the


46

CONSIDERATIONS

Process

Electron ionization process e + He −→ 2e + He+

References

EBE

[Alves and Ferreira 1991]

Table 2.1: Mechanisms for charged particle production in helium.The label EBE means that the rate coefficient is calculated from the corresponding electron collision cross-section, using the EEDF obtained from the solution to the homogeneous EBE. solution to the corresponding non-equilibrium Boltzmann equation (cf. Sections 2.1 and 2.2). If the electron transport equations are obtained from the two-term approximation form of the EBE, then the eTP and several particle and energy rate coefficients can be defined as integrals of various cross-sections over the space-time dependent EEDF (cf. Section 2.2.3). The latter can be deduced by solving the stationary, space-independent (homogeneous) EBE (written under the classical two-term approximation) for various electric field values, and by using the LEA corresponding to the following closure condition for electrons F (u, r, z, t) ≡ F (u, ε(r, z, t))

.

The problem becomes completely closed with the definition of a kinetic model, accounting for the production of charged particles due to collisional processes. In such a model, each electron-neutral encounter is characterized by a collisional cross-section, which is used in Eq. 2.68 to calculate the corresponding rate coefficient. In general, all other processes (electronion and ion-ion recombination; ion-neutral collisions) are described through experimentally measured rate coefficients.

2.4.1 Kinetic model All kinetic data used in the fluid model to simulate ccrf discharges in pure helium and hydrogen and in silane-hydrogen mixture, is listed in Tables 2.1-2.3. We consider here electron collisions only with ground state neutrals, yielding electronic (including ionization and dissociation), vibrational and rotational excitations. We have adopted the electron cross-sections compiled in reference [Bordage 1995; Perrin et al. 1996] for SiH4 , in reference [Alves and Ferreira 1991] for He and in reference [Loureiro and Ferreira 1989; Tawara et al. 1990] for H 2 . The rate coefficients were mainly taken from the review works of [Kushner 1988; Perrin et al. 1996]. Although the plasma environment is extremely rich in kinetic processes, we have restricted ourselves to the description of charged particle, thus considering the following species only: + − + + + electrons and 6 ions (SiH+ 2 , SiH3 , SiH3 , H , H2 , H3 ) for discharges in silane-hydrogen mix-


47

CONSIDERATIONS

Rate coefficient(a) (cm3 s−1 )

References

2e + H+ 2

EBE

[Loureiro and Ferreira 1989]

2e + H+ + H

EBE


9.75 × 10−8 Te−0.5 5.66 × 10−8 Te−0.5 2.62 × 10−13 Te−0.5

[Perrin et al. 1996; Chan 1983] [Perrin et al. 1996; Chan 1983] [Perrin et al. 1996; Chan 1983]

2.10 × 10−9

[Theard and Huntress 1974]

Process

Electron ionization processes e + H2

0.93(b)

−→

0.07

(b)

−→

Electron-ion recombination e + H+ −→ H2 + H 3 e + H+ −→ H+H 2 e + H+ −→ H + hν Ion-neutral collisions −→ H+ H+ 3 + H + 1.71eV 2 + H2

Table 2.2: Mechanisms for charged particle production and destruction in hydrogen. Te is in eV. The label EBE means that the rate coefficient is calculated from the corresponding electron collision cross-section, using the EEDF obtained from the solution to the homogeneous EBE. (b) Branching ratio from [Šimko et al. 1997; Tawara et al. 1990]. (a)

tures; electrons and He+ for discharges in pure helium, electrons and three positive ions ( H + , + H+ 2 , H3 ) for discharges in pure hydrogen. The present work is not interested in studding the

evolution to steady-state of the different neutral plasma species, as they diffuse towards the reactor substrate, producing collisions between them and with the background gas. Therefore, the model does not take into account kinetic processes involving excited states or other neutral species different from the background gas, like silane radicals or atomic hydrogen. Electron-ion recombination occurs either by radiative recombination (for a single positive ion like H+ ), or through the very efficient dissociative recombination mechanism [Perrin et al. 1996]. X + + e− −→ X + hν XY + + e− −→ X + Y

radiative recombination

(2.88)

dissociative recombination

(2.89)

The corresponding rate constants are found to be a decreasing function of the electron kinetic temperature. krr ≈ 10−13 Te−1/2

(cm3 s−1 )

radiative recombination

(2.90)

kdr ≈ 10−7 Te−1/2

(cm3 s−1 )

dissociative recombination,

(2.91)

according to [Kushner 1988; Kline 1982; Chan 1983]. The rate constants Eqs. 2.90-2.91 are re-calculated after each iteration of the numerical code, using the updated values of the electron


48

CONSIDERATIONS

Rate coefficient(a) (cm3 s−1 )

References

2e + SiH+ 3 +H

EBE

[Bordage 1995; Chatham et al. 1984]

SiH+ 2 H+ 2

EBE

[Bordage 1995; Chatham et al. 1984]

EBE


EBE


Electron attachment e + SiH4 −→ SiH− 3 +H

EBE

[Haaland 1990; Bordage 1995]

Electron detachment e + SiH− −→ 2e + SiH3 3

EBE

[Haaland 1990; Bordage 1995]

Electron-ion recombination e + SiH+ −→ SiH + H 2 e + SiH+ −→ SiH2 + H 3 e + H+ −→ H2 + H 3 e + H+ −→ H+H 2 e + H+ −→ H + hν

1.69 × 10−7 Te−0.5 1.69 × 10−7 Te−0.5 9.75 × 10−8 Te−0.5 5.66 × 10−8 Te−0.5 2.62 × 10−13 Te−0.5

[Perrin et al. 1996; Kline 1982] [Perrin et al. 1996; Kline 1982] [Perrin et al. 1996; Chan 1983] [Perrin et al. 1996; Chan 1983] [Perrin et al. 1996; Chan 1983]

Ion-ion recombination − SiH+ + SiH −→ SiH3 + SiH3 3 3 + − SiH2 + SiH3 −→ SiH2 + SiH3 − H+ −→ SiH4 + H2 3 + SiH3 + − H2 + SiH3 −→ SiH3 + H2 H+ + SiH− −→ SiH4 3

7.22 × 10−6 (Tg /300)−0.5 7.10 × 10−6 (Tg /300)−0.5 1.27 × 10−6 (Tg /300)−0.5 8.76 × 10−7 (Tg /300)−0.5 4.51 × 10−7 (Tg /300)−0.5

[Perrin et al. 1996; Hickman 1979] [Perrin et al. 1996; Hickman 1979] [Perrin et al. 1996; Hickman 1979] [Perrin et al. 1996; Hickman 1979] [Perrin et al. 1996; Hickman 1979]

Ion-neutral collisions SiH+ + SiH −→ SiH+ 4 2 3 + SiH3 + SiH2 + SiH4 −→ Si2 H+ 4 + H2 + SiH+ + H −→ SiH 2 2 3 +H + H+ + SiH −→ SiH 4 3 3 + H2 + H2 + H2 + SiH4 −→ SiH+ 3 + H2 + H H+ + SiH4 −→ SiH+ 3 + H2 + H+ + H −→ H 2 2 3 +H

1.07 × 10−9 2.50 × 10−10 1.01 × 10−10 5.16 × 10−10 6.23 × 10−10 5 × 10−10 2.10 × 10−9

[Kushner 1988; Henis et al. 1972] [Chatham and Gallagher 1985] [Kushner 1988; Allen et al. 1977] [Kushner 1988; Allen et al. 1977] [Kushner 1988; Allen et al. 1977] [Kushner 1988] [Theard and Huntress 1974]

Process

Electron ionization processes e + SiH4

0.43(b)

−→

0.57

(b)

−→

e + H2

0.93(c)

−→

0.07(c)

−→

2e + 2e +

+ H2

2e + H+ + H

Table 2.3: Mechanisms for charged particle production and destruction in silane-hydrogen gas mixtures. (a) Te in eV and Tg in K. The label EBE means that the rate coefficient is calculated from the corresponding electron collision cross-section, using the EEDF obtained from the solution to the homogeneous EBE. (b) Branching ratio from [Kushner 1988]. (c) Branching ratio from [Šimko et al. 1997; Tawara et al. 1990].


49

CONSIDERATIONS

mean energy profile ε(r, z, t) - cf. Eq. 2.61. For the ion-ion recombination rate coefficients we have adopted the scaling formula proposed by [Hickman 1979] which is a function of the ionion reduced mass miir and the electron affinity EA of the neutral species parent of the negative ion, krec = 5.34 10−7 EA−0.4 miir Here, the reduced mass miir

−0.5

−0.5

(Ti /300)−0.5

(cm3 s−1 ).

(2.92)

is in a.m.u. and the electron affinity is in eV (1.406 for SiH3

[Nimlos and Ellison 1986]). The rate constants for ion-neutral collisions, in pure hydrogen and silane-hydrogen mixtures, have been obtained from [Kushner 1988; Henis et al. 1972; Allen et al. 1977; Theard and Huntress 1974; Chatham and Gallagher 1985].

2.4.2 Electron transport parameters Electron transport parameters play a key role in discharge modeling, which justifies all effort in improving their correct evaluation. In ccrf discharges the problem is mainly with the evaluation of eTP and rate coefficients within the discharge sheaths, especially if a description of the electron energy transport is intended. The question is closely related to the choice between the local electric field approximation or the local mean energy approximation as closure condition for electrons. The need for such approximations comes from the lack of information upon the space-time dependent EDF (cf. Sections 2.2.3 and 2.4). In earlier discharge models the electron transport parameters were considered either constant [Graves and Jensen 1986; Richards et al. 1987] or dependent on the reduced electric field [Boeuf 1987, 1988]. Additionally, the ionization and the energy-loss rate coefficients were supposed dependent on the electron energy via a fitting formula [Golant 1959] that gives the electron mean energy as a function of the reduced electric field E/P [Richards et al. 1987; Oh et al. 1990]

E [Vcm−1 Torr−1 ] . (2.93) p This is the frame-work of the local electric field approximation LFA, already mentioned in Secεe [eV] = 5.3 + 0.033

tion 1.5.3, based on the assumption that, at a given position and time, the energy gained by the electrons from the electric field is exactly balanced by their energy losses in collisions. This equality can be expressed through Eq. 2.64 by neglecting both its LHS time derivative and the −∇ · Γε terms in its RHS. The local field approximation is valid at high pressures, when electron collision frequencies are sufficiently high as to justify the assumption of a local equilibrium with the reduced electric field.


CONSIDERATIONS

50

One of the advantages in using LFA is the fact that the variation of the transport coefficients with the electric field can be directly obtained form Townsend dc experiments [Ward 1962; Richards et al. 1987]. However, the LFA does allow the closure of the fluid equation system by calculating the EDF (and thus the eTP and the source terms) as an exclusive function of the reduced electric field. This is surely a reasonable approximation in discharge regions characterized by uniform and slowly varying fields. In ccrf discharges, however, the electric field in the space-charge regions is not uniform and has a rapid variation, which means that the local drift, diffusion and ionization rate coefficients may not be in equilibrium with the local electric field. In fact, the LFA can lead to nonphysical results within the discharge sheaths. First, the LFA overestimates the ionization process in the sheaths due to the presence of the high electric fields, which enables the electrons to achieve high energies across the space-charge region towards the plasma. Consequently, the LFA can lead to an under-estimation of the self-consistent mean electron density, when in reality the electron energy is not locally balanced due to the high spatial and temporal variation of the electric field. Second, the good behavior of the electron diffusion coefficient within the discharge sheaths is also affected by the LFA: the very strong rf electric field within a sheath induces an increase of De as electrons diffuse against the field, towards the discharge wall. This problem is usually avoided in other works by imposing a constant characteristic energy uk ≡ De /µe of 1 or 2 eV within the sheath (sometimes everywhere in the discharge), and by calculating De from uk and µe , the latter deduced from the results of a Boltzmann solver [Punset 1998; Boeuf and Pitchford 1995b]. Third, when the LFA is used to deduce the eTP in ccrf discharges from the results of Townsend discharges, the E/N values of the latter must be extrapolated up to the reduced rf field values, which are at least ten times higher. Consequently, this extrapolation gives unrealistic results for a number of discharge parameters, like the electron mean energy [Meijer 1991]. The natural dependence of the ionization rate with the electron mean energy has suggested the use of the local mean energy approximation LEA [Nienhuis et al. 1997] as a judicious correction to the nonphysical results obtained in the framework of the LFA. The LEA assumes that the normalized EEDF and its related eTP are an exclusive function of the electron mean energy, whose local space-time profile is given by solving Eq. 2.64. The procedure adopted when using the LEA can be summarized as follows: • (i) the stationary, space-independent EBE (written under the classical two-term approximation [Delcroix and Bers 1994]) is solved at different electric fields (E/N )Boltz , for the gas or mixture of gases under study; • (ii) the results so obtained are used to construct a look-up table for the reduced diffusion coefficients and mobilities 2.47-2.50 and the particle and energy rate coefficients


51

CONSIDERATIONS

2.44, 2.52-2.53, the mean values hvi and hvui used in boundary conditions 2.82-2.83 as a function of the electron mean energy ε (cf. Table 2.4); • (iii) the solution to the fluid model equations yields the space-time profile of the electron mean energy ε(r, z, t). The latter can be used to obtain the space-time profiles of the eTP and rate coefficients by interpolating the results of Table 2.4. (E/N )Boltz

ε

De N

(10−16 Vcm2 )

(eV )

(1022 cm−1 s−1 )

1.3E+1 1.4E+1 1.5E+1

... 4.7E+0 4.9E+0 5.2E+0 ...

... 5.2E+0 5.7E+0 6.1E+0 ...

Dε N

µε N

< vu >

(1022 eVcm−1 s−1 ) (1022 cm−1 s−1 ) (106 eVcm s−1 )

5.0E+1 5.7E+1 6.4E+1

... 1.5E+1 1.6E+1 1.7E+1 ...

... 7.4E+2 8.0E+2 8.6E+2 ...

kdet

kiSiH4

SiH4 kdis

(10−10 cm3 s−1 )

6.2E-1 6.1E-1 6.0E-1

(10−10 cm3 s−1 ) (10−10 cm3 s−1 ) ... 1.1E+0 1.5E+0 1.8E+0 ...

... 2.3E+0 2.7E+0 3.2E+0 ...

µe N

(1022 V−1 cm−1 s−1 ) (106 cm s−1 ) ... 1.6E+0 1.6E+0 1.6E+0 ... kloss

1.1E+2 1.2E+2 1.2E+2

katt

(10−10 eVcm3 s−1 ) (10−10 cm3 s−1 ) ... 2.7E+2 3.2E+2 3.7E+2 ... kiH2

5.1E-4 4.9E-4 4.7E-4

H2 kdis

(10−10 cm3 s−1 ) (10−10 cm3 s−1 ) ... 8.9E-1 1.2E+0 1.6E+0 ...

9.4E+0 1.1E+1 1.3E+1

Table 2.4: Tabulation of electron transport parameters and rate coefficients. Line after line, the results on this table were obtained by running a homogeneous Boltzmann equation solver for a specific (E/N )Boltz value.

Although approximate, the LEA has proved to yield physically meaningful results provided the EBE is not solved for very high values of ε, in violation of the two-term approximation. However, during the rf cycle, the electron mean energy can present very high values for example near the corner (r = R, z = 0), where the rf electrode and the grounded grid meet (cf. Figure 2.1), due to the strong intensity of the rf field therein. These high values of ε correspond to values of (E/N )Boltz that are often beyond the validity limit of the two-term approximation.


CONSIDERATIONS

52

This limitation can be partially waved by adopting a time-dependent EBE, that allows a net electron density to build-up following the production of secondary electrons [Frost and Phelps 1962; Pitchford et al. 1996]. In this case, each ionization process distributes the available energy among the two electrons involved (the scattered and the secondary); as a consequence, the magnitude of the electron distribution function tail decreases, and the electron anisotropy is reduced. In this work, however, we have used a stationary, space-independent, two-term Boltzmann solver [Alves and Ferreira 1991] to calculate the EEDFs and the corresponding eTP and rate coefficients. These calculations where carried out for a set of electric fields that ensure a small electron anisotropy (typically, (E/N )Boltz . 10−15 V cm2 or, equivalently, ε . 10 − 20 eV). For electron mean energies above ∼ 10 − 20 eV (normally up to ∼ 100 eV), the eTP and rate coefficients were deduced from a Maxwellian EEDF at temperature Te (cf. Eq. 2.61). The transition value of ε is by no means unique, and depends mainly on the gas or mixture of gases considered. In general, it is chosen as the value of ε at which the majority of the Boltzmann eTP cross the corresponding Maxwellian parameters. Contrarily to other authors, when using the LEA we have imposed no ad-hoc assumption concerning a constant electron characteristic energy in the discharge sheaths. Note that the strong decrease observed for ε, as a function of time and within a discharge sheath, yields the expected decrease of De when electrons diffuse towards the discharge wall. However, the correct determination of ε near the boundaries can constitute a serious numerical problem: (i) the solution to Eq. 2.64 gives the electron energy density ne ε, which has very low values within a discharge sheath; (ii) in general, the electron density ne is vanishingly small near the discharge boundaries; (iii) the local values of the electron mean energy (which are used to define the local values of all eTP and rate coefficients) are calculated using ε = ne ε/ne , whose evaluation can lead to numerical oscillations and thus to a lack of convergence. In order to avoid this problem, some authors [Gogolides and Sawin 1992; Passchier and Goedheer 1993a; Lymberopoulos and Economou 1993b; Passchier and Goedheer 1993b; Lymberopoulos and Economou 1993a; Boeuf and Pitchford 1995a; Fiala et al. 1994; Boeuf and Pitchford 1996] have chosen to keep the assumption uk = const (associated to the LFA), even when adopting the LEA. This ad-hoc choice turns out to have another advantage: the use of linear relationships between the transport parameters for particle and energy transport (cf. Eqs. 2.36-2.37) drastically reduces the non-linearity of the problem, contributing to both a stability enhancement and a considerable reduction of calculation time. However, simulation results strongly depend on the set of eTP used, and thus on the approximations adopted in their evaluation. Figure 2.2 shows the eTP and the εTP De N and Dε N , as a function of ε, obtained from a


53

CONSIDERATIONS

180 -1

140

-1

DeN, DεN (10 cm s )

160 120

(a)

He B

22

100 80 60

A

40 20 0 0

20

40

60

80

100

120

140

160

(b)

50

22

-1

-1


60

40

SiH4 + H2

B

30 20

A

10 0 0

20

40

60

80

100

120

140

160

(c)

H2

250

-1

-1


300

22

200 150 100

B

50

A

0 0

20

40

60

80

100

120

140

160

ε (eV)

Figure 2.2: Particle (A) and energy (B) reduced electron diffusion coefficient in helium (a), in 7% SiH4 − 93% H2 mixture (b) and in hydrogen (c). Blue and red lines are obtained from Boltzmann and Maxwellian calculations, respectively.


54

CONSIDERATIONS

(a)

10

µeN, µεN (10

22

-1

-1

-1

cm V s )

12

He

8

B

6 4

A

2 0 0

20

40

60

80

100

120

140

160

(b)

15

µeN, µεN (10

22

-1

-1

-1

cm V s )

18

12

SiH4 + H2

9 6

B

3

A

0

20

40

60

80

100

120

140

12

-1

10

H2

-1 22

µeN, µεN (10

160

(c)

-1

cm V s )

0

8 6 4

B

2

A 0

20

40

60

80

100

120

140

160

ε (eV)

Figure 2.3: Particle (A) and energy (B) reduced electron mobility coefficient in helium (a), in 7% SiH4 − 93% H2 mixture (b) and in hydrogen (c). Blue and red lines are obtained from Boltzmann and Maxwellian calculations, respectively.


CONSIDERATIONS

55

combination of Boltzmann (blue lines) and Maxwellian (red lines) calculations a) in He; b) in 7% SiH4 − 93% H2 and c) in H2 . Similarly, Figure 2.3 a), b) and c) shows µe N and µε N , as a function of ε. We have performed several numerical tests in which the Boltzmann eTP were adopted far beyond the 10 eV transition value (see red curves in Figs. 2.2 and 2.3). In general, we have obtained erroneous simulation results (especially for the values of V dc and Weff ), and in some cases a lack of convergence. Under these circumstances, we have decided to compute the above parameters using a maxwellian EEDF for energies larger than ∼ 10-15 eV.

2.4.3 Ions transport parameters We assume in this work that all ion species i are thermalized at gas temperature (T i ' Tg ), in which case the reduced diffusion coefficient can be obtained from Eq. 2.87 by using different ion mobility laws depending on the reduced effective electric field. Note that all ion mobility expressions, presented in this section, are referred to a standard gas density of 2.69×10 19 cm−3 . For the simulations at low pressures in pure helium, we consider the presence of He + only. −17 eff V cm2 ), its mobility At low fields (EHe + /N < 30 Td [Delcroix and Bers 1994]; 1 Td = 10

was calculated from the expression proposed by [Chanin and Biondi 1957] µHe+ = (2.96 × 10−3

p Tg + 3.11 × 10−2 )−1 (V−1 s−1 cm2 ) .

(2.94)

eff At higher fields (EHe + /N > 30 Td), we have adopted [McDaniel and Mason 1973; Perrin et al.

1996] eff −1/2 µHe+ = bHe (EHe (V−1 s−1 cm2 ) + /N )

(2.95)

where bHe is a normalization factor ensuring an asymptotic convergence towards the low-field mobility 2.94. + + For silane and hydrogen mixtures we consider the presence of H+ , H+ 2 , H3 , SiH2 and − eff SiH+ 3 positive ions and SiH3 negative ions. At low fields (Ei /N < 156 Td or, equivalently,

Eieff /p < 50 V cm−1 Torr−1 [Perrin et al. 1996] at Tg = 300 K), we have adopted the prediction of Langevin’s theory [McDaniel and Mason 1973; Perrin et al. 1996] to calculate the mobility of ion species i in a gas component α µLi,α =

13.9 −1/2 Tg (δα miα (V−1 s−1 cm2 ) . r ) ai,α

(2.96)

Here, δα is the polarizability of the gas molecule. (δSiH4 = 4.62 Å3 [Haaland 1990] and δH2 = 0.805 Å3 [Weast 1986]), miα r is the ion-gas reduced mass (in a.m.u.), and a i,α is a correction factor that depends on the ion-gas pair (ai,SiH4 = 4 [Reents and Mandich 1990] for i = SiH+ 3


56

CONSIDERATIONS

, SiH+ 2 ; ai,α = 1 otherwise). The Langevin’s expression 2.96 was directly used to deduce the mobilities of all (silane and hydrogen) ions in SiH4 . For hydrogen ions in H2 , we have adopted the mobilities proposed in references [Bretagne et al. 1994; Šimko et al. 1997; Phelps 1990] (see table 2.5). For silane ions in H2 , we have adopted µi,H2 = µLi,H2 f , where f = µH+3 ,H2 /µLH+ ,H = 3

2

0.72 is a correction factor that measures the systematic deviation from Langevin’s theory, of the ion mobilities in H2 . Mobility [V−1 s−1 cm2 ] µH+ ,H2 14.6 µH+2 ,H2 11.6 10.3 µH+3 ,H2 Table 2.5: Low field mobilities for hydrogen ions in H2 We have adopted an expression similar to Eq. 2.95 to deduce the high-field (Eieff /N > 156 Td) mobilities of all silane and hydrogen ions, with the exception of µ H+2 ,H2 . The mobility of + + H+ 2 is strongly dependent on the ion conversion reaction H 2 + H2 −→ H3 + H, which basically

controls its kinetics. This ion conversion reaction is very effective at low fields, hence µH+2 ,H2 is calculated for EHeff+ /N < 156 Td using the H+ 2 -H2 collision frequency (estimated from the 2

cross-section proposed in [Phelps 1990]). However, the ion conversion reaction is less probable at higher fields, which makes H+ can be 2 the most abundant hydrogen ion. In this case, µ H+ 2 ,H2 calculated using the high-field expression proposed by [Ward 1962] for a "mean" hydrogen ion " # 1.06 × 10−23 c 1− (V−1 s−1 cm2 ) , (2.97) µH+2 ,H2 = (EHeff+ /N )1/2 (EHeff+ /N )3/2 2

where

EHeff+ /N 2

2

2

is in V cm , and c is a normalization factor to be calculated by imposing

µH+2 ,H2 = 11.6 V−1 s−1 cm2 at EHeff+ /N = 156 Td. 2

The effective mobility µi , of an ion species i in a gas mixture, was calculated from the mobilities µi,α of this species in each gas component α using Blanc‘s law X (Nα /N ) 1 = µi µi,α α

(2.98)

For ccrf discharges in pure hydrogen we adopt at low fields the of Table 2.5; at high fields

+ + we use Eq. 2.97 for the H+ 2 mobility and relations similar to Eq. 2.95 for the H and H3 mobil-

ities.

2.5 The discharge electrical parameters An important cross-checking comparison between model simulations and experimental measurements involves two electrical parameters which allow a global characterization of a ccrf


57

CONSIDERATIONS

discharge: the self-bias voltage Vdc (which measures the discharge asymmetry) and the electrical power coupled Weff (which measures the effective electrical power transferred to the discharge plasma). Both of these parameters are closely related to the main condition defining a ccrf discharge: a zero time-average current-flow through the electrodes.

2.5.1 Introduction The alternate voltage applied to a capacitive rf plasma reactor drives the electrons in their oscillatory movement between electrodes, over a quasi-static ion background. This movement can lead to a charge accumulation at the end of each rf cycle if, as it happens in most cases, the reactor has an asymmetric configuration, with more surfaces grounded than driven. In this case, an axial electron current develops when the electrons are pushed away from the (small surface) driven electrode, during the expansion (or contraction) phase of the rf (or grounded) sheath. Conversely, a larger axial electron current appears when electrons are collected from the (large surface) grounded electrode and walls, being pulled to the driven electrode, during the contraction (or expansion) phase of the rf (or grounded) sheath (cf. Section 1.4.2). Consequently, and in order to ensure that the rf reactor is operating in a pure capacitive mode (corresponding to a zero time-average current flowing through the electrodes), its external electric circuit must include a blocking capacitor CB , having negligible impedance at the driving frequency, in order to compensate for charge accumulation due to the asymmetric configuration. Figure 2.4 represents the schematic diagram and the equivalent electric scheme of an asymmetric ccrf reactor (see also Figure 1.2). The capacitances C d and Cg are associated with

CB

Driven electrode sheath Plasma region sheath

r Grounded grid

0

Ic

VC ~

Vrf cos ω t

Vd Vp Vg

Grounded electrode

Cd rf reactor

−1

Yp

Cg

z

a)

b)

Figure 2.4: Schematic diagram of a ccrf reactor (a) and echivalent electric scheme (b).

the driven and grounded sheaths, respectively, whereas Y p accounts for the plasma admittance.


58

CONSIDERATIONS

The sinusoidal rf current that flows in the circuit, Irf (t) = I˜max cos(ωt) (I˜max is the complex current amplitude), produces a voltage drop across the plasma given by: Irf (t) (2.99) Yp Eq. 2.99 gives a voltage that is linear in the rf current, yielding no harmonic generation. This Vp (t) =

is not the case for the potential drop Vs across the equivalent capacitance Cs = Cg Cd /(Cg +Cd ), which represents the series combination of the two sheath capacitances. In fact, C g and Cd depend on the sheath thickness, yielding a nonlinear equivalent C s which is responsible for the production of harmonics at the rf driving frequency. However, it is a good approximation to Fourier decompose the voltage Vs , taking its fundamental component as the main response to the sinusoidal sheath current, by introducing an average capacitance C s : Vs (t) ≡ Vg (t) − Vd (t) ' Vs0 + V˜s1 cos(ωt) (2.100) ˜ d[Vs1 cos(ωt)] (2.101) Irf (t) ≡ C s dt The rapid electron response to an instantaneous potential makes that I rf is mainly a conduction current, while is flowing through the plasma. In contrast to this, the current that flows through the reactor sheaths is almost entirely a displacement current, because the electron density is very small in these boundary regions. This justifies the electrical representation of the sheaths by pure capacitances. However, the global current that crosses the electrodes to flow in the external circuit is determined by the total net flux of electrons and ions over each electrode, being related to the small conduction currents in each sheath. Therefore, the total current density in each sheath, J t (r, z, t), should include its two components: the displacement current, J D (r, z, t), and the net conduction current, J c (r, z, t), respectively

Jt = JD + Jc ∂E J D = ε0 ∂t ! X X Jc = e Γp − Γn − Γe ' σE + J diff p

P

(2.102) (2.103) .

(2.104)

n

In equation 2.104, we have introduced the total plasma conductivity σ ≡ e(µ e ne + P eff ' E], and the net diffusion current (cf. Eqs. 2.63 p µp np + n µn nn ) [assuming E i

and 2.72)

J diff ≡ −e

X p

(Dp N )

X ∇ ∇ ∇ np + e (Dn N ) nn + e [(De N )ne ] N N N n

.

(2.105)


59

CONSIDERATIONS

From equation 2.104 it is possible to evaluate the global current that flows on the external circuit of the reactor, by integrating the axial component of the conduction current density for example over the surface of the driven electrode [see Figure 2.4(a)] Ic (t) = −Irf (t) =

Z

R

Jcz (r, 0, t)2πrdr .

