Multiphysics effects in quantum-dot structures

Mads Clausen Institute for Product Innovation University of Southern Denmark Alsion 2 Sønderborg, Denmark September, 2009

PhD Thesis

Multiphysics effects in quantum-dot structures

Candidate Daniele Barettin

Supervisors Prof. Morten Willatzen Dr. Benny Lassen

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This work is supported by the Danish Research Council for Technology and Production - QUEST.

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To my mother, who has made me come so far, and to my brother, who has made me come so far with a smile.

A mia madre, che mi ha fatto arrivare fin qui, e a mio fratello, che mi ci ha fatto arrivare sorridendo.

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There is a crack in everything, that is how the light gets in. Leonard Coen

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Contents Abstract

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Resum´ e

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List of Publications

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Introduction

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1 An 1.1 1.2 1.3

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overview of quantum dots From atoms to quantum dots Quantum dot applications . . QUEST project . . . . . . . . 1.3.1 Slow light . . . . . . . 1.4 Modelling quantum dots . . .

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2 Electromechanical fields 2.1 Fundamental equations . . . . . . . . . . . . . . . . . . 2.1.1 The strain tensor . . . . . . . . . . . . . . . . . 2.1.2 The stress tensor . . . . . . . . . . . . . . . . . 2.1.3 Free energy . . . . . . . . . . . . . . . . . . . . 2.1.4 Constitutive relations . . . . . . . . . . . . . . 2.2 Strain field in a quantum dot . . . . . . . . . . . . . . 2.2.1 Zincblende quantum dot . . . . . . . . . . . . . 2.2.2 Wurtzite quantum dot . . . . . . . . . . . . . . 2.3 Piezoelectric field in a quantum dot . . . . . . . . . . 2.3.1 Zincblende quantum dot . . . . . . . . . . . . . 2.3.2 Wurtzite quantum dot . . . . . . . . . . . . . . 2.4 Governing equations for electromechanical fields . . . . 2.4.1 Zincblende quantum dot . . . . . . . . . . . . . 2.4.2 Wurtzite quantum dot . . . . . . . . . . . . . . 2.5 Electromechanical fields in cylindrical coordinates . . . 2.5.1 Governing equations in cylindrical coordinates zincblende quantum dot . . . . . . . . . . . . . 9

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CONTENTS 2.5.2 2.6

Governing equations in cylindrical coordinates for a wurtzite quantum dot . . . . . . . . . . . . . . . . . . The valence force fields model . . . . . . . . . . . . . . . . . .

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3 Bandstructure ~k · p ~ theory 3.1 General theory for bulk crystals . . . . . . . . . . . . . . . . 3.2 Heterostructure quantum dots: formalism . . . . . . . . . . . 3.2.1 Exact envelope function theory . . . . . . . . . . . . . 3.3 Explicit Hamiltonians . . . . . . . . . . . . . . . . . . . . . . 3.3.1 One-band-model Hamiltonian for a zincblende quantum dot . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Inclusion of strain in one-band model . . . . . . . . . 3.3.3 Eight-band-model Hamiltonian for a zincblende quantum dot . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Spurious solutions in eight band ~k · ~p theory . . . . . . 3.3.5 Plane-wave cut-off method in polar coordinates . . . 3.3.6 Inclusion of strain in an eight-band model . . . . . . 3.3.7 Eight-band model in a rotational-symmetric system . 3.3.8 Six-band-model Hamiltonian for wurtzite quantum dot 3.3.9 Inclusion of strain in a six-band model . . . . . . . . .

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4 Optoelectronic properties 4.1 General theory . . . . . . . . . . . 4.2 Momentum matrix elements . . . . 4.2.1 Momentum matrix elements ric quantum dot . . . . . . 4.3 Dipole moments . . . . . . . . . .

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5 The GaN/AlN wurtzite quantum dot 5.1 GaN/AlN wurtzite cylindrical quantum dot . . . . . . . . . . 5.1.1 Two-dimensional rotational invariant continuum model 5.1.2 Comparison with a one-dimensional case . . . . . . . . 5.1.3 The three-dimensional fully-coupled continuum model 5.2 The GaN/AlN wurtzite hexagonal pyramid quantum dot . . .

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6 The InAs/GaAs zincblende quantum dot 97 6.1 Three-dimensional strain distributions due to anisotropic effects 98 6.2 Optical properties of strained quantum dots . . . . . . . . . . 102 6.2.1 Theory and system . . . . . . . . . . . . . . . . . . . . 102 6.2.2 Spurious solutions . . . . . . . . . . . . . . . . . . . . 104 6.2.3 Results: bandstructure and dipole moments . . . . . . 105 Conclusion

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Bibliography

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CONTENTS Acknowledgements

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CONTENTS

Abstract The first intent of this thesis is to present a description of the mathematical models for the electromechanical, electronic and optical properties of heterostructure quantum dots. I have focused mainly on the GaN wurtzite quantum dot and the InAs zincblende quantum dot embedded in a AlN and GaAs matrix, respectively, due to the relevance they have shown for optoelectronic applications. Besides these models, I present the related numerical results I have derived for these two structures. Presenting the continuum mechanical model I have implemented, and the results of the combined influence of lattice mismatch, piezoelectric effects, and spontaneous polarization for GaN/AlN quantum dot, I show that the governing equations for the electromechanical fields are axisymmetric. Therefore the electric- and mechanical-field solutions are axisymmetric as well, and the original three-dimensional problem can be solved with a twodimensional model in cylindrical coordinates. I compare these results for a cylindrical quantum dot both with analytical results for the one-dimensional case, and mainly with results given by a complete three-dimensional fullycoupled continuum model. We will find a very good qualitative and quantitative agreement between the two- and three-dimensional models, respectively, due to the rotational invariance of the governing equations for a wurtzite quantum dot. I have then studied a more realistic GaN/AlN quantum dot shape, an hexagonal pyramid. It is well known that the linear continuum mechanics models do not contain the full crystal symmetry, and therefore they show a higher degree of symmetry than atomistic models. So in order to quantitatively address the importance of strain, I present a comparison of the three-dimensional continuum mechanics and the atomistic valence-force field strain models, and study the implications for the electronic state energies for this latter structure using an effective-mass approximation to calculate the electronic groundstates for different models, volumes, and aspect ratio. Due to computational considerations, using an isotropic assumption I have developed a two-dimensional model in cylindrical coordinates also for a zincblende InAs/GaAs cylindrical quantum dot, even if its governing equations are not axisymmetric, and have compared results given by this model for the strain fields with the resulting strain field given by fully coupled 13

three-dimensional continuum model, to verify the validity of the assumption. I will demonstrate that these results lead to a good agreement with an appropriate choice for material parameters. Although the optoelectronic properties of InGaAs zincblende quantumdot with varying shape and size based on ~k · p~ theory have already been studied, only a small selection of all possible transitions have been examined. Thus, I will apply an eight-band ~k·~ p model to zincblende InAs/GaAs conical quantum dots, and study the impact of different size and shape and focus on the most relevant interband transitions. Furthermore, I will investigate the effect of strain and band-mixing between the conduction band and valence bands by comparing four different ~k · p~ models. Although previous studies have already included these effects, their impact on dipole moments have as yet not been investigated. In addition to the separation of the heavy and light holes due to the biaxial strain component, a general reduction in the transition strengths due to strain and energy crossings in the valence bands due to strain and band mixing effects is observed. Furthermore a non-trivial quantum dot size dependence of the dipole moments directly related to the biaxial strain component is observed. Due to the separation of the heavy and light holes the optical transition strengths between the lower conduction and upper most valence-band states computed using one-band model and eight-band model show general qualitative agreement, with some interesting exceptions which could be relevant for optoelectronic applications.

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Resum´ e Form˚ alet med denne afhandling er at præsentere en udledning af matematiske modeller for elektromekaniske, elektroniske og optiske egenskaber af heterostruktur kvanteprikker. Jeg fokuserer p˚ a GaN wurtzite og InAs zincblende kvanteprikker indlejret i AlN hhv. GaAs matricer fordi disse materialer har vist sig at være relevante i optoelektroniske anvendelser. Ud over de matematiske modeller vil jeg præsentere tilsvarende numeriske resultater, som jeg har beregnet for disse to strukturer. Ved at introducere en model baserende p˚ a kontinuert mekanik (continuum model), som jeg har implementeret, sammen med den kombinerede effekt af gitter-mismatch, piezoelektrisk effekt og spontan polarisering, viser jeg for GaN/AlN kvanteprikker at de grundlæggende ligninger for elektromekaniske felter er aksisymmetriske. Som følge vil løsningerne for det elektriske og mekaniske felt ogs˚ a være aksisymmetriske og det oprindelige tre dimensionale problem kan løses med en to dimensional model i cylindriske koordinater. Jeg sammenligner resultaterne for en cylindrisk kvanteprik b˚ ade med analytiske resultater for det en-dimensionale tilfælde og for resultater givet ved den tre-dimensionale, fuldt koblede model continuum model. Vi vil se en meget god kvalitativ og kvantitativ overenstemmelse mellem den to- og tre-dimensionale model pga. rotations uafhængigheden af de grundlæggende ligninger for en wurtzite kvanteprik. Jeg har ogs˚ a undersøgt en mere realistisk form for GaN/AlN kvanteprik, som er en hexagonal pyramide. Det er velkendt at den lineære continuum model ikke indeholder den fulde krystalsymmetri og derfor viser en højere symmetrigrad end atomistiske modeller. Med form˚ al at adressere den kvantitative vigtighed af strain, præsenterer jeg en sammenligning mellem den tre dimensionale continuum model og den atomistiske valens-kraft felt strain model. Samtidig undersøger jeg indvirkningen p˚ a de elektroniske tilstands energier for den hexagonale pyramidestruktur ved brug af en effektiv-masse tilnærmelse for at beregne de elektroniske grundtilstande for forskellige modeller, voluminer og side/længde forhold. Kvalitativ overenstemmelse mellem atomistiske strain beregninger og continuum modeller bliver demonstreret, dog bliver signifikante kvantitative diskrepanser op til 100 meV observeret. En mindre forskel p˚ a cirka 15 meV mellem fulde-og halvkoblede continuum modeller bliver fundet. 15

P˚ a grund af beregningsmæssige overvejelser, har jeg brugt en isotropisk antagelse til at udvikle en to-dimensional model i cylindriske koordinater for cylindriske zincblende InAs/GaAs kvanteprikker, selv om dets grundlæggende ligninger ikke er aksisymmetriske. Vi har sammenlignet resulterende strain felter givet ved denne model med strain felter givet ved den fuldt koblede tre dimensionale continuum model for at verificere gyldigheden af vores antagelse. Jeg demonstrerer at disse resultater fører til god overenstemmelse med et passende valg af materialeparametre. Selv om de optoelektroniske egenskaber af InGaAs zincblende kvanteprikke med varierende form og størrelse allerede er blevet undersøgt med ~k· p~ teorien, er kun et lille udvalg af de mulige overgange blevet undersøgt. Derfor vil jeg anvende en otte-b˚ ands ~k·~ p model p˚ akoniske zincblende InAs/GaAs kvanteprikker og undersøge udvirkningen af forskellige størrelser og former med focus p˚ a de mest relevante interb˚ ands overgange. Herudover vil jeg undersøge effekten af strain og b˚ and-mixing mellem konduktions og valensb˚ andende ved at sammenligne fire forskellige ~k · p~ modeller. Selvom tidligere undersøgelser allerede har taget hensyn til disse effekter, er deres indflydelse p˚ a dipolmomenter ikke blevet undersøgt endnu. Udover adskillelsen mellem tunge og lette huller pga. den biaksiale strain komponent, blev en generel reduktion i overgangsstyrke pga. strain, samt energioverlap i valensb˚ andende pga. af strain og b˚ andblandings effekter, iagttaget. Herudover observeres en ikke-triviel kvanteprik størrelsesafhængighed af dipolmomentet, som er direkte forbundet med den biaksiale strain komponent. P˚ a grund af adskillelsen mellem tunge og lette huller viser de optiske overgangsstyrker mellem den nedre konduktionsb˚ andstilstand og den øverste valensb˚ andstilstand, beregnet b˚ ade med en-b˚ ands og otte-b˚ ands modellen, generel kvalitativ overenstemmelse, med nogle interessante undtagelser, som kunne være relevante for optoelektroniske anvendelser.

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List of Publications The publications listed below have been written during my PhD.

List of Journal Publications • B. Lassen, D. Barettin, M. Willatzen, and L.C. Lew Yan Voon, Piezoelectric models for semiconductors quantum dots, Microelectronics Journal, 39 (11), 1226 (2008). • D. Barettin, J. Houmark, B. Lassen, M. Willatzen, T. R. Nielsen, J. Mørk, and A.-P. Jauho, Analysis of optical properties of strained semiconductor quantum dots for electromagnetically induced transparency, submitted to Phys. Rev. B.

List of Conference Proceeding Publications • D. Barettin, B. Lassen, and M. Willatzen, Electromechanical fields in GaN/AlN Wurtzite Quantum Dots, J. Phys. Conf. Ser. 107, 012001 (2008). • B. Lassen, M. Willatzen, D. Barettin, R. V. N. Melnik, and L. C. Lew Yan Voon, Piezoelectric effect and spontaneous polarization in GaN/AlN quantum dots, J. Phys. Conf. Ser. 107, 012008 (2008). • M. Willatzen, B. Lassen, D. Barettin, and L.C. Lew Yan Voon Piezoelectric models for semiconductors quantum dots, W28.00009, Proceeding of the American Physics Society, New Orleans, Louisiana, USA, March 10-14 (2008). • L.C. Lew Yan Voon, B. Lassen, D. Barettin, and M. Willatzen, Semiconductor quantum dots and piezoelectric models, Proceeding of CLACSA XIII (Santa Maria, Colombia 2008). • D. Barettin, S. Madsen, B. Lassen, and M. Willatzen, Comparison of wurtzite atomistic and piezoelectric continuum strain models: Implications for electronic bandstructure, Proceeding of PLMN09 to appear in Superlattices and Microstructures. 17

• B. Lassen, M. Willatzen, and D. Barettin, Band-mixing and strain effects in InAs/GaAs quantum ring, Proceeding of PLMN09 to appear in Superlattices and Microstructures.

List of Conference Contributions • D. Barettin, B. Lassen, and M. Willatzen, Electromechanical fields in GaN/AlN Wurtzite Quantum Dots, Poster, Physics-Based Mathematical Models of Low-Dimensional Semiconductor Nanostructures: Analysis and Computation (Banff 2007). • B. Lassen, M. Willatzen, D. Barettin, R. V. N. Melnik, and L. C. Lew Yan Voon, Piezoelectric effect and spontaneous polarization in GaN/AlN quantum dots, Talk, Physics-Based Mathematical Models of Low-Dimensional Semiconductor Nanostructures: Analysis and Computation (Banff 2007). • L.C. Lew Yan Voon, B. Lassen, D. Barettin, and M. Willatzen, Semiconductor quantum dots and piezoelectric models, Talk, CLACSA XIII (Santa Maria, Colombia 2007). • M. Willatzen, B. Lassen, D. Barettin, and L.C. Lew Yan Voon Piezoelectric models for semiconductors quantum dots, Talk, APS March Meeting (New Orleans, Louisiana 2008). • D. Barettin, B. Lassen, M. Willatzen, R.V.N. Melnik, and L.C. Lew Yan Voon, Three-dimensional strain distributions due to anisotropic effects in InGaAs semiconductor quantum dots, Talk, WCCM8-ECCOMAS (Venice 2008). • J. Houmark, D. Barettin, B. Lassen, T. R. Nielsen, J. Mørk, A.-P. Jauho, and M. Willatzen, Analysis of quantum dot EIT based on eightband k · p theory, Poster, ICPS (Rio de Jeneiro, 2008). • B. Lassen, D. Barettin, and M. Willatzen, Cylindrical symmetry and spurious solutions in 8 band k · p theory, Poster, ICPS (Rio de Janeiro, 2008). • D. Barettin, S. Madsen, B. Lassen, and M. Willatzen, Comparison of wurtzite atomistic and piezoelectric continuum strain models: Implications for optical properties, Poster, PLMN09 (Lecce 2009). • B. Lassen, M. Willatzen, and D. Barettin, Band-mixing and strain effects in InAs/GaAs quantum ring, Talk, PLMN09 (Lecce 2009).