(2.106)

0

The voltage VC , across the blocking capacitor CB , can now be calculated using Ic (t) = CB

dVC (t) . dt

(2.107)

2.5.2 Self-bias voltage In order to ensure that the rf reactor operates in a steady-state pure capacitive mode, the timeaverage current flowing through its electrodes, I c (t), must be zero [Meijer 1991; Lieberman and Lichtenberg 1994; Goedheer et al. 1995] Z t+Trf 1 Ic (t0 )dt0 ≡ 0 I c (t) ≡ Trf t

(2.108)

or, by using equation 2.107, dVC (t) =

1 Ic (t)dt = 0 =⇒ VC (t) = const . CB

(2.109)

Furthermore, the blocking capacitor CB has negligible impedance at the driving frequency (CB ' 50 pF), which in the steady-state limit gives dVC (t) =

1 Ic (t)dt ' 0 =⇒ VC (t) ' VC (t) ≡ Vdc = const CB

(2.110)

where Vdc is the dc self-bias voltage that develops in the case of an asymmetric configuration reactor. In fact, the steady-state voltage drop across the rf reactor is [see Figure 2.4(b) and Eq. 2.100] Vs (t) + Vp (t) = VC (t) + Vrf cos(ωt) ' Vdc + Vrf cos(ωt)

(2.111)

and by calculating the time-average of Eq. 2.111, over the rf period, we obtain (cf. Eq. 2.99 and 2.100) Vs0 = V g − V d = Vdc . Starting from a given value Vdc0 , the corrections to the dc self-bias voltage can be calculated from Eq. 2.106 and Eq. 2.110, within the rf period, dVdc = or, after each rf period, Vdc = Vdc0 +

Z

1 Ic (t)dt CB t+Trf

t

1 Ic (t0 )dt0 . CB

(2.112)

(2.113)


60

CONSIDERATIONS

For dominant "heavy"-ion species, the self-bias potential can be updated from Eq. 2.112, after each iteration time step. This procedure is used for example by [Boeuf and Pitchford 1995a], and can be applied to discharges in silane dominated SiH 4 +H2 mixtures and even in helium. However, for discharges in pure hydrogen or in hydrogen dominated SiH 4 +H2 mixtures (or generally, when "light"-ion species are dominant), this procedure is not always stable. The "light" ions are more sensitive to the rf field variations, which can induce the self-bias potential to be over-corrected within the rf period. In this case, we prefer the technique adopted by other authors [Kushner 1988; Goedheer et al. 1995], updating V dc from Eq. 2.113 after each rf period. An alternative procedure to calculate the dc self-bias voltage has been proposed by [Birdsall 1991], by applying Gauss’s law to the system plasma-electrodes and by imposing charge conservation. In this way, the build-up of Vdc is gradually taken into account, thus enhancing solution stability. The model prediction of the self-bias voltage is strictly dependent on the self-consistent calculation of the global current Ic (t), flowing on the external circuit of the reactor (cf. Eq. 2.106). Obviously, the latter current cannot be computed in a onedimensional model, which shows the importance of two-dimensional fluid simulations in order to fully characterize a ccrf reactor.

2.5.3 Coupled electrical power The above discussion shows that the total current flowing to the driven electrode, I t , can be obtained by integrating the axial component of the total current density over the surface of the electrode (cf. Eqs. 2.102-2.106) Z It (t) = Ic (t) + ID (t) =

R

Jcz (r, 0, t)2πrdr + 0

Z

R

JDz (r, 0, t)2πrdr .

(2.114)

0

Equation 2.114 can be used to evaluate the effective electrical power coupled to the plasma Z Trf 1 Weff ≡ [Vdc + Vrf cos(ωt)]It (t)dt . (2.115) Trf 0 The total current flowing along an electrode or a discharge wall, I k , is obtained by integrating the parallel component of the total current density over the boundary surface. For example, at a driven electrode (along which the rf potential is assumed constant, and thus E r =0 (see Section 2.3.5) we have (cf. Eqs. 2.102-2.105) Z R Z R Z Ik (t) = Jcr (r, 0, t)2πrdr + JDr (r, 0, t)2πrdr = 0

0

R

eJdiff r (r, 0, t)2πrdr

(2.116)

0

which shows that for long field wavelengths (λ ' 22 m L ' 0.5 cm, where L is a characteristic gradient length) Jdiff r (r, 0) ' 0 =⇒ Ik ' 0, or Γβr (r, 0) ' 0 charged particle fluxes.

(β = e, i) in terms of

Chapter 3 Numerics In this chapter, we discuss the numerical aspects which were used to solve the equation system derived in Chapter 2. We have chosen to discretize the partial differential equations, ensuring an adequate space-time representation of the discharge phenomena and an efficient reduction of computational time in obtaining the steady-state solution. With this end in mind, we have adopted a finite-difference method incorporating the exponential discretization scheme [Scharfetter and Gummel 1969] in the spatial treatment of the equation system, and a semi-implicit algorithm to advance the equations in time. The latter option overcomes the strong restrictions over the integration time-step, usually imposed by explicit algorithms. Similarly, Poisson’s equation was also solved using a semi-implicit technique. The steady-state solution is defined by imposing a convergence criterion upon the spacetime changes of the main plasma parameters.

3.1 General considerations The charged particle transport equations coupled with the electron mean energy transport equations and Poisson’s equation form a system of partial differential equations (PDEs) subject to specific boundary conditions in space and time (cf. Section 2.3.5). The solution to this equation system gives the space-time description of the charged-particle densities n e,p,n(r, z, t) and fluxes Γe,p,n(r, z, t) (for electrons, positive and negative ions, respectively), the electron mean energy ε(r, z, t), and the electric field distribution E(r, z, t) inside the discharge. The problem can be solved by using numerical discrete methods, which start by dividing the physical spatial domain D(r, z) (see Figure 2.1), into a set of discrete regions, limited by a finite network of points (i, j) called computational mesh or grid. The initial set of PDEs is then approximated in each discrete point/region by a set of equations describing, at each time, the various physical variables at different points within the domain D. This discretization procedure results in a huge number of algebraic equations, which correspond to the formulation

61

C HAPTER 3. N UMERICS

62

of the original problem. In contrast with analytical methods that solve the problem in a continuous range of (r, z, t) variables, the numerical discretization ensures a solution in a finite number of points, only. Obviously, an increase in the number of points of the computational mesh will result in a better description and, consequently, in a more accurate solution. When choosing a discretization method the following conditions are to be satisfied: • it must preserve the conservation laws of physical quantities; • it must ensure the physical meaning of the solution (e.g., its positivity); • it must be accurate (the accuracy quantify how much information is lost when a continuous function is discretized); We have adopted here the finite difference method (FDM) of discretization [Lapidus and Pinder 1982; Ferziger and Perić 1999]. It is relatively easy to implement and it can be applied to a rectangular discretization domain, since we have assumed an azimuthal symmetry for the cylindrical plasma reactor. The discretization domain is plotted in Figure 3.1 where the various finite-difference mesh lines, parallel to the coordinate axis, were also represented. The numerical solution of the discrete equation system advances in time, starting at t = 0 from an imposed set of initial conditions, by successively computing all physical variables at discrete time-step intervals ∆t. This iterative process continues until the steady-state convergence criterion is achieved. Throughout this chapter, the (i, j) subscripts indicate the mesh-point locations, while the superscript k is associated to an estimate at the time tk . Previous and subsequent times are given by tk−1 = tk − ∆t and tk+1 = tk + ∆t.

3.1.1 Computational mesh There are three kinds of regular polygons that can be used to divide the spatial domain in regular regions: rectangles, triangles and hexagons. Hexagonal lattice is seldom used because is difficult to implement. A triangular structure is recommended for very complicated geometries (often non-planar) and is usually adopted in the finite-element method of discretization [Lapidus and Pinder 1982]. Since we are dealing with a simple geometry, a rectangular mesh has been adopted in this work (see Figure 3.1). The computational mesh or grid is a finite set of mesh-points (nodes) Pi,j ∈ D(r, z), where i, j ∈ Z and i = 0, nr, j = 0, nz. The location of the nodes are ordered to form a regular, fixed grid or lattice. The coordinate (ri , zj ) of a mesh-point is then given by ri = ih and zj = jg,


63

respectively, where the quantities h = ri − ri−1 > 0 g = zj − zj−1 > 0 represent the grid steps in the radial r and axial z directions. The values r nr and znz correspond to the domain boundaries: the lateral grid wall is at rnr = R (R is the rf electrode radius) and the lower electrode is placed at distance znz = d from the rf electrode (located at z0 = 0). The computational grid should not be confused with the physical lateral grid of the plasma reactor. The selection of adequate spatial mesh steps (h, g) is particularly important, since the choice of high values for h or g can lead to a lack of spatial resolution, i.e., to an information loss on the short wavelength features of the numerical solution. Thus, the grid spacing is linked to the variation of variables inside the calculation domain. There is no need to employ a fine grid where the variables change slowly, and it could be necessary to use a nonuniform grid to describe regions where the variation is steep. To establish the number of points and, eventual, their distribution, exploratory calculations should be performed. Even coarse grids should give physically meaningful solutions. We have adopted first a uniformly spaced 2D grid, constituted by 33x17 = 561 mesh-points (nr = 32, nz = 16, corresponding to h = 2 mm and g = 2 mm). After the preliminary calculations, the grid was establish at 33x33 = 1089 mesh-points (nr = nz = 32, corresponding to h = 2 mm and g = 1 mm). Both grids give a reasonable description of the solution into the computation domain. Driven electrode i

i=nr

r

Grounded grid

i,j=0

j j=nz Grounded electrode

z Figure 3.1: Schematic representation of the computational mesh. The vertical boundaries are determined by the reactor axis (at r0 = 0) and the lateral grid (at rnr = R). The horizontal boundaries are defined by the rf electrode (at z0 = 0) and the grounded electrode (at znz = d).

If we try to calculate all the variables on the same grid, a particular difficulty arise. For example, if we have the potential profile at mesh nodes, in order to express the electric field values at mesh node locations, we have to consider the potential difference between two alternate


64

grid points, and not between adjacent ones. The first implication is evident. Since the potential gradient is taken from a coarser grid than we actually use, the solution accuracy decreases. The other implication, with more important consequences, is that an absurd zigzag profile for potential (e.g. Vi−1 = 100, Vi = 500, Vi+1 = 100, Vi+2 = 500 for 1D problems), can be seen at grid nodes as a zero electric field profile. Thus, a highly nonuniform potential distribution is felt by the momentum equation as a zero electric field region and such a behavior may seriously damage the solution [Patankar 1980]. The difficulty can be removed introducing a displaced grid for vectorial quantities. The immediate advantage is that the relevant values of such physical quantities are calculated now without any interpolation. Thus, the regular mesh of Figure 3.1 is used to define a staggered mesh, where different variables are calculated at different locations according to their physical meaning (see Figure 3.2). Basically, scalars quantities such as the charged particle densities ne,p,n, the electron mean energy ε, or the plasma potential V are calculated at node locations (i, j), while the axial and radial components of vector quantities such as the electric field Eq (q = r, z) or the fluxes Γqe,p,n are defined at half-distance between nodes (i + 1/2, j) and (i, j + 1/2), respectively. Therefore, the value of a scalar variable X defined in mesh-nodes Pi,j is represented as Xi,j , whereas the value of a vector quantity A defined in “half-points” (i + 1/2, j) is represented as Ari+1/2,j . Unlike the scalar variables, defined in the mesh-nodes (filled circles in Figure 3.2), which are calculated in [(nr + 1)x(nz + 1)] points, the radial vector components, defined in the radial half-points (open squares) are calculated in [(nr + 2)x (nz + 1)] points, and the axial vector components, defined in the axial half-points (filled squares) are calculated in [(nr+1)x(nz +2)] points. The network of points thus defined, covering the entire spatial domain D(r, z), is formed by internal points where the PDEs system is “projected” and by boundary points where the boundary conditions have to be respected. The adoption of a staggered grid should be linked to the conservation properties requested for a discretization sheme. For example, to ensure the particle conservation for the whole domain D(r, z), the flux of particles leaving a control volume (the computation domain can be covered with non-overlaping control volums such that there is one control volume surrounding each grid point; in Figure 3.2, the control volume around (i, j) point is defined by (i − 1/2, j − 1/2), (i+1/2, j −1/2), (i−1/2, j +1/2) and (i+1/2, j +1/2) locations) across a certain face must be identical to the flux of particles entering in the adjacent control volume through the same face. A consistent manner to do this is to define for fluxes one and the same expression in adjacent control volumes. Thus, when summed over the entire domain, for an overall balance, only the flux boundary conditions remain. A consistent flux formulation ensures that the conservation


65

r i−1

z

i i−1/2

i+1 i+1/2

j−1 mesh nodes radial half−points axial half−points

j−1/2 j g

j+1/2 j+1 h

Figure 3.2: The staggered mesh. Scalar variables are calculated at mesh-node locations (marked with filled circles), while vector components are defined at half-distance between nodes (empty squares and empty circles).

of the physical quantities such as particle, momentum, and energy is exactly satified over the whole calculation domain [Versteeg and Malalasekera 1995].

3.1.2 Finite-difference method After defining the computational mesh, the FDM is applied to the PDEs system, writing the various continuous terms as discretized finite-difference approximations. Since time integration will be treated in Section 3.2, we consider here the space derivative approximations only. According to the FDM, the first derivatives in space may be calculated using forward-, backward- or centered-difference (FD, BD or CD) expressions. For example, the axial derivative of variable X at node (i, j) can be written as

 ∂X   ∂z i,j

 X(z + ∆z) − X(z)   ≡ lim  ∆z→0  ∆z

where g is the grid step along z.

= i,j

              

Xi,j+1 −Xi,j g

(FD)

Xi,j −Xi,j−1 g

(BD)

Xi,j+ 1 −Xi,j− 1 2

2

g

(3.1)

(CD) ,

Expressions 3.1 can be derived from a truncated Taylor series, where the order of the first term to be eliminated from the development represents the truncation error. This is an estimate of the amount by which the numerical approximation deviates from the analytic solution. The order of an algorithm refers to the highest power of the expansion parameter that is retained in


66

the solution method. From this point of view, the FD and BD expressions are only first-order accurate, while the CD is second-order accurate. For this reason, the model will generally use CD expressions to discretize the first derivatives in space, the exception being in the treatment of boundary conditions. The implementation of the FDM at boundaries is made as follows. When boundary conditions are Dirichlet-type, the value of a variable at boundary points is directly used in the model. This occurs, for example, when solving Poisson’s equation, in which case the potentials of rf and grounded electrodes are directly imposed as boundary conditions (cf. Section 2.3.5). However, when boundary conditions are Neumann-type, we need to calculate the gradient of a variable at boundary points, in which case first-order FD or BD expressions are used for left/top or right/bottom boundaries, respectively. This occurs, for example, when calculating the symmetry boundary conditions at discharge axis (cf. Section 2.3.5). The model uses the first derivatives in space calculated at some half-points (i + 21 , j) or (i, j + 12 ). For a variable X we can write the following second-order accurate expressions  Xi+1,j − Xi,j ∂X   = (3.2)  ∂r i+ 1 ,j h 2  ∂X  Xi,j+1 − Xi,j  . (3.3) =  ∂z g 1 i,j+ 2

Finally, it is also possible to deduce second-order CD expressions for the second derivatives

in space. For example, the second axial derivative of variable X at node (i, j) can be written as  ∂2X   = Xi,j+1 − 2Xi,j + Xi,j−1 . (3.4) 2 ∂z i,j g2

3.2 Advancing equations in time

Time integration starts at t = 0 from an imposed set of initial conditions and proceeds at discrete time-step intervals ∆t, towards the final steady-state situation. Typical initial conditions correspond to particles densities of about 104 -105 cm-3 , an electron mean energy of about 3 eV and a constant potential profile. The fluxes and the electric field are normally set to zero. The applied rf voltage begins at its highest value and follows a cosine evolution. In general, the initial conditions can be arbitrarily chosen within a large range of values. However, for very-low pressure and high rf voltage conditions, a sufficiently high electron density must be used in the initial conditions set, in order to ensure "discharge ignition”. The electron density acts as a very sensitive parameter which ensures the development of a physical solution. In fact, a small number of electrons can be quickly eliminated from the discharge volume due to a combination of factors: high electric fields, the high electron mobility, and the


67

high flux boundary conditions. In these cases, the discharge breakdown can be achieved if the electron density starts from a higher value of about 106 -107 cm-3 . The model uses an efficient method to advance the equation set in time, in order to give a quick prediction of the steady-state solution. The different equations are solved successively, as indicated in the flow chart of the fluid model represented in Figure 3.3. This figure also shows the most important output physical quantities calculated on each module of the model. Note that the stationary, space-independent Boltzmann equation module runs (for a certain gas or mixture of gases) only once, prior to the fluid model, in order to construct the look-up table showing the dependence of the electron/energy transport parameters and collision terms on the electron mean energy. The table values are recorded and accessed whenever is needed (usually at the beginning of each iteration). The iterative procedure starts by using the values of charged particle densities to solve Poisson’s equation and calculate the electric field distribution in the discharge. This electric field profile is used to compute the new charged particle densities, the electron mean energy and the corresponding charged particle/electron energy fluxes. Notice that, for ions, an intermediate step is required in order to calculate the effective electric field involved in the ion transport equations. The charge particle fluxes are used to update the self-bias voltage potential, at the end of each iteration or rf period. The new values of densities are inserted again in Poisson’s equation and the process is repeated until convergence. Stationary, space−independ. Boltzmann Eq. solver Initial conditions

Look−up tables: eTP, ε TP and collision terms NO t=0

Convergence criterion

Poisson’s Eq. E(r,z,t)

YES

Results

t = t + ∆t Electron transport Eqs. n e (r,z,t)

Current conservation

Γe (r,z,t)

Electron energy transport Eqs.

ε (r,z,t) Γε (r,z,t)

Vdc

Effective electric field Eq. E

eff

(r,z,t)

Ion transport Eqs. n i (r,z,t) Γi (r,z,t)

Figure 3.3: The general computational flow chart of the model.

The selection of adequate time-step intervals ∆t is particularly important, since the choice of


68

high ∆t values can act as a filter for the high-frequency structure of the solution. This question is strongly related to the method used to advance the equation system in time: explicit, implicit or semi-implicit. In this work, the transport equations for electrons, ions and energy, as well as Poisson’s equation, are integrated in time using a semi-implicit method, which can ensure the stability of algorithms yet avoiding the large run times typical of explicit schemes.

3.2.1 Explicit, implicit and semi-implicit numerical schemes To point out the differences between explicit, implicit and semi-implicit time integration methods, we consider the electron continuity equation 2.62 ∂ne 1 ∂(rΓer ) ∂Γez =− − + Se ∂t r ∂r ∂z

.

When using a FD expression to discretize the time derivative term, the electron continuity equation can be formally written as nk+1 − nke = f (tk , nke )∆t e

,

(3.5)

if the electron flux (given by Eq. 2.63) and the source term are calculated at time t k . Equation 3.5 illustrates the explicit or forward time integration scheme, where the fluxes and source term (in the above example included in function f ) are evaluated using the solution values obtained at an earlier time tk , usually one time-step before the current integration time tk+1 . If the flux and source term estimations involve the solution values obtained at time t k+1 , one obtains nk+1 − nke = f (tk+1 , nk+1 )∆t e e

,

(3.6)

corresponding to the implicit or backward time integration method. A middle way between explicit and implicit schemes is given by the Crank-Nicholson algorithm, which describes the time evolution of variables using estimations of flux and source terms obtained by the trapezoidal rule

1 nk+1 − nke = [f (tk , nke ) + f (tk+1 , nk+1 )]∆t . (3.7) e e 2 The choice of a semi-implicit method, based on the Crank-Nicholson algorithm to integrate the PDEs in time can avoid important restrictions on the integration time step, which are normally associated with explicit methods. In principle, the time step ∆t is to be defined as large as possible in order to substantially reduce the computational effort in time integration, keeping however a reasonable time description of the rf period. This procedure is strongly limited when adopting an explicit method, which can lead to unstable algorithms for large ∆t. The problem can be controlled by monitoring the so-called Courant-Friedrichs-Lewy (CFL) restriction that limits the maximum value of the integration time step.


69

3.2.2 The Courant-Friedrichs-Lewy restriction The physical meaning of the CFL restriction can be easily understood referring to a onedimensional convective problem ∂ϕ(x, t) ∂ϕ(x, t) = −v ∂t ∂x

,

(3.8)

where ϕ(x, t) is a variable transported with the velocity v, and (x, t) are the spatial coordinate variable and time, respectively. The explicit treatment (cf. Eq. 3.5) of the time derivative term in Eq. 3.8, and the FD discretization of its spatial derivative term lead to ϕk+1 = ϕki − N (ϕki+1 − ϕki ) i

,

(3.9)

where N ≡ v∆t/∆x represents the number of grid cells that the variable flow crosses during the time-step ∆t. Since the new value ϕk+1 depends only on the old values ϕk and the known value of v, the explicit algorithm can be used very efficiently with a small computational cost. However, the numerical solution resulting from such an algorithm becomes inaccurate if the value of N is larger than unity, in which case the variable flow travels more than one cell per time-step. The CFL stability restriction states that v∆t . The convergence error of a given variable, say V , is then defined as V =

< V (i, j) >k − < V (i, j) >k

0

,

k +k0 2

(3.66)

where k and k 0 designate two different times separated by one rf period. Finally, the convergence criterion can be formally defined as c < 0.1%

,

(3.67)

where c = min(ne , np , nn , ε , V , Vdc )

,

(3.68)

with ne , np , nn , ε , V and Vdc the convergence errors associated to the discharge parameters under monitoring. In general, condition 3.67 is met after a few hundred rf periods (corresponding to 3-5 hours on a Pentium III - 500 MHz) for the electron density and mean energy, the plasma potential and the dc self-bias voltage. In helium and silane-hydrogen discharges with small SiH − 3 population, the above criterion is easily achieved also by the ion particle densities. However, in silane dominated discharges, convergence is controlled by the high concentration of negative ions as a consequence of the very small recombination rate when compared to the attachment rates, combined with the absence of transport losses for this species (see Chapter 5).

Chapter 4 Simulation results for He discharges This chapter presents calculation results for discharges operating in pure helium. We emphasis the general features of ccrf discharges either by computing the time-average values of the different physical quantities, or by following their time variation during the rf cycle. The changes of electrical parameters with both gas pressure and rf applied voltage are analysed. The influence on results of the different terms defining the effective electric field is discussed. We also provide comparisons of the computed electrical parameters with earlier calculations and experimental measurements.

4.1 General considerations The model is used here to describe the transport of charged particles in ccrf cylindrical discharges operating in pure helium. The experimental setup configuration is schematically shown in Figure 4.1, and has been described in detail by [Kae-Nune et al. 1994; Leroy 1996; Leroy et al. 1998]. The discharge chamber is similar to the Gaseous Electronics Conference (GEC) reference Cell [Pak and Riley 1992; Hargis et al. 1994], which is used as a standard configuration setup in many rf studies, both experimental and computational [Merad 1998]. Shortly, the adopted GEC Cell is a parallel plate cylindrical setup with total radius R = 6.4 cm (6 cm electrode radius; 0.2 cm electrode-grid separation; 0.2 cm grid thickness) and inter-electrode distance d = 3.2 cm (see Figure 4.1), which uses a lateral grid in order to confine the plasma between the electrodes. The rf power is applied to the upper (driven) electrode, whereas the lower electrode and the lateral grid are both grounded. The reactor configuration shown in Figure 4.1 is normally used in PECVD processes, which justifies that the lower electrode (where the substrate is placed) can be heated up to 1200 K in order to favour deposition. The model calculations are restricted to the volume between the electrodes and the lateral grid. Hence, the calculated 2d profiles correspond to simulations inside the rectangle delimited by the discharge axis (r = 0), the lateral grid (r = R), and the upper and lower electrodes

86

C HAPTER 4. S IMULATION

RESULTS FOR

HE

87

DISCHARGES

gas

rf electrode lateral grid substrate grounded electrode pumping

pumping

Figure 4.1: Schematic configuration of the rf PECVD reactor under study. (z = 0 and z = d, respectively) (cf. Figure 3.1). The axial distributions of physical quantities are usually presented at discharge axis. Table 4.1 summarizes the set of parameters used for simulations in pure helium. The rf frequency f and the coupling capacitance CB have been kept constant at 13.56 MHz and 50 pF, respectively. Furthermore, we assume a constant pressure throughout the reactor, and a constant gas temperature at 300 K. Parameter Value Electrode radius R (cm) 6.4 Electrode separation d (cm) 3.2 Frequency f (MHz) 13.56 Rf voltage Vrf (V) 100-350 Pressure p (Torr) 0.35-10.0 Gas temperature Tg (K) 300 Coupling capacitance CB (pF) 50 Table 4.1: Set of parameters used for simulations in helium.

The typical features of rf discharges are outlined here by considering a helium discharge operating at p = 1 Torr and Vrf = 217 V. For calculations in helium, the results were obtained by imposing a general convergence error εc < 8 × 10−4 (see Section 3.7). The typical time evolution of the convergence error, for various plasma parameters, is shown in Figure 4.2. One observes that a few hundreds rf cycles are enough to satisfy the convergence criterion, after which the discharge electrical parameters no longer evolve significantly, and the system can be


RESULTS FOR

HE

88

DISCHARGES

convergence error, εc

0.1

(1) (2) (3) (4) (5)

0.01

1E-3

1E-4 0

50

100

150

200

rf periods Figure 4.2: Time evolution of the convergence error, for the following time-average parameters: electron density (1), ion density (2), electron mean energy (3), plasma potential (4), and selfbias voltage (5). The curves were obtained for a helium discharge operating at p = 1 Torr and Vrf = 217 V.

considered to be in steady-state situation. In the next sections, the main features of rf discharges in helium are presented, by analysing the typical space-time distributions of different physical quantities.

4.2 Electric field and charged-particle distributions Figure 4.3 shows a typical distribution of the axial and radial components of the electric field, for a rf discharge operating in helium at p = 1 Torr and Vrf = 217 V. In this figure, the negative values indicate that the axial electric field orientation is in opposite direction to the z axis. As expected, either in terms of time-average values, or in terms of space-time dependent values, the magnitudes of the axial and radial reduced electric field components are small in the discharge bulk (about 10−17 Vcm2 and 10−18 Vcm2 for the axial and radial components, respectively), becoming higher in sheath regions. Since the discharge configuration is asymmetric, the magnitude of the axial electric field is greater near the rf electrode than in the sheath region of the grounded electrode [1.6 × 10−14 Vcm2 in comparison with only 0.9 × 10−14 Vcm2 , for its timeaverage values, see Figure 4.3(a)]. In the vicinity of the grid, the time-average radial reduced electric field, Er /N , has the same order of magnitude as the axial reduced electric field, Ez /N , near the grounded electrode [see Figure 4.3(a) and (b)]. The alternate contraction and expansion of sheaths during the rf cycle is shown in Fig-


RESULTS FOR

HE

89

DISCHARGES

ure 4.3(c), were the space-time distribution of Ez /N between electrodes is plotted for r = 0. At the beginning of the rf cycle, the grounded sheath experiments the higher values of electric (b)

Vcm )

2

100

60

Er /N (10

0 -50 -100 -150 0.0

80

-16

50

Ez/N (10

-16

2

Vcm )

(a) 100

1.6

3.2

r (cm)

4.8

6.4

40 20

3.2 2.4 1.6 0.8 z (cm) 0.0

0 0.0

3.2 2.4 0.0 1.6 18.5 0.8 z (cm) 37.0 55.5 t (ns) 74.0 0.0

-16

2

-200

6.4

(d)

Er /N (10

-16

E z/N (10

-100

4.8

180

Vcm )

2

Vcm )

(c)

0

3.2

r (cm)

200 100

1.6

3.2 2.4 1.6 0.8 z (cm) 0.0

150 120 90 60 30 0 0.0

6.4 4.8 3.2 18.5 r (cm) 1.6 37.0 55.5 t (ns) 74.0 0.0

Figure 4.3: Time-average, 2d steady-state profile of axial Ez /N (a) and radial E r /N (b) reduced electric field components, for a helium discharge operating in conditions of Fig. 4.2. Time evolution (during one rf cycle) of axial E z /N (at r = 0) (c) and radial E r /N (at z = d/2) (d) reduced electric field components, for the same conditions as before.

field, while the rf sheath has almost vanished (hence, the electric field has small values there) due to the incoming flux of electrons that recovers quasineutrality. At later time, the grounded sheath begins its contraction (|Ez /N | decreases at z ∼ 3.2 cm), whereas the rf sheath starts its expansion (|Ez /N | increases at z ∼ 0). At half-cycle (t = T /2 ' 37 ns) the situation is completely reversed, with the rf sheath now attaining its maximum extension, the magnitude of the axial reduced electric field being about 2.7 × 10−14 Vcm2 . As expected, the second half of the rf cycle drives the sheath regions towards the initial situation. Since the grid is grounded, the behavior of the corresponding sheath region is in opposition with that of the powered electrode. The radial reduced electric field Er /N , computed at z = d/2 as a function of time and space, is shown in Figure 4.3(d). The time modulation of the grid sheath is fairly well represented by a cosinusoid, having its minimum value at t ' 37 ns. The time-average profile of the plasma potential V p (r, z) is shown in Figure 4.4(a). This plot shows a flat potential region at about 73 V (corresponding to a zero electric field domain),


RESULTS FOR

HE

90

DISCHARGES

(a) 100 75

Vp (V)

50 25 0 -25 -50 0.0

1.6

3.2

r (cm)

4.8

3.2 2.4 1.6 0.8 z (cm) 6.4 0.0

(b) 180

Vp (V)

90 0 -90 -180 -270

3.2 2.4 0.0 1.6 18.5 0.8 z (cm) 37.0 55.5 t (ns) 74.0 0.0

Figure 4.4: Time-average, 2d steady-state profile of the plasma potential V p (a), for a helium discharge operating in conditions of Fig. 4.2. Time evolution (during one rf cycle) of plasma potential (at r = 0) (b), for the same conditions as before.