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Introduction Quantum dot heterostructures have recently received much attention due to their relevance for optoelectronic devices [89]. The electron spectrum of an ideal quantum dot is a set of discreet levels material-, shape- and size-dependent, with a density of states given by a delta function. These properties could significantly improve some characteristics of laser devices, such as obtaining an increasing of the optical confined Γ factor and of the optical gain [10]. Although an inhomogeneous broadening of spectra is usually caused by some size and shape distribution of quantum dots, the possibility of changing growth regimes leads to quantum dots with different size, shape and density depending on the requirements. These structures have been studied for the fabrication of lasers devices working in 1,3-1,55 micron wavelength range, and interesting results have been achieved using active regions based on the arrays of quantum dots, which show superior laser characteristics such as ultra-low threshold current densities [12, 13], ultrahigh temperature stability [14], very high differential efficiency [15], small α-factor (temperature independent linewidth enhancement factor) and chirp (shift of the lasing wavelength with current) [16]. Besides, their typical fast carrier capture to the exited state in combination with slow relaxation from exited to the ground state (about 7 ps) has been exploited for mode-locking technology [17]. In particular the InGaAs quantum dot has been proposed as a component in devices for controlling the emission pattern of phased array antennas [94], as ultrafast optical amplifiers or in all-optical switches [95], e.g., as active media in buffers based on slow-light phenomena, utilizing electromagnetically induced transparency (EIT) or population oscillation [96]. Slow-down effects are generally induced by a narrow spectral resonance in the medium, which leads to a rapid variation of the refractive index and subsequent modification of the group velocity of a propagating light beam. These effects were studied for the first time in the experiments of Hau et al. [19]: a control beam was injected in an atomic gas of Sodium atoms along with the signal to be slowed-down, and it coherently coupled discrete levels in the medium, showing a strongly absorbing line where the signal frequency is split, and making the medium opaque in a narrow spectral range and inducing a rapid variation of the refractive index. The speed of light was reduced to only 17 19

m/s cooling the atomic gas to the nanokelvin range, and since then light has even been brought to a complete stop [21]. On the other hand, it has been shown that using GaAlN quantum dots, a intense room-temperature visible luminescence can be obtained on Si 111 substrate, and that this emission energy can be continuously tuned from blue to orange by simply controlling the quantum dot size. Moreover, the mixing of properly chosen quantum dot sizes leads to white light emission [88]. A fundamental aspect in this context is to have accurate models for the optical properties of the quantum dots, including the influence of the strain field and its effects on the bandstructure. The strain field is generated by the differences of the lattice constants of the dot and of the substrate material, and it is relevant for both its selforganization mechanism and the impact of strain on bandstructure. The electromechanical fields in quantum dot heterostructures are usually determinate either with the continuum mechanical model [26], or with the valence force field model [27], or using density functional calculations [28]. For the results presented in this thesis I have implemented different versions of the continuum mechanical model, namely a semi- and a fully-coupled threedimensional model, and also a two-dimensional rotational invariant model, both for wurtzite GaN/AlN wurtzite and zincblende InAs/GaAs quantum dots, the latter by using an isotropic assumption [99]. A Keating valence force field model for wurtzite GaN/AlN quantum dot is also presented, derived from a lattice-mismatched isovalent semiconductor zincblende alloy valence force field model [46], which is used for some comparisons with the continuum model. We also show the impact of strain fields on electronic structure for hexagonal pyramid wurtzite GaN/AlN quantum dots. The calculation of electronic states in quantum dots has been achieved with different methods, for example the simple effective mass approximation [8], the multiband ~k · p~ theory [29], or other more atomistic theoretical approaches, like the tight-binding calculations [30], and the pseudopotential method [31]. In this thesis, ~k · p~ theory [51, 52] is used to determine the bandstructure of quantum dots. One of the first popular ~k·~p multiband calculation schemes for bulk materials is due to Luttinger and Kohn [55, 54] which later on was extended to heterostructures by an ad hoc symmetrization procedure [82]. In order to overcome this ad hoc procedure, Burt formulated the so-called exact envelope function method [32, 86] and soon after Foreman [69] used this method to derive a six-band model for the valence bands of zincblende heterostructures. In 2001 Pokatilov et al. [71] extended this to an eight-band model for the conduction and the valence bands. They studied spherical quantum dots using a spherical approximation and compared their model, based on exact envelope-function theory, against the usual symmetrized ap20

proach. The asymmetry parameter present in the Burt-Foreman formalism but not in the Luttinger-Kohn formalism is shown to lead to changes of approximately ±25 meV in the electronic bandstructures of InAs/GaAs [71]. The optoelectronic properties of InGaAs zincblende quantum-dot with varying shape and size based on ~k · p~ theory have already been studied by Schliwa et al. [97] and Veprek et al. [98]. However, so far, only a small selection of all possible transitions have been studied. Application of eight-band model based on the Burt-Foreman formalism derived in Ref. [71] to zincblende InAs/GaAs conical quantum dots, studying the impact of different size and shape and focus on the interband transitions, has shown great relevance for optoelectronic applications. In particular, I show that some of the as yet unexplored transitions are highly relevant for optical applications, as EIT. Furthermore, we investigate the effect of strain and band-mixing between the conduction band and valence bands by comparing four different ~k · p~ model. Although the previous studies do include these effects their impact on dipole moments have as yet not been investigated. This thesis is part of the QUEST project, whose main goal is the fabrication and modelling of quantum dots enabling the slowdown of light. The project brings together three groups from the Technical University of Denmark (DTU) and The University of Southern Denmark (SDU) with strong and complementary research experience [18]. Chapter 1 gives an overview of the quantum dots, explaining some of their fabrication techniques, the most relevant crystal structures, and some of their most interesting applications. I also give more details concerning the QUEST project, and its aims. In Chapter 2 I give a complete derivation of the continuum models for electromechanical calculations. Starting from a classical model for solids, it is shown how this applied to the heterostructure quantum dot, and also how the piezoelectrical effect is included. A relevant part is dedicated to two-dimensional models in cylindrical coordinates, which has been widely used for some of the numerical results. An overview of ~k · p~ theory is presented in Chapter 3. I give a description both of the general bulk theory and of the exact envelope theory as derived by Burt [32, 86], and show how it has been applied to quantum dot heterostructures. I also present namely the Hamiltonians I have implemented for the models. Chapter 4 is dedicated to optical properties: a derivation for the absorption coefficients is presented, and an explanation for how dipole moments for optical transitions have been derived from momentum matrix elements of cylindrical symmetric quantum dots. In Chapter 5 results for electromechanical fields for wurtzite GaN quantum dot embedded in AlN matrix are presented. In addiction, some results for the impact of the electromechanical fields on the electronic structures for these dots are given. 21

Chapter 6 covers results for zincblende InAs quantum dot embedded in GaAs matrix. Results for electromechanical fields and bandstructure are presented with a focus on the impact of strain fields and band mixing on optoelectronic properties.

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Chapter 1

An overview of quantum dots 1.1

From atoms to quantum dots

When Bloch introduced the concept of electronic bandstructure for an ideal crystal in 1928 [1], it appeared as something completely new in the world of physics, which was mainly dominated in that period by atomic physics, with its discrete levels of the electronic bound states. In solid state physics the electron energy is a multivalued function of momentum, energy bands with a continuous density of states divided by gaps, and a correspondent wave function completely delocalized in real space. If we could limit the motion of a carrier in a layer with a thickness comparable with the carrier de Broglie wavelength, we could observe the effect of size quantization in one dimension. We know that the de Broglie wavelength λ depends on the effective mass mef f of the carrier and on the temperature T : λ= p

h , 3mef f kT

(1.1)

where h and k are the Planck and Boltzmann constants, respectively. Since mef f can be much smaller than the free electron mass, these quantization effects can already be observed at a thickness ten to one hundred times larger than the lattice constant. The first studies of size quantization using also films of metals and semiconductors started between the late 1950s and early 1960s [2], and the effect of an increasing effective band gap due to decreasing film thickness was shown. At that time experimental studies were limited by technological restraints, but in the late 1960s with the advent of novel epitaxial techniques and metal organic chemical vapor deposition it was possible to insert coherent layers with thickness of a few lattice constants of a semiconductor of lower bandgap in a matrix with a larger bandgap, restricting carrier movement to only two dimensions (two-dimensional heterostructures: quantum well) [3]. 23

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CHAPTER 1. AN OVERVIEW OF QUANTUM DOTS

The optical researches of the quantum well and superlattices showed a step-like character of the absorption spectrum which is related to the twodimensional density of states in a quantum well (see Figure 1.1, 2D) [4], and a decrease of layer thickness leads to a shift of the steps toward higher photon energy. Also exciton effects, with zero-dimensional properties in potential fluctuations of the quantum well, were demonstrated [5]. In the late 1980s, after having understood the main properties of the quantum well, researchers started to look for a further reduction of dimensionality, first with quantum wires (see Figure 1.1, 1D) [6], with the carrier free to move only in one dimension, and finally to a complete localization of the carrier in all the three dimensions (see Figure 1.1, 0D), with a consequent breakdown of the classical bandstructure with a continuous dispersion of energy: the quantum dot.

Figure 1.1: Nature of electronic states in bulk material (3D), quantum well (2D), quantum wire (1D), and quantum dot (0D). A quantum dot has a discrete energy levels structure as we have in atomic physics, and many of its properties resemble those of an atom in a cage, and this also leads to the common definition for artificial atom, which is quite often adopted to indicate a quantum dot. A qualitative measure of the appearance of size quantization effects in the three dimensions and of energy separation of the levels is still given by the de Broglie wavelength of equation (1.1). Usually the size of quantum dot is around 10 nm, and it contains about 104 atoms. The quantum dots immediately showed interesting properties which made them good candidates for applications for use in such novel devices as single electron transistors, or quantum dot lasers. In the last few years heterostructure quantum dots have been successfully realized by means of the so-called self-organized (or self-ordering or self-assembly) effects, which occur during the growth of strained heterostructures: thermodynamic and kinetic ordering effects which can create a unique three-dimensional island within a matrix. While in other fields of science, such as biology, a self-organized system is usually a non-equilibrium state, for quantum dot fabrication it is generally an equilibrium process. Among the materials shown to give the most interesting results for optical applications we have the III-V systems InGaAs/AlGaAs, and the group

1.1. FROM ATOMS TO QUANTUM DOTS

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III nitrides-like, such as (In, Ga, Al)N. In the last two chapters of this thesis the original results for GaN/AlN wurtzite and InAs/GaAs zincblende quantum dots will be presented, respectively. The wurtzite crystal structure, as illustrated in Figure 1.2, is named after the mineral wurtzite, and it is a crystal structure for various binary compounds and an example of hexagonal crystal system. In most of these compounds, the bulk crystal does not normally take the wurtzite form, but it can be favored in some nanocrystal forms of the materials.

Figure 1.2: A wurtzite structure [7]. The zincblende structure, named after the mineral zincblende (sphalerite) is showed in Figure 1.3. As in a rock-salt structure, the two atom types form two interpenetrating face-centered cubic lattices, but with a difference in how the two lattices are positioned relative one another. The arrangement of atoms is the same as a diamond cubic, but with alternating types of atoms at the different lattice sites. A quantum dot which could be useful for a device operating at room temperature should have a deep localizing potential, a small size, a high uniformity and coherence without defects like dislocations. Concerning the size, the lower limit is given by the condition that at least one energy level for an electron or an hole or both is given. If we consider

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Figure 1.3: A zincblende structure [7]. for the sake of simplicity a spherical quantum dot, the critical diameter Dmin depends strongly on the band offset of the corresponding bands of the materials. For example, with a confinement potential defined by the conduction band offset given by ∆Ec ≈ 0.3 eV, we have for the minimum diameter [8]: Dmin = p

π¯h = 4 nm, 2m∗e ∆Ec

(1.2)

where m∗e is the effective electron mass and ¯h is the Planck constant divided by 2π. In such a quantum dot with this size the separation between the electron and hole levels is too small, since even at finite temperatures the thermal evaporation of the carriers results in their depletion. In InAs/GaAs quantum dots the conduction band offset is much larger, while the electron effective mass is smaller, and the critical diameter is around 3-5 nm, depending on non parabolicity effects. On the other hand, we can also consider a limit for the maximum size of a quantum dot, since for example a thermal population of higher-lying energy levels is not efficient for many optical devices, such as lasers, for example. If we want to limit the the thermal population of higher-lying levels to 5%,

1.2. QUANTUM DOT APPLICATIONS we can write the condition for the dot as [8] : 1 kT ≤ (E2QD − E1QD ), 3

27

(1.3)

where E1QD and E2QD are the energies of the first and second level of the quantum dot, respectively. This limit is a function of the operating temperature, and at room temperature equation (1.3) gives a maximum size between 12 and 20 nm, depending on the materials. The quantization for the holes is more complicated, especially for InAs/GaAs system, because of the large electron/hole mass ratio, while in group III nitride, where we have more similar masses, a strong localization is easier to achieve.

1.2

Quantum dot applications

We have seen that the reduction of the dimension leads to the narrowing of carrier density of the state (DOS) (see Figure 1.1) and in such a way provides a higher number of carriers on the energy level corresponding to the edge of dimensional quantization band. The electron spectrum of an ideal quantum dot is a set of discreet levels which depends on the material system, shape and size of the quantum dot. The density of states of an ideal quantum dot is a delta function. These properties could significantly improve some characteristics of laser devices due to increasing of optical confined Γ factor and of the optical gain [10]. Even if in a real grown structure there is some size and shape distribution of quantum dots which leads to inhomogeneous broadening of spectra, an optimized growth conditions can minimize such an effect. Moreover, changing growth regimes it is possible to get quantum dots with different size, shape and density depending of the requirements. Self-assembled quantum dots could be fabricated during the epitaxial growth [molecular beam epitaxy (MBE) or metallorganic vapor phase epitaxy (MOVPE)] under certain growth conditions. Quantum dots in material systems InAs/GaAs and InAs/InP are fabricated in the so-called StranskiKrastanow regime (see Figure 1.4).

Figure 1.4: The so-called Stranski-Krastanow process for the fabrication of quantum dots.

Basically, the InAs, which is significantly mismatched by the lattice parameter to the substrate material, is depositing layer-by-layer on the substrate (Figure 1.4 b). The strain in the structure leads to an increase in

28


the elastic energy of the system, so when the thickness of planar layer of InAs exceeds the critical value, which in case of InAs on top of GaAs is 1.7 monolayer, part of this material forms an array of “identical” islands with diamond-like shape on top of planar wetting layer (Figure 1.4 c). After formation quantum dots can be overgrown by planar layer(s) of the material with a lattice parameter close to the substrate one. In the first work on InAs/GaAs the possibility for the formation of three-dimensional, coherently strained, i.e., dislocation-free, islands was established [9]. In Figure 1.5 there is a schematic drawing of the geometry of InAs/GaAs quantum dots [10].

Figure 1.5: Schematic geometry of InAs/GaAs quantum dots (from Ref. [10]). Typical sizes of this kind of quantum dots are 10-60 nm in diameter and about 5-10 nm in height. Figure 1.6 shows TEM microphotographs of InAs quantum dots on InP substrate taken from Ref. [11]. These material systems are interesting from with regards the fabrication of lasers devices working in 1,3-1,55 micron wavelength range. Fabrication of active region based on the arrays of quantum dots due to their density of states allows superior laser characteristics such as ultra-low threshold current densities [12, 13], ultrahigh temperature stability [14], very high dif-

1.3. QUEST PROJECT

29

Figure 1.6: TEM microphotographs of InAs quantum dot on InP substrate, grown by MOVPE [11]. (a) Close view of a single quantum dot, which presents a slight elongation along the [1¯ 10] direction, and the edges are oriented along the [3¯10] and [1¯30] directions. (b) TEM cross view of a quantum dot. The average height is 2.1 nm, and the thickness of the wetting layer is 0.8 nm. The dark contrast around the quantum dot corresponds to the strain field surrounding the nanostructure (from Ref. [11]). ferential efficiency [15], small α-factor (temperature independent linewidth enhancement factor) and chirp (shift of the lasing wavelength with current) [16]. Besides, using GaN quantum dots on Si 111 substrate an intense visible luminescence at room temperature has been obtained, and it was shown that the emission energy can be continuously tuned from blue to orange by simply controlling the quantum dot size (see Figure 1.7). Moreover, the mixing of properly chosen quantum dot sizes leads to white light emission [88]. This thesis is part of a bigger project also related to fabrication and characterization of quantum dots: the QUEST project [18], whose main goal is the fabrication and modelling of quantum dots enabling the slowdown of light. The QUEST project is presented in details in the following section.

1.3

QUEST project

Quest is a research project exploring the use of semiconductor quantum dot technology for realizing practical slow-light devices and integrated op-

30


Figure 1.7: Photographs of the light emitted at room-temperature from GaN/AlN quantum dots on Si 111 excited by a 10 mW unfocused HeCd laser 0.3 W/cm2. (From Ref. [88]). tical amplifiers. Such devices find important applications within information, communication and sensor technology and the project targets practical demonstrations within these areas, leading to possibilities of commercial exploitation. From a wider perspective, the proposed project contributes to the ongoing evolution of the information society. The project brings together three groups from the Technical University of Denmark (DTU) and the University of Southern Denmark (SDU) with strong and complementary research experience.

1.3.1

Slow light

When Lene Hau and others published their article [19], claiming that the speed of light could be reduced to “cyclist speed”, they attracted a huge interest in the field of slow-light propagation. This was not only due to the vast possibilities opened up by this new fundamental discovery but specifically because of its potential applications in new opto-electronic devices. For example it could allow for the temporary storing of light [20], and an adjustable delay or phase-shifter could be used for optically fed phased-array antennas as well as optical clock distribution on high-speed electronic chips. A reduced speed of light also implies greater localised intensity, an effect which could be applied to sensory and measurement technology to increase

1.3. QUEST PROJECT

31

its sensitivity towards an optical beam and/or reduce the size of the device. In the experimental demonstrations in Ref. [19] an ultra-cold gas was used, but if slow-light propagation was used for a practical optical device, it would be possible to work with a solid-state device operating at room temperature. The semiconductor technology has the big advantage of being well-established and cheap in terms of production. To obtain a slow-down effect we need a medium with a discrete resonance as understood in the physics of the atom. As mentioned previously semiconductor quantum dots are often considered as artificial atoms, so their possible use in this field was immediately taken recognized. In addition, the discrete resonances can be controlled by changing the dot size and engineering the band gap composition. Only two electrons are required in order to fully populate each of the discrete levels. These features allow for low current densities for lasers and a high-power and high-speed operation for amplifiers, and this explains the great theoretical and experimental interest in strain-relaxed quantum dots (see Figure 1.8).

Figure 1.8: A strain-relaxed quantum dot. There are several physical effects which can lead to a slow-down of light. Usually a narrow spectral resonance is induced in the medium, leading to a rapid variation of the refractive index and subsequent modification of the group velocity of a propagating light beam. In the experiments of Hau et al. [19], a control beam was injected in an atomic gas of Sodium atoms along with the signal to be slowed-down, and it coherently coupled discrete levels in the medium. As a consequence of this coupling, a strongly absorbing line at the signal frequency was split, making the medium opaque in a narrow spectral range, inducing at the same time a rapid variation of the refractive index. It was proved that by cooling the atomic gas to the nanokelvin range, the speed of light was reduced to only 17 m/s and then brought to a complete stop [21]. In order to obtain the EIT it is fundamental to have coherence between the different energy levels, thus all the diffusive factors which could lead to a rapid dephasing are strongly disadvantageous. It is important to underline that even though the use of quantum dots to realize EIT has been theoretically postulated, it has not been experimentally

32


demonstrated yet [20]. Figure 1.9 shows one of the possible schemes, where a signal beam ωs involves a transition from the valence to the conduction band. As should be clear the control beam ωp1 couples discrete levels within the conduction band.

Figure 1.9: Use of quantum dots in EIT application. EIT is not the only phenomenon which can generate slow-down effects, for example mention can be made of the coherent population oscillations (CPO)[22], which is basically a wave mixing effect, and it is usually easier to achieve than EIT. By using CPO Bigelow et al. succeeded in reducing the speed of light to 57 m/s in a ruby crystal at room temperature [22]. Unfortunately these experiments were performed at very low bandwidths, typically less than a MHz, while the most interesting applications are in the GHz range. The CPO-induced light slow-down has been demonstrated in semiconductor quantum wells, while in Ref. [23] a speed of 200 m/s was achieved in a narrow bandwidth in Ref. [23]. Research into the use of quantum dot semiconductors heterostructures to improve lasers and optical amplifiers is making significant progress. It is notable that quantum dot lasers have the record for low threshold current density [24], while it was recently reported that the realization of a quantum dot optical amplifier could improve the standard semiconductor optical amplifiers in terms of output power and operation bandwidth [25].

1.4

Modelling quantum dots

Knowledge of the geometric structures and of chemical composition of a quantum dot is a prerequisite for a development of a mathematical model of its elastic, electronic and finally optical properties, which is the main goal of this thesis.

1.4. MODELLING QUANTUM DOTS

33

Strain caused by the differences of the lattice constants of the dot and substrate (matrix) material is decisive for both the self-organization mechanism and electro-optical properties, and for the impact of strain on bandstructure. The strain distribution in solids is usually treated with the continuum mechanical model [26], or with the valence force field model [27], or using density functional techniques [28]. In Chapter 2 a complete account is given of how the continuum model has been derived, which has been widely used for the numerical results, as well as an explanation of the most important features of the valence force fields model, which has been applied in order to make some comparisons with the continuum model. For the calculation of electronic states in quantum dots several schemes have been used with different levels of sophistication, can be made mention for example of the simple effective mass approximation [8], the multiband ~k · ~p theory [29], or other more atomistic theoretical approaches, like the tight-binding calculations [30] and the pseudopotential method [31]. Since for the numerical results ~k · ~ p theory has been used, in Chapter 3 I present a brief exposition of this theory, from the simple bulk case to the more complex exact envelope function theory given by Burt [86, 32], and applied by Foremann to heterostructures [69], showing explicitly the different Hamiltonians used for the different quantum dots I have studied. The theory of some optical properties is presented in Chapter 4, where some important concepts such as the absorption, the interband and intraband optical transitions are introduced. In addition, a number of interesting theoretical reasons are outlined to explain why it has been decided to use the momentum matrix elements as fundamental quantities in order to calculate all the other optical quantities of a quantum dot. These three chapters will give the theoretical basis to introduce and understand the numerical results for wurtzite GAN/AlN quantum dot in Chapter 5, and for zincblende InAs/GaAs quantum dot in Chapter 6.

34


Chapter 2

Electromechanical fields 2.1 2.1.1

Fundamental equations The strain tensor

Under the action of applied forces, solid bodies exhibit a deformation, changing their shape and volume [26]. If the position of any points of the body before the deformation is given by the position vector ~r (with components x1 = x, x2 = y, x3 = z), after the deformation we have a new value for the position ~r ′ (with components x′i ). So we can define the displacement vector as follows: ~u = ~r ′ − ~r,

(2.1)

with components ui = x′i − xi . The distance between q two points of the p ′ ′ ′ ′ 2 2 2 body is given by dl = dx1 + dx2 + dx3 and dl = dx12 + dx22 + dx32 before and after the deformation, respectively. Using dx′i = dxi − dui and substituting dui = (∂ui /∂xk )dxk we can write using the general summation rule: ′

dl 2 = dl2 + 2

∂ui ∂ui ∂ui dxi dxk + dxk dxl . ∂xk ∂xk ∂xl

(2.2)

Rearranging the second and third term we can finally write: ′

dl 2 = dl2 + 2εik dxi dxk , where we have defined a strain tensor: 1 ∂ui ∂uk ∂ul ∂ul εik = + + . 2 ∂xk ∂xi ∂xi ∂xk

(2.3)

(2.4)

In almost all general cases (for exceptions see Ref. [26]) if a body is subjected to a small deformation all the components of the strain tensor are 35

36

CHAPTER 2. ELECTROMECHANICAL FIELDS

small, so neglecting the last term in (2.4) as being of the second order of smallness we can write: ∂uk 1 ∂ui + . (2.5) εik = 2 ∂xk ∂xi

2.1.2

The stress tensor

For a body in mechanical equilibrium the resultant of all the forces on every single portion of the body is equal to zero. In case of deformation, some internal molecular forces, defined as internal stresses, tend to return the body to equilibrium. These forces, in absence of macroscopic electric fields, are near-action forces, which, acting on the considered portion of the body by the surrounding parts, effect only the surface of the portion itself. If we want to express these forces, we have to consider the sum of all the R ~ forces on all the volume elements dV of that portion of bodies f dV , where f~ is the force per unit volume. According to Newton’s third law, they are equal to the sum of the forces exerted on the portion by the surrounding parts, i.e., from what we said above, equal to an integral over the surface of the portion. From a theorem of vector analysis we know that an integral of a vector over an arbitrary volume can be transformed into an integral over the surface of the volume if the vector is the divergence of a tensor of rank two. This leads to: I Z Z ∂σik ~ dV = σik dAk , (2.6) fi dV = ∂xk with ∂σik . f~i = ∂xk

(2.7)

The tensor σik is called the stress tensor, and σik dAk is the ith component of the force on the surface elements dAk . It is also possible to show that the stress tensor is symmetric [26]: σik = σki .

(2.8)

At the mechanical equilibrium the internal stresses in every volume element must be balanced, so that f~i = 0. Thus the equilibrium equations for a deformed body can be written as: ∂σik = 0. ∂xk

(2.9)

2.1. FUNDAMENTAL EQUATIONS

2.1.3

37

Free energy

We want to derive now an expression of the free energy as a function of the strain tensor. ik Multiplying the force f~i = ∂σ ∂xk by the displacement δui and integrating over the volume V the work δW done by the internal stress per unit volume can be calculated: Z Z ∂σik δui dV, (2.10) δW dV = ∂xk integrating by parts and considering an infinite medium not deformed at infinity we have [26]: δW = σik δεik ,

(2.11)

so the expression for the internal energy dU at the thermodynamic equilibrium for a reversible process at temperature T is: dU = T dS − dW = T dS + σik dεik ,

(2.12)

where S is the entropy of the system. Introducing the Helmholtz free energy of the body F = U − T S we obtain: dF = −SdT + σik dεik , which for a constant temperature leads to: ∂F σik = . ∂εik

(2.13)

(2.14)

The idea is to expand F in powers of εik with an assumption of small deformations, and, if an isotropic deformed body at a constant temperature is considered, its undeformed state is a state in absence of any external forces, i.e., σik = 0. Because of (2.14) this implies that there are no linear terms in the expansion of F , which in terms of second order can be written as: 1 F = F0 + λεii 2 + µεik 2 . 2

(2.15)

The constant term F0 is the free energy of the undeformed body, and it will be omitted in the following equations. The quantities λ and µ are called Lamè coefficients. The expression of the free energy for a crystal under a compression at constant temperature is , as is the case for isotropic bodies, still a quadratic function of the strain tensor, but with a larger number of coefficients. The general form for a deformed crystal is given by [26]: 1 F = Ciklm εik εlm , 2

(2.16)

38


where Ciklm is the rank four elastic modulus tensor, with the following symmetric properties: Ciklm = Ckilm = Cikml = Clmik .

(2.17)

With (2.14) the stress tensor for a crystal in terms of strain tensor is given by: ∂F = Ciklm εlm . (2.18) σik = ∂εik The elastic modulus tensor is usually expressed also as Cαβ , with α and β taking values from 1 to 6 in correspondence with xx, yy, zz, yz, zx, xy.

2.1.4

Constitutive relations

For a dielectric material there is an additional contribution given by the electric field to equation (2.12): ~ · dD ~ = Ei dDi , dU elec. = E

(2.19)

~ and D ~ are the electric field and electric displacement vectors, rewhere E spectively. So the total internal energy reads: dU = T dS + σik dεik + Ei dDi ,

(2.20)

and it is also possible to define in a similar way another thermodynamic potential, the enthalpy: dH = T dS − εik dσik − Di dEi .

(2.21)

For isentropic processes by applying the chain rule to the derivates of U and H it is possible to express some thermodynamic identities: ∂σkl ∂Di (piezoelectric coefficient) (2.22) =− eikl = ∂εkl ∂Ei ǫik = ˆ

Ciklm =

∂Di ∂Ek

(permittivity tensor)

(2.23)

∂σik ∂εlm

(elastic modulus tensor).

(2.24)

For a small isentropic variation we can therefore write: ∂Di ∂Di dEk + dεkl = ǫîk dEk + eikl dεkl dDi = ∂Ek ∂εkl

2.2. STRAIN FIELD IN A QUANTUM DOT 0000000 1111111 0000000 1111111 1111111 0000000 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111

39

ε (0) ε (u) 0000 1111 1111 0000 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111

000 111 111 000 000 111 000 111

Figure 2.1: Tensor of local intrinsic strain and local strain tensor.

dσkl =

∂σkl ∂σkl dEi + dεmn = −eikl dEi + Cklmn dεmn , ∂Ei ∂εmn

(2.25)

so the constitutive relations for the relative finite quantities can be set down as: Di = ǫîk Ek + eikl εkl σkl = −eikl Ei + Cklmn εmn ,

2.2

(2.26)

Strain field in a quantum dot

The lattice constants in semiconductor heterostructures vary with coordinates, and the lattice mismatch between the quantum dot structure and the matrix material in which it is embedded generates an intrinsic local strain field different from zero [40]. The free elastic energy can be written as: Z 1 d~r Ciklm (~r)εik (~r)εlm (~r), F = (2.27) 2 V where V is the total volume of the system. To take into account lattice mismatch (see Figure 2.1), the strain tensor is represented as (i, j = x, y, z): (u)

(0)

εij = εij + εij , (0)

(2.28) (u)

where εij is the tensor of local intrinsic strain and εij is the local strain tensor dependent on positions, given by (2.5).

2.2.1

Zincblende quantum dot

The elastic energy density for a crystal with zincblende symmetry reads [40]: 1 F = [C11 (ε2xx + ε2yy + ε2zz ) + 2C12 (εxx εyy + εxx εzz + εyy εzz ) + 4C44 (ε2xy + ε2xz + ε2yz )], 2 (2.29)

40


since the only linearly independent elastic constants for a zincblende structure are given by: C1111 ≡ C11

C1122 ≡ C12

C2323 ≡ C44 .

(2.30)

The intrinsic strain tensor is given by: (0)

εij = δij a, a

(2.31)

−a

QD in the dot and zero otherwise. Here amatrix and aQD with a = matrix amatrix are the lattice constants of the matrix and the quantum dot, respectively.