RESULTS FOR

HE

91

DISCHARGES

from which the potential drops to zero towards the grounded walls, and to V p = −63.7 V towards the powered electrode. The latter value is the so-called self-bias voltage. Figure 4.4(b) shows, as a function of axial position and time, the evolution of the potential distribution at r = 0. In consistency with the time variation of Ez /N , the highest potential drop (' 280 V) across the rf sheath occurs at t ' 37 ns. The 2d profiles of charged particle densities are shown in Figure 4.5. For helium discharges, we consider electrons and He+ ions only. In the bulk, the time-averaged values of both charged-particle densities are 109 cm−3 , but space-time averages throughout the discharge volume indicate a lower value for the electron density (1.7 × 109 cm−3 ) than for the ion density (1.9 × 109 cm−3 ), the difference being due to the higher ion density within the sheaths. At steady-state situation, the maxima (4.8 × 109 cm−3 ) of time-average electron and ion densities appear to be located both at half distance between electrodes, and in radial direction at (a)

(b) 0.5 -3

0.3

0.4

10

cm )

0.4

0.3

np (10

ne (10

10

-3

cm )

0.5

0.2 0.1 0.0 1.6

3.2

r (cm)

4.8

6.4

3.2 2.4 1.6 0.8 z (cm) 0.0

0.2 0.1 0.0

-3

cm )

10

np (10

-3

cm )

10

ne (10

0.0

18.5

37.0

t (ns)

55.5

74.0

3.2 2.4 1.6 z (cm) 0.8 0.0

4.8

(d)

0.3

0.1

3.2

r (cm)

(c) 0.4

0.2

1.6

3.2 2.4 1.6 0.8 z (cm) 6.4 0.0

0.4 0.3 0.2 0.1

3.2 2.4 0.0 1.6 18.5 z (cm) 0.8 37.0 55.5 t (ns) 74.0 0.0

Figure 4.5: Time-average, 2d steady-state profiles of the electron density ne (a) and ion density np (b), for a helium discharge operating in conditions of Fig. 4.2. For one rf cycle, the corresponding time variations at r = 0 are plotted in (c) and (d), respectively.

r ∼ 4.8 cm [see Figure 4.5(a) and (b)]. The location of these peaks, out of discharge axis, has


RESULTS FOR

HE

DISCHARGES

92

been observed in experiments carried out in similar geometries [Overzet and Hopkins 1993]. Also, other 2d discharge models working in configuration which use lateral walls, predict the maximum of charge particle densities at a certain radial position [Young and Wu 1993b; Dalvie et al. 1993]. Note that the development of a maximum average electron density, in the discharge bulk, is a direct consequence of the confining effects induced by the sheath electric fields. The movement of electrons causes the modulation of sheath thickness. In Figures 4.5(c) and (d) we compare, as a function of position and time, the electron and ion axial distributions. One observation of this figure shows that in the first half cycle, electrons are pushed away from the rf sheath by the increasing electric field, while in the second half (when the field decreases) they return back to the vicinity of the rf electrode. Actually, they are evacuated by conduction and return when diffusion dominates, each of these process controlling about 50% of the rf cycle. Contrarily to electrons, the ions continuously flow towards the walls, thus disregarding the time evolution of the sheath electric field. Therefore, the ion density presents no important modulation in time, and the ions can be considered to form a more or less fixed background on which the electron movement, in and out of the sheaths, occurs.

4.3 Electron mean energy and ionization rate distribution When evacuated from the sheath regions, electrons are accelerated by the electric field to gain momentum and energy. Moving with high energy towards the discharge bulk, they are involved in ionization collisions at sheath-bulk interface and within the bulk (a small fraction of high energy electrons penetrate into the bulk with sufficient energy to cause ionization of neutrals). After entering in the plasma bulk (where the electric field becomes quite small), the electron Joule heating is considerably reduced and we can expect the electron mean energy to present a quite flat profile. These global features are reproduced in Figures 4.6 and 4.7. The time-average profile of the electron mean energy is shown in Figure 4.6(a). The electron mean energy in the bulk is about 7.3 eV, but higher values are observed in the sheaths (10.5 eV near the rf electrode and 7.5 eV close to the grounded electrode). Electrons acquire energy also in front of the lateral grid, due to the radial electric field existing in this region. The highest energy value observed (ε ' 13 eV) occurs in the corner where the rf electrode meets the lateral grid. There, the potential drop across a very small distance is responsible for the high values of the electric field and the electron mean energy. Figure 4.6(b) shows the (z, t) contour-plot of the mean energy density (ne ε) between electrodes. The maximum value of ne ε coincides in time with the moment when electrons experience their maximum acceleration, but it is located in space at a different position with respect to the electron density maximum. In fact, the small values


RESULTS FOR

HE

93

DISCHARGES

15

ε (eV)

12 9

(a)

6 3 6.4 4.8 3.2

r (cm)

1.6 2.4

1.6 0.0 3.2

0.8

0.0

z (cm)

3.2

2.8E10 2.6E10 2.3E10 1.9E10 1.4E10 1.1E10 7E9 3.5E9 0

z (cm)

2.4 1.6

(b)

0.8 0.0 0.0

18.5

37.0

55.5

74.0

t (ns)

Figure 4.6: Time-average, 2d steady-state profile of the electron mean energy ε (a), for a helium discharge operating in conditions of Fig. 4.2. For one rf cycle, the space-time variation of the mean energy density ne ε (in eVcm−3 ) at r = 0 is plotted in (b).


RESULTS FOR

HE

94

DISCHARGES

of ne within the sheath and the reduced values of ε inside the plasma bulk set the maximum of ne ε at a location slightly closer to the rf electrode although, for the conditions of Figure 4.6, this maximum is still located near the half-distance between electrodes. Notice that the n e ε maximum appears to extend towards the rf electrode (due to the electron injection from the rf sheath to the plasma bulk), and that this extension stops just before the first half period, most probably because, at that moment, the rf sheath has already lost the majority of its hot electrons.

3.2

1.1E15 9.6E14 8.3E14 6.9E14 5.5E14 4.1E14 2.8E14 1.4E13 0

z (cm)

2.4 1.6

(a) 0.8 0.0 0.0

1.6

3.2

4.8

6.4

r (cm) 3.2

3E15 2.6E15 2.3E15 1.9E15 1.5E15 1.1E15 7.5E14 3.7E13 0

z (cm)

2.4 1.6

(b) 0.8 0.0 0.0

18.5

37.0

55.5

74.0

t (ns)

Figure 4.7: Spatial contour plot of the time-average, steady-state, helium ionization rate (in cm−3 s−1 ) (a), for a discharge operating in conditions of Fig. 4.2. For one rf cycle, the space-time variation of helium ionization rate at r = 0 is plotted in (b).


RESULTS FOR

HE

DISCHARGES

95

The contour plot of the time-average, steady-state helium ionization rate is shown in Figure 4.7(a). The model predicts the largest ionization rate at r = 5.0 cm and z = 0.8 cm. This maximum is located between the peaks of the electron mean energy and the electron density, being closer to the corner where the electric field has its highest values. In the discharge center and in front of the grounded electrode the ionization rate is relatively small, approximately one order of magnitude below its maximum value. Near the lateral grid, where electrons can also achieve enough energy, the ionization rate is once again comparable to that observed in the rf sheath. The space-time variation of the helium ionization rate (at r = 0) is plotted in Figure 4.7(b). This figure shows an ionization peak at the sheath-bulk interface, when the increasing electric field pushes the electrons into the discharge. For a discharge maintenance in α regime, primary electrons must accelerate to energies above the ionization threshold during the sheath expansion process, in order to ensure the required volume ionization. Very few ionization events are observed at the end of the rf period, very close to the rf electrode. We associate their presence to the small fraction of electrons coming from the bulk with enough energy to ionize the neutrals. For the simulation conditions considered here, the energy gained by electrons at the grounded sheaths seems to be insufficient to yield a strong ionization rate.

4.4 Discharge currents and particle fluxes The space-time variation of voltages, currents and particle fluxes are presented in this section. Figure 4.8(a) shows the rf applied potential, the potential of the driven electrode and the potential difference between the plasma and the driven electrode, as a function of time, for one rf cycle. Due to the presence of self-bias voltage (Vdc = −63.7 V for the conditions shown in this figure), the powered electrode potential is shifted towards negative values. From this figure we can also observe that plasma potential is always positive with respect to the driven electrode potential [see curve (3)]. As expected, the largest potential drop across the rf sheath occurs at half-period, and its lowest value is observed at the beginning/end of each period, allowing the escape of electrons to the rf electrode. The total discharge current computed at the rf electrode is plotted in Figure 4.8(b), together with the rf applied voltage. As expected, the capacitive coupling mode in which the discharge operates is reflected by a current-voltage phase shift of about π/2. Note that the small deviation of this phase shift from π/2 is associated to the power absorption by the plasma. The total discharge current consists in the conduction current (driven by both electrons and ions) and the displacement current (see Eq. 2.102). We note that the total current is sinusoidal-like, thus


RESULTS FOR

HE

96

DISCHARGES

revealing that its major contribution at the rf electrode comes from the displacement current (which follows the sinusoidal evolution of the electric field within the rf sheath).

300

(a)

(1)

V (V)

150

0

(2)

(3)

-150

-300

0

18.5

37

55.5

74

t (ns) 450

450

(b) 300

300

(1)

V (V)

150

0

0

-150

-150

-300

-300

-450

0

18.5

37

55.5

74

I (mA)

150 (2)

-450

t (ns)

Figure 4.8: (a) Time variation, during one rf cycle, of steady-state voltages for a helium discharge operating in conditions of Fig. 4.2: the rf applied potential (1); the potential at driven electrode (2); the potential difference between the plasma and the driven electrode (3). (b) Time variation, during one rf cycle, of the rf applied potential (1) and the discharge current (2), for a helium discharge operating in conditions of Fig. 4.2.

The time variation of the total-current different components is shown in Figure 4.9(a), during one rf cycle. From this figure one can confirm that the displacement current gives the most important contribution to the total current. In opposition, the ion current is extremely small, being almost constant over the period. The electron current at z = 0 is nearly zero during the


RESULTS FOR

HE

DISCHARGES

97

entire cycle, with only one important exception: approximately 10 ns around the beginning of each period, a massive quantity of electrons escapes from the discharge, crosses the reduced potential barrier [see Figure 4.8(a)] and reaches the electrode, which accounts for the electron current peaks at the beginning of the rf period. For a capacitively coupled discharge, running in steady-state situation, there is an almost perfect balance between the number of electrons and ions that strike the rf electrode, during one rf cycle. This means that the electron current peaks, at the beginning of each rf period, must be compensated by an almost constant ion bombardment of the driven electrode [see curves (1)-(2) in Figure 4.9(a)], which is ensured by the development of the dc self-bias voltage. This situation is completely different within the plasma volume. For comparison, Figure 4.9(b) plots the discharge current components computed at the rf electrode (z = 0, curves 1-3) and inside the discharge bulk (at z = d/2, curves 1’-3’). At the discharge center the electric field goes to zero and the displacement current drastically decreases, making the electron conduction current the dominant contribution to the total discharge current. Note that the ion current is still low at the discharge center. The total current [not plotted in Figure 4.9(b)] continues to present a sinusoidal shape, with a π/2 phase difference with respect to voltage. The space-time contour plots, at r = 0, of the charge particle and energy axial fluxes are shown in Figure 4.10, during one rf period. The positive values of fluxes correspond to an oriented transport towards the grounded electrode, while negative values indicate an oriented transport towards the driven electrode. In Figure 4.10(a) we note that, as time advances, electrons are first pushed away from the driven electrode due to the increasing potential barrier. When the potential barrier starts to decrease (after half period), the electron flux changes its sign and, at the end of period, a noticeable flux of electrons strikes the powered electrode. The axial energy flux accompanies the electron movement, which can be confirmed by analyzing Figure 4.10(b). For the ions, Figure 4.10(c) shows that the orientation of their axial flux appears to be split between the driven and grounded electrodes, following the orientation of the corresponding electric fields in either sheaths. The small ion mobility justifies the continuous ion bombardment on both electrodes, during the entire rf cycle, and the fact that there is not net change in the orientation of the axial ion flux.

4.5 The effect of gas pressure and applied voltage A few simulations were performed to outline the effect of gas pressure and rf applied voltage on discharge parameters. Figure 4.11 shows the spatial contour plot of the time-average electron density, produced at Vrf = 217 V and pressures: p = 0.5, 1.0 and p = 3.0 Torr. An observation of this figure


RESULTS FOR

HE

DISCHARGES

98

Figure 4.9: (a) Time variation, during one rf cycle, of discharge currents at the rf electrode (z = 0) for a helium discharge operating in conditions of Fig. 4.2: electron current (1); ion current (2); displacement current (3); total current (as a sum of the previous currents) (4). (b) Time variation, during one rf cycle, of discharge current components, calculated at z = 0 (curves 1-3) and z = d/2 (curves 1’-3’), for a helium discharge operating in conditions of Fig. 4.2. The labels 1-3 (and the corresponding prime labels) are for the same current components as in (a).


RESULTS FOR

HE

99

DISCHARGES

3.2

2E16 1.5E16 1.1E16 5.9E15 1.3E15 -2.4E15 -7.1E15 -1.3E16 -1.8E16

z (cm)

2.4 1.6

(a) 0.8 0.0 0.0

18.5

37.0

55.5

74.0

t (ns) 3.2

4E17 3.3E17 2.5E17 1.8E17 7E16 -3.5E16 -1E17 -1.4E17 -2E17

z (cm)

2.4 1.6

(b) 0.8 0.0 0.0

18.5

37.0

55.5

74.0

t (ns) 3.2

4E14 2.6E14 1.1E14 -3E13 -1.7E14 -3.2E14 -4.6E14 -6.1E14 -7.5E14

z (cm)

2.4 1.6

(c) 0.8 0.0 0.0

x10

18.5

37.0

55.5

74.0

t (ns)

Figure 4.10: Space-time contour plots at r = 0 of the axial electron flux (in cm −2 s−1 ) (a), the axial energy density flux (in eVcm−2 s−1 ) (b), the axial ion flux (in cm−2 s−1 ) (c), during one rf cycle, and for a helium discharge operating in conditions of Fig. 4.2. The fluxes are positive when transport is oriented towards the grounded electrode and negative when transport is oriented towards the driven electrode.


RESULTS FOR

HE

100

DISCHARGES

3.2

2.8E9 2.5E9 2.1E9 1.8E9 1.4E9 1.1E9 7E8 3E8 0

z (cm)

2.4 1.6

(a) 0.8

0.0 0.0

1.6

3.2

4.8

6.4

r (cm)

3.2

5E9 4.4E9 3.8E9 2.8E9 1.4E9 1.1E9 7E8 3E8 0

z (cm)

2.4 1.6

(b) 0.8 0.0 0.0

1.6

3.2

4.8

6.4

r (cm)

3.2

9E9 7.9E9 6.8E9 5E9 3.8E9 2.8E9 7E8 3E8 0

z (cm)

2.4 1.6

(c)

0.8 0.0 0.0

1.6

3.2

4.8

6.4

r (cm)

Figure 4.11: Spatial contour plot of the time-average, steady state, electron density (in cm −3 ), for helium discharges operating at Vrf = 217 V and the following pressures: p = 0.5 Torr (a), p = 1.0 Torr (b), and p = 3.0 Torr (c).


RESULTS FOR

HE

101

DISCHARGES

shows that the volume occupied by the plasma increases with the gas pressure (check the first contour line at 3 × 108 cm−3 , which gives an indication of sheath thickness reduction). Between the two extreme pressures, the electron density maximum increases from 2.8 × 10 9 cm−3 to 9 × 109 cm−3 , yielding a general upwards shift of the entire density profile (check the evolution of the green region in the contours of Figure 4.11). At p = 3.0 Torr, the electron density maximum approaches the driven electrode-grid corner region, where the ionization rate becomes more intense. In comparison with condition p = 0.5 Torr, for which this maximum is located approximately at half distance between electrodes, at p = 3.0 Torr it moves closer to the rf electrode, at one quarter of the inter-electrode distance. The radial position of the electron density maximum also changes by approximately 1.0 cm, between 0.5 and 3.0 Torr. The influence of pressure on the axial component of the time-average reduced electric field E z /N is shown in Figure 4.12. When the pressure goes from 0.5 Torr to 3.0 Torr, the zero elec-

Ez/N (10

-16

2

Vcm )

200 100 0 -100

(3) (2)

-200 (1)

-300

0.0

0.8

1.6

2.4

3.2

z (cm) Figure 4.12: Axial profile (at r = 0) of the time-average axial reduced electric field Ez /N , for heliumm discharges operating at Vrf = 350 V and the following pressures: p = 0.5 Torr (1), p = 1.0 Torr (2), and p = 3.0 Torr (3).

tric field region extends towards both electrodes, with the consequent reduction in the intensity of the electric field in the sheath regions. In consequence, the electron heating in sheath regions decreases and the average value of the electron mean energy in the discharge bulk also follows this trend, changing from 8.4 eV at p = 0.5 Torr to 6.5 eV at p = 3.0 Torr. The space-time average value of the ionization rate, calculated throughout the discharge volume and for one rf cycle, varies from 1.9 × 1014 cm−3 s−1 at p = 0.5 Torr to 5.9 × 1014 cm−3 s−1 at p = 3.0 Torr. The larger charged-particle production at higher pressures enhances the ion current collected at the rf electrode, yielding a decrease in the magnitude of the self-bias volt-


RESULTS FOR

HE

102

DISCHARGES

age. The higher discharge currents and the smaller phase-shifts between the discharge current and the applied voltage, contribute to an enhancement of coupled electrical power at high pressures. The effect is considerable: Weff varies from 3.6 W at 0.5 Torr to 19.1 W at 3.0 Torr. Note that this effect dominates over the reduction of the displacement current, which contributes to decrease the coupled power. Figure 4.13 shows the variation, with the gas pressure, of the self-bias voltage V dc , and the coupled electrical power Weff . This figure also shows the variation of each of these electrical parameters with the applied rf voltage, for a fixed gas pressure p = 0.5 Torr. While the coupled 0 -50

-20

-55

Vdc (V)

Vdc (V)

-40 -60 -80

-60 -65

-100

-140

-70

(a)

-120 100

200

300

-75

400

(b) 0.5

1.0

2.0

2.5

3.0

p (Torr)

Vrf (V) 20

12

16

Weff (w)

9

Weff (w)

1.5

6 3

(c)

0 100

200

300

Vrf (V)

400

12 8 4 0

(d) 0.5

1.0

1.5

2.0

2.5

3.0

p (Torr)

Figure 4.13: The self-bias voltage Vdc , as a function of the applied rf voltage (at p = 0.5 Torr) (a) and gas pressure (at Vrf = 217 V) (b), for helium discharges. Coupled electrical power Weff , as a function of the applied rf voltage (at p = 0.5 Torr) (c) and gas pressure (at V rf = 217 V) (d), for a helium discharge.

power increases with both the applied voltage and the gas pressure, the absolute value of the self-bias voltage increases with the applied voltage decreasing with gas pressure. The difference is due to the fact that, for the coupled power, the total discharge current is important, while for self-bias voltage, only the balance between the ion and the electron conduction currents is determinant. The total discharge current increases with both Vrf and p, but due to different reasons: (i) as a consequence of the increase of displacement current, for the variation with


RESULTS FOR

HE

DISCHARGES

103

Vrf ; (ii) as a consequence of the increase of charged particle current, for the variation with gas pressure. At a given pressure, a larger rf voltage accentuates the discharge asymmetry and, in consequence, the balance between the charged-particle currents is only set for higher Vdc absolute values. Contrarily, for a given Vrf , an increase in the gas pressure favours the production of charged particles as a consequence of collisional encounters, which contributes to a reduction in the configuration asymmetry.

4.6 The influence of the ion inertia term and the effective electric field The aim of the simulations presented in this section is to give an insight over the importance of the ion inertia term and the effective electric field on discharge parameters. Calculations have been performed for the work conditions indicated in Table 4.2. For each of these conditions, Condition p (Torr) C1a-He 0.5 C1b-He 0.5

Vrf (V) 108.0 217.0

C2a-He C2b-He

1.0 1.0

108.0 217.0

C3a-He C3b-He

3.0 3.0

108.0 217.0

Table 4.2: Work conditions for helium discharge simulations.

three different simulations (A)-(C) were performed. Simulations (A) assume a constant ion mobility and neglect the ion effective electric field equation 2.76. Simulations (B) and (C) adopt the variation of the ion mobility with the electric field proposed in Section 2.3.3. In (B), the effective electric field equation is written neglecting the ion inertia term; in (C), the full equation 2.76 is considered. Table 4.3 shows, for each condition of Table 4.2, the predicted results of discharge electrical parameters and the maximum value of the time-average electron density. When conditions (B)(C) are adopted to describe the ion transport, the time-average electron densities increase by a factor of 1.3 − 1.8 (this factor is higher at low pressures), with respect to the values calculated in condition (A). This difference is a direct consequence of the fact that, for a field-dependent mobility, the ions flow to the walls with smaller velocities, which increases the charged particle populations


RESULTS FOR

HE

104

DISCHARGES

in the discharge bulk. A reduction in the ion velocity within the sheaths yields a reduction in the ion current collected at the rf electrode, which causes the self-bias voltage to adjust to more negative values between simulations (A) and (B)-(C). Moreover, the fact that the ion density increases at the discharge center leads to a rise in the plasma potential (for example, the voltage increase by about 4 V between simulations (A) and (C) at condition C2b-He). This effect, combined with the reduction observed in the sheath thickness, contributes to an enhancement of the electric field (hence of the displacement current) within the sheaths, thus explaining the observed increase in the power coupled to the plasma for simulations (B) and (C). Condition Parameter C1a-He

C1b-He

C2a-He

C2b-He

C3a-He

C3b-He

Vdc Weff ne Vdc Weff ne Vdc Weff ne Vdc Weff ne Vdc Weff ne Vdc Weff ne

(A) −32.06 0.98 0.63 −73.30 3.65 1.49 −21.94 1.70 1.31 −58.13 6.21 3.33 −11.31 3.58 2.03 −48.46 16.20 6.24

Simulations (B) −33.98 1.21 1.12 −74.16 3.72 2.71 −25.13 2.23 1.91 −62.79 7.01 4.81 −12.04 4.58 2.70 −50.79 19.46 8.25

(C) −34.44 1.19 1.14 −75.63 3.66 2.75 −24.41 2.19 1.91 −63.67 6.89 4.83 −12.58 4.57 2.74 −51.18 19.14 8.28

Table 4.3: Electrical parameters, Vdc (in V), Weff (in W) and maximum time-average electron density ne (in 109 cm−3 ), for helium discharges operating in conditions of Table 4.2. Simulation results (A)-(C) were obtained assuming different ion transport conditions: (A), constant ion mobility and no effective electric field; (B), field-dependent mobility and effective electric field calculated without inertia term; (C), field-dependent mobility and effective electric field calculated with inertia term.

The introduction of field-dependent ion mobilities and the adoption of an ion effective electric field brings changes in Vdc and Weff up to 10% and 28%, respectively. The influence of the ion inertia term on the electrical parameters is relatively small, about 3%. Note that the predicted values for Weff decrease when the inertia term is set on, due to the reduction of the sheath electric field (hence of the displacement current) when we go from simulations (B) to (C).


RESULTS FOR

HE

105

DISCHARGES

Vcm ) (10

+

eff

E z /N, E

-16

He , z 2

/N

75 0

(2)

-75 -150 -225 -300

(1) (3)

(4,5)

0.0

0.8

1.6

2.4

3.2

z (cm) Figure 4.14: Axial component of different reduced electric fields affecting the ion transport, at r = 0 and t = 8T /20, for a helium discharge operating in conditions of Fig. 4.2. The curves correspond to: Ez /N , calculated assuming a constant ion mobility and no effective electric field eff (1); EHe + ,z /N , calculated for a field-dependent mobility, with and without the ion inertia term (2 and 3, respectively); Ez /N , for the same conditions as curves 2 and 3 (4 and 5, respectively).

The different axial reduced electric fields, affecting the ion transport in simulations (A)-(C), are plotted in Figure 4.14, at r = 0 and t = 8T /20, corresponding to the time at which the rf sheath has its maximum width. Therefore, when analysing this figure, one should focus on the rf sheath region, located at the vicinity of z ' 0. An observation of Figure 4.14 reveals that: (i) the effective electric field has a smaller absolute value than the rf electric field, which comes from the fact that the latter controls the intensity of the former (cf. Eq. 2.76); (ii) the inclusion of the ion inertia term in the effective field equation leads to a small decrease of Eieff , as the ions

react even slower to the field variations and thus charge separation becomes less important; (iii) the imposition of a constant (higher) ion mobility yields a decrease of the effective field (which in this case is identified to the rf field), in order to limit the variation in the ion drift velocity (see Eq. 2.75). The study of the phase shift between the effective electric field and the rf electric field will be carried out in the next chapter, devoted to the description of silane-hydrogen discharges. The variety of ions in these gas mixture discharge will then be used to clarify this analysis.

4.7 Comparison with experimental results Table 4.4 compares simulation results and measured values of the self-bias potential V dc , and the effective electrical power coupled to the plasma Weff , obtained for helium discharges under


RESULTS FOR

HE

106

DISCHARGES

the following work conditions: p = 1.0 Torr, Vrf = 217 V (C2b-He) and p = 3.0 Torr, Vrf = 108 V (C3a-He). Simulation results (A)-(C) were obtained in the present work, assuming the same ion transport conditions as in Section 4.6. Simulation results (C’) were obtained for the same conditions as (C), by updating the self-bias voltage at each time step (see Section 2.5.2). Simulation results (D) were reported in a previous work [Leroy 1996], and are based on an earlier model version (which adopted field-dependent ion mobilities but no effective electric field). Condition Parameter C2b-He C3a-He

Vdc (V) Weff (W) Vdc (V) Weff (W)

Simulations (A) (B) (C) (C’) −58.1 −62.8 −63.7 −66.8 6.2 7.0 6.9 6.9 −11.3 −12.0 −12.6 −18.8 3.6 4.6 4.6 4.6

Experiment (D) −64.0 5.9 −18.0 4.0

−92.0 7.5 −36.0 5.0

Table 4.4: Comparison of computed and measured values of electrical parameters for helium discharges working conditions (C2b-He) and (C3a-He) (see Table 4.2). Simulation results (A)(C) were obtained in this work, assuming the same ion transport conditions as in Table 4.3. Simulation results (C’) were obtained for the same conditions as (C), by updating V dc at each time step. Simulation results (D) were obtained in a previous work using an earlier model version [Leroy 1996]. There is at least a 40% uncertainty in measured values of V dc .

The magnitude of the self-bias voltage was measured near the driven electrode using a technique described by [Kae-Nune et al. 1995]. The subtractive method (cf. Section 1.4.3) was used to measure the effective power coupled to the plasma. In both conditions C2b-He and C3a-He, the results on this table show a better agreement between simulation results and experimental values for Weff than the one obtained before [Leroy 1996]. In particular, the simulations which adopt the effective electric field equation 2.76 reduce the deviation between predicted and measured values of Weff to only 8%, while the previous work of [Leroy 1996] gives a systematic deviation of about 20%. This correction is the result of an improved description of the discharge electric field distribution. The experimental values of Vdc reported here are the result of some preliminary electrical measurements in rf discharges carried out by the group of Dr. J. Jolly in its former laboratory Plasmas Réactifs en Intéraction Avec des Matériaux (PRIAM). The practical difficulties and the embedded errors associated to these measurements justify a 40% minimum uncertainty in the experimental values of Vdc , thus limiting the use of this quantity in model validation. Nevertheless, we can still try to explain the deviations between predictions and measurements of Vdc , which are particular important in condition C3a-He. The systematic over-estimation of the calculated self-bias potentials (less negative than the


RESULTS FOR

HE

DISCHARGES

107

measured ones) has been previously [Leroy et al. 1998] related to the fact that the grounded lateral grid is assumed here to be an opaque wall. In reality, electrons can partially "see" the very extensive grounded surfaces outside the plasma chamber, which enhances the discharge asymmetry, thus leading to a decrease on Vdc . Another possible contribution to this systematic deviation is the presence of helium molecular ions He+ 2 , which are not considered in the model. We expect these species to be produced in the discharge sheath by associative ionization He ? ? + He −→ He+ 2 + e, where He represents an excited state of helium (metastable or higher).