2.2.2

Wurtzite quantum dot

The elastic energy density for a crystal with wurtzite symmetry is given by [40]: 1 F = [C11 (ε2xx + ε2yy ) + C33 ε2zz + 2C12 εxx εyy + 2C13 εzz (εxx + εyy ) 2 + 4C44 (ε2xz + ε2yz ) + 2(C11 − C12 )ε2xy ], (2.32) because in a crystal with wurtzite symmetry the linearly independent elastic constants are: C1111 ≡ C11 ;

C3333 ≡ C33 ;

C1122 ≡ C12 ;

C1133 ≡ C13 ;

C2323 ≡ C44 ;

C2121 ≡ (C11 − C12 )/2.

(2.33)

The tensor of local intrinsic strain is given by: (0)

εij = (δij − δiz δjz )a + δiz δjz c, a

−a

c

(2.34)

−c

QD QD and c = matrix the lattice mismatch in the hexagwith a = matrix amatrix cmatrix onal plane and the c axis, respectively. The parameters amatrix , cmatrix , and aQD , cQD are the lattice constants of the matrix and the quantum dot, respectively. The c axis is the axis of the sixfold rotational symmetry of the wurtzite material, taken coincident with the z axis.

2.3

Piezoelectric field in a quantum dot

Under an applied stress some semiconductors develop an electric moment with magnitude proportional to the stress [40]. The induced polarization is related to the strain tensor by the piezoelectric coefficients (2.22): Pi (~r) = eilm (~r)εlm (~r),

(2.35)

2.3. PIEZOELECTRIC FIELD IN A QUANTUM DOT

41

the index ilm run over the spatial coordinates, namely: 

 e111 e122 e133 e123 e113 e112  e211 e222 e233 e223 e213 e212  . e311 e322 e333 e323 e313 e312

(2.36)

For some wurtzite nitrides we have also to take into account a spontaneous polarization Psp , whose polarity depends on the last anion or cation at the surface. The total polarization generates a piezoelectric field EP , which in the absence of external charges can be evaluated by solving the MaxwellPoisson equation: ~ · D(~ ~ r) = 0, ∇

(2.37)

~ is given by the first of equations (2.26). where the displacement vector D

2.3.1


Converting from tensor notation to matrix notation by: eilm =

eik , 1 e 2 ik ,

k = 1, 2, 3; , k = 4, 5, 6;

(2.38)

the independent piezoelectric coefficients for a zincblende structure are given by: 

 0 0 0 e14 0 0  0 0 0 0 e14 0  , 0 0 0 0 0 e14

(2.39)

so the polarization in terms of components can be expressed: Px = e14 εyz , Py = e14 εxz , Pz = e14 εxy ,

(2.40)

while for the permittivity tensor we have:

ǫˆZb



 ǫˆ 0 0 =  0 ǫˆ 0  , 0 0 ǫˆ

where ˆǫ is a constant value material dependent.

(2.41)

42


2.3.2


In a crystal with wurtzite symmetry the only nonzero component for the spontaneous polarization is along the z-axis (the axis of the sixfold rotational symmetry, as already mentioned), while the independent piezoelectric coefficients are:   0 0 0 0 e15 0  0 0 0 e15 0 0  , (2.42) e31 e31 e33 0 0 0

so nominally the components of the polarization are:

Px = e15 εxz , Py = e15 εyz , Pz = e31 (εxx + εyy ) + e33 εzz + Psp , and the permittivity tensor is given by:   ǫˆ11 0 0 ǫW z =  0 ǫˆ11 0  , ˆ 0 0 ǫˆ33

(2.43)

(2.44)

with ǫˆ11 and ǫˆ33 constant values material dependent.

2.4

Governing equations for electromechanical fields

The governing equations for the electromechanical fields in an heterostructures quantum dot are given by the above mentioned equilibrium equation (2.9) - also known as Navier’s static equation - and Maxwell-Poisson equations (2.37) which are repeated here: ∂σij =0 ∂xj ∇ · D = 0,

(2.45)

From equations (2.45) it is possible to obtain a set of four coupled equations in the electromechanical fields. The expression for the the stress tensor and the electric displacement are given by the constitutive relations (2.26), which are expressed in a more convenient way here: σik = Ciklm εlm + eikn

∂V , ∂xi

Di = −ǫin

∂V + eilm εlm + Psp,i , (2.46) ∂xi

where V is the electric potential. In the following two sections we derive the governing equations for zincblende and a wurtzite heterostructure quantum dots, equations which will be used in Chapter 5 and Chapter 6 for the three-dimensional models.

2.4. GOVERNING EQUATIONS FOR ELECTROMECHANICAL FIELDS43

2.4.1


Equations (2.45) using (2.5) for a zincblende structure read nominally: ∂uy ∂uy ∂ux ∂ ∂ux ∂ ∂ux ∂ ∂ ∂ C11 + C44 + C44 + C12 + C44 ∂x ∂x ∂y ∂y ∂z ∂z ∂x ∂y ∂y ∂x ∂ ∂uz ∂ ∂uz ∂ ∂V ∂ ∂V ∂ + C12 + C44 + e14 + e14 + C11 + 2C12 a = 0 ∂x ∂z ∂z ∂x ∂y ∂z ∂z ∂y ∂x ∂uy ∂uy ∂uy ∂ ∂ux ∂ ∂ux ∂ ∂ ∂ C44 + C12 + C44 + C11 + C44 ∂x ∂y ∂y ∂x ∂x ∂x ∂y ∂y ∂z ∂z ∂ ∂uz ∂ ∂uz ∂ ∂V ∂ ∂V ∂ + C12 + C44 + e14 + e14 + C11 + 2C12 a = 0 ∂y ∂z ∂z ∂y ∂x ∂z ∂z ∂x ∂y ∂uy ∂uy ∂ux ∂ ∂ux ∂ ∂ ∂ ∂uz ∂ C44 + C12 + C44 + C12 + C44 ∂x ∂z ∂z ∂x ∂y ∂z ∂z ∂y ∂x ∂x ∂ ∂uz ∂ ∂uz ∂ ∂V ∂ ∂V ∂ + C44 + C11 + e14 + e14 + C11 + 2C12 a = 0 ∂y ∂y ∂z ∂z ∂x ∂y ∂y ∂x ∂z ∂ux ∂ ∂ux ∂ ∂uy ∂ ∂uy ∂ ∂uz ∂ e14 + e14 + e14 + e14 + e14 ∂y ∂z ∂z ∂y ∂x ∂z ∂z ∂x ∂x ∂y ∂uz ∂ ∂V ∂ ∂V ∂ ∂V ∂ e14 + εˆ + εˆ + εˆ =0 (2.47) + ∂y ∂x ∂x ∂x ∂y ∂y ∂z ∂z

2.4.2


The explicit expression for equations (2.45) using (2.5) is given by:

∂uy ∂ ∂ux ∂ (C11 − C12 ) ∂ux ∂ ∂ux ∂ C11 + + C44 + C12 ∂x ∂x ∂y 2 ∂y ∂z ∂z ∂x ∂y ∂ (C11 − C12 ) ∂uy ∂ ∂uz ∂ ∂uz + + C13 + C44 ∂y 2 ∂x ∂x ∂z ∂z ∂x ∂V ∂ ∂V ∂ ∂ e31 + e15 + C11 + C12 a + C13 c = 0 + ∂x ∂z ∂z ∂x ∂x ∂ (C11 − C12 ) ∂ux ∂ ∂ux ∂ (C11 − C12 ) ∂uy + C12 + ∂x 2 ∂y ∂y ∂x ∂x 2 ∂x ∂ ∂uy ∂ ∂uy ∂ ∂uz ∂ ∂uz + C11 + C44 + C13 + C44 ∂y ∂y ∂z ∂z ∂y ∂z ∂z ∂y ∂V ∂ ∂V ∂ ∂ e31 + e15 + C11 + C12 a + C13 c = 0 + ∂y ∂z ∂z ∂y ∂y

44

CHAPTER 2. ELECTROMECHANICAL FIELDS ∂uy ∂uy ∂ux ∂ ∂ux ∂ ∂ ∂ C44 + C13 + C44 + C13 ∂x ∂z ∂z ∂x ∂y ∂z ∂z ∂y ∂uz ∂ ∂uz ∂ ∂uz ∂ ∂V ∂ C44 + C44 + C33 + e15 + ∂x ∂x ∂y ∂y ∂z ∂z ∂x ∂x ∂ ∂V ∂ ∂V ∂ + e15 + e33 2C13 a + C33 c = 0 ∂y ∂y ∂z ∂z ∂z ∂uy ∂uy ∂ ∂ux ∂ ∂ux ∂ ∂ e15 + e31 + e15 + e31 ∂x ∂z ∂z ∂x ∂y ∂z ∂z ∂y ∂ ∂uz ∂ ∂uz ∂ ∂uz ∂ ∂V + e15 + e15 + e33 + εˆ11 ∂x ∂x ∂y ∂y ∂z ∂z ∂x ∂x ∂Psp ∂V ∂ ∂V ∂ =0 (2.48) εˆ11 + εˆ33 + + ∂y ∂y ∂z ∂z ∂z

2.5

Electromechanical fields in cylindrical coordinates

Due to computational reasons, it is useful to derive an expression for the strain tensors and the free elastic energy in cylindrical coordinates:   x = ̺cosϕ y = ̺sinϕ ,  z=z

(2.49)

so first we consider the general transformation rules for a tensor w in a new system of coordinates [41]: ′ wij = αik αjm wkm ′ wijk ′ wijkl

= αir αjs αkt wrst

(Second rank tensor),

(2.50)

(Third rank tensor),

= αim αjn αkp αlq wmnpq

(2.51)

(Fourth rank tensor),

(2.52)

where α are the direction cosines of the axes of the new system with respect to the old one. Using these rules the following relations between the strain tensor and the displacement vectors in a system of cylindrical coordinates are derived: ∂u̺ 1 ∂uϕ u̺ ∂uz , εϕϕ = + , εzz = , ∂̺ ̺ ∂ϕ ̺ ∂z 1 1 ∂u̺ ∂uϕ uϕ 1 ∂u̺ ∂uz = + − + , ε̺z = , 2 ̺ ∂ϕ ∂̺ ̺ 2 ∂z ∂̺

ε̺̺ = ε̺ϕ

εϕz

1 ∂uz 1 ∂uϕ + = , 2 ∂z ̺ ∂ϕ (2.53)

and also some useful identities between the strain tensor in cartesian and

2.5. ELECTROMECHANICAL FIELDS IN CYLINDRICAL COORDINATES45 cylindrical coordinates are established: εxx + εyy = ε̺̺ + εϕϕ , εzz = εzz , εxz 2 + εyz 2 = ε̺z 2 + εϕz 2 , εxx 2 + εyy 2 + 2εxy 2 = ε̺̺ 2 + εϕϕ 2 + 2ε̺ϕ 2 , εxx εyy − εxy 2 = ε̺̺ εϕϕ − ε̺ϕ 2 ,

(2.54)

which can be used to write an expression for the free elastic energy both for a zincblende structure: 1 F = C11 (ε2̺̺ + ε2ϕϕ + ε2zz ) + C12 (ε̺̺ εϕϕ + εϕϕ εzz + ε̺̺ εzz ) + 2C44 (ε2̺ϕ + ε2ϕz + ε2̺z ), 2 (2.55) and for a wurtzite structure: 1 1 C11 (ε2̺̺ + ε2ϕϕ ) + C33 ε2zz + C12 ε̺̺ εϕϕ + C13 (εϕϕ εzz + ε̺̺ εzz ) 2 2 (2.56) + 2C44 (ε2ϕz + ε2̺z ) + (C11 − C12 )ε2̺ϕ .

F =

2.5.1

Governing equations in cylindrical coordinates for a zincblende quantum dot

Zincblende materials are not axisymmetric, so in order to reduce the problem to a two dimensional model in cylindrical coordinates, it is necessary to make an isotropic assumption. This entails disregarding the piezoelectricity in this model. To arrive at this assumption, the elastic tensors are written in a new system of complex coordinates: ξ = x + iy , (2.57) η = x − iy and because of the (2.17), we have in this new system: Cξηξη = Cηξξη = Cξηηξ = Cηξηξ ,

(2.58)

Cξξηη = Cηηξξ .

(2.59)

and

Using the properties of transformation (2.52) for a fourth order tensor I get:   Cxxxx = 2Cξξξξ + 4Cξηξη + 2Cξξηη C = −2Cξξξξ + 4Cξηξη − 2Cξξηη . (2.60)  xxyy Cxyxy = −2Cξξξξ + 2Cξξηη

46


To impose an isotropic cylindrical symmetry a rotation to the coordinates is applied: ′ ξ = ξeiϕ , (2.61) η ′ = ηe−iϕ with the constrain that the elastic tensor must be independent on this rotation, which leads to:   Cξ ′ ξ ′ ξ ′ ξ ′ = ei4ϕ Cξξξξ C ′ ′ ′ ′ = Cξηξη . (2.62)  ξηξη Cξ ′ ξ ′ η′ η′ = Cξξηη The constrain is satisfied only if Cξξξξ = 0, so equations (2.60) become:

which gives:

  Cxxxx = 4Cξηξη + 2Cξξηη C = 4Cξηξη − 2Cξξηη ,  xxyy Cxyxy = 2Cξξηη Cxxxx − Cxxyy = 2Cxyxy ,

(2.63)

(2.64)

which using the compact notation for the elastic tensors can be written as: 2C44 + C12 = C11 ,

(2.65)

which is the cylindrical isotropic assumption looked for. The constitutive relations for a zincblende quantum dot in cylindrical coordinates can be derived by applying this assumption: σ̺̺ = C12 (ε̺̺ + εϕϕ + εzz + 3a) + 2C44 (ε̺̺ + a) σϕϕ = C12 (ε̺̺ + εϕϕ + εzz + 3a) + 2C44 (εϕϕ + a) σzz = C12 (ε̺̺ + εϕϕ + εzz + 3a) + 2C44 (εzz + a) σ̺ϕ = 4C44 ε̺ϕ σϕz = 4C44 εϕz σ̺z = 4C44 ε̺z ,

(2.66)

Using the isotropic assumption (2.65) one can derive a set of rotational invariant governing equations for the strain fields for a zincblende quantum dot in cylindrical coordinates, and to separate the problem into a (̺, z) part and a ϕ part. With adequate boundary conditions, which will be specified in section (6.1) together with the presentation of the numerical results, it is possible to remove the angular dependence. In terms of displacement this leads to

2.5. ELECTROMECHANICAL FIELDS IN CYLINDRICAL COORDINATES47 uϕ = 0 and therefore we write here just the equations ̺ - and z -dependent used in the two-dimensional models: ∂u̺ ∂ ∂ ∂uz ∂ ∂uz ∂ ∂uz (C12 + 2C44 ) + C44 + C12 + C44 ∂̺ ∂̺ ∂z ∂z ∂̺ ∂z ∂z ∂̺ 2C44 ∂u̺ ∂ 2C44 ∂ C12 u̺ − 2 u̺ + + (3C12 + 2C44 )a = 0 + ∂̺ ̺ ̺ ̺ ∂̺ ∂̺ ∂u̺ ∂u̺ ∂ ∂ ∂ ∂uz ∂ ∂uz C44 + C12 + C44 + (C12 + 2C44 ) ∂̺ ∂z ∂z ∂̺ ∂̺ ∂̺ ∂z ∂z ∂ C12 C44 ∂u̺ C44 ∂uz ∂ + u̺ + ++ + (3C12 + 2C44 )a = 0. ∂z ̺ ̺ ∂z ̺ ∂̺ ∂z (2.67)