The latter reaction constitutes an additional ionization mechanism, which will probably induce lower (more negative) self-bias potentials, in order to compensate for the increase in the electron current, and so ensuring a pure capacitive operation mode. Furthermore, between 1 and 3 Torr, the population of helium molecular ions (& 10%ne ) is practically proportional to the gas pressure [Alves et al. 1992], which could explain the higher discrepancy between simulation results and measurements for Vdc , observed at higher pressures (condition C3a-He). Note, finally, the systematic over-estimation of the calculated Vdc values in column (C’) with respect to those in column (C). The more negative self-bias voltage obtained when the update of this parameter is performed at each time step is the mere result of its over-correction within the rf period. As mentioned before, (cf. Section 2.5.2) this numerical diffusion problem is particularly relevant in the presence of “light” ions, which are more sensitive to the rf field variations, in which case the Vdc value should be updated only after each rf period.

Chapter 5 Simulation results for H2 and SiH4-H2 discharges This chapter presents calculation results obtained for discharges operating in pure hydrogen and in silane-hydrogen mixtures. These simulations are used to emphasize the influence of geometry dimensions, pressure, applied voltage and silane dilution on different plasma parameters. We focus on variations of the rf electric field in H2 discharges, and silane dissociation rate in Si4 -H2 discharges. The results are tested against experimental measurements and compared to previous simulations.

5.1 General considerations The model is used here to describe the transport of charged particles in ccrf cylindrical discharges operating in pure hydrogen and in silane-hydrogen mixtures. The experimental setup configuration is similar to the one used in helium discharges (cf. Chapter 4), which is schematically represented in Figure 4.1. Similarly to what was indicated in Chapter 4, model calculations are restricted to the volume between the electrodes and the lateral grid, which yield a rectangular 2d work space, delimited by the discharge axis (r = 0), the lateral grid (r = R), and the upper and lower electrodes (z = 0 and z = d, respectively) (see Figure 3.1). Table 5.1 summarizes the set of parameters used for simulations in this chapter. As for helium discharges, the rf frequency and the coupling capacitance have been kept constant at 13.56 MHz and 50 pF, respectively. We assume a constant pressure throughout the reactor, although adopting different gas temperature profiles. For hydrogen, a constant gas temperature profile at 300 K is considered; for silane-hydrogen mixtures, we consider an axial linear profile from 300 − 400 K (at z = 0) to 560 K (at z = d, due to the grounded electrode heating cf. Section 4.1), as obtained from a thermal model [Leroy 1996; Leroy et al. 1998]. For SiH 4 H2 mixtures, we assume a constant gas composition by imposing the partial pressures p α of the

108


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

109

different gas components α. These restrictions are to be eliminated in the future, by coupling the transport model for charged particles with a transport model for neutral species, in order to perform a self-consistent calculation of the density profiles of all radical and background gas components. Parameter

Value H2 SiH4 -H2 Electrode radius R (cm) 1.6-12.8 6.4 Electrode separation d (cm) 1.6-6.4 3.2 Frequency f (MHz) 13.56 13.56 Rf voltage Vrf (V) 200-450 160-500 Pressure p (Torr) 0.15-5.0 0.07-0.3 Gas temperature Tg (K) 300 300-500 Coupling capacitance CB (pF) 50 50 Table 5.1: Set of parameters used for simulations in pure hydrogen and silane-hydrogen mixtures.

5.2 Simulations in pure hydrogen In this section we present simulation results for discharges operating in pure hydrogen. The re+ sults are obtained by taking into account the presence of electrons and H + , H+ 2 and H3 positive

ions. In this section, we first compare simulation results, obtained under similar conditions, for helium and hydrogen discharges. Then, we consider a H2 discharge operating at p = 0.5 Torr pressure and Vrf = 200 V applied voltage, in order to summarize its main features. This general characterization is done by using the complete form of the effective electric field equation 2.76 (including the ion inertia term) and by adopting field-dependent ion mobilities. These work conditions will be also adopted to analyse the influence of geometric dimensions on different plasma parameters. A systematic study of the rf electric field variations with changes in geometric dimensions, pressure and applied voltage is also carried out. These variations are shown to follow an universal similarity curve, if an adequate normalization is used when plotting the rf electric field as a function of pressure. The model is finally tested against earlier calculations and experimental measurements, for the work conditions indicated in Table 5.2. The influence on results of the effective electric field and the inertia term is also analysed for these conditions. The model convergence for H2 discharges is relatively fast, with convergence errors below 0.1% after less than 100 rf cycles. The typical time evolution of the convergence error c for the electron and positive ion densities, the electron mean energy, the plasma potential and the


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

110


1

(a) (1) (2) (3) (4)

0.1 0.01

(5) (6) (7)

1E-3 1E-4 0

20

40

60

80

100

120

rf periods -38

(b)

Vdc (V)

-40 -42 -44 -46 -48 -50 0

20

40

60

80

100 120

rf periods (c)

Weff (w)

6.3 6.0 5.7 5.4 5.1 0

20

40

60 80 rf periods

100 120

Figure 5.1: (a) Typical time evolution of convergence errors (obtained for a hydrogen discharge operating at p = 0.5 Torr and Vrf = 217 V), for the following time-average parameters: electron density (1), H2+ , H3+ and H + ion densities (2-4), electron mean energy (5), plasma potential (6), and self-bias voltage (7). The time evolution of the self-bias voltage V dc and the coupled electrical power Weff are shown in (b) and (c), respectively.


RESULTS FOR

H2

AND

S I H4 -H2

Condition p (Torr) C1-H2 0.5 C2-H2 1.0

DISCHARGES

111

Vrf (V) 217.0 217.0

Table 5.2: Work conditions for pure hydrogen discharge simulations.

self-bias voltage is shown in Figure 5.1. In this figure, the convergence criterion was taken as c < 8 × 10−4 (cf. Section 3.7). Figure 5.1 also shows the time evolution of the self-bias voltage and the coupled electrical power towards their steady-state values. The convergence time is controlled by the H+ 2 density because its value is subject to large variations within the rf cycle. Notice that the first converged parameters are the plasma potential and the self-bias voltage, indicating that the electrical configuration is rapidly established (after approximately 80 rf cycles, the deviation of Vdc and Weff from their steady-state values is less than 1%). For the initial conditions adopted here one observes that: (i) the self-bias voltage gradually becomes more negative, in order to avoid the escape of electrons from the discharge; (ii) the coupled electrical power has a sudden increase due to important conduction currents at early discharge times. The value of this quantity becomes stable once the electric-field distribution within the discharge is established.

5.2.1 General characteristics This section analyses the qualitative differences between ccrf discharges produced in hydrogen and helium. Simulations performed in pure hydrogen discharges reveal that the global space-time features of the different plasma quantities are qualitatively similar to those presented in Chapter 4 for helium discharges. Thus, the time-average electron density profile has an out-axis peak, located at half distance between electrodes, similar to the one shown in Figure 4.5(a). At steady-state, and during one rf cycle, the charged particle densities between electrodes feature an usual modulation of the electron population over a more or less fixed positive ion background, similarly to the results plotted in Figure 4.5(c,d). Also, the steady-state spatial profile of the time-average electron mean energy presents the same characteristics of Figure 4.6, with low energy values in the bulk and high energy values in the sheath regions. The time-average steady-state, spatial profiles of the plasma potential for H2 discharges are similar to those observed in Figure 4.4(a) for helium discharges. Table 5.3 compares the results obtained for helium and hydrogen discharge simulations, under equivalent operation conditions: p = 1 Torr and Vrf = 217 V. In these simulations we have adopted constant ion mobilities and no effective electric field. The time-average values


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

112

of the electron mean energy ε and the plasma potential V p were calculated on the discharge axis, at z = d/2. The quantities (ne )max and (S i )max represent the maximum values of the time-average electron density and ionization rate profiles, respectively. Parameter

Value Helium Hydrogen Vdc (V) -58.1 -39.5 V p (V) 70.1 53.6 max 9 −3 (ne ) (10 cm ) 3.3 1.6 max 15 −3 −1 (S i ) (10 cm s ) 1.0 2.1 ε (eV) 7.6 2.6 Weff (W) 6.2 11.9 Table 5.3: Plasma and electric parameters for discharge simulations in helium and hydrogen at p = 1 Torr and Vrf = 217 V.

Also for comparison purposes, Figures 5.2 and 5.3 represent, respectively, the axial profile of the time-average electron density (calculated at r = 0), and the spatial countour-plot of the time-average ionization rate for helium and hydrogen discharges, operating in the conditions of Table 5.3. The table indicates that, under the same work conditions, the model predicts a smaller absolute value of the self-bias voltage for hydrogen discharges, i.e. |V dc |H2 < |Vdc |He . This result shows that the lighter and highly mobile hydrogen ions are collected at the rf electrode more efficiently than helium ions, yielding a more positive value for the self-bias voltage. From this point of view, the substitution of He by H2 is equivalent to a reduction in the reactor asymmetry. Since hydrogen ions flow more rapidly towards the walls, the bulk potential falls by about 20% for a H2 discharge. The combination of a more positive Vdc value with the reduction of the plasma potential contributes to a reduction of the electric field (hence the sheath thickness), near the rf electrode. These observations can explain why (ne )max is two times smaller in H2 than in He. The weaker confinement observed in hydrogen discharges leads to flatter electron density profiles, thus to smaller ne values in discharge bulk (see Figure 5.2), even if the ionization potential of hydrogen is only 15.4 eV, in comparison with 24.5 eV for helium. The latter remarks account for the following results: (i) in hydrogen, the peak of the time-average ionization rate has a higher value and is located at a shorter distance from the rf electrode (see Table 5.3 and Figure 5.3), when compared to the corresponding results for helium discharges, under similar operating conditions; (ii) the time-average electron mean energy is at least ∼ 3 times higher for helium than for hydrogen discharges, under similar operating conditions. The results of Table 5.3 for Weff show that hydrogen discharges favour a transmission of power, which can be explained by an enhancement of the conduction current collected at the rf


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

113

Figure 5.2: Axial profiles (at r = 0) of the time-average, steady-state, electron density ne , for a ccrf discharge operating at p = 1 Torr and Vrf = 217 V in helium (red curve) and hydrogen (black).

electrode and a reduction of the phase-shift between the total discharge current and the rf applied voltage. The overall effect is an important rise of Weff (by a factor 2 for the work conditions of Table 5.3) for H2 discharges in comparison with He discharges. The above observations can be confirmed by analysing Figure 5.4, which compares the electron current Ie , the ion current Ii and the total discharge current It , collected at rf electrode for hydrogen and helium discharges. The corresponding rf electrode potential is also plotted to show the current-voltage phase-shift variation.

5.2.2 Positive charge distribution Figure 5.5 presents the contour plots of the time-average, steady-state, ion densities, for a hydrogen discharge operating at p = 0.5 Torr and Vrf = 200 V. From this figure one concludes + that the dominant ion is H+ 3 , according to model predictions. In fact, the maximum value of H 3 + density is more than one order of magnitude higher than the H + 2 or H densities. This result is

a direct consequence of the very efficient conversion of H+ 2 ions (produced by electron impact ionization of hydrogen molecules) into H+ 3 ions, by non-resonant charge exchange collisions + + H+ 2 +H2 → H3 + H + 1.71eV. Although this process becomes less efficient when the H2 ion

energy is greater than 10 eV (above this energy, collisions with hydrogen molecules result in


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

3.2

1E15 8.75E14 7.5E14 6.25E14 5E14 3.75E14 2.5E14 1.25E14 0

z (cm)

2.4

1.6

(a)

0.8

0.0 0.0

114

1.6

3.2

4.8

6.4

r (cm) 3.2

2.2E15 1.93E15 1.65E15 1.38E15 1.1E15 8.25E14 5.5E14 2.75E14 0

z (cm)

2.4

1.6

(b)

0.8

0.0 0.0

1.6

3.2

4.8

6.4

r (cm) Figure 5.3: Spatial contour-plots of the time-average, steady-state, ionization rate (in cm 3 s−1 ) of helium (a) and hydrogen (b), for a ccrf discharge operating in conditions of Figure 5.2.


RESULTS FOR

H2

AND

S I H4 -H2

100

DISCHARGES

115

(a )

I (m A )

80

e

60 40 20 0 0

1 8 .5

37

5 5 .5

74

t (n s ) (b )

0

i

I (m A )

-1 0 -2 0 -3 0 0

1 8 .5

37

5 5 .5

74

(c ) 3 0 0

150

150

0

0

V (V )

300

t

I (m A )

t (n s )

-1 5 0

-1 5 0

-3 0 0

-3 0 0 0

1 8 .5

37

5 5 .5

74

t (n s )

Figure 5.4: Time evolution of the electron conduction current (a), the ion conduction current (b), and the total discharge current (c), for hydrogen (black lines) and helium (red lines) discharges operating at p = 1 Torr and Vrf = 217 V. In (c), the time evolution, during one rf period, of the corresponding rf electrode potential is also plotted (dashed lines).


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

116

symmetric charge transfer processes), the model assumes the same efficiency for this reaction across the whole reactor. Hence, one can anticipate an over-estimation of the H + 3 density, and a subsequent under-estimation of the H+ 2 density, within the sheath regions. Furthermore, be+ cause the mobility of H+ 3 ions is slightly inferior to that of H2 ions the over-estimation of the

H+ 3 density leads to a reduction of the ion current at the rf electrode and, consequently, to more negative Vdc values. An observation of Figure 5.5 shows that the time-average spatial profiles of the H+ 3 and H+ ion densities present the same kind of features previously presented in Figure 4.5. However, the H+ 2 ion density does not follow this typical profile structure, with a single maximum value located at half distance between electrodes. In fact, the time-average, steady-state, H + 2 population presents two maxima located at the vicinity of the reactor corners (where the grid + meets the upper and lower electrodes). The rapid conversion of H + 2 into H3 ions strongly limits

the distribution of the former across the discharge volume. Therefore, the time-average spatial profile of H+ 2 density gives an accurate indication of the regions where this ion is actually produced. Consequently, the H+ 2 time-average density profile should be very similar to that of the time-average ionization rate, as it can be confirmed from Figures 5.5 and 5.6.

5.2.3 Ionization and dissociation rates The spatial contour plot of the typical time-average hydrogen ionization rate, for a discharge operating at p = 0.5 Torr and Vrf = 200 V is shown in Figure 5.6(a). The model predicts that most of the ionization processes occur at the bulk-sheath interface and only a few occur within the bulk, where the ionization rate is at least one order of magnitude smaller. An examination of Figure 5.6(a) shows that the hydrogen ionization rate presents two peaks, situated at positions (r = 5.0 cm, z = 0.6 cm) and (r = 5.6 cm, z = 2.2 cm), approximately and these maxima do not coincide with neither the maximum of the time-average electron density n e (located at half distance between electrodes) or that of electron mean energy ε. As seen before, the maxima of ne and ε are located at different positions in the reactor, which illustrate the very high non-local effects associated with energy transport. Furthermore, the peaks on the ionization rate appear to be located at positions where the time-average electron mean energy density n e ε is higher, for which it is still possible to find enough high energy electrons. Notice that the highest peak of the ionization rate can be found in front of the rf electrode, where the sheath potential drop is also higher. The space-time variation at r = 0 and during one rf period, of the hydrogen ionization rate is shown in Figure 5.6(b). The origin of the four peaks shown in this figure can be explained as follows. As indicated in Section 4.2 (see also Figure 5.8), at the beginning/end of each rf period the field barrier in front of the rf electrode becomes weaker, leading to an increase


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

3.2

2.6E7 2.3E7 2E7 1.6E7 1.3E7 9.8E6 6.5E6 3.5E6 0

2.4

z (cm)

117

1.6

(a)

0.8 0.0 0.0

1.6

3.2

4.8

6.4

r (cm) 3.2

1.2E9 1E9 9E8 7E8 6E8 4E8 3E8 2.5E8 0

z (cm)

2.4 1.6

(b)

0.8 0.0 0.0

1.6

3.2

4.8

6.4

r (cm) 3.2

8E7 7E7 6E7 5E7 4E7 3E7 2E7 1.5E7 0

z (cm)

2.4 1.6

(c)

0.8 0.0 0.0

1.6

3.2

r (cm)

4.8

6.4

Figure 5.5: Spatial contour plot of the time-average, steady-state, densities (in cm −3 ) of the + + following hydrogen ions: H+ 2 (a), H3 (b) and H (c). The results were obtained for a hydrogen discharge operating at p = 0.5 Torr and Vrf = 200 V.


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

3.2

1E15 8.8E14 7.5E14 6.3E14 5E14 3.8E14 2.5E14 1.3E14 0

2.4

z (cm)

118

1.6

(a)

0.8 0.0 0.0

1.6

3.2

4.8

6.4

r (cm) 3.2

3.5E15 3.1E15 2.6E15 2.2E15 1.8E15 1E15 5E14 1.2E14 0

z (cm)

2.4 1.6

(b)

0.8 0.0 0.0

18.5

37.0

55.5

74.0

t (ns) Figure 5.6: Spatial contour plot of the time-average, steady-state, hydrogen ionization rate (in cm−3 s−1 ) (a), for a discharge operating in conditions of Fig. 5.5. For one rf cycle, the space-time variation of hydrogen ionization rate at r = 0 is plotted in (b).


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

119

of the electron diffusion towards the electrode. Some of these electrons enter the rf sheath having acquired enough energy (within the opposite sheath) to ionize the background gas. The ionization processes occurring simultaneously with the diffusion of electrons into the rf sheath yield the peak located in Figure 5.6(b) at z ' 0.4 cm and t ' 0 or 70 − 74 ns. Later, electrons are heated by the increasing rf sheath electric field and, while they are pushed back into the bulk, they are able to produce a second ionization peak (more intense than the first one), located in Figure 5.6(b) at z ' 0.8 cm and t ' 25 ns. Since we are not considering here the production of secondary electrons following the ion bombardment of electrodes, the ionization collisions involving primary electrons reflected by the sheath are crucial for the discharge maintenance. The predicted value for the maximum ionization rate is about 3×10 15 cm−3 s−1 . Simultaneously with the increase of the rf electric field, it occurs a decrease of the electric field in the opposite sheath. Thus, as we approach half period, the electron diffusion towards the grounded electrode is intensified and another important ionization region can be observed at z ' 2.8 cm and t ' 28 ns. Similarly to what happens within the rf sheath, the electron population very close to the grounded electrode will be subject to Joule heating when its sheath electric field starts to increase (after t = 37 ns). Thus, at z ' 2.4 cm and t ' 65 ns a new ionization peak can be observed, produced by the electron acceleration within the grounded electrode sheath. A few final observations can be made about this issue: (i) the ionization processes produced in opposite sheaths by the positive (towards the electrode) and negative (towards the discharge bulk) electron flow occur almost simultaneously; (ii) for the same sheath, a time interval of about 30 ns separates the two ionization maxima; (iii) for the same sheath, the two ionization maxima occur at different distances from the electrode. For the rf sheath, the ionization peak due to electron heating occurs close to the bulk-sheath interface at z ' 0.8 cm, whereas the ionization peak due to electron diffusion is located at z ' 0.4 cm; (iv) other fluid and PIC-MC models for hydrogen discharges predict similar double ionization structures in front of electrodes [Leroy et al. 1995]; moreover, space-time resolved excitation measurements in hydrogen rf discharges identify this kind of double structures with the Balmer-alpha excitation (transition 3p-2s, at wavelength 653.3 nm) [Graham and Mahoney 1998]. It is interesting to note that, for a helium discharge operating under similar conditions, the space-time representation of the ionization rate shown no such double ionization structure [see Figure 4.7(b)], probably due to the higher ionization potential (24.6 eV for helium, in comparison with 15.4 eV for hydrogen). Note that the space-time resolved measurements of the 587 nm helium excitation (corresponding to the 2p-3d transition) reported in [Graham and Mahoney 1998] also shows no double emission structures for a rf helium plasma. We now focus on the hydrogen dissociation rate by electron impact, S d . Typical results for its time-average profile and its space-time variation during one rf period at discharge axis are


RESULTS FOR

H2

AND

S I H4 -H2

120

DISCHARGES

quite similar to the ones shown in Figure 5.6 for the hydrogen ionization rate. In fact, the timeaverage, steady-state, profile of Sd presents a maximum value of about 1016 cm−3 s−1 , located at z ' 1 cm and r ' 5.0 cm, being approximately constant and equal to 6 10 15 cm−3 s−1 within a radial distance of 4 cm from the discharge axis. The latter observation can be important when analysing the operation of plasma reactors for deposition, since it provides an indication of the region where we can expect to find film uniformity. The highest peak of the space-time variation (during one rf period, at discharge axis) of the hydrogen dissociation rate by electron impact has an intensity of about 1.6 1016 cm−3 s−1 and it occurs during the rf sheath expansion at z ' 1 cm and t ' 27 ns. The presence of hydrogen atoms in plasma reactors for deposition have a considerable influence over the structure of deposited films, and so a set of simulations has been performed in order to define the operating conditions that enhance H production. Figure 5.7 shows the varia tion of the space-time averaged, hydrogen dissociation rate, S d , as a function of gas pressure

and for different applied voltages. The examination of this figure reveals that the production of hydrogen atoms increases with the applied voltage, presenting a maximum for a certain pressure

4

-3

3

16

-1

(10 cm s )

value which also increases with Vrf .

2 VRF = 450 V VRF = 350 V VRF = 217 V

1 0 0

1

2

3

4

5

6

7

8

9

p (Torr) Figure 5.7: Space-time averaged hydrogen dissociation rate, as a function of gas pressure and for different applied voltages.


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

121

5.2.4 The electric field distribution. Field inversion and field reversal Figure 5.8 presents a typical distribution of the axial reduced electric field component Ez /N , at different times, for a hydrogen discharge operating at p = 0.5 Torr and V rf = 200 V. As expected, this distribution features a zero electric field region in the discharge center and two high intensity electric field regions in the discharge sheaths. Let us focus on the rf electrode sheath. The Ez /N value at z = 0 is generally negative (thus oriented towards the electrode) due to the presence of a negative self-bias voltage. However, during the time period when the magnitude of the electric field decreases (between t = 5T /10 and t = 9T /10), we can observe the formation of a small positive value region for E z /N , near the rf electrode. The massive presence of positive charges in the sheath and the electron accumulation at sheath border are responsible for the positive values of E z /N (at t = 8T /10 and t = 9T /10, for example). As the electron flow towards the rf electrode increases and more electrons are allowed to enter the sheath, the local charge separation vanishes and the intensity of this Ez /N positive value region decreases. Later, at the early beginning of the next period and before the rf sheath starts its expansion, when the value of the (still positive) rf potential decreases, a second (very weak) space-charge barrier develops to further limit the entrance of electrons in the sheath region. Consequently, the field near the rf electrode shifts negative (the so-called field inversion) for axial positions just after its positive value region. The formation of this “double sheath”is depicted in Figure 5.8(b) at time t = T /20, for hydrogen discharges operating at 0.5 Torr pressure and different applied voltages of 200 V and 400 V (the use of a higher Vrf allows an amplification of this phenomenon). An observation of this figure clearly shows that the values of Ez /N alternate twice between negative and positive from z = 0 to z ' 2.4 cm (the latter corresponding to the grounded sheath threshold). Simulations performed in hydrogen, under conditions favouring the development of strong electric fields in the sheath regions (e.g., p = 0.35 Torr and Vrf = 350 V), adopting constant ion mobilities and neglecting the inertia term in the effective field equation 2.76 (thus enhancing the ion flow), show the occurrence of a so-called field reversal near the rf electrode. This phenomenon is associated to the enlargement of the Ez /N positive value region, for a very short time interval between 9T /10 and 0T , during which the orientation of the instantaneous E z /N value is directed towards the discharge bulk (even at the rf electrode!). This effect comes as a response to the enhanced ion flow directed towards the rf electrode, and is rapidly canceled by the decrease of the rf potential which finally inverts the field direction towards the rf electrode, once again. The field reversal phenomenon has been experimentally observed in hydrogen discharges by [Graham and Mahoney 1998], and is often used to account for the intense H 2 excitation near the rf electrode. The fact that current fluid simulations can only predict a field


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

122

225

(a)

Ez/N (10

-16

2

Vcm )

150 75 0 -75

0 T/10 2T/10 3T/10 4T/10

-150 -225 -300

0.0

0.8

1.6

5T/10 6T/10 7T/10 8T/10 9T/10 2.4

3.2

z (cm)

10

Ez/N (10

-16

2

Vcm )

(b) (1)

5

0

(2)

-5

-10

0.0

0.8

1.6

2.4

z (cm)

Figure 5.8: (a) Axial profile (at r = 0) of the axial reduced electric field Ez /N , at different times along the rf period. The results are for a hydrogen discharge operating at p = 0.5 Torr and Vrf = 200 V. (b) Axial profile components (at r = 0) of the axial reduced electric field Ez /N , at t = T /20. The results are for a hydrogen discharge operating at p = 0.5 Torr and (1) Vrf = 200 V; (2) Vrf = 400 V.


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

123

reversal under very particular (and extreme) operating conditions is probably an indication that the hydrogen ion transport description within rf sheaths is still to be improved.

5.2.5

Comparison with experimental results. The influence of the ion inertia term

In Table 5.4 we compare simulation results and measured values for the self-bias voltage V dc and the coupled electrical power Weff . The results were obtained for hydrogen discharges operating under the work conditions indicated in Table 5.2, for various ion transport conditions. In particular, simulation results (A)-(E) shown in Table 5.4 were obtained assuming: (A), constant ion mobilities (low field values, see Section 2.4.3) and no effective electric field; (B), field-dependent ion mobilities (cf. Eqs 2.95 and 2.97) and effective electric field calculated from equation 2.76 without inertia term; (C), field-dependent ion mobilities and effective electric field calculated from equation 2.76 with inertia term; (C’), same conditions as (C), but updating the self-bias voltage after each time step (instead of at the end of the rf period); (D), same conditions as (C), but considering only one positive ion (H + 2 ); (E), simulation results obtained in a previous work using an earlier model version [Leroy 1996] (which adopted field-dependent ion mobilities, no effective electric field, and an ion kinetics which considers only the "mean" species H+ 2 ). When the ion transport is described using conditions (B) instead of (A), the changes in the electrical parameters are about 1.5% for Vdc and 7 − 8 % for Weff . These variations are considerably lower than the ones predicted for helium discharges (cf. Section 4.6), due to the smaller changes now observed in the magnitude of the electric field, between conditions (A) and (B). Note, however, that the transition from simulations (A) to (B) involve the presence of two competing effects: the decrease of the ion mobilities (which contributes to reduce the ion fluxes), and the enhancement of the sheath electric field and ion densities (which favors an ion flux increase). At low pressure condition C1-H2 , the enhancement of the electric field magnitude dominates over the reduction of ion mobilities. In this case, the ion current collected at the rf electrode increases between simulations (A) and (B), justifying a decrease on the absolute


Cond.

Parameter

C1-H2

Vdc (V) Weff (W) Vdc (V) Weff (W)

C2-H2

RESULTS FOR

H2

(A) (B) −47.5 −46.7 7.4 6.7 −39.5 −39.9 11.9 11.1

AND

S I H4 -H2

Simulations (C) (C’) −49.0 −63.2 6.1 6.0 −41.5 −59.4 10.6 10.6

124

DISCHARGES

(D) (E) −49.8 −51.0 7.1 8.9 −40.2 −45.0 12.2 16.4

Experiment −59.0 5.0 −60.0 9.0

Table 5.4: Calculated and measured values of electrical parameters for hydrogen discharges working in conditions (C1-H2 ) and (C2-H2 ) (see Table 5.2). Simulation results (A)-(D) were obtained in this work assuming different ion transport conditions: (A), constant ion mobility and no effective electric field; (B), field-dependent ion mobilities and effective electric field calculated without inertia term; (C), field-dependent ion mobilities and effective electric field calculated with inertia term; (C’), same conditions as (C), but updating the self-bias voltage after each time step; (D), same conditions as (C), but considering only the H + 2 ion. Simulation results (E) were obtained in a previous work using an earlier model version [Leroy 1996]. There is at least a 40% uncertainty in measured values of Vdc .

value of the self-bias voltage. The opposite situation occurs at higher pressure condition C2-H 2 , where the reduction of the ion mobilities accounts for a slightly more negative V dc . Contrarily to the different variations observed for Vdc , the effective power coupled to the plasma is always lower in simulations (B) than in (A), for both conditions C1-H 2 and C2-H2 . Furthermore, the effective electric field formulation leads to opposite effects in helium and hydrogen, as Weff was found to be higher in simulations (B) than in (A), when performed for He discharges. This is an important result, which corrects the predictions for W eff obtained in previous calculations reported by [Leroy 1996]. In fact, when compared to experimental values, the latter predictions were found to be under-estimated in He discharges, and over-estimated in H2 discharges (cf. Tables 4.4 and 5.4). The reason for the different behavior of Weff in He and H2 , as we go from simulations (A) to (B), is the result of two combined effects. First, the enhancement of the conduction current, due to the introduction of the effective electric field, is more intense in helium than in hydrogen; second, the higher mobility of hydrogen ions lead to an additional phase-shift between the total current It and the total potential at the rf electrode Ut = Vdc + Vrf cos(ωt), with the consequent reduction in the effective power coupled to the plasma. Figure 5.9 shows, for one rf cycle, the total current and the total potential at the rf electrode in simulations (A) and (B), for helium and hydrogen discharges. From Figure 5.9(a) one confirms that simulation (B) leads to a important increase of the total discharge current in helium, but keeping the phase difference between It and Ut (note the crossing point between curves 1 and 3, which occurs approximately at half-period). From Figure 5.9(b) one confirms that simulation (B) leads to only a very small increase of the total discharge current in hydrogen,


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

150

150

0

0

Ut (V)

(a) 300

It (mA)

300

125

(1) (2) -150 (3) (4) -300

-150 -300 0

18.5

37

55.5

74

t (ns)

150

150

0

0

Ut (V)

(b) 300

It (mA)

300

(1) (2) -150 (3) (4) -300

-150 -300 0

18.5

37

55.5

74

t (ns) Figure 5.9: (a) Time variation, during one rf cycle and for simulations (A) and (B), of the total current It (curves 1 and 3, respectively) and the total potential at the rf electrode U t (curves 2 and 4, respectively), for a helium discharge operating at p = 0.5 Torr and V rf = 217 V. (b) As in (a), but for a hydrogen discharge operating at same pressure and applied voltage conditions.


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

126

which is accompanied by the introduction of an additional phase shift between I t and Ut (note the crossing point between curves 1 and 3, which occurs before half-period). The latter facts contribute to a reduction on the value of Weff , as we go from simulations (A) to (B) in hydrogen. Note finally that this effect is also present in silane-hydrogen discharges, operating in hydrogen dominated conditions (see conditions C2 and C3 in Table 5.8). In order to separate between the contribution of variable ion mobilities and that of effective electric field in the predicted values of Weff , for conditions C1-H2 and C2-H2 , we have performed calculations which used Eq. 2.76, while keeping constant ion mobilities. The results obtained are very close to the ones shown in column (C), which indicates that the correction in Weff here obtained is mainly due to the use of the effective electric field. The influence of the ion inertia term, on the values of electrical parameters, can be analysed from the results presented in columns (B) and (C) of Table 5.4. In general, the inclusion of an inertia term in Eq. 2.76 contributes to a decrease in the ion current flowing towards the rf electrode, thus reducing the incoming positive charge and increasing the absolute value of the self-bias potential [Vdc changes by about 4−5 % from (B) to (C)]. Furthermore, the introduction of ion inertia leads to a small decrease in the magnitude of the sheath electric field (more noticed at low pressures, see condition C1-H2 ), with the consequent reduction in both conduction and displacement currents, and thus in the power coupled to the plasma [W eff decreases by about 5 − 9 % from (B) to (C)]. Simulations (C’) were obtained using the same ion transport conditions as in (C), but adopting a different calculation method for the self-bias voltage. In (C’), V dc was updated after each time step iteration, thus forcing the capacitive coupling condition (cf. Section 2.5.2) to be valid at all times. The imposition of a continuous correction for Vdc produces only small changes in most discharges parameters, like the electron and ion densities, the electron mean energy or the power coupled to the plasma [Weff changes less than 2 % from (C) to (C’)]. However, the value of the self-bias voltage is strongly modified [Vdc becomes 30-40 % more negative from (C) to (C’)], as a result of a global negative shift observed for the time-average plasma potential profile. The new potential distribution enhances the electric field within the rf sheath, while reducing its value within the grounded sheath. Hence, a Vdc overcorrection within the rf period can lead to the artificial introduction of an additional discharge asymmetry, especially if applied to a light ion system at high pressures (see condition C2-H2 ), which exhibits a strong response to any rf field variation (cf. also Sections 2.5.2 and 4.7). A simplified discharge kinetics has been adopted in simulations (D), disregarding the presence of H+ and H+ 3 ions in the discharge and thus considering, as in [Leroy 1996], only a “mean” positive ion (H+ 2 ). The values of Vdc and Weff obtained under (D) are compared to (standard) simulation results (C), which use the same transport conditions but assume the full ion kinetics.


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

127

Although simulations (D) produce minor changes in the self-bias voltage (less than 3 %), the predicted values of the power coupled to the plasma are considerably higher (by about 16 %) in (D). We can thus conclude that the simplified kinetics adopted in earlier calculations (E) was responsible for the large and systematic over-estimation of Weff with respect to experimental measurements. The results in Table 5.4 further show that the predicted values of W eff , obtained with the present model, are considerably closer to experiment.

5.2.6 Effects of reactor dimensions Since geometry plays an important role in plasma technologies, it is relevant to investigate the scaling of plasma parameters with reactor dimensions. This section analyses the variation of the plasma potential, the rf electric field, the electron density and the electron mean energy with changes in the reactor radius R and its inter-electrode distance d. The “standard” dimensions are R = 6.4 cm and d = 3.2 cm, corresponding to those of the experimental setup presented in Section 5.1. The study begins with the influence of the electrode radius on a hydrogen discharge operating at p = 0.5 Torr pressure and Vrf = 200 V applied voltage. The simulations used a constant inter-electrode distance d = 3.2 cm, adopting four alternative values for R: 1.6 cm, 3.2 cm, 6.4 cm and 12.8 cm. Figure 5.10(a) plots the time-average distributions of plasma potential and axial component of the rf electric field at discharge axis, for the different values of R adopted here. As expected, results show that the discharge symmetry is strongly favored by the increase of the reactor radius, as in this case the ratio of the grounded area (the surface of the lower electrode plus that of the lateral grid) to the rf electrode area approaches unit. The magnitude of the potential drop between the plasma and the electrodes is a crucial parameter in deposition/etching systems, because it controls the flux of charged particles through the sheath, in front of the wafer. With this respect, Figure 5.10(a) shows that the discharge symmetrization, following an increase in the reactor radius, involves a simultaneous increase of the bulk potential and a reduction of the self-bias voltage. In fact, between R = 1.6 cm and R = 12.8 cm, the absolute value of V dc is reduced by 60 V, with the potential drop across the rf sheath decreasing by a factor 1.3, while the potential drop across the grounded sheath increases by a factor of 2. Note that the enhancement of the ion current (hence the less negative self-bias voltages), which occurs when R increases, is a strict consequence of the larger area over which the ions are collected. This statement can be confirmed by analysing Figure 5.10(b), where we observe a reduction of the electric field intensity near the driven electrode, as R increases. Note, conversely, that the electric-field intensity near the grounded sheath increases with R, thus meaning that the ions will receive more energy when crossing that sheath.


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

60

128

(a)

Vp (V)

30 0 -30

R R R R

-60

= = = =

1.6 cm 3.2 cm 6.4 cm 12.8 cm

2.4

3.2

-90 0.0

0.8

1.6

z (cm)

Ez/N (10

-16

2

Vcm )

100

(b)

50 0 -50 -100

R R R R

-150 -200 -250

0.0

0.8

1.6

2.4

= = = =

1.6 cm 3.2 cm 6.4 cm 12.8 cm

3.2

z (cm) Figure 5.10: Axial profile (at r = 0) of the time-average, steady-state plasma potential V p (a) and axial reduced electric field Ez /N (b), for hydrogen discharges operating in conditions of Fig. 5.5, at different values of the rf electrode radius R.


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

129

The effective power coupled to the plasma discharge also increases with R, once again due to the enlarged area over which the charged particles are collected. The predicted values of Weff show a considerably increase from 0.18 W at R=1.6 cm to 20.4 W at R=12.8 cm. For the standard radius R=6.4 cm, we have Weff = 5.2 W. The impact on the electron density of variations in the discharge radius is analysed in Figure 5.11. The figure shows the spatial contour plots of the time-average, steady-state electron density for hydrogen discharges operating with three different radius: R = 3.2 cm, R = 6.4 cm (the “standard” value) and R = 12.8 cm. An observation of this figure reveals that: (i) the maximum electron density value is little affected by changes in R, remaining at approximately 1.2 × 109 cm−3 for all situations analysed here; (ii) the maximum electron density is located at approximately the same distance (∼ 1.6 cm) of the lateral grid, for all radius considered. Note that this maximum approaches the discharge axis as R decreases; (iii) the discharge symmetrization which occurs when R increases leads to electron density profiles whose maximum is located at half distance between electrodes. The latter result can also be confirmed by analysing Figure 5.12(a), which plots the axial profile of the time-average electron density for the different values of R adopted here. At low values of R the electron density presents a steeper profile near the rf electrode, due to the different confinement electric fields existing within the rf and grounded sheaths. As R increases, the simulation results become closer to a 1d case, when no self-bias voltage develops, and the electron density shows a more symmetric profile between electrodes. The small increase of the electron density at low values of R is related to an enhancement of the electron mean energy in the rf sheath region, as the radius decreases. The behavior of ε with changes in R can be observed in Figure 5.12(b), which plots the axial profile of the time-average mean electron energy, for different values of the rf radius. The variations of the mean energy intensity follow that of the reduced electric field Ez /N in sheath regions [see Figure 5.10(b)], showing an increase of ε (hence of ionization mechanisms) near the rf electrode, and a decrease near the grounded electrode, when R is reduced. Note that the discharge symmetrization leads to comparable values of ε in both sheaths, as R increases. To summarize, an increase of the electrode radius in rf discharges, operating under the same conditions of pressure and applied voltages, leads to a discharge symmetrization implying: • higher coupled electrical powers and lower self-bias voltages; • an increase in the energy of ions flowing towards the grounded substrate; • more symmetric profiles for the electron density and mean energy. We now examine the influence of the inter-electrode distance on the same hydrogen discharge operating at p = 0.5 Torr pressure and Vrf = 200 V applied voltage. The simulations


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

z (cm)

(a)

r (cm)

z (cm)

(b)

r (cm)

z (cm)

(c)

r (cm)

!

130

"

Figure 5.11: Spatial contour plot of the time-average, steady-state electron density (in cm −3 ), for hydrogen discharges operating in conditions of Fig. 5.5 and for the following values of the electrode radius: R = 3.2 cm (a), R = 6.4 cm (b) and R = 12.8 cm (c).


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

131

0.15

10

-3

ne (10 cm )

(a) R R R R

0.12 0.09

= = = =

1.6 cm 3.2 cm 6.4 cm 12.8 cm

0.06 0.03 0.00 0.0

0.8

1.6

2.4

3.2

z (cm) 25 R R R R

ε (eV)

20 15

= = = =

1.6 cm 3.2 cm 6.4 cm 12.8 cm

1.6

2.4

(b)

10 5 0 0.0

0.8

3.2

z (cm) Figure 5.12: Axial profile (at r = 0) of the time-average, steady-state electron density ne (a) and electron mean energy ε (b), for hydrogen discharges operating in conditions of Fig. 5.5 and for different values of the rf electrode radius.


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

132

used a constant electrode radius R = 6.4 cm, adopting three alternative values for d: 2.4 cm, 3.2 cm and 6.4 cm. Figure 5.13 presents the time-average distributions of plasma potential and axial component of the rf electric field at the discharge axis, for the different values of d adopted here. The results show that an increase in the inter-electrode distance has three main consequences: (i) the self-bias voltage becomes more negative (see the potential value at z = 0), thus indicating that the discharge is more asymmetric (the absolute value of Vdc increases 18% when d is multiplied by 2); (ii) the plasma bulk potential tends to smaller values (when d changes from 2.4 cm to 6.4 cm, the bulk potential decreases 14 V); (iii) the potential drop across the grounded sheath is considerably reduced. Once again, these results can be explained using symmetry arguments. When d increases, the grounded surface of the lateral grid becomes larger, which enhances the discharge asymmetry. The more negative Vdc values so produced force the rf potential to shift negative during most of the time cycle, thus yielding a reduction of the bulk potential as an overall effect. Note finally that an increase in R or in d is responsible for totally opposite effects in terms of the potential drop across the sheaths, as can be checked by comparing Figures 5.10 and 5.13. This comes as a direct consequence of discharge symmetrization (tending to balance both potential drops), which is favored by large electrode radii and small inter-electrode distances. Another important consequence of a large separation between electrodes is the increase in the value of the effective power coupled to the plasma (Weff increases around 30%, from 5.2 W to 6.9 W, when d varies between 3.2 cm and 6.4 cm). Althought larger values of d and R both lead to an increase of Weff , the physical reasons which justify these variations are quite different. In the first case, the power enhancement is due to a stronger electric field near the rf electrode (see Figure 5.13), while in the second case it is justified by a larger collection surface for the discharge current. The impact on the electron density of variations in the inter-electrode distance are analysed in Figure 5.14(a). This figure shows the time-average axial profile of ne , for hydrogen discharges operating at different d values. An observation of this figure reveals that: (i) the maximum electron density value is little affected by changes in d, increasing from 7 × 10 8 cm−3 at d = 2.4 cm to 1.1 × 109 cm−3 at d = 6.4 cm; (ii) at small d values the maximum electron density is symmetrically located between electrodes, but it deviates towards the rf electrode in asymmetric discharges produced at larger inter-electrode distances; (iii) the density profile near the rf electrode becomes steeper at large d values. The above results can be correlated with that one of Figure 5.14(b,c), which show the timeaverage axial profiles of the electron mean energy ε and energy density ne ε, for hydrogen discharges operating in the same conditions as before. Figure (b) reveals that by increasing d we


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

60

133

(a )

p

V (V )

30 0 d = 2 .4 cm d = 3 .2 cm d = 6 .4 cm

-30 -60

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4

z (c m ) 2

Vcm )

1 20

(b )

80 40

z

E /N (1 0

-1 6

0 -40 d = 2 .4 cm d = 3 .2 cm d = 6 .4 cm

-80 -120 -160

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4

z (c m ) Figure 5.13: Axial profile (at r = 0) of the time-average, steady-state plasma potential V p (a) and axial reduced electric field Ez /N (b), for hydrogen discharges operating in conditions of Fig. 5.5, at different values of the inter-electrode distance d.


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

134

0.12 -3

0.09

10

ne (10 cm )

(a)

0.06

d = 2.4 cm d = 3.2 cm d = 6.4 cm

0.03 0.00 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4

ε (eV)

z (cm) 16 14 12 10 8

(b) d = 2.4 cm d = 3.2 cm d = 6.4 cm

6 4 2 0 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4

(c)

0.3

d = 2.4 cm d = 3.2 cm d = 6.4 cm

0.2

10

-3

neε (10 cm eV)

z (cm)

0.1 0.0 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4

z (cm) Figure 5.14: Axial profile (at r = 0) of the time-average, steady-state electron density ne (a), electron mean energy ε (b), and energy density ne ε (c), for hydrogen discharges operating in conditions of Fig. 5.5, at different values of inter-electrode distance d.


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

135

produce an overall reduction of ε within the discharge. Note that the bulk mean energy decreases by a factor two, when d increases from 2.4 cm at 6.4 cm, and that simultaneously the differences between the calculated mean energies near rf and grounded electrodes are intensified. It is interesting to observe that the electron mean energy does not follow the path of the rf sheath electric field, which slowly increases with d (see Figure 5.13). The explanation can be obtained from Figure (c), which shows a global, small increase of the energy density n e ε with d. This behavior is mainly controlled by the electron density, and it demonstrates the physical interest in the monitoring of the ne ε (more that ε), which is actually what is done in our numerical calculations. Figure 5.14(c) shows another interesting feature. The energy density ne ε presents a small ripple in both transition regions between bulk and sheaths. These slope variations are induced by the very steep decrease of the electron mean energy in these regions (where ionization events mainly occur), combined with non-locality effects which produce the maxima of ε and ne at different locations. Figure 5.15 presents the spatial contour plots of the time-average, steady-state electron density for hydrogen discharges operating with the inter-electrode distances considered here. From this figure one can observe that, as d increases, the region of higher electron density expands towards the discharge axis while it gets closer to the rf electrode, as a result of discharge asymmetry. For the operating conditions of Figures 5.15(a,b), the contour plot of the time-average hydrogen ionization rate is similar to that shown in Figure 5.6. However, for large d’s, the value of mean energy decreases within the reactor domain and the ionization processes concentrate close to the rf electrode, where there is still enough energy available. To summarize, an increase of the inter-electrode distance in rf discharges, operating under the same conditions of pressure and applied voltages, enhances the discharge asymmetry implying: • higher values of coupled electrical power and self-bias voltage; • a decrease in the energy of ions flowing towards the grounded substrate; • more asymmetric profiles for the electron density and mean energy.

5.2.7 Similarity law This section presents a systematic study of the rf field variations in hydrogen ccrf discharges, with changes in geometric dimensions, pressure and applied voltages. The aim of this study is to show that these variations follow an universal similarity law, if an adequate normalization is used when plotting the rf electric field as a function of pressure. The simulation conditions covered a large domain of pressures (0.25 - 8.0 Torr), for three values of applied voltage Vrf (217, 350 and 450 V), and three values of inter-electrode distance


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

z (cm)

(a)

r (cm)

z (cm)

(b)

r (cm)

z (cm)

(c)

%

r (cm)

(

! " !'&" &%" % $ !

(

" #!" !"

136

" #!" !"

Figure 5.15: Spatial contour plot of the time-average, steady-state electron density (in cm −3 ), for hydrogen discharges operating in conditions of Fig. 5.5 and for the following values of inter-electrode distance: d = 2.4 cm (a), d = 3.2 cm (b) and d = 6.4 cm (c).


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

137

p Weff E rms /N nmax e (Torr) (W) (10−16 Vcm2 ) (109 cm−3 ) 0.25 3.22 147.33 0.62 0.5 5.98 84.89 1.25 217.0 1.0 10.48 47.87 1.74 1.5 13.11 32.60 1.84 2.0 13.28 23.54 1.79 3.0 6.10 12.83 1.5 1.0 27.07 80.37 3.37 1.5 37.22 58.20 3.91 350.0 2.0 45.52 45.65 4.19 3.0 51.59 30.54 4.23 4.0 45.66 21.25 3.94 5.0 25.61 14.66 3.51 1.5 64.58 76.38 5.6 2.0 85.49 61.92 6.3 3.0 114.40 44.60 7.1 450.0 3.25 114.52 41.08 7.04 4.0 108.30 32.40 6.6 5.0 95.30 24.30 6.1 6.0 76.80 18.60 5.6 Vrf (V)

Table 5.5: Calculated

values of the coupled electric power Weff , the space-average, rms reduced electric field E rms /N and the maximum electron density value nmax , for ccrf hydrogen dise charges operating at different applied voltages Vrf and gas pressures p. In these simulations, the inter-electrode distance was kept constant at d = 3.2 cm.


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

138

d (1.6, 3.2 and 6.4 cm). Each discharge condition was represented using a single value of

the rf electric field, E rms , obtained by calculating the root-mean-square (rms) in time, and the

average value in space, of the electric field space-time distribution. Calculations were performed adopting variable ion mobilities and the ion effective field equation 2.76, without its inertia term.

The results obtained for the effective electric power Weff , the space-averaged, rms reduced

are summarized in electric field E rms /N , and the maximum electron density value nmax e Tables 5.5 and 5.6.

For the first set of simulations (cf. Table 5.5), p was varied at different Vrf values by keeping a constant inter-electrode distance of d = 3.2 cm. For a second set of simulations (cf. Table 5.6), p was varied at different d values by keeping a constant rf applied voltage of V rf = 350 V. Note that the adopted pressures ensure similar values of the N ×d product, for the three inter-electrode distances chosen. d p (cm) (Torr) 2.0 3.0 1.6 4.0 6.0 8.0 1.0 1.5 3.2 2.0 3.0 4.0 5.0 0.5 1.0 6.4 1.5 2.0 3.0

Weff E rms /N (W) (10−16 Vcm2 ) 29.55 57.99 39.14 40.98 45.72 31.39 50.98 20.38 46.32 13.84 27.07 80.37 37.22 58.20 45.52 45.65 51.59 30.54 45.66 21.25 25.61 14.66 23.50 109.31 40.54 65.08 40.10 43.36 36.81 31.40 25.46 18.46

nmax e (109 cm−3 ) 4.10 4.65 4.81 4.62 3.93 3.37 3.91 4.19 4.23 3.94 3.51 2.52 3.35 3.25 3.10 2.75

Table 5.6: Calculated

values of the coupled electric power Weff , the space-average, rms reduced electric field E rms /N and the maximum electron density value nmax , for ccrf hydrogen dise charges operating at inter-electrode distances d and gas pressures p. In these simulations, the rf applied voltage was kept constant at Vrf = 350 V.

Figure 5.16 plots the space-average, rms reduced electric field E rms /N as a function of

N d, for the different values of applied voltages Vrf and inter-electrode distances d presented

in Tables 5.5 and 5.6. Note that the observed decrease of E rms /N with the increase of N d

(at constant d and Vrf ) is related to an enlargement of the very low electric field region, at the

discharge center. The latter enlargement is associated to a compression of sheaths, which is


RESULTS FOR

H2

S I H4 -H2

AND

DISCHARGES

139

favored by the increase of both pressure and inter-electrode distance. Note also that pressure effects dominate over inter-electrode distance effects in controlling the electric field profile, as

E rms /N increases with d, at constant N d [hence for decreasing N values, see Figure 5.16(b)].

Note finally that, as expected, the space-average, rms reduced electric field E rms /N increases

-16

2

Vcm )

with the rf applied voltage Vrf , at constant N d [see Figure 5.16(a)].

150

(a)

120

V rf = 217 V V rf = 350 V V rf = 450 V

/N (10

90 60 30 0 0

10

20

30 16

40

50

60

-2

120

(b)

90

d = 1.6 cm d = 3.2 cm d = 6.4 cm

/N (10

-16

2

Vcm )

Nd (10 cm )

60 30 0 0

10

20

30 16

40

50

60

-2

Nd (10 cm )

Figure 5.16: Plot of the space-average, rms reduced electric field E rms /N , as a function of N d, for different values of the rf applied voltage Vrf (at d = 3.2 cm) (a), and for different values of inter-electrode distance d (at Vrf = 350 V) (b).

Figure 5.17 represents the effective electrical power coupled to the plasma W eff as a function of N d, for the different values of applied voltages Vrf and inter-electrode distances d presented


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

140

in Tables 5.5 and 5.6. The variation of Weff with N d is similar to the one observed for nmax e

120

(a)

Weff (w)

100 80 60 40

V rf = 217 V V rf = 350 V V rf = 450 V

20 0 0

15

30

45

60 16

75

90

-2

Nd (10 cm ) (b)

Weff (w)

50

d = 1.6 cm d = 3.2 cm d = 6.4 cm

40 30 20 0

15

30

45 16

60

75

90

-2

Nd (10 cm ) Figure 5.17: Effective electrical power coupled to the plasma Weff , as a function of N d, for different values of the rf applied voltage Vrf (at d = 3.2 cm) (a), and for different values of inter-electrode distance d (at Vrf = 350 V) (b).

with p (cf. Tables 5.5 and 5.6), presenting a maximum for both fixed inter-electrode distance and applied voltage. At low pressures, the strong decrease of the rf electric field with increasing gas density [see Figure 5.16] is responsible for a significant increase in the electron density (hence, in the coupled electric power), in order to ensure the discharge maintenance. Note that this variation was already observed when studying helium discharges (see Figure 4.13). At high pressures, the charged particle losses are strongly limited by diffusion, and thus the


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

141

discharge operating point can be obtained for increasingly smaller electron densities (hence, smaller coupled electric powers). Note that this result is consistent with the one presented in Figure 5.7 for the hydrogen dissociation rate. A further inspection of Figure 5.17 shows that: (i) when Vrf increases, the maximum value of Weff is strongly enhanced, being shifted towards higher pressures [see Figure 5.17(a)]; pressure effects dominate over inter-electrode distance effects in controlling the coupled electric power profile, as Weff decreases with d, at constant N d [hence for decreasing N values, see Figure 5.17(b)]. Remember that, at low pressures, an increase of either the gas density or the inter-electrode distance yields an increase of Weff (cf. also 5.2.6). As stated before, the main purpose here is the introduction of adequate variables, in order to align the different sets of points in Figures 5.16-5.17 as an universal similarity curve. The interest with this representation is that the “working point” of a reactor, operating at given rf voltage, gas pressure and inter-electrode distance, can then be uniquely defined by using its

( E rms /N , N d) similarity curve. In what follows, the operation conditions corresponding to Vrf = 350 V and d = 3.2 cm will

be considered as the “reference” conditions. The strategy adopted here involves the introduction of two similarity parameters, E ? /N and N Λ, to be defined through an adequate normalization of the space-average, rms reduced electric field and the gas pressure, in such a way that all plots of E ? /N as a function of N Λ yield a single universal characteristic curve. In other words, the different sets of points in Figures 5.16-5.17 are to collapse with the (red) one, corresponding to the above defined “reference” condition, if these figures are re-plotted using the similarity parameters as the new coordinates.

For a fixed value of d = dref = 3.2 cm, the variations of E rms /N with pressure are

accompanied by strong and unequal variations of Weff . Since the coupled power is proportional to the square of the electric field, the similarity parameter representing the space-average, rms electric field, normalized to power 1 W, is defined as

E rms /N E? = 1 N W2

.

(5.1)

eff

The new plot of E ? /N as a function of N d, for different Vrf at fixed dref , is represented in Figure 5.18. The universal feature of this representation can be randomly tested. For arbitrary

p = 0.5 Torr and Vrf = 280 V operating conditions, simulation results yield E rms /N = 115.67 10−16 Vcm2 and Weff = 9.78 W, corresponding to similarity parameters

1 ( E rms /N )(Weff )− 2 = 36.99 × 10−16 Vcm2 N d = 5.15 × 1016 Vcm−2

which fairly fits the similarity curve of Figure 5.18.

,


RESULTS FOR

H2

AND

S I H4 -H2

142

DISCHARGES

2

Vcm )

80 Vrf = 217 V; d = dref Vrf = 350 V; d = dref Vrf = 450 V; d = dref

*

E /N (10

-16

60 40 20 0 0

10

20

30

Nd (10

16

40

50

60

-2

cm )

1 Figure 5.18: Similarity curve of E ? /N = ( E rms /N )(Weff )− 2 vs. N d, for hydrogen discharges operating at dref = 3.2 cm for different Vrf values. We are now left with the definition of the second similarity parameter, N Λ, allowing a representation that includes different inter-electrode distances. The quantity Λ corresponds to an effective length defined as dref /lref , (5.2) d/l where l represents the axial thickness of the rf sheath, lref being the corresponding thickness Λ=d

obtained for d = dref . The values of the rf sheath thickness l are estimated as the distance between the rf electrode and the axial position where the time-average electric field Ez goes to zero [see, for example, Figure 4.12 or 5.13(b)]. With this definition, Λ can be seen as the correction of the “reference” inter-electrode distance, according to the relative variation of the rf sheath thickness. In fact, equation 5.2 can be rewritten as

l

, lref thus showing that the similarity parameter Λ acts as follows Λ = dref

Λ < d,

for d > dref

⇒

l < lref

,

Λ > d,

for d < dref

⇒

l > lref

,

Λ = d,

for

d = dref

⇒

l = lref

.

Figure 5.19 represents the new plot of E ? /N as a function of N Λ, for the various values of Vrf and d adopted in Figure 5.16. The figure confirms that the similarity parameters E? /N


RESULTS FOR

H2

AND

S I H4 -H2

143

DISCHARGES

90 Vrf = Vrf = Vrf = Vrf = Vrf =

70 60 50 40

350 350 350 450 217

V; V; V; V; V;

d d d d d

= = = = =

dref/2 dref 2dref dref dref

30

*

E /N (10

-16

2

Vcm )

80

20 10 0 0

10

20

NΛ (10

30 16

40

50

60

-2

cm )

Figure 5.19: Similarity curve of E ? /N vs. N Λ [Λ = d (dref /lref ) (d/l)−1 ], for hydrogen discharges operating at various rf applied voltages and inter-electrode distances.

and N Λ were adequately defined, as they allow to group very different discharge conditions under the same characteristic curve. The universal feature of this representation was tested for arbitrary p = 0.5 Torr, Vrf = 200 V and d = 2.4 cm operating conditions. In this case simulation

results yield E rms /N = 89.76 10−16 Vcm2 , Weff = 4.17 W and Λ = 3.36 cm, corresponding to similarity parameters

1 ( E rms /N )(Weff )− 2 = 43.95 × 10−16 Vcm2 N Λ = 5.4 × 1016 Vcm−2

,

which again fits the similarity curve of Figure 5.19 quite well. The similarity law pictured by Figure 5.19 can be extremely useful in defining the working points of ccrf discharges in hydrogen, as well as in featuring their reduced electric field, coupled power, and sheath thickness for given pressure and inter-electrode distance.

5.3 Simulations in SiH4-H2 mixtures In this section we present the main simulation results and outline some specific features of discharges operating in SiH4 -H2 mixtures. Results are obtained by taking into account the + + + − + presence of electrons, SiH+ 2 , SiH3 , H , H2 , H3 positive ions, and the SiH3 negative ion, and

by adopting the kinetic processes presented in Table 2.3 [Salaba¸s et al. 2002b].