2.5.2

Governing equations in cylindrical coordinates for a wurtzite quantum dot

In cylindrical coordinates the constitutive relations (2.46) for a wurtzite quantum dot take the form: ∂V , ∂z ∂V σϕϕ = C11 εϕϕ + C12 ε̺̺ + C13 εzz + (C11 + C12 )a + C33 c + e31 , ∂z ∂V , σzz = C33 εzz + C13 (ε̺̺ + εϕϕ ) + 2C13 a + C33 c + e33 ∂z σ̺ϕ = (C11 − C12 )εrϕ , 1 ∂V , σϕz = 2C44 εϕz + e15 ̺ ∂ϕ ∂V , σ̺z = 2C44 ε̺z + e15 ∂̺ ∂V D̺ = 2e15 ε̺z − εˆ11 , ∂̺ 1 ∂V Dϕ = 2e15 εϕz − εˆ11 , ̺ ∂ϕ ∂V + Psp , (2.68) Dz = e31 (ε̺̺ + εϕϕ ) + e33 εzz − εˆ33 ∂z σ̺̺ = C11 ε̺̺ + C12 εϕϕ + C13 εzz + (C11 + C12 )a + C33 c + e31

and inserting these relations in equations (2.45) a set of coupled equilibrium equations for the strain and the electrical fields in cylindrical coordinates

48


are obtained: ∂σ̺̺ ∂̺ ∂σ̺ϕ ∂̺ ∂σ̺z ∂̺ ∂D̺ ∂̺

1 ∂σϕ̺ ∂σz̺ 1 + + + (σ̺̺ − σϕϕ ) = 0, ̺ ∂ϕ ∂z ̺ 1 ∂σϕϕ ∂σzϕ 2 + + + σ̺ϕ = 0, ̺ ∂ϕ ∂z ̺ 1 ∂σϕz 1 ∂σzz + + + σ̺z = 0, ̺ ∂ϕ ∂z ̺ 1 ∂Dz + = 0. D̺ + ̺ ∂z

(2.69)

These equations are invariant with respect to rotations around the z axis (in spite of the lack of axisymmetry of the underlying wurtzite lattice, refer to detailed discussions in Ref. [42]), hence solutions can be separated into a (̺, z) part and a ϕ part. If these equations are applied to a cylindrical symmetric wurtzite quantum dot, with adequate boundary conditions (see section (5.1.1) where they are introduced with the numerical results), the axisymmetry of the system (equations and geometry) leads to the absence of angular dependence, therefore uϕ = 0. The remaining equations can be expressed in the following instructive form: LU = f,

(2.70)

where U ≡ (ur , uz , V ), L is a second-order differential operator given by: 

  L= 

1 ∂ ̺ ∂̺ C12

∂ 1 + C11 ∂̺ ̺

∂ ∂ ∂̺ C11 ∂̺

+

∂ ∂ ∂z C44 ∂z

+

∂ ∂ ∂̺ C44 ∂z

+

∂ ∂ ∂z C13 ∂̺

∂ + ∂z C13 ̺1 + C44 ̺1 ∂z

∂ ∂ ∂̺ e15 ∂z

∂ + e15 ̺1 ∂z +

∂ ∂ ∂z e31 ∂̺

+

∂ 1 ∂z e15 ̺

∂ ∂ ∂̺ C13 ∂z ∂ ∂ ∂̺ C44 ∂̺ ∂ ∂ ∂̺ e15 ∂̺

+

+

∂ ∂ ∂z C44 ∂̺

∂ ∂ ∂z C33 ∂z

∂ + C44 ̺1 ∂̺

∂ + e15 ̺1 ∂̺ +

∂ ∂ ∂z e33 ∂z

and f is the source term given by   ∂ [(C11 + C12 )a + C33 c] − ∂̺ ∂ . f = − ∂z [2C13 a + 2C13 c] ∂ − ∂z Psp

∂ ∂ ∂̺ e31 ∂z ∂ ∂ ∂̺ e33 ∂z

∂ ∂ ∂z e15 ∂̺

∂ ∂ ∂z e15 ∂̺

(2.72)

Because the functions appearing in the source term are piecewise constant what we have in reality are surface sources for the strain and the electric potential (the discontinuities appear at the interfaces).

2.6



  ∂ + e15 ̺1 ∂̺ ,  ∂ ∂ 1 ∂ − ∂z εˆ33 ∂z − εˆ11 ̺ ∂̺ (2.71)

+

∂ ∂ − ∂̺ εˆ11 ∂̺

+

The valence force fields model

The atomistic valence force field model was first applied to study the lattice dynamics of diamond by Musgrave and Pople [43]. Later, Nusimovici

2.6. THE VALENCE FORCE FIELDS MODEL

49

and Birman [44] developed a model for wurtzite semiconductors with eight adjustable parameters. The most popular valence force fields scheme is probably due to Keating [27], where two parameters α and β are used. This model applies to covalent semiconductors. From an atomistic point of view it is possible to divide the requirements the elastic strain energy Fs is subjected to into two groups: 1. General conditions: rotational and displacement invariance. 2. Conditions imposed by the symmetry of the crystal structure. For the sake of simplicity in considering any general type of deformation it is assumed that the elastic strain depends only on the positions of the nuclei. The latter is only valid in nonmetallic crystals, since in this case the Born-Oppenheimer approximation ensures that the electrons completely follow the nuclei. So the following models of valence force fields hold only on condition that the forces on the electrons are always negligible [45]. The requirement that the energy is an invariant under any arbitrary displacement of the lattice is satisfied if Fs depends only on the difference between nuclear positions: Fs = Fs (~xk − ~xl ) = Fs (~ukl ),

(2.73)

where ~ukl = ~xk − ~xl and ~xk is the position vector of the kth nucleus after deformation. But Fs must be invariant under a transformation in which the atoms are displaced by a rigid rotation of the crystal, and ~ukl are not invariant under such transformation, since they transform as vectors. We can form an invariant from scalar products of ~ukl and functions of such products: ~ kl · U ~ mn )/2a, λklmn = (~ukl · ~umn − U

(2.74)

~ kl = X ~k − X ~ l and X ~ k is the position vector where a is the lattice constant, U of the kth nucleus in the undeformed crystal. The final term is included so that the invariant is equal to zero when the deformation is removed. The energy Fs is a function of a large number of λklmn , and since these are small, we can use them as a basis for a series expansion of Fs . The constant term is disregarded, while the linear terms vanish if the energy is an extremum at equilibrium. So for small strain we can write using the summation convention: 1 pqrs λklmn λpqrs + O(λ3 ). Fs = Bklmn 2

(2.75)

pqrs The coefficients Bklmn must be positive definite, in order to ensure that Fs is a definite minimum. There are more terms λklmn than necessary, because pqrs most of the coefficients Bklmn are not independent. It was shown that there

50


are only 3N − 6 independent invariants [27], which have to be determinate from the crystal structure. In considering a slightly deformed primitive structure, which it can be envisioned as a large number of parallelepipeds with atoms at each of the eight corners. Without deformations, all parallelepipeds are identical unit cells. The arrangement of the eight atoms on the corners of a cell is given by 18 scalar products, and a convenient set of these is obtained by taking the squares of the lengths of the 12 edges of the cell and the 6 off-diagonal products, i.e., the angles between vectors, represented by arcs in Figure 2.2 (a).

Figure 2.2: (a) The unit cell with the 6 off-diagonal products. (b) The same cell with four of its six neighboring cells. (See Ref. [27]).

The four atoms of the adjacent cell which are not already fixed by the above scalar products are determined by the 8 remaining edge lengths of this cell and by 4 more angles, as we can see in Figure 2.2 (b). The rest of the crystal is included by adding cells and using only the necessary scalar products for the atom positions. We have three different types of lattice points according to the number of necessary scalar products. First, the points lying along three lines passing through point 0 [see again Figure 2.2 (a)] in the


51

initial cell and parallel to the three basis vectors of the undistorted lattice, which are associated with three diagonal products (edge lengths) and three off-diagonal products (angles), as shown in Figure 2.3 (a).

Figure 2.3: Types of lattice points according the number of associated scalar products: (a) points on the references lines, (b) points in the references planes but off the references lines, (c) other points. (See Ref. [27]). Second, we have the points in the reference planes, but off the reference lines, associated with three diagonal products but only one off-diagonal product, as shown in Figure 2.3 (b). Finally, we have the points which do not lie in the above mentioned planes, associated with three diagonal scalar products but no off-diagonal products, as shown in Figure 2.3 (c). This nonuniformity in the distribution of the scalar products is clearly undesirable, and it can be removed it by invoking the invariance of a crystal under the operations of the relevant translation subgroup, and by the assumption

52


that interactions over distances of the order of the crystal dimensions are negligible. This assumption is necessary especially if is not to be considered a bulk material, such as in an heterostructural quantum dot. It is possible to write ~x1 (l), ~x2 (l), ~x3 (l) as the position vectors of three neighbors of the atom cell (l) relative to this latter atom and which become the lattice basis vectors when the distortion is removed, so that (2.75) can be rewritten in the following way: Fs =

1X 2 ′ l,l

3 X

m,n,m′ ,n′ =1

Bmnm′ n′ (l − l′ )λmn (l)λm′ n′ (l′ ) + · · ·,

(2.76)

where ~m · X ~ n )/2a, λmn (l) = (~xm (l) · ~xn (l) − X

(2.77)

and is symmetric in (m, n), the sums over l, l′ run over all the unit cells, and Bmnm′ n′ (l − l′ ) is invariant under all the operations of the space group, it is positive defined and falls off rapidly as (l − l′ ) increases, in order to have a convergent expression. This formulation can be extended to deal with nonprimitive structures: for diatomic structures one suitable set of scalar products consists of a set like the previous one but using the atoms on one sublattice together with three extra scalars per unit cell of the diatomic structure necessary to locate the B atom relative to the A atom. However, a more convenient set is given by using ~x1 (l), ~x2 (l), ~x3 (l) as the position vectors of the B atoms in the neighboring unit cells relative to the A atom of cell (l) and ~x4 (l) as the position vectors of the atom B in the cell (l) relative to the A atom there. Thus Fs can be expressed as: 1X Fs = 2 ′

4 X

l,l m,n,m′ ,n′ =1

Bmnm′ n′ (l − l′ )λmn (l)λm′ n′ (l′ ) + · · ·.

(2.78)

In the same way it is possible to extend the formulation to structures with a greater numbers of atoms per unit cells. Within the harmonic approximation equation (2.78) becomes: 1 mn n (l − l′ )um Fs = Kab a ub , 2

(2.79)

where um a is the ath component of the displacement of the mth nucleus mn are linear combinations of the B and the Kab mnm′ n′ , a form suitable for calculation purposes, which has been applied to the calculation of the elastic constants of the diamond structure [27]. This model includes only two types of interaction, a nearest-neighbor term and a noncentral second-neighbor term. The basic unit cell of the diamond structures is a rhombohedron with


53

Figure 2.4: Crystal model for a diamond structure. The atoms on the two different sublattice are represented by open and filled circles. (See Ref. [27]).

two atoms (atoms 1 and 0 in Figure 2.4) on its major axis, which is directed along the [111] direction. The three neighboring unit cells of interest contain atoms 2 and 5, 3 and 6, 4 and 7, respectively. It is derived from equation (2.79) the following expression for the strain energy with two constants by including only diagonal products of the λ’s: 4 1X X Fs = Bmnmn (O)λ2mn (l) 2 l

m,n=1

4 4 β X 1X α X 2 2 2 2 2 (x0i (l) − 3a ) + 2 (x0i (l) · x0j (l) + a ) , = 2 4a2 2a l

i=1

i,j>i1

(2.80)

where the atomic labeling is as in Figure 2.6 and the required symmetry has been imposed by Bmmmm (O) = α (for all m), Bmnmn (O) = β (all m, n, m 6=

54


n), and by including the term in λ234 only if the symmetry B3434 = B1212 , etc. is satisfied. Recently physical properties of the semiconductor alloy A1−x Bx C have been studied using the valence force fields model [46]. Lattice-mismatched zincblende semiconductor alloy ground state configurations have been determinate [47], and also groundstate search of a group of lattice-mismatched IIIV semiconductor alloys, such as GaInN, GaInP, GaInAs, GaInSb, InAsSb, and InPAs have been performed. A valence force fields model for latticemismatched isovalent semiconductor zincblende alloys has been derived in Ref.[46], where the strain energy is given by: FV F F

4 XX 2 3αlm 2 rlm − d2lm = 2 8dlm m=1 l

+

2 X 6 XX l

s=1 k=1

3βlsk [rlsk1 rlsk2 cos(Θlsk ) − dlsk1 dlsk2 cos(Θ0 )]2 , 8dlsk1 dlsk2 (2.81)

where l runs through all the lattice sites in the unit cell, s = 1, 2 denotes the two sublattice sites in in the zincblende cell, m runs through the four different bonds, and k runs through the six angles with the vertex at site ls. The two bonds that form the angle k at the site ls are represented by lsk1 and lsk2, while dlm (similarly for dlsk1 and dlsk2 ) is the ideal bond length for bond lm, and rlm (similarly for rlsk1 and rlsk2 ) is the correspondingly calculated bond length. The angle formed between lsk1 and lsk2 is given by Θlsk , while Θ0 = 109.5◦ is the ideal tetrahedral bond angle. Martin [48] utilized bond-stretching and (α) and bending (β) parameters similar to Keating but added point-ion Coulombic forces to the free energy. The model given by equation (2.81) was used to develop our Keating-like model for wurtzite quantum dot heterostructures. The free energy of the elastic part is given by a sum over all atoms i:   " X X X 3αij 3β 2 ijk  rij · rij − d2ij + (rij · rik − dij dik cos(Θijk ))2  , FV F F = 8dij dik 8d2ij i

j

k6=j

(2.82)

and the sums over j and k run over the nearest neighbor atoms, d and r are the bulk and distorted distances between neighbor atoms, Θijk is the ideal unrelaxed tetrahedral bond length, and α, β are empirical materialdependent elastic parameters as mentioned above. At present, the piezoelectric effect has not been included in the energy expression for the valence force fields model.

Chapter 3

Bandstructure ~k · p~ theory As already anticipated in Chapter 1, the three most important empirical methods for bandstructure calculations are the tight-binding [30], the pseudopotential [31], and the ~k·~ p method [29], which basically differ in the choice of basis functions for the Schr¨ odinger equation: atom-like for tight-binding, plane-wave for pseudopotential, and Bloch states for the ~k · p~ method. The bandstructure results presented in this thesis have been obtained using different models of the ~k · ~ p theory, so in this chapter some of the most relevant features of this theory will be discussed. For a complete and detailed exposition of the ~k · p~ method I refer readers to the book by Lew Yan Voon and Willatzen [29]. In section (3.1) there is a presentation of the general theory for the bulk crystal, basically following that derived by Kane [51]. Section (3.2) concerns the application of the theory to a class of perturbed nonperiodic crystals, i.e., the heterostructure quantum dots. Here the exact envelope function theory as derived by Burt [32], and then applied by Foreman to quantum dot heterostructures [63] is outlined. Finally in section (3.3) the Hamiltonians and the parameters used for the models and results presented in this thesis will be dealt with.

3.1

General theory for bulk crystals

An interesting observation which can explain the development of the ~k · ~p theory was formulated by Bir and Pikus [70]: since the the physics of semiconductors is mainly governed by the carriers in the extrema of the various energy bands, only the neighborhoods of the band extrema should be relevant, and the qualitative physics should be governed by the shape of these energy surfaces. The first models of the ~k · p~ theory as a perturbative theory were presented in the work of Dresselhaus et al. [52], and Kane [51], while an approach based on the symmetry analysis was given by Luttinger with the 55

56

~ · P~ THEORY CHAPTER 3. BANDSTRUCTURE K

methods of invariants [55]. Cardona and Pollak showed that it was possible go beyond the neighborhood of band extrema without using perturbation theory obtaining realistic band structures for Si and Ge using a so-called full-zone ~k · p~ theory [56]. As mentioned previously, the ~k · p~ theory is an empirical bandstructure method based on the Bloch form of the wave function for a bulk crystal [51, 52]: ~

ψnk (~r) = eık·~r unk (~r),

(3.1)

where ~k is the wave vector and n is the band index. Here unk is a function with the periodicity of the crystal. If the Hamiltonian has the form: H≡

p2 + V (~r), 2m

(3.2)

we can write for the Schr¨ odinger equation [53]: Hψnk (~r) = Enk ψnk (~r), which gives explicitly in terms of unk : h~ ¯ ¯h2 k2 H + k · p~ + unk (~r) = Enk unk (~r). m 2m

(3.3)

(3.4)

For any given ~k, e. g., ~k = k~0 , the set of functions unk0 (~r) is a complete set with respect to the periodicity of the crystal. The idea of ~k · p~ theory is to use this completeness to expand the wave function in terms of unk0 (~r). Usually the zone-center Bloch functions un (~k0 = 0) is chosen, because of the interest in the electronic structure around the Γ point. With the following definition: Hk~0 ≡ H +

¯ ~ h ¯ 2 k02 h , k0 · p~ + m 2m

(3.5)

the equation (3.4) can be rewritten as: H~k unk (~r) = Enk unk (~r),

(3.6)

with H~k ≡ Hk~0 +

¯ ~ ~ h ¯2 2 h (k − k0 ) · p~ + (k − k02 ). m 2m

(3.7)

h~ ¯ For the Γ point (~k0 = 0) the second term is m k · p~, giving rise to the name of the theory, and Equation (3.6) is the ~k · p~ equation.

3.2. HETEROSTRUCTURE QUANTUM DOTS: FORMALISM

57

Using as basis the complete set unk0 (~r) then Hk becomes the following matrix: h2 2 ¯ ¯h k δnl + ~k · p~nl , (3.8) Hnl = En + 2m m where ~pnn′ are the momentum matrix elements: Z p~nl = u∗n (~r)~ pul (~r)d~r,

(3.9)

with the integral over the unit cell and En = hun0 |H|un0 i. For optical properties interest is usually only directed at the behavior around the Fermi level, so in diagonalizing matrix (3.8) only the conduction and the valence bands are dealt exactly. These band are coupled to higher and lower bands by the momentum matrix elements ~ pnn′ , and in order to remove these couplings either an unitary transformation can be found that to first order removes the coupling terms [54], or the perturbation theory introduced by L¨ owdin can be used [57]: the bands are divided into set A, which includes the bands treated exactly, and set B, which contains all the other bands. So in the lowest order the coupling between sets A and B is removed by introducing the perturbed functions: u′i = ui +

B X k

Hki uk , Hii − Hkk

(3.10)

where i is in A and k is in B, and the renormalized interaction between u′i and u′j are given by: ′ Hij

= Hij +

B X k

Hik Hkj , Hii +Hjj − Hkk 2

(3.11)

and this will result in a finite set of equations.

3.2

Heterostructure quantum dots: formalism

The theory of Wannier-Luttinger-Kohn with its effective equation for the Bloch electron in the presence of a slowly varying perturbation for graded semiconductors and semiconductor inversion layers [58, 60], was afterward applied to atomically sharp heterojunctions, which were fabricated in the 1970’s using molecular-beam epitaxy and metal-organo chemical vapor depositation techniques. It can be used in this case even though the perturbation is no longer slowly varying [82, 84]. This theory leads to an effective Schr¨ odinger equation for an external potential which, due to the difference

58


in band edges, entails a problem similar to the particle-in-a-box problem in textbook quantum mechanics. There are, though, at least three differences: the mass is now the effective mass, the treatment of the valence band requires a multiband description, and finally the material properties are position dependent and do not generally commute with the spatial differential operators which appear in the Wannier-like equation. One of the most used formalisms for the bandstructure calculations of heterostructures is the exact envelope function theory as derived by Burt [32, 36, 33, 34, 35, 39, 38], and then further developed by Foreman [63, 69, 64, 65, 66, 67, 68]. This will be presented in the following section. Burt did not propose a heuristic differential equation asking what kind of solutions were permitted, but instead imposed a constrain on the envelope function, i.e., that they have to be continuous and infinitely differentiable, and asked what kind of differential equations they satisfy. Even though his one-band formulation gave an Hamiltonian very similar to the heuristic one, his theory was largely ignored until Foreman used it to derive an explicit six-band Hamiltonian and showed that it can lead to notable differences compared to the symmetrized Luttinger-Kohn Hamiltonian for the heavy-hole mass in (001) quantum wires structure [54]. Afterwards Foreman also extended his model to an eight-band model, and this is generally known as Burt-Foreman model.

3.2.1

Exact envelope function theory

In this section the exact envelope function theory (EEFT) for ~k·~ p bandstructure calculations in nanoscale semiconductor heterostructures is derived. For a complete explanation of this theory I refer the reader to the articles of Burt and Foreman [33, 32, 63]. The strategy is to arrive at a derivation of an effective-mass equation starting from the Schr¨ odinger equation: energies E and wave functions ψ are found by solving the single particle Schr¨ odinger eigenvalue equation: Hψ = Eψ,

(3.12)

where the Hamiltonian is given by: H=

p2 + V (~r), 2m

(3.13)

~ m is the electron mass and V (~r) is the crystal potential. here ~ p = ¯hi ∇, Subject to some boundary conditions, V (~r) reflects the periodicity of the material, but in a semiconductor heterostructure we face two main difficulties in solving equation the (3.12) for the bandstructure. Firstly, we have only a local periodicity, since there are at least two different crystal structures in two different materials. Then, the crystal potential V (~r) is usually unknown, so that an exact solution can not be evaluated.

3.2. HETEROSTRUCTURE QUANTUM DOTS: FORMALISM

59

The exact envelope function theory simplifies these problems using the local periodicity of the crystal. The idea is to expand the wave function ψ in a complete set of functions un , the periodic eigenfunctions of a homogeneous system: X ψ(~r) = φn (~r)un (~r), (3.14) n

where the functions φn (~r) are a set of unknown envelope functions varying slowly over a lattice unit cell, which are solutions of an infinite set of coupled differential equations. The basic idea is to insert the expansion (3.14) into both sides of the Schr¨ odinger equation, rearranging the expression into an envelope expansion form, and finally equate the coefficients of un (~r) on both sides. This entails: X i¯ hX h2 2 ¯ ∇ φn (~r) − p~nl · ∇φl (~r) + Hnl φl (~r) = Eφn (~r), (3.15) − 2m m l

l

where ~pnl is defined as in (3.9) and Hnl is given by: Z Hnl = u∗n (~r)Hul (~r)d~r.

(3.16)

In order to reduce system (3.15) to a finite set, it is necessary to choose a finite set of periodic functions un and treat the rest as perturbation. So to derive an effective-mass-type equation from equation (3.15) the small envelope functions have to be eliminated in favor of some dominant ones. This entails a set of equations called a multiband equation. Hence, the envelope functions need to be divided in two groups φs and φr . The finite set φs (and φs′ ) corresponds to the periodic functions us that are of interest and treated exactly. The envelope functions φr (and φr′ ), correspondent to periodic functions ur , form the rest of the envelope functions, and will be treated in a perturbative approach as small compared with functions in set φs . For slowly varying envelope functions with n = r we have approximately [32]: X i¯h (3.17) (− p~rs′ · ∇φs′ + Hrs′ φs′ ). φr = (E − Hrr )−1 m ′ s

If one substitutes for φr in equation (3.15) using equation (3.17), one obtains: ¯h2 X i¯h X (r) − ∇ · [γss′ · ∇φs′ (~r)] − p~ss′ · ∇φs′ (~r) 2m ′ m ′ s s X (2) i¯ hX psr · ∇[(E − Hrr )−1 Hrs′ ]φs′ (~r) ~ + Hss′ φs′ (~r) − m ′ ′ s ,r

s

i¯ hX − (E − Hrr )−1 (~ psr Hrs′ + Hsr p~rs′ ) · ∇φs′ (~r) = Eφs (~r), m ′ s ,r

(3.18)


60

(r)

where the contribution γss′ from remote bands in set φs is: (r)

γss′ = δss′ +

2 X p~sr (E − Hrr )−1 p~rs′ , m r

(3.19)

and (2)

Hss′ = Hss′ +

X r

Hsr (E − Hrr )−1 Hrs′ ,

(3.20)

is the effective second order Hamiltonian. The fourth term on the right hand of equation (3.18), namely −

i¯ hX p~sr · ∇[(E − Hrr )−1 Hrs′ ]φs′ (~r), m ′

(3.21)

s ,r

is non-zero only near interfaces and involves large energy denominators, so it can be safely ignored. Hence, it is finally possible to write the following for the effective-mass-type equation: −

X (2) i¯h X ¯2 X h (r) ∇ · [γss′ · ∇φs′ (~r)] − ~pss′ · ∇φs′ (~r) + Hss′ φs′ (~r) 2m ′ m ′ s s s′ i¯ hX − (E − Hrr )−1 (~ psr Hrs′ + Hsr p~rs′ ) · ∇φs′ (~r) = Eφs (~r). (3.22) m ′ s ,r

One of the most interesting features of this method is that it is not restricted to a particular type of Hamiltonian, hence it can treat spin-orbit interaction as well as stressed structures. The spin-orbit Hamiltonian is given by: HSO =

¯ h ~ ) · p~, (~σ × ∇V 4m2 c2

(3.23)

where ~σ are the Pauli spin matrices and c is the speed of light. In terms of envelope expansion it is necessary to add to the right hand of equation (3.22): X (3.24) [HSO ]ss′ φs′ (~r), s

where [HSO ]nl =

Z

u∗n (~r)HSO ul (~r)d~r.

(3.25)

The application of the theory has been to many different types of nanostructures, in particular to quantum wells, superlattices, quantum wires and nanowires, and quantum dots.

3.3. EXPLICIT HAMILTONIANS

61

In the following sections I will review in detail the three different Hamiltonians which have been utilized for the bandstructure calculations presented in this thesis. The one-band model for zincblende quantum dot, where only a band is treated exactly both for valence and conduction band. The eight-band model for zincblende quantum dot, where the s-like conduction band and the three fold degenerate p-like valence bands, all twice degenerate with respect to spin, are included. Finally, the six-band model for wurtzite quantum dot, where for the valence band structure, due to a larger band gap, it is possible to reduce the set of periodic functions to only the three fold degenerate p-like valence bands.

3.3 3.3.1

Explicit Hamiltonians One-band-model Hamiltonian for a zincblende quantum dot

In the following, the heterostructure solutions are expanded in the Bloch cell-functions basis set: |S ↑i, |X ↑i, |Y ↑i,|Z ↑i,|S ↓i,|X ↓i, |Y ↓i,|Z ↓i. The one-band model for electrons in the conduction band is given by [40]: He φe = Ee φe ,

(3.26)

where He , φe and Ee are the electron Hamiltonian, the envelope wave function and the energy, respectively. The electron Hamiltonian is: He = Hs (~ re ) + Heǫ (~ re ) + Ec (~ re ) + eVp .

(3.27)

re ) is the strain dependent part for the Here Hs (~re ) is the kinetic part, Heǫ (~ electron and Ec (~ re ) is the energy of the unstrained electron band edge, e is the electron charge and Vp is the piezoelectric potential as defined in Chapter 2. The kinetic part is given by: h2 ~ 1 ~ ¯ k k , (3.28) Hs (~re ) = 2m0 me (~r) where ¯h is Planck’s constant, m0 is the is the free-electron mass, ~k is the wave vector operator and me is the electron effective masses in units of m0 . The solutions are spin degenerate in this one band model: ψe↑ = φe |S ↑i . ψe↓ = φe |S ↓i

(3.29)

The√one-band model for the heavy-hole √ states in the valence band |hh ↑i = (1/ 2)(|X ↑i+i|Y ↑i) , |hh ↓i = (i/ 2)(|X ↓i−i|Y ↓i) is found in a similar

62


way: Hhh φhh = Ehh φhh ,

(3.30)

where Hhh , φhh and Ehh are the heavy-hole Hamiltonian, the envelope wave function and the energy, respectively. The two wave functions are given by: ↑ ψhh = φhh |hh ↑i . ↓ ψhh = φhh |hh ↓i

(3.31)

The heavy-hole Hamiltonian Hhh is: 0 ǫ Hhh = Hhh (~rhh ) + Hhh (~rhh ) + Ev (~rhh ) + eVp ,

(3.32)

0 is the kinetic part, H ǫ (~ where Hhh hh rhh ) is the strain dependent part for the heavy hole and Ev (~rhh ) is the energy of the unstrained hole band edge, e is the electron charge and Vp is the piezoelectric potential. The kinetic part is given by: 1 ~ ¯h2 ~ 0 k k , (3.33) Hhh (~rhh ) = 2m0 mhh (~r)

where mhh is the heavy-holes effective mass, given as in Ref. [103], but assuming for the Luttinger parameters γ2 = γ3 . For the meaning of this parameters and the reason of this choice refer to the following (3.3.3) and (3.3.7) sections, respectively. The piezoelectric potential in (3.27) and (3.32) is found solving Maxwell-Poisson equation, namely the second equation of equations (2.45).

3.3.2

Inclusion of strain in one-band model

In the one-band model the strain dependent part of the electron Hamiltonian for a zincblende crystal structure is given by [40]: Heǫ (~re ) = ac (~r )(εxx (~r ) + εyy (~r ) + εzz (~r )) = ac (~r )εH (~r ),

(3.34)

while for the heavy-hole Hamiltonian it is: ǫ (~rhh ) = −av (~r )(εxx (~r ) + εyy (~r ) + εzz (~r )) Hhh b + (εxx (~r ) + εyy (~r ) − 2εzz (~r )) 2 b = −av (~r )εH (~r ) + εB (~r ), 2

(3.35)

where ac (av ) and b are the conduction (valence)-band deformation hydrostatic potential and the shear deformation potential, respectively [70], and εH (~r ) = εxx (~r ) + εyy (~r ) + εzz (~r ),

(3.36)


63

and εB (~r ) = εxx (~r ) + εyy (~r ) − 2εzz (~r ),

(3.37)

are the hydrostatic and biaxial strain component, respectively.