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

144

The model is tested against previous calculations and experimental measurements [Leroy 1996; Leroy et al. 1998], for the work conditions indicated in Table 5.7. Because C3 is a typical Condition p (Torr) C1 0.068 C2 0.068 C3 0.3

Vrf (V) 160.0 490.0 450.0

% SiH4 83 60 7

% H2 17 40 93

Table 5.7: Work conditions for discharge simulations in silane-hydrogen mixtures.

condition for the PECVD of a-Si:H, most of the results presented here were obtained under this condition. Figure 5.20 shows the typical time evolution of the convergence error c for the electron and ion densities, the electron mean energy, the plasma potential and the self-bias voltage of a SiH4 -H2 discharge operating in condition C3. In this figure, the convergence criterion was taken as c < 10−3 (cf. Section 3.7). As in hydrogen (see Figure 5.1) the electric features of silanehydrogen discharges are rapidly established, after only a few ten rf cycles, as can be confirmed by analysing the time evolution of c for the plasma potential and the self-bias voltage. This behavior is also followed by the coupled electric power which, for condition C3 and after 100 rf cycles, varies less than 1%. Figure 5.20 also shows that the convergence time is controlled here by the SiH− 3 density, for which the convergence error is roughly two orders of magnitude above that of the electrical discharge parameters. The very slow convergence of the negative ion density is a direct consequence of the relatively small cross-sections associated to its creation and destruction mechanisms: electron attachment and ion-ion recombination, respectively. Note that this feature is commonly observed in many electronegative gas discharge simulations.

5.3.1 Typical results Figure 5.21 represents the axial profile of the time-average positive and negative charge particle densities, electron mean energy, plasma potential and axial reduced electric field Ez /N , for a SiH4 -H2 rf discharge operating in condition C3. As expected, in the discharge bulk the total density of negative ions and electrons equals that of positive ions, and both present a maximum value of 1.4 × 109 cm−3 at half distance between electrodes. The absence of charge separation is consistent with the observation of a constant plasma potential ('125 V) and a zero electric field within this discharge region (often termed the plasma region) [see Figures 5.21(c) and 5.21(d)]. Contrarily, near each electrode a positive space-charge region develops (usually termed the sheath region), leading to very strong confinement electric fields. The electron mean


RESULTS FOR

H2

AND

S I H4 -H2


1

DISCHARGES

(1) (2) (3) (4) (5)

0.1 0.01

145

(6) (7) (8) (9) (10)

1E-3 1E-4 1E-5 0

50

100

150

200

250

rf periodes Figure 5.20: Typical time evolution of the convergence error (obtained for a silane-hydrogen discharge operating in condition C3 ), for the following time-average parameters: electron den+ − + + + sity (1), SiH+ 3 , SiH2 and SiH3 ion densities (2-4), H3 , H2 and H ion densities (5-7), electron mean energy (8), plasma potential (9), and self-bias voltage (10).

energy varies from 5 eV in the plasma region to much higher values (between 35 and 65 eV) in the sheath regions, where electrons can participate in high-energy threshold processes. The electron mean energy is two times higher in the rf sheath than in the grounded sheath [see Figure 5.21(b)], and the same factor 2 seems to relate the potential drops across the sheaths [see Figure 5.21(c)]. However, the electric field variations across both sheaths are quite similar because the rf sheath thickness is larger than the grounded sheath thickness [see Figure 5.21(d)]. Figures 5.22 and 5.23 show, respectively, the time-average, steady-state, axial distribution (at r = 0) and radial distribution (at z ' 0) of ion densities, for a silane-hydrogen discharge operating in condition C3. In Figures 5.22(a) and 5.23(a) we have plotted the densities of silane + − ions SiH+ 3 , SiH2 and SiH3 , whereas Figures 5.22(b) and 5.23(b) represent the densities of hy+ + + + + drogen ions H+ 3 , H2 and H . An observation of these figures shows that SiH2 , H and H2 have

negligible densities when compared to other ion species. Note that the density of the negative ion SiH− 3 is also very small [see Figure 5.22(a)], as it is expected for a hydrogen-dominated condition like C3. In silane-dominated conditions C1 and C2 the SiH − 3 density becomes more important, with values comparable to those of the electron density (6 × 10 8 cm−3 , for condition C1). For all conditions, the SiH− 3 ion density is always vanishing small at z ' 0 (and near all other physical boundaries), as a consequence of boundary condition 2.85. From Figure 5.22 we can further observe that: (i) in the discharge bulk, the total density of silane ions is higher


RESULTS FOR

AND

(a)

0.09 (2)

0.03 0.00

(b)

40 30 20 10

(1)

0.0

0 0.8

1.6

2.4

0.0

3.2

0.8

z (cm)

1.6

2.4

3.2

Ez/N (10

-16

50

2

(c)

Vcm )

z (cm)

100

Vp (V)

146

DISCHARGES

50

0.12

0.06

S I H4 -H2

60

ε (eV)

10

-3

nn,p (10 cm )

0.15

H2

0 -50 -100

(d)

600 300 0 -300 -600

0.0

0.8

1.6

z (cm)

2.4

3.2

0.0

0.8

1.6

2.4

3.2

z (cm)

Figure 5.21: Axial profile (at r = 0) of the time-average, steady-state negative nn (1) and positive np (2) charged particle densities (a), electron mean energy ε (b), plasma potential V p (c), and axial reduced electric field Ez /N (d), for silane-hydrogen discharges operating in condition C3.


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

-3

0.06

ni (10

0.09

10

cm )

0.12

147

(a)

0.03 x10 x10

0.00 0.0

0.8

1.6

2.4

3.2

z (cm)

-3

0.02

ni (10

0.03

10

cm )

0.04

(b)

x3

0.01 x3 0.00

0.0

0.8

1.6

2.4

3.2

z (cm) Figure 5.22: Axial profile (at r = 0) of time-average, steady-state ion densities for a silane+ hydrogen discharge operating in condition C3 . Silane ions (a): SiH + 3 (solid curve); SiH2 + − (dashed); SiH− 3 (dotted). The SiH2 and SiH3 ion densities are multiplied by 10. Hydrogen ions + + + + (b): H3 (solid curve); H2 (dashed); H (dotted). The H+ 2 and H ion densities are multiplied by 3.


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

(a)

ni (10

10

-3

cm )

0.020

148

0.015 x10

0.010 0.005 0.000 0.0

1.6

3.2

4.8

6.4

r (cm) (b)

0.024 0.018

ni (10

10

-3

cm )

0.030

0.012 x3

0.006

x3

0.000 0.0

1.6

3.2

4.8

6.4

r (cm) Figure 5.23: Radial profile (at z ' 0) of time-average, steady-state ion densities for a silane+ hydrogen discharge operating in condition C3. Silane ions (a): SiH + 3 (solid curve); SiH2 + + (dashed). The SiH2 ion density is multiplied by 10. Hydrogen ions (b): H3 (solid curve); + + + H+ 2 (dashed); H (dotted). The H2 and H ion densities are multiplied by 3.


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

149

than the total density of hydrogen ions, due to the higher ionization cross-section and the lower ionization potential of silane (11.6 eV) with respect to hydrogen (15.4 eV); (ii) as usual, the maximum charged-particle density value occurs near the midpoint between the electrodes; (iii) + the asymmetric profile of H+ 3 ion comes as a combined effect of the large production of H 2 (its

precursor) near the rf electrode (see Figure 5.23) and its higher mobility with respect to silane ions. Figure 5.23 also shows that the radial profiles of ion densities present maxima near the corner (r = R, z ' 0), where the rf field has its strongest intensity. The predicted ion density radial profiles are relatively constant within a 3 cm distance from the discharge axis, with a small maximum located near r ' 4.8 cm. Figure 5.24 represents the contour plot of the time-average, steady-state silane dissociation rate by electron impact, ne NSiH4 (νdSiH4 /NSiH4 ) = ne νdSiH4 (where νdSiH4 /NSiH4 is the dissociation rate coefficient of silane, see equation 2.68), for silane-hydrogen discharges operating in conditions C1-C3. The figure shows that the dissociation rate exhibits a maximum in radial direction, near the grounded grid. In silane-dominated conditions and for constant pressure (conditions C1 and C2), the silane dissociation rate is controlled by the electron density. Between conditions C1 (Vrf = 160 V) and C2 (Vrf = 490 V), ne increases causing the enhanced maximum of ne νdSiH4 to shift towards the midpoint between the electrodes. In the hydrogendominated condition C3, the silane dissociation rate presents a (smaller) double peak as the combined result of two shifted maxima: the ne maximum, located at the discharge bulk, and the ε maximum (which influences νdSiH4 ), located at the rf sheath. The dissociation rate of silane is an important parameters in the modeling of a-Si:H PECVD. The dissociation rates obtained here were found to be a factor 3-50 smaller than in previously reported simulations [Leroy 1996; Leroy et al. 1998] for conditions C1-C3 respectively. This fact will cause a reduction in the calculated deposition rates of a-Si:H thin films which, in these earlier works, were overestimated with respect to experimental measurements [Leroy et al. 1998]. We now analyse the effects of ion inertia in silane-hydrogen discharges, by using Eq. 2.76 to calculate the effective electric field of ion species i, Eeff i , with and without its inertia term. Figure 5.25 presents the axial distribution, at r = 0 and t = 8T /20, of the axial reduced ef+ eff fective electric field, Ei,z /N , for i = SiH+ 3 and H3 positive ions. For comparison purposes,

the axial profile of the corresponding rf electric field Ez /N is also plotted. As before (see Figure 4.14) the presence of the ion inertia term in Eq. 2.76 reduces the effective electric intensity (especially in sheath regions), as the ions react slower to the field variations leading to a smaller charge separation. In terms of instant values, the intensity of the effective electric field: (i) is smaller than the intensity of the corresponding rf field, as the former is controlled by the latter, eff cf. Eq. 2.76; (ii) is smaller for heavier ion species (like SiH+ 3 ), thus confirming the idea that Ei


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

3.2

1.1E15 9.6E14 8.3E14 6.9E14 5.5E14 4.1E14 2.8E14 1.4E14 0

2.4

z (cm)

150

1.6

(a)

0.8 0.0 0.0

1.6

3.2

4.8

6.4

r (cm) 3.2

2.4E15 2.1E15 1.8E15 1.5E15 1.2E15 9E14 6E14 3E14 0

z (cm)

2.4 1.6

(b) 0.8 0.0 0.0

1.6

3.2

4.8

6.4

r (cm) 3.2

1.8E14 1.6E14 1.4E14 1.1E14 9E13 6.8E13 4.5E13 2.3E13 0

z (cm)

2.4 1.6

(c) 0.8 0.0 0.0

1.6

3.2

4.8

6.4

r (cm)

Figure 5.24: Spatial contour plot of the time-average, steady-state, silane dissociation rate by electron impact (in cm−3 s−1 ), for silane-hydrogen discharges operating in conditions C1 (a), C2 (b) and C3 (c).

RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

151

600 300 0 -300

eff

Ez/N, E z /N (10

-16

2

Vcm )


-600 -900

-1200

(a)

0.0

0.8

1.6

2.4

3.2

-300 -400 -500

eff

Ez /N (10

-16

2

Vcm )

z (cm)

-600

(b) 0.0

0.1

0.2

0.3

0.4

0.5

z (cm)

Figure 5.25: Axial component of different reduced electric fields, at r = 0 and t = 8T /20, for a silane-hydrogen discharge operating in condition C3 (a). The curves correspond to: E z /N eff eff (black), ESiH + /N (red) and E + /N (blue), calculated for a field-dependent mobility with H3 ,z 3 ,z and without the ion inertia term (dot and straight lines, respectively). A zoom of the near rf electrode region is given in (b).


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

152

corresponds to the electric field with which ion species i is in equilibrium. The plots, as a function of z, of the time-average rf and effective electric fields (associated to SiH+ 3 ion) are presented in Figure 5.26. We see that, contrarily to what was observed in Figure 5.25 obtained at a given time instant, both fields feature very similar time-averaged axial distributions, with the effective field presenting a smaller intensity than the corresponding rf field, in the sheath regions. The results of Figures 5.25 and 5.26 clearly suggest that it exists an important phase shift between these electric fields.

2

Vcm )

600

Ez/N (10

-16

300 0 -300 -600 0.0

0.8

1.6

2.4

3.2

z (cm) Figure 5.26: Axial profile (at r = 0) of different time-average, steady-state reduced electric fields, for a silane-hydrogen discharge operating in condition C3. The curves correspond to: eff Ez /N (solid) and ESiH + /N (dotted). ,z 3

To confirm this hypothesis, we plot in Figure 5.27 the space-time contour-plot (at r = 0) of the same axial reduced electric fields shown in Figure 5.25. In analysing this figure we concentrate on the maximum negative values of the electric fields, within the rf sheath. As expected, Figure 5.27(a) shows that the rf electric field has its maximum intensity at half-period (t ' 37 ns), while Figures 5.27(b)-(c) show that the intensity peaks of the effective fields for + + H+ 3 and SiH3 ions occur at later times. For the heavier SiH3 ion a phase shift of about π/2

is observed, the latter value being reduced for the lighter H+ 3 ion, which reacts more rapidly to electric field variations. The effective electric field plays also an important role in the calculation of electrical parameters. Table 5.8 compares simulation results and measured values of the self-bias voltage V dc and the coupled electrical power Weff , for silane-hydrogen discharges operating in conditions C1-C3, and for various ion transport conditions. In particular, simulation results (A)-(D) shown in Table 5.8 were obtained assuming:


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

3.2

1.2E-13 9E-14 6E-14 3E-14 0 -3E-14 -6E-14 -9E-14 -1.2E-13

2.4

z (cm )

153

1.6

(a) 0.8 0.0 0.0

18.5

37.0

55.5

74.0

t (ns) 3.2

8E-14 5E-14 3E-14 1E-14 -1E-14 -3E-14 -5E-14 -8E-14 -1E-13

z (cm )

2.4 1.6

(b) 0.8 0.0 0.0

18.5

37.0

55.5

74.0

t (ns)

3.2

6E-14 4E-14 2E-14 8E-15 -8E-15 -2E-14 -4E-14 -6.5E-14 -8E-14

z (cm )

2.4 1.6

(c) 0.8 0.0 0.0

18.5

37.0

55.5

74.0

t (ns)

Figure 5.27: Space-time contour plots at r = 0 of different reduced electric fields (in Vcm2 ), during one rf cycle, and for a silane-hydrogen discharge operating in condition C3. E z /N (a); eff EHeff+ ,z /N (b); and ESiH + /N (c). ,z 3

3


Condition Parameter C1 C2 C3

Vdc (V) Weff (W) Vdc (V) Weff (W) Vdc (V) Weff (W)

RESULTS FOR

H2

AND

S I H4 -H2

Simulations (A) (B) (C) −42.4 −45.9 −46.9 1.4 1.5 1.5 −132.6 −147.6 −147.9 9.1 5.3 5.3 −99.1 −119.5 −120.0 19.7 11.9 11.9

154

DISCHARGES

Experiment (D) (E) −46.4 −43.0 1.5 1.7 -147.4 −172.0 5.5 9.5 -119.2 −141.0 13.5 20.0

−69.5 1.5 −197.5 6.0 −171.5 15.0

Table 5.8: Calculated and measured values of electrical parameters for silane-hydrogen discharges operating in conditions C1-C3 (see Table 5.7). Simulations results (A)-(D) were obtained in this work, assuming different ion transport conditions: (A), three types of ions (H + 2, + − SiH3 and SiH3 ), with constant mobilities and no effective electric field; (B), same ions as in (A), with field-dependent mobilities and effective electric field calculated with inertia term; + − + + (C), six types of ions (H+ , H+ 2 , H3 , SiH2 , SiH3 and SiH3 ), with field-dependent mobilities and effective electric field calculated with inertia term; (D), same ions as in (C), with fielddependent mobilities and effective electric field calculated without inertia term. Simulation results (E) were obtained in previous works using an earlier model version [Leroy 1996; Leroy et al. 1998]. There is at least a 40% uncertainty in measured values of V dc .

+ − • (A), three types of ions (H+ 2 , SiH3 and SiH3 ), with constant mobilities (low field values,

see Section 2.4.3) and no effective electric field (Eieff = E); • (B), same ions as in (A), with field-dependent mobilities (cf. Eqs 2.95 and 2.97) and Eeff i calculated from Eq. 2.76 with inertia term; + + + − • (C), six types of ions (H+ , H+ 2 , H3 , SiH2 , SiH3 and SiH3 ), with field-dependent mobili-

ties and E eff i calculated from Eq. 2.76 with inertia term; • (D), same ions as in (C) , with field-dependent mobilities and Eeff calculated i from Eq. 2.76 without inertia term. • (E), simulation results obtained in previous works using an earlier model version [Leroy 1996; Leroy et al. 1998] (which adopted field-dependent ion mobilities, no effective electric field, and a positive ion kinetics which considers only the "mean" species H+ 2 and SiH+ 3 ). The results in Table 5.8 show that there is here a better agreement between simulations and experiment for Weff than the one obtained before by [Leroy 1996; Leroy et al. 1998]. However, we can still observe a 20% − 30% systematic overestimation of the calculated V dc with respect to measurements, even if a 40% uncertainty is associated to the latter ones. Notice that the disagreement between calculated and measured Vdc values increases as we go from silanedominated condition C1 to the hydrogen-dominated condition C3, which reflects the enhanced


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

155

difficulties in obtaining a correct estimation of the charged-particle current for dominating light hydrogen ions. With that respect, measurements of Weff can be considered as more reliable. Comparison of simulation results (A), (B) and (E) with experimental values for W eff indicates that the systematic over-estimation of this parameter in earlier works may be related to a poor description of ion transport. In fact, the results obtained here show that the introduction of the effective electric field significantly improves the predictions of Weff . For condition C2, for example, earlier calculations (E) yielded a 58% deviation with respect to experiment, while predictions of simulation (B), that uses a similar ion kinetics, are only 11% apart. From simulation results (A)-(D) in Table 5.8 we can observe that the self-bias potential increases (becoming less negative) in the following situations: (i) by reducing to only three the number of ion species [(C) to (B)]; (ii) by imposing a constant (higher) ion mobility [(B) to (A)]; (iii) by eliminating the ion inertia term from Eq. 2.76 [(C) to (D)]. In all situations (i)-(iii) the ion flux (hence the ion conduction current) will directly increase; moreover, in situations (i) and (ii) the electron current is also expected to decrease, due to the suppression of the ionization + + + mechanisms leading to the production of SiH+ 2 , H and H3 . Note that while H2 is the sole

hydrogen ion considered in simulations (A) and (B), H+ 3 becomes the dominant hydrogen ion + in simulations (C), due to the very effective ion conversion reaction H + 2 + H2 −→ H3 + H (see

also Figures 5.22 and 5.23). Situation (ii) is by far the one that leads to more significant changes in the self-bias potential and the coupled electrical power (20% for Vdc in condition C3; up to 40% for Weff in conditions + + C2 and C3), which also reveals that ion species SiH+ 2 , H and H2 have only little influence on

electrical parameters Vdc and Weff (the highest change of Vdc is ∼ 2% between simulations (B) and (C), for condition C1). In the silane-dominated condition C1 (see Table 5.7), the introduction of the effective electric field [from simulation (A) to simulations (B) and (C) in Table 5.8] leads to variations of Vdc and Weff which are similar to the ones observed in helium rf discharges (see Section 4.7 and Table 4.4): the self-bias voltage becomes more negative and the coupled electrical power experiences a small increase, due to the reduction of the total current collected at the rf electrode and the enhancement of the electric field within the sheaths. However, in the hydrogen-dominated conditions C2 and C3, we observe the decrease of Weff for increasing self-bias potentials. This behavior is similar to the one observed in hydrogen discharges at high pressures (see Section 5.2.5 and Table 5.4), being associated to a decrease in the phase shift between the total rf current and potential. Note, finally, that the elimination of the ion inertia term in the effective electric field Eq. 2.76 is responsible for a small increase of the coupled electrical power, due to the enhancement of the discharge current [in condition C3, W eff increases by 13% between simulations (C) and (D)].


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

156

5.3.2 Effects of SiH4 dilution The deposition rate of a-Si:H films, produced by plasma deposition techniques, is controlled by a balance between deposition and etching processes at the substrate. This fact explains the variations of the deposition rate with changes in the SiH4 - H2 mixture composition. In particular, a reduction of silane concentration yields a significant drop in the corresponding deposition rates, not only as a direct consequence of silane depletion but mostly because of the enhanced etching by hydrogen. Typically, the PECVD deposition rate is 220 Å/min for undiluted silane, to become only 10 Å/min for 99% SiH4 relative concentration [Tsai 1988]. In CVD, the etching produced by strong H2 concentrations can reach the point where an amorphous deposition gives place to a micro/nano-crystalline configuration [Solomon et al. 1993; Godet et al. 1995; Matsuda 1999; Cicala et al. 2001], justifying the use of silane dilution as a key parameter to select between different film structures. Moreover, the percentage of hydrogen atoms incorporated into the deposited film affects its properties. Secondary Ion Mass Spectrometry indicates that this percentage presents a maximum value for a composition mixture of 5% SiH 4 and 95% H2 , being relatively constant for silane concentrations above 20% [Tsai 1988]. This section analyses the influence of silane dilution on various rf discharge parameters. Simulations are carried out in Si4 -H2 discharges, operating at p = 0.3 Torr pressure, Vrf = 400 V applied voltage and 5 different composition mixtures, corresponding to silane relative concentrations of 1%, 5%, 10%, 30% and 50%. Under these work conditions, the maximum + 9 −3 densities of H+ and 2.4 × 3 and SiH3 (the most abundant ion species) range from 10 cm

108 cm−3 (for 1% SiH4 ) to 2 × 107 cm−3 and 1.4 × 1010 cm−3 (for 50% SiH4 ), respectively. 7 −3 Furthermore, the maximum value of SiH− at 1% SiH4 , rising 3 density is smaller than 10 cm

to 109 cm−3 at 50% SiH4 . Figure 5.28 shows typical time-average, 2d profiles of the electron density and mean energy, for a 5% SiH4 - 95% H2 mixture. These spatial profiles present the same general characteristics as the ones obtained for helium or hydrogen discharges: (i) the electron density exhibits a maximum value near the midpoint between the electrodes and very steep gradients within the sheath regions, as a direct consequence of the electric field confinement; (ii) the electron mean energy profile presents a maximum near the corner where the rf and the grounded grid meet; (iii) a flat, low energy region (∼ 4 eV for this dilution) is observed within the discharge bulk, indicating that most of the available rf power concentrates in the sheaths. When the concentration of SiH4 increases, the electron density peak comes closer to both the rf electrode and the grounded grid, as can be seen in Figure 5.29 representing the spatial contour plot of ne for the working conditions considered here. This shift in the position of the electron-density maximum is related to a modification of the electron energy distribution. First, when the concentration of SiH4 increases the electron production is enhanced, due to the low


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

157

(a)

e

10

-3

n (10 cm )

0.020 0.015 0.010 0.005 6.4

4.8

3.2

1.6

r (cm )

0.0

0.0 0.8 1.6 2.4 z (cm ) 3.2

(b )

100

ε (eV )

80 60 40 20 6.4

4.8

3.2

r (cm )

1.6

0.0

0.0 0.8 1.6 2.4 z (cm ) 3.2

Figure 5.28: Time-average, 2d steady-state profiles of the electron density ne (a) and the electron mean energy ε (b), for a silane-hydrogen discharge (5% SiH4 ) operating at p = 0.3 Torr and Vrf = 400 V.

RESULTS FOR

3.2

1.4E9 1.2E9 1E9 8.7E8 7E8 5.2E8 3.5E8 1.7E8 0

2.4

z (cm)

H2

1.6

(a)

AND

2.8E9 2.4E9 2E9 1.4E9 1E9 7E8 3.5E8 1.7E7 0

1.6

(b)

0.8

1.6

3.2

4.8

0.0 0.0

6.4

1.6

r (cm)

4.8

6.4

1.6

(c)

0.8

3.2

4.5E9 4E9 3.6E9 3E9 2E9 1E9 3.5E8 1.7E7 0

2.4

z (cm)

4E9 3.6E9 3.2E9 2.8E9 2E9 1E9 3.5E8 1.7E7 0

2.4

z (cm)

3.2

r (cm)

3.2

0.0 0.0

158

DISCHARGES

2.4

0.8 0.0 0.0

S I H4 -H2

3.2

z (cm)


1.6

(d)

0.8

1.6

3.2

r (cm)

4.8

6.4

0.0 0.0

1.6

3.2

4.8

6.4

r (cm)

Figure 5.29: Spatial contour plot of the time-average, steady-state electron density (in cm −3 ), for silane-hydrogen discharges operating in conditions of Fig. 5.28, at different SiH 4 relative concentrations: 5% (a), 10% (b), 30% (c), and 50% (d).


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

159

ionization potential of silane (the maximum value of ne varies from 1.3 × 109 cm−3 at 1% SiH4 , to 4.1 × 109 cm−3 at 50% SiH4 ). Second, the increase in electron density leads to a decrease of the electron mean energy, mainly in the sheath regions and for SiH 4 percentages above 10%. Third, the electron density profile becomes highly asymmetric (with its peak shifted towards the rf sheath), in order to compensate for variations in the electron energy density. The previous results can be confirmed in Figures 5.30(a)-(b), which represent the axial profiles (at r = 0) of the time-average electron density and mean energy, for the working conditions considered here. One can observe that sheath thickness is highly reduced as the silane concentration increases, and that this modification is associated to higher electron densities and lower electron mean energies in the sheath region. The axial profiles shown in Figures 5.30(a)-(b) are a good example of a transition from a non-local to a local discharge behavior, controlled by silane dilution. This sheath contraction is also reflected in Figure 5.30(c), which plots the axial profile (at r = 0 of the time-average plasma potential, for the same conditions. From this figure we observe that the sheath thickness reduces by at least a factor 2, when the relative concentration of silane increases from 1% to 50%. Notice that the potential drop across the sheath regions is almost the same for all dilutions (∼215 V for the rf sheath), which leads to very intense confinement electric fields in these regions. These enhanced electric fields ensure a limitation of the strong spatial losses, normally associated to highly-distributed electron densities [see Figure 5.29(d)]. The silane dilution in SiH4 -H2 mixtures has also an important influence on the values of electrical parameters. Figure 5.31 plots the variations of the self-bias voltage and the coupled electrical power, as a function of silane relative concentration. From this figure, one observes a 7% decrease of Vdc and a 300% increase of Weff as the relative concentration of silane varies between 1% and 50%. The shift of the self-bias voltage towards more negative values is a direct consequence of an increase in the electron current, due to the enhanced electron production in silane-dominated conditions. However, the strong increase of the coupled electrical power with silane concentration is not only due to the increase of the global conduction current, but is mainly associated to the high displacement currents resulting from an enhanced intensity of the sheath electric fields in silane-dominated conditions. The variation, with silane relative concentration, of the space-time-averaged SiH 4 and H2 dissociation rates is given in Figure 5.32 [Salaba¸s et al. 2002a]. As expected, the SiH 4 dissociation rate considerably increases with silane concentration (varying from 10 12 at 1% SiH4 to 1015 cm−3 s−1 at 50% SiH4 ), while the H2 dissociation rate decreases by a factor 2 with only a 30% increase in SiH4 concentration. Notice the very small increase in the H2 dissociation rate, following a 5% increase in SiH4 concentration. This result agrees with the report of [Tsai 1988], which indicates a maximum hydrogen content in deposited films at this silane dilution.


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

0 .5

% S iH 4 1 % 5 % 10 % 30 % 50 %

(a ) 0 .4

-3

n (1 0 c m )

160

10

0 .3

e

0 .2 0 .1 0 .0 0 .0

0 .8

1 .6

2 .4

3 .2

2 .4

3 .2

2 .4

3 .2

z ( cm )

ε ( eV )

7 0 (b ) 60 50 40 30 20 10 0 0 .0

% S iH 4 1 % 5 % 10 % 30 % 50 %

0 .8

1 .6

z ( mm ) 150

(c )

50

p

V ( V )

100

% S iH 4 1 % 5 % 10 % 30 % 50 %

0 -5 0

-1 0 0 -1 5 0

0 .0

0 .8

1 .6

z (c m ) Figure 5.30: Axial profiles (at r = 0) of the time-average, steady-state electron density ne (a), electron mean energy ε (b) and plasma potential V p (c), for silane-hydrogen discharges operating in conditions of Fig. 5.28, at various SiH4 dilutions.


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

161

-104

Vdc (v)

-106 -108 -110 -112

(a) 0

10

20

30

40

50

60

% SiH 4 30

Weff ( w )

25 20 15 10

(b) 5 0

10

20

30

40

50

% SiH4

Figure 5.31: Self-bias voltage Vdc (a) and coupled electrical power Weff (b) as a function of silane relative concentration, for silane-hydrogen discharges operating in conditions of Fig. 5.28.


RESULTS FOR

H2

AND

S I H4 -H2

DISCHARGES

162

16

-3 -1

( 10 cm s )

0.24 0.20 0.16 0.12 0.08 0.04 0.00

0

10

20

30

40

50

% SiH 4 Figure 5.32: Space-time-averaged dissociation rates of SiH4 (squares) and H2 (circles), as a function of silane relative concentration, for silane-hydrogen discharges operating in conditions of Fig. 5.28.

From the point of view of simulations, an increase in silane concentration leads to longer calculation times. The effect is due to the increasing presence of negative ions, whose creation and destruction mechanisms have very small cross-sections. At 1% SiH 4 , the electron density is a few orders of magnitude higher than the SiH− 3 density, and so the latter has very little influence in the calculation time. However, at 50% SiH4 , the density of negative ions equals that of electrons (∼ 1.7 × 109 cm−3 for the space-time average densities), in which case the time evolution of the convergence error is controlled by the SiH− 3 density. This result is shown in Figure 5.33, which plots the time evolution of c for SiH− 3 density at various silane dilutions. As observed, the time for convergence is four times higher at 50% SiH 4 than at 1% SiH4 , which justifies the introduction of speed-up procedures to accelerate convergence, at high negative ion densities [Yan and Goedheer 1999]. These precautions can however be avoided at moderate and high silane dilutions, like the ones adopted in this work.