3.3.3

Eight-band-model Hamiltonian for a zincblende quantum dot

The eight-band model includes the electron, heavy-holes, light-holes and spin-orbit split-off bands around the Γ point of the Brillouin zone, and treats all the other bands as remote. This model, as well as the other multiband models, e.g., four-band and six-band models, have been derived for homogeneous bulk material, under the assumption that all the parameters are constant, but for a quantum dot heterostructure at the heterointerfaces these parameters have an abrupt change from their values in one material to those in the adjacent one. Foreman has explicitly derived an 8 × 8 effectivemass Hamiltonian for planar heterostructures from the Burt’s exact envelope function theory introduced in section (3.2.1)[69]. Here the nonsymmetrized eight-band Hamiltonian for a heterostructure following Ref. [71] is presented. The eight-band model wave function is given by a linear combination of the eight Bloch parts weighted by the respective envelope functions:

ψ=

8 X

φi ui ,

(3.38)

i=1

where φi are the envelope function and ui are the Bloch states. We use the following Bloch basis for the eight-band Hamiltonian : u1 = |S ↑i

u2 = |S ↓i,

(3.39)

i 1 u3 = √ (|X ↑i + i|Y ↑i) u4 = √ (|X ↓i + i|Y ↓i − 2|Z ↑i) 2 6 i 1 u5 = √ (|X ↑i − i|Y ↑i + 2|Z ↓i) u6 = √ (|X ↓i − i|Y ↓i) 6 2 −i 1 u7 = √ (|X ↓i + i|Y ↓i + |Z ↑i) u8 = √ (|X ↑i − i|Y ↑i − |Z ↓i), 3 3 (3.40) The eight-band Hamiltonian is in this basis given by [71]: ′ ˆ 8S , Hˆ8 = Hˆ8 + H

(3.41)


64

ˆ S is the strain dependent part and Hˆ8 ′ is given by: where H 8 

ǫc + T

0

q

iV1

2 3 V0

√i V−1 3 q i 23 V0

0

√i V0 q3 i 23 V1 −i √ S 2

q

2 3 V−1

  −1 −1 √ √ V V0 0 ǫc + T 0 −V−1  3 1  √3 †  −iV 0 ǫv − P − Q −S −R 0 i 2R  q 1 q  √ † † 2 3 −1 † 2  √ V −S ǫ − P + Q −C −R i 2Q −i V v ¯h  ′ 1 3 0 2Σ 3 ˆ q q H8 =  √ −i † 2 † 2m0  √ −R† −C † ǫv − P ∗ + Q∗ ST i 32 Σ∗ i 2Q∗  3 V−1 −i 3 V0 √  †  √i S ∗ 0 −V−1 0 −R† S∗ ǫv − P ∗ − Q∗ i 2R†  2 q q √ √  −i † i 2 † 3 T † ′  √ V0 √ −i 2R −i 3 V1 S −i 2Q C −i 2 Σ ǫv − P 2  q3 q √ √ † † −i T 2 −1 √ √ V −i 2R† S C† ǫ′v − P ∗ i 32 Σ† −i 2Q∗ 3 V−1 3 0 2 (3.42)

with the following given expressions: ~k = −i∇, ~

kˆx + ikˆy kˆ+ = √ , 2

kˆx − ikˆy kˆ− = √ , 2

1 1 1 ˆ (p1 kz + kˆz p2 ), V1 = (p1 kˆ+ + kˆ+ p2 ), V−1 = (p1 kˆ− + kˆ− p2 ), 2 2 2 T = kˆ+ αkˆ− + kˆ− αkˆ+ + kˆz αkˆz , P = kˆ+ (γ1 − 2χ)kˆ− + kˆ− (γ1 + 2χ)kˆ+ + kˆz γ1 kˆz , Q = kˆ+ (γ2 − χ)kˆ− + kˆ− (γ2 + χ)kˆ+ − 2kˆz γ2 kˆz , √ R = 3(kˆ+ (γ2 − γ3 )kˆ+ + kˆ− (γ2 + γ3 )kˆ− ), √ S = −i 6(kˆ− (γ3 + χ)kˆz + kˆz (γ3 − χ)kˆ− ), √ χ ˆ χ ˆ Σ = −i 6 kˆ− γ3 − kz + kˆz γ3 + k− , 3 3 √ (3.43) C = −i2 6(kˆ− χkˆz − kˆz χkˆ− ),

V0 =

where 2m0 2m0 ǫv = 2 Ev , 2 Ec , ¯h ¯h 2m0 ′ δ = 2 ∆, ǫv = ǫv − δ, ¯h

ǫc =

(3.44)

m0 is the free electron mass, ∆ is the spin-orbit splitting of the valence band and remote band contributions are accounted for by use of modified Luttinger parameters [73]: γ1 = γ1L −

Ep 3Eg

γ2 = γ2L −

Ep 6Eg

γ3 = γ3L −

Ep , 3Eg

(3.45)



         ,        


65 ˆ2

P is the Kane energy where Eg = Ec − Ev is the energy gap, Ep = 2m 0 L [81], and γi (i = 1, 2, 3) are the Luttinger parameters of the valence band. Furthermore:

p1 = Pˆ + ξ,

p2 = Pˆ − ξ,

(3.46)

where ξ is a dissymmetry parameter due to the heterostructure [69], and pz |Zi. In the equations it is assumed that ξ = 0. For a discussion Pˆ = − 2i h hS|ˆ ¯ of the influence of ξ readers are referred to [71]. A second dissymmetry parameter is given as [69]: χ = (2γ2 + 3γ3 − γ1 )/3.

(3.47)

The parameter α appearing in T can be evaluated using the relation: Ep 2 1 1 1 = + α+ , (3.48) mc m0 3 Eg Eg + ∆ where mc is the conduction band effective mass at Γ point. The above eight-band model has the well known problem of giving rise to spurious solutions resulting from the inclusion of large ~k components where the perturbative approach of ~k · p~ breaks down. In the following section it is shown that this problem has been circumvented by changing the Kane energy Ep as suggested by Foreman [74].

3.3.4

Spurious solutions in eight band ~k · ~p theory

Spurious solutions appear in eight band ~k · p~ theory for semiconductor heterostructures due to the presence of Fourier wave components which are far away from the ~k · ~ p expansion point (usually the Γ point). Figure 3.1 shows the conduction and valence dispersion curves for bulk InAs along the [111]-direction found using tight-binding [62] and eight band ~k · p~ theory. Set 1 Set 2

mc [me ] 0.025 0.025

γ1 19.16 19.16

γ2 8.18 8.18

γ3 8.69 8.69

Eg [eV ] 0.370 0.370

∆so [eV ] 0.393 0.393

Ep [eV ] 18.88 17.87

Table 3.1: ~k · ~ p parameters used in Figure 3.1. me is the free electron mass. It can be seen that close to the Γ point the two models are in agreement, however, further away from the Γ point the ~k · ~p model deviates from the more accurate tight-binding results. When ~k · p~ theory is used for heterostructures it is necessary to require that only Fourier components close to the Γ point are used. Here we focus on the two most common approaches to achieve this: the plane-wave cut-off method where the problem is solved in Fourier space and only components


66

2 1 0

Energy [eV]

−1 −2 −3 −4 −5 −6 −7 0

0.2

0.4

0.6

0.8 1 k [π/a]

1.2

1.4

1.6

0.2

0.4

0.6

0.8 1 k [π/a]

1.2

1.4

1.6

2 1 0

Energy [eV]

−1 −2 −3 −4 −5 −6 −7 0

Figure 3.1: Plots showing the conduction and valence bands of InAs along the [111]-direction. Top: Using ~k · ~p parameter set 1, bottom: using parameter set 2 from Table 3.1. The solid black lines are tight-binding results (see Ref. [72]) and the dashed blue lines are ~k · p~ results. a is the lattice constant of InAs.


67

close to the Γ point are used and the approach proposed by Foreman [74, 98] where the Kane energy Ep is chosen so that large k-vector Fourier components have energy far away from the energy range of interest, see Figure 3.1 (bottom). In the next section it is shown how the plane-wave cut-off method using polar coordinates for the one band model is used, however, the approach is easily extended to multiband models.

3.3.5

Plane-wave cut-off method in polar coordinates

For a heterostructure of cubic materials the one band model is given by: ˆ = [∇A(~r)∇ + V (~r)]ψ(~r) = Eψ(~r), Hψ

(3.49)

2

where A(~r) = 2m¯h∗ (~r) , m∗ is the effective mass, V is the effective potential (the conduction-band edge), and ∇ is the gradient. The problem is cylindrical symmetric as long as the effective mass m∗ , the effective potential V and the domain of our system Ω are cylindrical symmetric. Solutions are sought of the form: Z ˆ ~k)eih~k·~ri d3 k, ψ( (3.50) ψ(~r) = Ωk

where Ωk is a small region around ~k = 0 to be defined later (it needs to reflect the cylindrical symmetry). Using the cylindrical symmetry it is possible to show that solutions can be put to the form: ˜ kz ), ψl (~k) = eilθk ψ(k,

(3.51)

where l is an integer, kx = k cos(θk ) and ky = k sin(θk ). Assuming that the region in k space Ωk is given by kz,min < kz < kz,max , 0 ≤ k < kmax and 0 ≤ θk < 2π we can integrate the angular dependence of equation (3.50) out, giving: ˜ ′ , k′ ) = E ψ(k z Z kz,max Z kmax ˜ Al+1 (k′ , kz′ , k, kz ) + A˜l−1 (k′ , kz′ , k, kz ) ′ ˜ k kψ(k, kz )dkdkz 2 0 kz,min Z kz,max Z kmax ˜ kz )dkdkz A˜l (k′ , kz′ , k, kz )kz′ kz ψ(k, (3.52) + kz,min kz,max

+

Z

kz,min

0

Z

kmax

0

˜ kz )dkdkz , V˜l (k′ , kz′ , k, kz )ψ(k,


68 where

A˜l (k′ , kz′ , k, kz ) = Z zmax Z rmax (z) k ′ A(r, z)Jl (kr)Jl (k′ r)ei(kz −kz )z rdrdz, 2π zmin 0

V˜l (k′ , kz′ , k, kz ) = Z zmax Z rmax (z) k ′ V (r, z)Jl (kr)Jl (k′ r)ei(kz −kz )z rdrdz. 2π zmin 0

(3.53)

(3.54)

The problem has now been reduced to a two dimensional problem while ensuring that the Cartesian plane wave expansion is restricted to small ~k components. It is not surprising that the radial part should be expanded in Bessel functions as these functions are the solutions to the Laplace eigenvalue equation in cylindrical coordinates. It is, however, not obvious that restricting to small k components of the Bessel functions (and kz components) is equivalent to the restriction to small Fourier components which has been shown to be the case.

3.3.6

Inclusion of strain in an eight-band model

To derive the strain-dependent part of the eight-band Hamiltonian we follow Ref. [75] with only a slight difference in the matrix formulation due to a different choice of basis:



       ˆS =  H  8       

a0 0 0 0 a0 0 0 0 −p + q −is†

0

0

0

0

r†

0

0

0

0

0

0

0

− √12 s† √ i 2r †

0 0 is −p − q 0 r† √ i 2q q − 32 s†

0 0 r

0 0 0

0 0

0 0 √ − √12 s −i 2r q √ 0 r −i 2q − 32 s† q √ −p − q −is − 32 s −i 2q √ is† −p + q i 2r † − √12 s† q √ − 32 s −i 2r z 0 √ i 2q 0 z − √12 s



        ,       

(3.55)


69

with a0 = ac (~r )(ǫxx (~r ) + ǫyy (~r ) + ǫzz (~r )) = ac (~r )εH (~r ), p = −av (~r )(ǫxx (~r ) + ǫyy (~r ) + ǫzz (~r )) = −av (~r )εH (~r ), b b q = (ǫxx (~r ) + ǫyy (~r ) − 2ǫzz (~r )) = εB (~r ), 2 2 r 3 r= b(ǫxx (~r ) − ǫyy (~r )) − idǫxy (~r ), 2 s = −d(ǫxz (~r ) − iǫyz (~r )),

z = av (~r )(ǫxx (~r ) + ǫyy (~r ) + ǫzz (~r )) = av (~r )εH (~r ),

(3.56)

where ac (av ) and b, d are the conduction (valence) band hydrostatic deformation potential and shear valence band deformation potentials, respectively [70], εH (~r ) and εB (~r ) are the are the hydrostatic (3.36) and biaxial (3.37) strain component, respectively, while the coupling between the conduction and the valence band has been disregarded.

3.3.7

Eight-band model in a rotational-symmetric system

The equations for the eight-band model for a zincblende quantum dot are not cylindrical √ symmetric. However, using the spherical approximation γ2 = γ3 [76] and 3b = d, the above equations are cylindrical symmetric and the envelope functions can be expressed in cylindrical coordinates as: φi (θ, r, z) = φi (x(θ, r), y(θ, r), z) =

1 ı(Fz −Jzi )θ ˆ e φi (r, z) , N

(3.57)

where Fz is the total angular moment in the z-direction, Jzi is the angular moment in the z-direction of ui and N is a normalization factor ensuring that hψ | ψi = 1. Because the system is cylindrical symmetric, Fz is a good quantum number, so each energy Ej has a well-defined Fzj . In the implementation of the cylindrical symmetric eight-band problem φi (r, z) is found in terms of coefficients ai (l, m) such that X φî (r, z) = ai (l, m)eıkz,m z Jm (kr,l r), (3.58) l,m

where Jm (kr,l r) is the Bessel function of the first kind of order m.

3.3.8

Six-band-model Hamiltonian for wurtzite quantum dot

The conduction band state can be found solving a one-band model equation analogous to (3.27), where for a wurtzite structure the kinetic part is given by [40]: 1 1 h2 ¯ kz k kz + ~kz⊥ ⊥ ~kz⊥ , (3.59) Hs (~re ) = 2m0 me (~r) me (~r)


70

k ~⊥ where me and m⊥ e are electron effective masses in units of m0 and kz = ~k − ~kz . The valence hole states are eigenstates of the six-band envelopefunction equation:

Hh φh = Eh φh ,

(3.60)

where Hh , φh and Eh are the 6 matrix of the hole Hamiltonian, the envelope wave function and the energy, respectively. The six wave functions are given by: ψh = (|Xi| ↑i, |Y i| ↑i, |Zi| ↑i, |Xi| ↓i, |Y i| ↓i, |Zi| ↓i) · φh ,

(3.61)

The hole Hamiltonian Hh is: Hh =

0 HXY Z (~rh ) + Hhǫ (~rh ) 0 HXY Z (~rh ) + Hhǫ (~rh

+ Ev (~rh ) + eVp , (3.62)

where HXY Z is a 3 × 3 matrix of the kinetic part, Hhǫ (~rh ) is the strain dependent part for the hole, Ev (~rh ) is the energy of the unstrained hole band edge, e is the electron charge and Vp is the piezoelectric potential. The matrix for the kinetic part is given by [40]: h2 ¯ HXY Z = 2m0   kx N2 kz + kz N2′ kx kx L1 kx + ky M1 ky + kz M2 kz kx N1 ky + ky N1′ kx , kx M1 kx + ky L1 ky + kz M2 kz ky N2 kz + kz N2′ ky ky N1 kx + kx N1′ ky × ′ ′ kx M3 kx + ky M3 ky + kz L2 kz − δcr kz N2 ky + ky N2 kz kz N2 kx + kx N2 kz (3.63) where L1 = A2 + A4 + A5 ,

L2 = A1 ,

M1 = A2 + A4 − A5 ,

M2 = A1 + A3 ,

N1 = 3A5 − (A2 + A4 ) + 1, √ N2 = 1 − (A1 + A3 ) + 2A6 , δcr = 2m0 ∆cr /¯h2 ,

N1′

M3 = A2 ,

= −A5 + A2 + A4 − 1,

N2′ = A1 + A3 + −1,

(3.64)

and Ak k = 1, ..., 6 are the Rashba-Sheka-Pikus parameters of the valence band and ∆cr is the crystal-field splitting energy [40]. The piezoelectric potential is always found solving the Maxwell-Poisson equation [second of equations (2.45)].


3.3.9

71

Inclusion of strain in a six-band model

In the one-band model the strain dependent part of the electron Hamiltonian for a wurtzite crystal structure is given by [40]: Heǫ (~re ) = akc (~r )εzz (~r ) + a⊥ r )(εxx (~r ) + εyy (~r )), c (~

(3.65)

k

where ac and a⊥ c are the conduction-band deformation hydrostatic potentials. The strain-dependent part of the hole wurtzite Hamiltonian is:   l1 εxx + m1 εyy + m2 εzz n1 εxy n2 εxz , n1 εxy m1 εxx + l1 εyy + m2 εzz n2 εyz Hhǫ (~rh ) =  n2 εxz n2 εyz m3 (εxx + εyy ) + l2 εzz (3.66) where l1 = D2 + D4 + D5 ,

l2 = D1 ,

m1 = D2 + D4 − D5 , m2 = D1 + D3 , √ n1 = 2D5 , n2 = 2D6 ,

m3 = D2 , (3.67)

and Dk (k = 1, ..., 6) are the valence band deformation potential [40]. All the potential and the strain tensors are position dependent.