RESULTS FOR

H2

AND

S I H4 -H2

0.1

DISCHARGES

163

% SiH4 1% 5% 10 % 30 % 50 %

0.01

1E-3 0

150

300

450

600

750

900

rf periods Figure 5.33: Time evolution of the convergence error for the time-average SiH − 3 density at different silane dilutions. The curves were obtained for silane-hydrogen discharges operating in conditions of Fig. 5.28.

Chapter 6 Conclusions 6.1 Conclusions This work presented a two-dimensional self-consistent fluid model describing the charged particle transport in capacitively coupled radio-frequency discharges. The model has solved the continuity and momentum transfer equations for electrons and ions, coupled with the transport equation for the electron mean energy and Poisson’s equation for the plasma potential. The set of fluid equations has been closed using the local mean energy approximation. The latter comes to assume that the dependence of the electron energy distribution function is exclusively handled by the electron mean energy, thus allowing the introduction of a space-time dependence for the electron parameters (including those for energy transport) via the mean energy profile. For a given gas or gas mixture, the local mean energy approximation involves the solution of the stationary, space-independent Boltzmann equation (written under the classical two-term approximation) and its articulation with the space-time profile of the electron mean energy, as obtained from the fluid model. The coherent formulation adopted here improved earlier fluid models. In particular: 1. The electron transport equations were directly obtained from the moments of the two-term EBE, which was also used in implementing the local mean energy approximation. 2. The procedure mentioned in 1. yielded a drift-diffusion form for both the electron particle and energy fluxes, which includes gradient terms for the electron particle and energy diffusion coefficients. 3. The electron-flux boundary conditions were reviewed with respect to previous works. An alternative expression was proposed also in the framework of the EBE two-term development. 4. The ion momentum transfer equation was written in its most general form, neglecting

164

C HAPTER 6. C ONCLUSIONS

165

only its viscous tensor term. In particular, the non-linear ion inertia term was kept in this equation by generalizing the earlier concept of effective electric field. 5. The so-called effective electric field was interpreted as the electric field with which the ions are in equilibrium, and so it was used in writing the ion flux boundary conditions. 6. The work has adopted (effective) field-dependent ion mobilities, in order to limit the ion flow in regions of high electric field. 7. The silane and hydrogen kinetics was updated with respect to previous works, in particu+ + lar by further considering the presence of SiH+ 2 , H and H3 ions.

The model updates some numerical procedures usually adopted in fluid models. A semiimplicit splitting method was used to advance equations in time, by adopting a Crank-Nicholson algorithm with an integration time step controlled by the Courant-Friedrichs-Lewy condition. As an improvement with respect to other works, Poisson’s equation was numerically solved by direct matrix methods (thus yielding a high-precision solution for the plasma potential), adopting a semi-implicit technique which writes this equation at time t by using a space-time estimation at time t + ∆t. The Scharfetter-Gummel exponential scheme was employed in the spatial discretization of the charged particle and energy fluxes, thus ensuring an adequate transition between convective and diffusive transport regimes. The combination of the above algorithms resulted in a stable and robust numerical code, whose convergence takes only a few hours on a Pentium III at 500 MHz. The adopted convergence criterion checks the main plasma parameters, assuming that a steady state solution was reached when their relative changes, between two consecutive periods, are less than 0.1%. The model was applied to a cylindrical parallel-plate configuration, producing ccrf discharges in helium, hydrogen and silane-hydrogen mixtures, at fixed frequency (13.56 MHz), for a wide range of pressures and applied voltages. The model validation was made against previous calculations and experimental measurements. The results obtained are summarized as follows. • The radial transport inside the reactor chamber has a significant effect on the chargedparticle profiles, and thus cannot be neglected. In particular, the maximum value of the plasma density profile develops in radial direction, and is located near the half distance between electrodes. • The maximum value of the ionization rates is located radially at the plasma-sheath interface. Inside the sheaths, a double ionization structure was observed during one rf period, for discharges produced in hydrogen and silane-hydrogen mixtures.


166

• The spatial profile of the time-average electron mean energy gives a good description of electron heating within sheath regions. • The introduction of field-dependent ion mobilities and an effective electric field for each ion species results in a better description of the charged particle coupled dynamics. The substantial correction in the predicted values of the coupled electrical power gives a clear indication of an improved ion transport description. In particular, the present model predicts deviations of about 8 %, 20 % and 11 % with respect to experimental measurements, for discharges produced in He, H2 and SiH4 -H2 , respectively. Earlier simulations [Leroy 1996] reported, in the same conditions, deviations of about 20 %, 80 % and 58 %, respectively. • The influence of the ion inertia term on the discharge parameter values is estimated to be responsible for deviations up to a few percent. • The results for hydrogen discharges showed that the over-simplified ion kinetics adopted in previous works, was the main responsible for the systematic over-estimation of the coupled electrical power. In silane-hydrogen discharges, and for the tested dilutions, the update of the hydrogen kinetics brought only small changes on the discharge parameter values. • Changes in the geometric dimensions of the reactor produce the following effects. The increase of the electrode radius and the decrease of the inter-electrode distance yield more symmetric discharges (hence smaller Vdc absolute values), as the ratio between driven and grounded reactor areas tends to unit. As expected, the coupled electrical power increases for both larger radius and inter-electrode distances. In the first case, the power enhancement is due to a stronger electric field near the rf electrode, while in the second case is justified by a larger collection surface for the discharge current. The more symmetric spatial profiles of the time-average electron density, obtained for larger electrode radius and smaller inter-electrode distances, lead to a slight decrease in the plasma density at discharge axis. • In hydrogen discharges, a systematic study of the rf electric field variations with changes in geometric dimensions, pressure and applied voltages was carried out. These variations were shown to follow a universal similarity law, if an adequate normalization was used when plotting the rf electric field as a function of pressure. • The main characteristics of silane-hydrogen rf discharges can be summarized in terms of relationship between different electrical parameters and gas pressure, applied voltage or


167

silane dilution. As with any model, one must remember that reliable results can only be obtained in the framework of the approximation set adopted. For example, the present model has some limitations to work at low gas pressure (less than 100 mTorr) and high rf applied voltages (higher than 450 V), as for these conditions the discharge electric field becomes extremely high and the small anisotropy approximation is clearly violated. Calculations performed in such situations yield unrealistic positive self-bias voltages, thus indicating a strong depletion of the electron population in the discharge. Nevertheless, we have shown that the present code is able to give a good electric description of ccrf discharges operating in a wide range of conditions, with a low computational effort.

6.2 Future research In future, the present model is likely to become part of a predictive tool for the modeling of PECVD techniques. For this, the charged-particle transport code must be coupled (via the dissociation rates of silane and hydrogen) to two other calculation codes: (i) a chemical module, describing the dissociation of background gases by electron impact to create reactive species that diffuse towards a substrate; (ii) a surface module, describing the surface kinetics in order to account for the plasma-wall interaction and the thin film deposition. Within this modular code it will be possible to eliminate the restriction of assuming a constant background gas composition, by self-consistently calculating the partial pressures of the different silane and hydrogen components. Fluid models normally assume that electrons have the leading role in non-local energy transport, thus neglecting the energy balance equation for both ions and neutrals. This approximation is based on two assumptions: (i) the ions cannot respond to the rapid oscillations of the rf electric field: (ii) the ions have practically the same energy as the neutral atoms, due to the very efficient energy transfer between ions and neutrals with similar mass. These assumptions are certainly correct in the discharge bulk, where the time average electric field is nearly zero. However, the high values of the rf electric field in the discharge sheaths can be responsible for an increase in the ion energy far above the thermal energy of neutrals [Heintze and Zedlitz 1996; Hamers 1998], which incidentally can account for the important role of ions in surface processing. Moreover, the development of very intense electric fields in discharge sheaths constitutes a strong limitation to the use of a mere drift-diffusion form for the electron flux, obtained by adopting the small anisotropy approximation. These points are to be improved in future upgrades of this model. Vibrational excited species can have an important role in rf discharges produced in molecu-


168

lar gases, as a significant part of the coupled power can be transferred to vibrational excitation. In the particular case of H2 , vibrational excited species may constitute important channels of atomic hydrogen and charged-particle production. In order to clarify the role of these species, the present charged-particle transport model should be self-consistently coupled to a kinetic module for H2 , describing the diffusion of vibrational excited hydrogen molecules and excited hydrogen atoms.

Appendix A

A.1 We provide here the derivation of expressions 3.36 involved in the resolution of the radial part of the electron continuity equation 3.13 at boundary positions (0, j). An overall balance for the near-boundary electron continuity equations may be obtain by integrating through each boundary control volume and taking into account the flux boundary conditions. These integration yield the set of boundary discretized equations. The coefficents in 3.36 can be deduced after the integration of the electron continuity equation 2.62 radially from 0 to h/2, and axially between z −g/2 and z +g/2. In fact, only the radial integration is responsible for formulas 3.36, the axial integration providing analog coefficients for 3.14. For simplicity, we present only the radial integration Z h/2 Z h/2 ∂ne 1 ∂(rΓer ) 1 + r dr = Se r dr ∂t r ∂r 2 0 0

,

(A.1)

which gives, k k  h i h i h2 1  t h2 nte − nte  1 k k t t t t  + Se  . (A.2) (rΓer ) 1 ,j − (rΓer )0,j + (rΓer ) 1 ,j − (rΓer )0,j = 2 2 8 ∆t 0,j 2 8 2 0,j

It is clear from the above equation that the integration performed is equivalent with a FD dis-

cretization scheme at the boundary (0, j). The discretized equation obtained via integration expresses the particle conservation principle just as the differential form used in finite differences. Note now that, in the second and third terms of Eq. A.2, the values computed at (0, j) are zero for two reasons. First, r|0,j = 0 and, second, the boundary condition 2.81 imposes a zero electric field in points located on the discharge axis. The radial electron flux, as it results form the exponential scheme, writes as (cf. Eq. 3.30) h i Γr 1 ,j = zpr 1 ,j De1,j ne1,j − De0,j ne0,j exp(−zer 1 ,j h) 2

2

2

where the relations 3.34 were used.

169

,

(A.3)

A PPENDIX A

170

Now, by multiplying Eq. A.2 with 8∆t/h2 and introducing Eq. A.3 in Eq. A.2, we obtain h it 4∆t t ne0,j + 2 zpr 1 ,j De1,j ne1,j − De0,j ne0,j exp(−zer 1 ,j h) = 2 2 h ∆t tk 4∆t k k = nte0,j + Se0,j − 2 (rΓer )t0,j . (A.4) 2 h Grouping the terms according to the unknown vector of electron densities we have (cf. Eq. 3.32) 4∆t 4∆t t t r 1 (zpr) 1 ,j De1,j = 0 + ne0,j 1 − 2 r 1 (zpr) 1 ,j De0,j exp(−zer 1 ,j h) + ne1,j 2 2 2 h 2 h2 2 k i tk ∆t Set0,j 4∆t h tk = ne0,j + . (A.5) − 2 (rΓer ) 1 ,j 2 2 h Relations 3.36 can be now obtain by identifying the coefficients in Eq. A.5.

A.2 The resolution of Poisson’s equation at boundary points, for example (0, j), is made in the the following way. Equation 3.47 is integrated radially from 0 to h/2 and axially between z − g/2 and z + g/2 yielding Z h/2 Z

z+g/2

1 ∂(rEr ) ∂Ez r dr dz + r dr dz 2 ∂r ∂z z−g/2 0 −1 Z h/2 Z z+g/2 ρ ∆t r dr dz − 1+ = τD z−g/2 0 0 −1 Z h/2 Z z+g/2 ∆t ∆t − 1+ ∇ · J diff r dr dz 0 τD 0 z−g/2

=

.

(A.6)

For the LHS terms the integration gives h/2 r 2 h/2 z+g/2 h2 g(rEr )0 + 0 Ez z−g/2 = gr 1 ,j Er 1 ,j + Ez0,j+ 1 − Ez0,j− 1 = 2 2 2 2 2 8 V1,j − V0,j h2 V0.j+1 − V0,j − V0,j + V0,j−1 + , = gr 1 ,j 2 h 8 g where Eqs. 2.77 and 3.1 were used. The first term in the RHS is −1 −1  Z h/2 Z z+g/2  ρ h2 ∆t ∆t  r dr dz = gρ 1 + 1+   τ 8 τ 0 D D 0 z−g/2

and the second writes as ∆t 1+ 0 ∆t = 1+ 0 ∆t 1+ = 0

∆t τD

−1 Z

h/2

0

Z

,

(A.8)

0,j

z+g/2

∇ · J diff r dr dz = z−g/2

−1 h/2 h2 z+g/2   g(rJrdiff ) 0 + Jzdiff z−g/2 = 8 −1 2   ∆t h g(rJrdiff ) 1 ,j + Jzdiff 0,j+ 1 − Jzdiff 0,j− 1 2 2 2 τD 8 ∆t τD

(A.7)

(A.9)

A PPENDIX A

171

where the boundary conditions 2.81 were used. Multiplying Eq. A.6 by (− h8g2 ), taking into account the above expressions A.7-A.9, and using 3.55- 3.56, the integration of Poisson’s equation can be written as −1  8g 2 ρ ∆t 8g 2  1+ 0 − 2 + 3 r 1 ,j V0,j + 3 r 1 ,j V1,j + V0,j−1 + V0,j+1 =  + H0,j 2 2 0,j h h 0 τD

, (A.10)

where H0,j

−1 2 ∆t ∆t 8g e 1 = 1+ r D n − D n − D n + D n + e e e e p p p p 1,j 1,j 0,j 0,j 1,j 1,j 0,j 0,j 0 τD h3 2 ,j + e De0,j+1 ne0,j+1 − 2De0,j ne0,j + De0,j−1 ne0,j−1 − − Dp0,j+1 np0,j+1 + 2Dp0,j np0,j − Dp0,j−1 np0,j−1 . (A.11)

l Thus, from Eqs. 3.51 and A.10, the corresponding coefficients C0,j , l = 1, 6 given by ex-

pressions 3.54 follow. Note that their expressions are deduced here considering in Poisson’s equation electrons and positive ions only.

A.3 The derivation of near boundary equations for the effective electric field follows the integration method mentioned in Section 3.6. For example, the axial component (q = z) of the equation 2.73 can be calculated in half-points (0, j + 12 ) by integrating the equation radially from 0 to h/2, and axially between zj and zj+1 . Aditionally, the boundary conditions 2.81 were used. With these observations, the different terms of Eq. 2.73 are computed as • the LHS term Z h/2 Z 0

zj+1 zj

k ∂viz hg  k+1 dr dz = viz 0,j+ 1 − viz 0,j+ 1 2 2 ∂t 2∆t

,

(A.12)

• the inertia term in the RHS Z

h/2

Z

zj+1

∂viz dr dz = ∂r 0 zj Z zj+1 h ∂viz  ∂viz  k k   ) 1 ,j + (vir ) 0,j = (vir = 2 4 ∂r ∂r zj ∂v  k hg  k iz  k = vir  1 ,j + vir  1 ,j+1 = 1 ,j+ 21 2 2 2 4 2 ∂r  k k k g k = vir  1 ,j + vir  1 ,j+1 viz 1,j+ 1 − viz 0,j+ 1 . 2 2 2 2 8

IT 1 =

v ir

(A.13)

A PPENDIX A

172

IT 2 =

Z

h/2 0

1 = 2

Z

Z

zj+1

v iz zj

h/2 0

Z

Z

zj+1

zj zj+1

∂viz dr dz = ∂z

∂(vikz )2 dr dz = ∂z

∂(vikz )2 dz = ∂z zj k 2 ∂(v )  1 h g ∂(vikz )2  i z  = = + 0,j+1 0,j 2 2 2 ∂z ∂z k k k 1 2 k = h viz 0,j+ 3 − vi2z 0,j+ 1 + vi2z 0,j+ 1 − vi2z 0,j− 1 = 2 2 2 2 8  1 2 k k h viz 0,j+ 3 − vi2z 0,j− 1 = . 2 2 8 1h = 2 2

(A.14)

• the second term in the RHS Z

h/2 0

Z

zj+1 zj

1 νik vikz dr dz = h g νik 1 vikz 1 0,j+ 2 0,j+ 2 2

.

(A.15)

• the third term in the RHS

Z

h/2 0

Z

zj+1

Z zj

 e h k+1  e k+1 k+1 eff k+1   g E − E . Ez − Eieff dr dz = Z z iz z 0,j+ 21 0,j+ 21 mi mi 2 (A.16)

Finally, the boundary equation (r = 0) for the evolution of the axial components of effective electric field reads as (cf. relations A.12-A.16 and Eq. 2.75) k+1  Eieff = z 0,j+ 1 2

k k+1 A B  Ez 0,j+ 1 + Eieff − z 0,j+ 21 2 A+B A+B 1 2 IT − νik 1 vikz 1 − , A + B 0,j+ 2 0,j+ 2 hg A +B

(A.17)

where IT = IT 1 + IT 2 Ze A ≡ mi Z µi B ≡ . ∆t

(A.18)

Bibliography W. N. Allen, T. M. H. Cheng and F. W. Lampe, “Ion-molecule reactions in SiH 4 -D2 mixtures”, J. Chem. Phys. 66 (8) (1977) 3371. K EY: [Allen et al., 1977] L. L. Alves and C. M. Ferreira, “Electron kinetics in weakly ionized helium under dc and hf applied electric fields”, J. Phys. D 24 (1991) 581. K EY: [Alves and Ferreira, 1991] L. L. Alves, G. Gousset and C. Ferreira, “Self-contained solution to the spatially inhomogeneous electron Boltzmann equation in a cylindrical plasma positive column”, Phys. Rev. E 55 (1) (1997) 890. K EY: [Alves et al., 1997] L. L. Alves, G. Gousset and C. M. Ferreira, “A collisional-radiative model for microwave discharges in helium at low and intermediate pressures”, J. Phys. D 25 (1992) 1713. K EY: [Alves et al., 1992] M. S. Barnes, T. J. Colter and M. E. Elta, “Large-signal time-domain modeling of low-pressure rf glow discharges”, J. Appl. Phys. 61 (1) (1987) 81. K EY: [Barnes et al., 1987] C. K. Birdsall, “Particle-in Cell Charged Particle Simulations, plus Monte Carlo Collisions with neutral atoms, PIC-MCC”, IEEE Trans. on Plasma Science 19 (2) (1991) 65. K EY: [Birdsall, 1991] C. K. Birdsall, E. Kawamura and V. Vahedi, “Particle simultation methods for glow discharges: Past, present and future, with applications”, in U. Kortshagen and L. Tsendin, editors, “Electron Kinetics and Applications of Glow Discharges”, (NATO ASI Series B: Physics vol. 367, Plenum Press, New York, 1998), pp. 59–74. K EY: [Birdsall et al., 1998]

173

B IBLIOGRAPHY

174

J.-P. Boeuf, “Numerical model of rf glow discharges”, Phys. Rev. A 36 (6) (1987) 2782. K EY: [Boeuf, 1987] J.-P. Boeuf, “A two-dimensional model of dc glow discharges”, J.Appl.Phys. 63 (5) (1988) 1342. K EY: [Boeuf, 1988] J.-P. Boeuf, P. Belenguer and T. Hbit, “Plasma particle interaction”, Plasma Sources Sci. Technol. 3 (1994) 407. K EY: [Boeuf et al., 1994] J.-P. Boeuf and L. Pitchford, “Two-dimensional model of a capacitively coupled rf discharges and comparisons with experiments in the Gaseous Electronics Conference reference reactor”, Phys. Rev. E 51 (2) (1995a) 1376. K EY: [Boeuf and Pitchford, 1995] J.-P. Boeuf and L. C. Pitchford, “Pseudospark discharges via computer simulation”, IEEE Trans. on Plasma Science 19 (2) (1991) 286. K EY: [Boeuf and Pitchford, 1991] J. P. Boeuf and L. C. Pitchford, “Field reversal in the negative glow of a dc glow discharge”, J. Phys. D: Appl. Phys. 28 (1995b) 2083. K EY: [Boeuf and Pitchford, 1995] J. P. Boeuf and L. C. Pitchford, “Siglo-2d, a 2d user-friendly model for glow discharge simulation (kinema software, [email protected], http://www.csn.net/siglo, monument, co)”, (1996). K EY: [Boeuf and Pitchford, 1996] M.-C. Bordage, Contribution a l’étude de la cinétique électronique dans un gaz faiblement ionise., Ph.D. thesis, Université Paul Sabatier de Toulouse (1995), nr 1455. K EY: [Bordage, 1995] J. Bretagne, G. Gousset and T. Šimko, “Ion transport simulation in a low-pressure hydrogen gas at high electric fields by a convective-scheme method”, J. Phys. D 27 (1994) 1866. K EY: [Bretagne et al., 1994] J. W. Butterbaugh, L. D. Baston and H. H. Sawin, “Measurements and analysis of radio frequency glow discharge electrical impedance and network power loss”, J. Vac. Sci. Technol. A 8 (2) (1990) 916.

B IBLIOGRAPHY

175

K EY: [Butterbaugh et al., 1990] D. E. Carlson and C. R. Wronski, “Amorphous silicon solar cells”, Appl. Phys. Lett. 28 (11) (1976) 671. K EY: [Carlson and Wronski, 1976] A. Catalano, “Solar cells made of amorphous and microcrystalline semiconductors”, in J. Kanicki, editor, “Amorphous and microcrystalline semiconductor devices”, (Artech House, Inc, 1991), pp. 9–76. K EY: [Catalano, 1991] C. F. Chan, “Reaction cross-sections and rate coefficients related to the production od positive ions”, Tech. Rep. LBID-632, Lawrence Berkley Lab. (1983). K EY: [Chan, 1983] J. S. Chang and G. Cooper, “A practical difference scheme for Fokker-Planck equations”, J. Comput. Phys. 6 (1970) 1. K EY: [Chang and Cooper, 1970] L. M. Chanin and M. A. Biondi, “Temperature dependence of ion mobilities in helium, neon and argon”, Phys. Rev 106 (1957) 473. K EY: [Chanin and Biondi, 1957] B. N. Chapman, Glow Discharge Processes (John Wiley & Sons, Inc., 1980). K EY: [Chapman, 1980] H. Chatham and A. Gallagher, “Ion chemistry in silan dc discharges”, J. Appl. Phys. 58 (1) (1985) 159. K EY: [Chatham and Gallagher, 1985] H. Chatham, D. Hils, R. Robertson and A. Gallagher, “Total and partial electron collisional ionization cross sections for CH4 , C2 H6 , SiH4 and Si2 H6 ”, J. Chem. Phys. 81 (4) (1984) 1770. K EY: [Chatham et al., 1984] R. C. Chittick, J. H. Alexander and H. F. Sterling, J. Electrochemical Soc. 116 (1969) 77. K EY: [Chittick et al., 1969]

B IBLIOGRAPHY

176

S. J. Choi and M. J. Kushner, “The role of negative ions in the formation of particles in lowpressure plasmas”, J. Appl. Phys. 74 (2) (1993) 853. K EY: [Choi and Kushner, 1993] G. Cicala, P. Capezzuto and G. Bruno, “Microcrystalline silicon by plasma enhanced chemical vapor deposition from silicon tetrafluoride”, J. Vac. Sci. Technol. A 19 (2) (2001) 515. K EY: [Cicala et al., 2001] J. W. Coburn, “Some fundamental aspects of plasma-assisted etching”, in R. Shul and S. Pearton, editors, “Handbook of Advanced Plasma Processing Techniques”, (Springer, 2000), pp. 1–33. K EY: [Coburn, 2000] M. Dalvie, M. Surendra and G. S. Selwyn, “Self-consistent fluid modeling of radio frequency discarges in two dimensions”, Appl. Phys. Lett. 62 (24) (1993) 3207. K EY: [Dalvie et al., 1993] J.-L. Delcroix and A. Bers, Physique des plasmas (InterEditions et CNRS Editions, 1994). K EY: [Delcroix and Bers, 1994] J. H. Ferziger and M. Perić, Computational methods for fluid dynamics (Springer, 1999). K EY: [Ferziger and Perić, 1999] A. Fiala, L. C. Pitchford and J. P. Boeuf, “Two-dimensional, hybrid model of low-pressure glow discharge”, Phys. Rev. E 49 (6) (1994) 5607. K EY: [Fiala et al., 1994] L. S. Frost and A. V. Phelps, “Rotational excitation and momentum transfer cross sections for electrons in h2 and n2 from transport coefficients”, Phys. Rev. 127 (5) (1962) 1621. K EY: [Frost and Phelps, 1962] A. Gallagher, “Neutral radical deposition from silane discharges”, J. Appl. Phys. 63 (7) (1988) 2406. K EY: [Gallagher, 1988] C. Godet, N. Layadi and P. R. i Cabarrocas, “Role of mobile hydrogen in the amorphous silicon recrystallization”, Appl. Phys. Lett. 66 (23) (1995) 3146. K EY: [Godet et al., 1995]

B IBLIOGRAPHY

177

V. A. Godyak and A. S. Khanneh, “Ion bombardment secondary electron maintenance of steady RF discharge”, IEEE Trans. Plasma Sci. PS-14 (2) (1986) 112. K EY: [Godyak and Khanneh, 1986] V. A. Godyak and R. B. Piejak, “In situ simultaneous radio frequency discharge power measurements”, J. Vac. Sci. Technol. A 8 (5) (1990) 3833. K EY: [Godyak and Piejak, 1990] C. G. Goedde, A. J. Lichtenberg and M. A. Lieberman, “Self-consistent stochastic electron heating in radio frequency discharges”, J. Appl. Phys. 64 (9) (1988) 4375. K EY: [Goedde et al., 1988] W. J. Goedheer, P. M. Meijer, J. Bezemer, J. D. P. Passchier and W. G. J. H. M. van Sark, “Frequency effects in capacitively coupled radio-frequency glow discharges: a comparison between a 2-d fluid model and experiments”, IEEE Trans. Plasma Science 23 (4) (1995) 644. K EY: [Goedheer et al., 1995] E. Gogolides, J. P. Nicolai and H. H. Sawin, “Comparison of experimental measurements and model predictions for radio-frequency Ar and SF6 discharges”, J. Vac. Sci. Technol. A 7 (1989) 1001. K EY: [Gogolides et al., 1989] E. Gogolides and H. H. Sawin, “Continuum modeling of radio-frequency glow discharges.i. theory and results for electropositive and electronegative gases”, J. Appl. Phys. 72 (9) (1992) 3971. K EY: [Gogolides and Sawin, 1992] V. E. Golant, Sov. Phys. Tech. Phys. 4 (1959) 680. K EY: [Golant, 1959] V. E. Golant and A. P. Z. I. E. Sakharov, Fundamentals of plasma physics (John Wiley & Sons, Inc., 1980). K EY: [Golant and Sakharov, 1980] W. G. Graham and C. M. O. Mahoney, “Experimental studies of rf sheath”, in U. Kortshagen and L. Tsendin, editors, “Electron Kinetics and Applications of Glow Discharges”, (NATO ASI Series B: Physics vol. 367, Plenum Press, New York, 1998), pp. 503–510. K EY: [Graham and Mahoney, 1998]

B IBLIOGRAPHY

178

D. B. Graves, “Fluid model simulations of a 13.56 MHz rf discharge: Time and space dependence of rates of electron impact excitation”, J. Appl. Phys. 62 (1987) 88. K EY: [Graves, 1987] D. B. Graves and K. F. Jensen, “A continuum model of DC and RF discharges”, IEEE Trans. Plasma Sci. PS-14 (2) (1986) 78. K EY: [Graves and Jensen, 1986] P. Haaland, “Dissociative attachment in silane”, J. Chem. Phys. 93 (6) (1990) 4066. K EY: [Haaland, 1990] G. J. M. Hagelaar, F. J. Hoog and G. M. W. Kroesen, “Boundary conditions in fluid models of gas discharges”, Phys. Rev. E 62 (1) (2000) 1452. K EY: [Hagelaar et al., 2000] E. A. G. Hamers, Plasma deposition of hydrogenated amorphous silicon, Ph.D. thesis, Utrecht University (1998). K EY: [Hamers, 1998] P. J.-J. Hargis, P. A. M. K.E. Greenberg, J. B. Gerardo, J. R. Torczynski, J. R. Roberts, j. K. Olthoff, J. R. Whetstone, R. V. Brunt, M. A. Sobolewki, M. Anderson, M. P. Splichal, J. L. Mock, P. Bletzinger, A. Garscadden, R. A. Gottscho, Selwyn, M. Dalvie, J. E. Heindenreich, J. W. Butterbaugh, M. L. Brake, M. L. Passow, J. Pender, A. Lujan, M. E. Elta, D. B. Graves, H. H. Sawin, M. J. Kushner, J. T. Verdeyen, R. Horwath and T. R. Turner, “The gaseous electronics conference radio-frequency cell: a defined parallel-plate radio-frequency system for experimental and theoretical studies of plasma-processing discharges”, Rev. Sci. Instrum. 1 (1994) 140. K EY: [Hargis et al., 1994] M. Heintze and R. Zedlitz, J. Non-Cryst. Solids 198-200 (1996) 1038. K EY: [Heintze and Zedlitz, 1996] J. M. S. Henis, G. W. Stewart, M. K. Tripodei and P. P. Gaspar, “Ion-molecule reactions in silane”, J. Chem. Phys. 57 (1) (1972) 389. K EY: [Henis et al., 1972] A. P. Hickman, “Approximate scaling formula for ion-ion mutual neutralization rates”, J. Chem. Phys. 11 (1979) 4872.

B IBLIOGRAPHY

179

K EY: [Hickman, 1979] E. H. Holt and R. E. Haskell, Foundations of plasma dynamics (The Macmillan Company, New York, 1965). K EY: [Holt and Haskell, 1965] A. A. Howling, L. Sansonnens, J.-L. Dorier and C. Hollenstein, “Negative hydrogenated silicon ion clusters as particle precursors in rf silane plasma depostion experiments”, J. Phys. D: Appl. Phys. 26 (1993) 1003. K EY: [Howling et al., 1993] http://www.pv.unsw.edu.au/eff/, “http://www.pv.unsw.edu.au/eff/”, (2001). K EY: [http://www.pv.unsw.edu.au/eff/, 2001] P. Kae-Nune, J. Perrin, J. Guillon and J. Jolly, “Mass spectrometry detection of SiH m and CHm radicals from SiH4 -CH4 -H2 discharges under high temperatures deposition conditions of silicon carbide”, Jpn. J. Appl. Phys. 33 (1994) 4303. K EY: [Kae-Nune et al., 1994] P. Kae-Nune, J. Perrin, J. Guillon and J. Jolly, “Mass spectrometry detection of radicals in SiH4 -CH4 -H2 glow discharge plasmas”, Plasma Sources Sci. Technol. 4 (1995) 250. K EY: [Kae-Nune et al., 1995] F. J. Kampas and M. J. Kushner, “Effect of silane pressure on silane-hydrogen rf glow discharges”, IEEE Trans. Plasma Sci. PS-14 (2) (1986) 173. K EY: [Kampas and Kushner, 1986] J. Kanicki, Amorphous and microcrystalline semiconductor devices (Artech House, Inc, 1991). K EY: [Kanicki, 1991] H. Kausche and R. D. Plättner, “Effect of a-Si:H powder formation in silane plasma on solar cell processing”, in “Proc. 11-th E. C. Photovoltaic Solar Energy Conference”, (Harwood Academic Publishers, 1992), pp. 195–198. K EY: [Kausche and Plättner, 1992] A. von Keudell and J. R. Abelson, “Direct insertion of SiH3 radicals into strained Si-Si surface bonds during plasma deposition of hydrogenated amorphous silicon films”, Phys. Rev. B 59 (8) (1999) 5791.

B IBLIOGRAPHY

180

K EY: [von Keudell and Abelson, 1999] L. E. Kline, “Performance predictions for electron-beam controlled on-off switches”, IEEE Trans. Plasma Sci. PS-10 (4) (1982) 24. K EY: [Kline, 1982] S. Kolodinski, J. H. Werner, U. Rau, J. K. Arch and E. Bauser, “Thin film silicon solar cells from liquid phase epitaxy”, in “Proc. 11-th E. C. Photovoltaic Solar Energy Conference”, (Harwood Academic Publishers, 1992), pp. 53–56. K EY: [Kolodinski et al., 1992] M. J. Kushner, “A model for the discharge kinetics and plasma chemistry during plasma enhaces chemical vapor deposition of amorphous silicon”, J. Appl. Phys. 63 (8) (1988) 2532. K EY: [Kushner, 1988] L. Lapidus and G. F. Pinder, Numerical solution of partial differential equations in science and engineering (John Wiley & Sons, Inc., 1982). K EY: [Lapidus and Pinder, 1982] P. G. LeComber and W. E. Spear, “Electronic transport in amorphous silicon films”, Phys. Rev. Lett. 25 (1970) 509. K EY: [LeComber and Spear, 1970] O. Leroy, Diagnostics et modélisation physico-chimique de décharges radio-fréquence dans les mélanges SiH4 /H2 pour le dépot de silicium, Ph.D. thesis, Université de Paris-Sud Orsay (1996), nr. 4561. K EY: [Leroy, 1996] O. Leroy, G. Gousset, L. L. Alves, J. Perrin and J. Jolly, “Two-dimensional modelling of SiH 4 H2 radio-frequency discharges for a-Si:H deposition”, Plasma Sources Sci. Technol. 7 (1998) 348. K EY: [Leroy et al., 1998] O. Leroy, P. Stratil, J. Perrin, J. Jolly and P. Belenguer, “Spatiotemporal analysis of the double layer formation in hydrogen radio frequency discharges”, J. Phys. D: Appl. Phys. 28 (1995) 500. K EY: [Leroy et al., 1995]

B IBLIOGRAPHY

181

S. M. Levitskii, “An investigation of the sparking potential of a HF discharge in a gas in the transition range of frequencies and pressures”, Zh. Tekh. Fiz. 27 (1957) 1001. K EY: [Levitskii, 1957] M. Li, H. Date and D. B. Graves, “Fluid kinetic and hybrid simulation strategies for modeling chemically complex inductively coupled plasmas”, in U. Kortshagen and L. Tsendin, editors, “Electron Kinetics and Applications of Glow Discharges”, (NATO ASI Series B: Physics vol. 367, Plenum Press, New York, 1998), pp. 349–366. K EY: [Li et al., 1998] R. L. Liboff, Introduction to the theory of kinetics equations (John Wiley & Sons, Inc., 1969). K EY: [Liboff, 1969] M. A. Lieberman and A. J. Lichtenberg, Principles of plasma discharges and materials processing (John Wiley & Sons, Inc., 1994). K EY: [Lieberman and Lichtenberg, 1994] J. Loureiro and C. M. Ferreira, “Electron and vibrational kinetics in the hydrogen positive column”, J. Phys. D 22 (1989) 1680. K EY: [Loureiro and Ferreira, 1989] D. P. Lymberopoulos and D. J. Economou, “Fluid simulation of radio frequency glow discharges: Two-dimensional argon discharge including metastables”, Appl. Phys. Lett. 63 (18) (1993a) 2478. K EY: [Lymberopoulos and Economou, 1993] D. P. Lymberopoulos and D. J. Economou, “Fluid simultaions of glow discharges: effect of metastable atoms in argon”, J. Appl. Phys. 73 (1993b) 3668. K EY: [Lymberopoulos and Economou, 1993] D. P. Lymberopoulos and D. J. Economou, “Modeling ans simulation of glow discharge plasma reactors”, J. Vac. Sci. Technol. A 12 (4) (1994) 1229. K EY: [Lymberopoulos and Economou, 1994] D. P. Lymberopoulos and D. J. Economou, “Spatiotemporal electron dynamics in radiofrequency glow discharges: fluid versus dynamic Monte-Carlo simulations”, J. Phys. D: Appl. Phys. 28 (1995) 727. K EY: [Lymberopoulos and Economou, 1995]

B IBLIOGRAPHY

182

T. Makabe, N. Nakano and Y. Yamaguchi, “Modeling and diagnostics of the structure of rf glow discharges in Ar at 13.56 MHz”, Phys. Rev. A 45 (4) (1992) 2520. K EY: [Makabe et al., 1992] A. Matsuda, “Formation kinetics and control of microcryslalite in µc-Si:H from glow discharge plasma”, J. Non-Crystallin Solids 59&60 (1983) 767. K EY: [Matsuda, 1983] A. Matsuda, “Growth mechanism of microcrystalline silicon obtained form reactive plasmas”, Thin Solid Films 337 (1999) 1. K EY: [Matsuda, 1999] A. Matsuda, T. Kaga, H. Tanaka and K. Tanaka, “Lifetime of dominant radicals for the deposition of a-Si:H from SiH4 and Si2 H6 glow discharges”, J. Non-Crystalline Solids 59&60 (1983) 687. K EY: [Matsuda et al., 1983] A. Matsuda, M. Matsumura, S. Yamasaki, H. Yamamoto, T. Imura, H. Okushi, S. Iizima and K. Tanaka, “Boron doping of hydrogenated silicon thin films”, Jap. J. Appl. Phys. 20 (3) (1981) L183. K EY: [Matsuda et al., 1981] A. Matsuda, S. Yamasaki, K. Nakagawa, H. Okushi, K. Tanaka, S. Iizima, M. Matsumura and H. Yamamoto, “Electrical and structural properties of phosphorous-doped glow-discharge Si:F:H and Si:H films”, Jap. J. Appl. Phys. 19 (6) (1980) L305. K EY: [Matsuda et al., 1980] M. Matsumura, “Amorphous silicon charged coupled devices”, in J. Kanicki, editor, “Amorphous and microcrystalline semiconductor devices”, (Artech House, Inc, 1991), pp. 141–166. K EY: [Matsumura, 1991] E. W. McDaniel and E. A. Mason, The Mobility and Diffusion of Ions in Gases (John Wiley & Sons, Inc., 1973). K EY: [McDaniel and Mason, 1973] P. M. Meijer, The electron dynamics of RF discharges, Ph.D. thesis, Utrecht University (1991). K EY: [Meijer, 1991]

B IBLIOGRAPHY

183

A. Merad, Modélisation fluide des décharges radio fréquence basse pression, a couplage capacitif, et consideration de la presence de poudres, Ph.D. thesis, Université Paul Sabatier de Toulouse (1998), nr. 2932. K EY: [Merad, 1998] M. Meyyappan and T. R. Govindan, “Radio frequency discharge modeling: moment equation approach”, J. Appl. Phys. 74 (4) (1993) 2250. K EY: [Meyyappan and Govindan, 1993] G. J. Nienhuis, Plasma models for silicon deposition, Ph.D. thesis, Utrecht University (1998). K EY: [Nienhuis, 1998] G. J. Nienhuis, W. J. Goedheer, E. A. G. Hamers, E. G. J. H. M. van Stark and J. Bezemer, “A self-consistent fluid model for radio-frequency discharges in SiH 4 -H2 compared experiments”, J. Appl. Phys. 82 (5) (1997) 2060. K EY: [Nienhuis et al., 1997] M. R. Nimlos and G. B. Ellison, J. Am. Chem. Soc. 108 (1986) 6522. K EY: [Nimlos and Ellison, 1986] T. E. Nitschke and D. B. Graves, “A comparison of particle in cell and fluid model simulations of low-pressure radio frequency discharges”, J. Appl Phys. 76 (10) (1994) 5646. K EY: [Nitschke and Graves, 1994] Y.-H. Oh, N.-H. Choi and D.-I. Choi, “A numerical simulation fo rf glow discharge containing an electronegative gas composition”, J. Appl. Phys. 67 (1990) 3264. K EY: [Oh et al., 1990] L. J. Overzet and M. B. Hopkins, Appl. Phys. Lett. 63 (1993) 2484. K EY: [Overzet and Hopkins, 1993] S. R. Ovshinski, “Solar electricity speeds down to earth”, in D. Adler, B. B. Schwartz and M. Silver, editors, “Disordered Materials”, (Plenum Press, New York, 1991), pp. 203–206, reprinted from New Scientist 80 (1131), 674, (1978). K EY: [Ovshinski, 1991] S. R. Ovshinski and A. Madan, “Amorphous photovoltaic cells”, in D. Adler, B. B. Schwartz and M. Silver, editors, “Disordered Materials”, (Plenum Press, New York, 1991), pp. 206– 209, reprinted from Solar Energy Symposia of the 1978 Annual Meeting pp. 69-73.

B IBLIOGRAPHY

184

K EY: [Ovshinski and Madan, 1991] H. Pak and M. E. Riley, “paper BB-5”, in “45th Gaseous Electronics Conference”, (1992). K EY: [Pak and Riley, 1992] J. D. P. Passchier and W. J. Goedheer, “Relaxation phenomena after laser-induced photodetachment in electronegative rf discharges”, J. Appl. Phys. 73 (3) (1993a) 1073. K EY: [Passchier and Goedheer, 1993] J. D. P. Passchier and W. J. Goedheer, “A two-dimensional fluid model for an argon rf discharge”, J. Appl. Phys. 74 (6) (1993b) 3744. K EY: [Passchier and Goedheer, 1993] V. S. Patankar, Numerical heat transfer and fluid flow (Hemisphere Publishing Corporation, 1980). K EY: [Patankar, 1980] J. Perrin, C. Böhm, R. Etemadi and A. Lloret, “Possible routes for cluster growth and particle formation in rf silane discharges”, Plasma Sources Sci. Technol. 3 (1994) 252. K EY: [Perrin et al., 1994] J. Perrin and T. Broekhuizen, “Surface reaction and recombination of the SiH 3 radical on the hydrogenated amorphous silicon”, Appl. Phys. Lett. 50 (8) (1987) 433. K EY: [Perrin and Broekhuizen, 1987] J. Perrin, O. Leroy and M. C. Bordage, “Cross-sections, rate constants and transport coefficients in silane plasma chemistry”, Contrib. Plasma Phys. 36 (1) (1996) 3. K EY: [Perrin et al., 1996] J. Perrin, A. Lloret, G. de Rosny and J. P. M. Schmitt, “Positive and negative ions in silane and disilane multipole discharges”, Int. J. Mass Spectrom. Ion Phys. 57 (1984) 249. K EY: [Perrin et al., 1984] J. Perrin, M. Shiratani, P. Nune, H. Videlot, J. Jolly and J. Guillon, “Surface reaction probabilities and kinetics of H, SiH3 , Si2 H5 , CH3 and C2 H5 during deposition of a-Si:H and a-C:H from H2 ,SiH4 and CH4 discahrges”, J. Vac. Sci. Technol. A 16 (1) (1998) 278. K EY: [Perrin et al., 1998]

B IBLIOGRAPHY

185

A. V. Phelps, “Cross sections and swarm coefficients for H+ , H2 + , H3 + , H, H2 and H− in H2 for energies from 0.1 eV to 10 keV”, J. Phys. Chem. Ref. Data 19 (3) (1990) 653. K EY: [Phelps, 1990] L. C. Pitchford, J. P. Boeuf and W. L. Morgan, “Bolsig, a 2 term, user-friendly solver for the electron boltzmann equation (kinema software, [email protected], http://www.csn.net/siglo, monument, co)”, (1996). K EY: [Pitchford et al., 1996] O. A. Popov and V. A. Godyak, “Power dissipated in low-pressure radio-frequency discharge plasmas”, J. Appl. Phys. 57 (1) (1985) 53. K EY: [Popov and Godyak, 1985] C. Punset, Modélisation bidimensionnelle fluide d’un écran a plasma, Ph.D. thesis, Université Paul Sabatier de Toulouse (1998), nr. 2979. K EY: [Punset, 1998] Y. P. Raizer, Gas discharge physics (Springer, 1991). K EY: [Raizer, 1991] S. Rauf and M. J. Kushner, “A self-consistent analytical model for non-collisional heating”, Plasma Sources Sci. Technol. 6 (1997) 518. K EY: [Rauf and Kushner, 1997] W. D. Reents and M. L. Mandich, “H/D isotope exchange reaction of SiH 3 + with SiD4 and SiD3 + with SiH4 : Evidence for hybride stripping reaction”, J. Chem. Phys. 93 (5) (1990) 3270. K EY: [Reents and Mandich, 1990] W. D. Reents and M. L. Mandich, “Water induced particle formation in the ion chemistry of silane”, Plasma Sources Sci. Technol. 3 (1994) 373. K EY: [Reents and Mandich, 1994] A. D. Richards, B. E. Thompson and H. H. Sawin, “Continuum modeling of argon radio frequency glow discharge”, Appl. Phys. Lett. 50 (9) (1987) 492. K EY: [Richards et al., 1987]

B IBLIOGRAPHY

186

P. J. Rodrigues, Discretização de operadores de Fokker-Planck unidimensionais em grelhas não-uniformes, Master’s thesis, Universidade Técnica de Lisboa - Instituto Superior Técnico (2001). K EY: [Rodrigues, 2001] A. Salaba¸s, G. Gousset and L. L. Alves, “Charged particle transport modeling in silanehydrogen rf capacitively coupled discharges”, Vacuum Elsevier 69 (1-3) (2002a) 213. K EY: [Salaba¸s et al., 2002] A. Salaba¸s, G. Gousset and L. L. Alves, “Two dimensional fluid modeling of charged particle transport in radio-frequency capacitively coupled discharges”, Plasma Sources Sci. Technol. 11 (2002b) 448. K EY: [Salaba¸s et al., 2002] D. L. Scharfetter and H. K. Gummel, “Large-signal analysis of a silicon read diode oscillator”, IEEE Trans. Electron Devices ED-16 (1) (1969) 64. K EY: [Scharfetter and Gummel, 1969] J. P. M. Schmitt, “Fundamental mechanisms in silane plasma decompositions and amorphous silicon deposition”, J. Non-Crystalline Solids 59&60 (649). K EY: [Schmitt, 1983] T. Šimko, V. Martišovitš, J. Bretagne and G. Gousset, “Computer simulation of H + and H3 + transport parameters in hydrogen drift tubes”, Phys. Rev. E 56 (5) (1997) 5908. K EY: [Šimko et al., 1997] I. Solomon, B. Drévillon, H. Shirai and N. Layadi, “Plasma deposition of microcrystalline silicon: the selective etching model”, J. Non-Crystallin Solids 164&166 (1993) 989. K EY: [Solomon et al., 1993] T. J. Sommerer and M. J. Kushner, “Numerical investigation on the kinetics and chemistry of rf glow discharge plasmas sustained in He, N2 , O2 , He/N2 /O2 , He/CF4 /O2 , and SiH4 /NH3 using a Monte Carlo-fluid hybrid model”, J. Appl. Phys. 71 (1992) 1654. K EY: [Sommerer and Kushner, 1992] W. E. Spear and P. G. LeComber, Solid State Commun. 17 (1975) 1193. K EY: [Spear and LeComber, 1975]

B IBLIOGRAPHY

187

W. E. Spear, R. J. Loveland and A. Al-Sharbaty, “The temperature dependence of photoconductivity in a-Si”, J. Non-Cryst. Solids 15 (1974) 410. K EY: [Spear et al., 1974] D. L. Staebler and C. W. Wronski, “Reversible conductivity changes in discharge-produced amorphous Si”, Appl. Phys. Lett. 31 (4) (1977) 292. K EY: [Staebler and Wronski, 1977] H. F. Sterling and R. G. Swann, Solid-State Electr. 8 (1965) 653. K EY: [Sterling and Swann, 1965] J. E. Stevens, “Plasma fundamentals for materials processing”, in R. Shul and S. Pearton, editors, “Handbook of Advanced Plasma Processing Techniques”, (Springer, 2000), pp. 33–68. K EY: [Stevens, 2000] R. A. Stewart, P. Vitello and D. B. Graves, “Two-dimensional fluid model of high density inductively coupled plasma sources”, J. Vac. Sci. Technol. B 12 (1) (1994) 478. K EY: [Stewart et al., 1994] R. A. Street, Hydrogenated amorphous silicon, Cambridge Solid State Science Series (Cambridge University Press, 1991). K EY: [Street, 1991] M. Surendra, “Radiofrequency discharge benchmark model comparison”, Plasma Sources Sci. Technol. 4 (1995) 56. K EY: [Surendra, 1995] K. Suzuki, “Flat panel display using amorphous and monocrystalline semiconductor devices”, in J. Kanicki, editor, “Amorphous and microcrystalline semiconductor devices”, (Artech House, Inc, 1991), pp. 77–139. K EY: [Suzuki, 1991] H. Tawara, Y. Itikawa, H. Nishimura and M. Yoshino, “Cross sections and related data for electron collisions with hydrogen molecules and molecular ions”, J. Phys. Chem. ref. Data 19 (3) (1990) 617. K EY: [Tawara et al., 1990] S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical recipies in Fortran (Cambrige University Press, 1992).

B IBLIOGRAPHY

188

K EY: [Teukolsky et al., 1992] L. P. Theard and W. T. Huntress, “Ion-molecule reactions and vibrational deactivation of H 2 + ions in mixtures of hydrogen and helium”, J. Chem. Phys. 60 (7) (1974) 2840. K EY: [Theard and Huntress, 1974] C. C. Tsai, “Plasma deposition of amorphous and crystalline silicon: the effect of hydrogen on the growth, structure and electronic properties”, in H. Fritzsche, editor, “Amorphous silicon and related materials”, (World Scientific Publishing Company, 1988), pp. 123–147. K EY: [Tsai, 1988] C. C. Tsai, J. C. Knights, G. Chang and B. Wacker, “Film formation mechanisms in the plasma deposition of hydrogenated amorphous silicon”, J. Appl. Phys. 59 (8) (1986) 2998. K EY: [Tsai et al., 1986] C. C. Tsai, J. G. Shaw, B. Wacker and J. C. Knights, “Film growth mechanisms of amorphous silicon in diode and triode glow discharge systems”, Mat. Res. Soc. Symp. Proc 95 (1987) 219. K EY: [Tsai et al., 1987] M. M. Turner, “Collisionless electron heating in an inductively coupled discharge”, Phys. Rev. Lett. 71 (12) (1993) 1844. K EY: [Turner, 1993] V. Vahedi, M. A. Lieberman, G. DiPeso, T. D. Rognlien and D. Hewett, “Analytic model of power deposition in inductively coupled plasma sources”, J. Appl. Phys. 78 (1995) 1446. K EY: [Vahedi et al., 1995] P. E. Vanier, F. J. Kampas, R. R. Cordeman and G. Rajeswaran, “A study of hydrogenated amorphous silicon deposited by rf glow discharge in silan-hydrogen mixtures”, J. Appl. Phys. 56 (6) (1984) 1812. K EY: [Vanier et al., 1984] G. A. Vawter, “Ion beam etching of compound semiconductors”, in R. Shul and S. Pearton, editors, “Handbook of Advanced Plasma Processing Techniques”, (Springer, 2000), pp. 507– 547. K EY: [Vawter, 2000]

B IBLIOGRAPHY

189

P. L. G. Ventzek, R. J. Hoekstra and M. J. Kushner, “Two-dimensional modeling of high plasma density inductively coupled sources for materials processing”, J. Vac. Sci. Technol. B 12 (1) (1994) 461. K EY: [Ventzek et al., 1994] P. L. G. Ventzek, T. Sommerer, R. Hoestra and M. Kushner, “Two-dimensional hybrid model of inductively coupled plasma sources for etching”, Appl. Phys. Lett. 63 (5) (1993) 605. K EY: [Ventzek et al., 1993] S. Vepˇrec, F.-A. Sarrott, S. Rambert and E. Taglauer, “Surface hydrogen content and passivation of silicon deposited by plasma induced chemical vapor deposition from silane and the impllications for the reaction mechanism”, J. Vac. Sci. Technol. A 7 (4) (1989) 2614. K EY: [Vepˇrec et al., 1989] H. K. Versteeg and W. Malalasekera, An introduction to computational fluid dynamics (Longman Scientific & Technical, 1995). K EY: [Versteeg and Malalasekera, 1995] P. Vidaud, S. M. Durrani and D. R. Hall, “Alpha and gamma RF capacitative discharges in N 2 at intermediate pressures”, J. Phys. D: Appl. Phys. 21 (1988) 57. K EY: [Vidaud et al., 1988] A. L. Ward, “Effect of space charge in cold-cathode gas discharge”, Phys. Rev. 112 (6) (1958) 1852. K EY: [Ward, 1958] A. L. Ward, “Calculations of cathode-fall characteristics”, J. Appl. Phys. 33 (9) (1962) 2789. K EY: [Ward, 1962] R. C. Weast, CRC Handbook of Chemistry and Physics (Chemical Rubber, Boca Raton F. L., 1986). K EY: [Weast, 1986] M. Yan and W. J. Goedheer, “A PIC-MC simulation of the effect of frequency on the characteristics of VHF SiH4 -H2 discharges”, Plasma Sources Sci. Technol. 8 (1999) 349. K EY: [Yan and Goedheer, 1999] N. A. Yatsenko, “Medium pressure, high-current high-frequency capacitive discharge”, Zk. Tekh. Fiz. 50 (1980) 2480.

B IBLIOGRAPHY

190

K EY: [Yatsenko, 1980] N. A. Yatsenko, “Relationship between the high constant plasma potential and the conditions in an intermediate-pressure rf capacitive discharge”, Zh. Tekh. Fiz. 51 (1981) 1195. K EY: [Yatsenko, 1981] N. A. Yatsenko, “The normal current-density effect in a moderate-pressure capacitive hf discharge”, Zh. Tekh. Fiz. 52 (1982) 1220. K EY: [Yatsenko, 1982] F. F. Young and C. H. Wu, “A comparative study between non-equilibrium and equilibrium models of rf glow discharges”, J. Phys. D: Appl. Phys. 26 (1993a) 782. K EY: [Young and Wu, 1993] F. F. Young and C. H. Wu, “Two-dimensional, self consistent, three moment simulation of rf glow discharges”, IEEE Trans. Plasma Sci. 21 (3) (1993b) 312. K EY: [Young and Wu, 1993]

Index cross-section

alpha regime, 15

helium, 49

amorphous materials, 4

hydrogen, 49

hydrogenated silicon, 5

silane, 49

anisotropy, 26 asymmetric configuration, 60

CVD, 3

BBGKY, 25

DDA, 22, 38

BD, 70

dielectric relaxation time, 84

Boltzmann equation, 25

discharge electrical parameters, 60 discharges

moments, 30

alpha and gamma, 15

solver, 55

modeling, 18

boundary conditions, 45

powder regime, 17

CAIBE, 2

radio-frequency, 8

ccrf, 8

discretization, 66

CD, 70

effective electric field, 89

CFL, 73

Poisson’s equation, 85

closure condition, 37, 40, 52

spatial, 77

electrons, 48

transport equations, 80

ions, 48

drift-diffusion approximation, 22

collisional regime, 38

validity, 38

conservation equation energy, 31

EBE, 22

momentum, 31

ECR, 2

particles, 30

EDF, 11

control volume, 69

EEDF, 35

convergence criterion, 65, 91

effective electric field, 40, 44, 89

convergence error, 92

electrical parameters

coupled electrical power, 14, 63

coupled power, 63

Courant-Friedrichs-Lewy condition, 73

self-bias voltage, 62 eRC, 21

Crank-Nicholson algorithm, 73, 76

191

I NDEX

192

eTP, 21

Maxwell’s time, 84

exponential scheme, 78

maxwellian distribution function, 26

FD, 70 FDM, 66 field inversion, 131

MC, 19 mean energy, 28 mean velocity, 28 mesh generation, 66

reversal, 131 finite difference method, 66, 69 fluid

staggered, 68 model equations boundary conditions, 45

description, 29

continuity, 41

models, 20

effective electric field, 44 electron flux, 41

gamma regime, 15

energy flux, 42

heat flow vector, 29

ion transport, 43

hybrid models, 22

mean energy, 42

hydrodynamic acceleration, 31 approach, 27 condition, 38 IC, 2 IDF, 27 inertia term, 40, 89 initial conditions, 71

Poisson, 45 model flow chart, 72 moment equations, 34 non-equilibrium conditions, 26 OES, 6 particle density, 28 PDE, 65

kinetic model, 49

PECVD, 4

kinetic pressure tensor, 28

PIC, 19

LEA, 21, 37, 53 LFA, 21, 52, 53

Poisson’s equation, 45, 83 powder regime, 17

LHS, 31

radio-frequency discharges, 8

LIF, 6

rf, 2

Liouville equation, 24

RHS, 30

LTP, 1

RIBE, 2

matrix

RIE, 2

pentadiagonal, 87

Scharfetter-Gummel algorithm, 78

tridiagonal, 82

self-bias voltage, 13, 62

I NDEX small anisotropy assumption, 47 solar cell, 6 sputtering, 3 statistical models, 19 thermal conductivity, 33 time integration methods, 72 transport parameters electrons, 52 expressions, 36 ions, 58 truncation error, 70 two-term expansion, 34 viscous stress tensor, 29, 32 VLSI, 2

193

Fluid model for charged particle transport in

Fluid model for charged particle transport in

Suggest Documents

CHARGED PARTICLE TRANsPORT IN MAGNETIC FIELD ...

Charged Particle Energization and Transport in

CHARGED PARTICLE TRANsPORT IN MAGNETIC FIELD ...

Simplified Charged Particle Beam Transport Modeling Using

Simplified Charged Particle Beam Transport Modeling Using ...

nonlinear energetic charged particle transport and ... - IOPscience

Charged particle transport in antidot lattices in the presence of

Generalised balance equations for charged particle transport via ...

Third-order transport coefficients for charged particle swarms - SCL

An integrated particle model for fluid-particle-structure ...

Studying charged particle optics

Thermodynamically consistent mesoscopic fluid particle model

Charged-coupled devices for charged-particle ... - PSFC - MIT

Radiation reaction for a massless charged particle

Cellular automata model in particle transport studies in ... - ANU

NOETHER SYMMETRIES FOR CHARGED PARTICLE MOTION ...

Charged Particle in an Electromagnetic Field

2 Charged Particle in a Magnetic Field

A minimal coupled fluid-discrete element model for bedload transport

Light Particle Tracking Model for Simulating Bed Sediment Transport ...

MOMENT METHOD IN CHARGED-PARTICLE BEAM DYNAMICS

Charged particle multiplicity measurements in NA60

Charged Particle in a Magnetic Field - Galileo

CHARGED PARTICLE MULTIPLICITIES IN ULTRA-RELATIVISTIC ...