72


Chapter 4

Optoelectronic properties 4.1

General theory

In the following some of the optical properties for the semiconductor quantum dot heterostructure are introduced, as the absorption coefficient for photons and the interband and intraband dipole moments between confined states in conduction and valence band induced by the electron-photon interaction. The emission rate W is first calculated for a simple two-energy level system, i.e., electrons in a crystal lattice, between state m with energy Em , and state n with energy En . This rate is given by the Fermi Golden Rule [77]: W =

2π | hn | Hint | mi |2 δ(En − Em − ¯hω), h ¯

(4.1)

where ¯hω is the photon energy and Hint is the time-dependent part of the interaction Hamiltonian responsible for emission of photons given by: Hint = −

e ~ A(~r, t) · ~p, m0

(4.2)

where e is the electron charge, m0 is the free electron mass, p~ is momentum ~ is related to the electric field according operator, and the potential vector A ~ ~ r , t) = − ∂ A . E(~ ∂t

(4.3)

The electric field is given by: ~ = ~ǫ E0 (eiωt + e−iωt ), E

(4.4)

with ~ǫ polarization vector and E0 =

r

2¯ hω , n ′ ǫ0 V 73

(4.5)

74

CHAPTER 4. OPTOELECTRONIC PROPERTIES

where n′ is the refractive index of the medium, ǫ0 is the vacuum permittivity, and V is the volume of the system. Using the above expressions and integrating the electric field we can write (4.2) as: Hint

ei =− m0

r

¯ h (eiωt + e−iωt )~ǫ · p~. 2ωǫ0 V

(4.6)

Inserting (4.6) in equation (4.1) gives the emission rate: W =

πe2 ¯ h | hn | ~ǫ · ~p | mi |2 δ(En − Em − ¯hω), 2 ′ n m0 hωV

(4.7)

and since the absorption is given by the number of photons absorbed per unit distance, it finally results in: α(¯ hω) =

πe2 ¯h | ~ǫ · ~pnm |2 δ(En − Em − ¯hω), n′ cm20 hωV

(4.8)

where c is the vacuum light velocity and ~pnm = hn | ~p | mi,

(4.9)

is the momentum matrix element. In order to generalize the expression (4.8) for absorption for discrete levels to a bulk semiconductor, it needs to be integrated over the occupied electron and hole states in a semiconductor, states given by the quasi-Fermi distribution for the conduction and valence band, respectively. A further generalization to a lower dimensional system, such as a quantum dot heterostructure, is straightforward [78, 79], and leads to: α(¯ hω) =

X πe2 ¯h v | ~ǫ · p~nm |2 δ(Enc − Em − ¯hω), n′ cm20 hωV n,m

(4.10)

where the indices n and m in the sum represent the electron and hole state. An alternative derivation of equation (4.10) can be achieved using the density matrix theory to compute the complex optical susceptibility χ for quantum dot [80]. The complex susceptibility correlates the electric field to the polarization ℘ and to the absorption: ℘(ω) = χ(ω)E(ω),

(4.11)

′

(4.12)

′′

ǫ(ω) = ǫ (ω) + iǫ (ω) = 1 + 4πχ(ω), ω ′′ ǫ (ω), α(ω) = nc

(4.13)

4.1. GENERAL THEORY

75

where ǫ(ω) is the dielectric function. The dynamic equation for the density matrix is given by: i¯ h

∂ ̺ = [H + Hint , ̺] + LR , ∂t

(4.14)

where LR models all dissipative processes, H is the total Hamiltonian of the electronic excitations in quantum dots and Hint is the interaction Hamiltonian describing the dipole coupling to the light field, respectively. By solving equation (4.14) at the first order, i.e., neglecting all the dissipative processes (for a complete derivation of the solution I refer readers to Ref. [80], page 345), it is possible finally arrive at:

χ(ω) =

1 iX ~ 1 | dnm |2 + , (4.15) h n,m ¯ γnm + i(ωnm − ω) γnm − i(ωnm + ω)

where γnm is a damping constant, ωnm = ωn − ωm is the energy difference between state n in conduction band and state m in valence band, and d~nm = hn | d~ | mi =

Z

d3 r ψn∗ (~r)d~ ψm (~r),

(4.16)

where d~ = e~r,

(4.17)

is the electric dipole matrix element, and ψn (~r) and ψm (~r) are the wave function of state n and m, respectively. So the absorption can be expressed as: α(ω) =

γnm 4πω X . | dnm |2 2 ′ hcn n,m ¯ γnm + (ωnm − ω)2

(4.18)

Neglecting the absorption linewidth in the Lorentzian lineshape gives: lim

γnm = πδ(ωnm − ω) = πδ(ωn − ωm − ω) 2 + (ω 2 γnm nm − ω) v =¯ hπδ(Enc − Em −¯ hω), (4.19)

γnm →0

which inserted in (4.18) leads to (4.10).

76


4.2

Momentum matrix elements

The momentum matrix element (4.2) between a state ψn and a state ψm in a quantum dot heterostructure is given by:

pnm = ~

=

Z

N X

(φi ui )∗ ~p (φj uj )dV =

V i,j=1

N Z X

u∗i p~uj dV

i,j=1

Z

V

φ∗i φj dV

+

N Z X i=1

V

(u) (φ) φ∗i p~ φi dV = p~nm + ~pnm ,

(4.20)

where p~ (u) and p~ (φ) are the Bloch and envelope parts of the momentum matrix element, respectively and N is the number of the band exactly treated in the ~k · p~ expansion, e.g., N = 8 in eight band model. The wave functions are defined as in (3.38) and ui are the Bloch-cell functions given by (3.39) and (3.40). The envelope functions are assumed to be slowly varying at the scale of the primitive cell [82]. It is usual to consider only the Bloch part P~ (u) since the envelope part (φ) P~ is usually of an order of magnitude smaller, as shown in Ref. [85], and as has also been confirmed by our results.

4.2.1

Momentum matrix elements in a cylindrical symmetric quantum dot

As explained in section (3.3.7), assuming that the geometry of our problem is cylindrical symmetric and using the spherical approximation [76], the envelope functions of a state n in cylindrical coordinates are given by (3.57) and (3.58), which can be rewritten here in a slightly different way: 1 ι(Fzn −Jzi )θ î e φn (r, z) , φ˜in (θ, r, z) = φin (x(θ, r), y(θ, r), z) = Nn X mz i i k φˆn (r, z) = an (l, m)e z JLi (kli r),

(4.21) (4.22)

l,m

where Fzn is the total angular moment in the z-direction, Jzi is the angular moment in the z-direction of | ui i, Nn is a normalization factor ensuring that hψn | ψn i = 1, Li = Fz − Jzi is the angular momentum of the envelope function and kli R are the zeros of the Li Bessel function, and R is the radius of a cylindrical symmetric quantum dot.

4.2. MOMENTUM MATRIX ELEMENTS

77

Cylindrical coordinates give : p~

(u)

=

N Z N X X i=1 j=1

φinc (x, y, z)∗ φjmv (x, y, z)dxdydz hui | p~ | uj i =

N N πZR2 X X X i an (l, m)∗ ajm (l, m)[JL′ i (kl R)]2 hui | p~ | uj i, (4.23) = Nn Nm i=1 j=1 l,m

where Z is the height of the quantum dot, and the allowed transitions follow the selection rule Li − Lj = 0. The ui , uj states are only S, P − type states, so, due to the symmetrical properties, for a zincblende quantum dot the only non-zero elements of hui | ~p | uj i are: hS | px | Xi = hS | py | Y i = hS | pz | Zi = pcv .

(4.24)

The factor pcv is given in terms of the so-called Kane energy [81]: Ep =

p2cv = 21.5 eV. 2m0

(4.25)

~ it gives in cylindrical coordinates: Since p~ = −i¯ h∇, ~ ≡ (cosθ ∂ − sinθ ∂ ∇ ∂r r ∂θ

;

sinθ

∂ cosθ ∂ + ∂r r ∂θ

;

∂ ), ∂z

(4.26)

(φ)

for the envelope part p~ab we have: (φ)

p~ab =

N Z X

φinc (x, y, z)∗ p~ φimv (x, y, z)dxdydz,

(4.27)

i=1

which is a three-components vector: pz(φ) =

N 2π 2 ¯ hR2 X X i an (l, m)∗ aim (l, m)m[JL′ i (kl R)]2 , Nn Nm

(4.28)

i=1 l,m

with the allowed transitions given by the selection rule Li − Lj = 0, and px(φ)

N i¯ hπZR X X i ′ kl kl′ =− an (l , m)∗ aim (l, m) 2 JL (kl R)JL′ i±1 (kl′ R), Nn Nm kl − kl2′ i±1 ′ i=1 l ,l,m

(4.29)

py(φ) =

N ¯ πZR X X i ′ h kl kl′ ′ ′ an (l , m)∗ aim (l, m) 2 2 JLi±1 (kl R)JLi±1 (kl R), Nn Nm k − k ′ l l′ i=1 l ,l,m

(4.30)

with selection rule Li − Lj ± 1 = 0.

78

4.3


Dipole moments

As shown in Ref. [86], the evaluation of the momentum matrix element p~nm is more correct than the dipole matrix element because dipole elements are not well defined between infinitely-extended Bloch states. In fact, if a two-band electronic structure approximation (see Ref. [83]) for the calculation of the dipole matrix for interband transitions in a quantum well is considered, the wavefunction is: ψ(x) = φc (x)uc (x) + φv (x)uv (x),

(4.31)

where uc and uv are the zone-centre eigenfunctions for the conduction and valence band minimum with energy Ec and Ev , respectively, and φc and φv are the envelope functions [82, 32, 84]. The zone-centre eigenfunctions uc and uv are periodic in the lattice constant a, and are normalized so that the squared modulus has mean value unity over a unit cell, namely: 1 a

Z

x0 +a x0

1 |uc | dx = a 2

Z

x0 +a x0

|uv |2 dx = 1,

(4.32)

with the integrals being independent of x0 . The conduction band envelope (c) function φc is dominant for low-lying conduction band states, and obeys the effective- mass equation [82, 32, 84]. If a wide deep well of width L is considered, for the conduction groundstate inside the well the conduction envelope function will be: φc(c)

=

r

πx 2 cos , L L

(4.33)

and is zero outside. The valence envelope function of the conduction band groundstate instead approximates to: (c)

(Ec − Ev )φv(c) ≈ −

dφc i¯h pvc , m dx

(4.34)

where pvc is the interband momentum matrix element: pvc =

x0 +a

dx ∗ u puc . a v

(4.35)

r 2 πx π sin , L L L

(4.36)

Z

x0

Which therefore gives: φv(c)

≈

i¯hpvc mEg

4.3. DIPOLE MOMENTS

79

with Eg = Ec − Ev . In the same way for the highest valence band state ψ (v) will have an envelope function as follows: r 2 πx (v) φv = cos , (4.37) L L φc(v)

i¯ hpvc ≈− mEg

r πx π 2 sin . L L L

(4.38)

This discussion will be limited to the two band expansion, since it has been shown that the remaining bands do not contribute to the matrix elements in the limit of a wide well [86]. It is now possible to evaluate the dipole matrix elements for this quantumwell model: Z ∞ (c) (v) ψ (c)∗ xψ (v) dx hψ |x|ψ i = Z−∞ ∞ (φc(c) uc + φv(c) uv )∗ x(φc(v) uc + φv(v) uv )dx. (4.39) = −∞

(c)

(v)

The dominant term in ψ (c) is φc uc , while in ψ (v) is φv uv , which gives approximately: Z ∞ (c) (v) (4.40) (φc(c) uc )∗ x(φv(v) uv )dx = Icv . hψ |x|ψ i ≈ −∞

In the limit of slowly varying envelope functions the rapidly varying part of the integral can be replaced by its mean value, which for the orthogonality of the zone-centre eigenfunctions is the same for all the unit cells regardless of (c) (v) their positions. Moreover as φc and φv are one and the same normalized function, it is possible to write: Z ∞ (c) (v) (v) ∗ hψ |x|ψ i ≈ uc xuv φ(c)∗ c φv dx −∞ Z x0 +a dx ∗ ∗ u x uv . (4.41) ≈ uc xuv = a c x0 There is a disturbing aspect to this result: it depends on the choice of x0 , and this can be shown not only for this two-band approximation, but also as a general feature. This suggests that the factorization of the integrand into slowly and rapidly varying parts has been misguided. A better factorization (c)∗ (v) is φc xφv as the slowly varying part and u∗c uv as the rapidly varying part. But the mean value u∗c uv is now zero, independently of the choice of x0 [86]. So this gives zero for the whole integral, forcing the consideration of the effects the effects of the second term in the envelope expansion, even if they

80


are small, in order to calculate the dipole matrix element. So for the integral having two conduction band envelope functions in the integrand, using the fact that |uc |2 is rapidly varying and has mean value unity over a unit cell, we have: Z ∞ Z ∞ (v) φ(c)∗ (4.42) (φc(c) uc )∗ x(φc(v) uc )dx ≈ Icc = c xφc dx, −∞

−∞

and in a similar way for the integral involving two valence band envelope functions: Z ∞ Z ∞ (4.43) φv(c)∗ xφv(v) dx. (φv(c) uv )∗ x(φv(v) uv )dx ≈ Ivv = −∞

−∞

Now using the explicit forms of the envelope functions (4.33, 4.36, 4.37, 4.38), it can be see that the two integrals are equal and together contribute Icc + Ivv =

pcv imωg

(4.44)

to the dipole matrix elements, with h ¯ ωg = Eg . (c) (v) ∗ Now the integral of (φv uv ) x(φc uc ) vanishes in the limit of a wide well, since u∗v uc = 0, so finally the dipole matrix element can be written as: hψ (c) |x|ψ (v) i =

pcv . imωg

(4.45)

These results hold even if the complete envelope expansion is used. On the other hand, although the evaluation of the momentum matrix element hψ (c) |p|ψ (v) i proceeds in a similar way, it is much simpler, even though the action of the derivate on the u− and φ−type products doubles the number of terms, because only one term contributes in the limit of a wide well. In fact, the term involving the derivate of an envelope function can be dropped immediately because they contribute terms of order 1/L to the matrix element. So only terms involving derivates of the periodic functions uc and uv survive. But u∗c p uc and u∗v p uv vanish, since uc and uv are Bloch functions for stationary points in the band structure. Hence, the momentum matrix element is given by: Z ∞ Z ∞ (v) ∗ (v) ∗pu φ(c)∗ φ u p u dx = hψ (c) |p|ψ (v) i = φ(c)∗ u v v c v c c φv dx = pcv , c −∞

−∞

(4.46)

which is the interband bulk matrix element. Taking together (4.45) and (4.46) leads to: hψ (c) |p|ψ (v) i = imωg hψ (c) |x|ψ (v) i, in accordance with the standard textbook result [87].

(4.47)

4.3. DIPOLE MOMENTS

81

Thus, for the results shown in section (6.2.3) p~nm is first calculated and then is used the relation between the momentum and the dipole matrix element as given by [87]: p~nm = hψn |~ p |ψm i = im0 ωnm hψn |~r |ψm i = im0 ωnm~rnm , d~nm = e~rnm ,

(4.48) (4.49)

where ¯hωnm = En − Em . In the case of the one-band model, for the intraband transitions only p~ (φ) are different from zero, because according to group theory in a zincblende crystal the intraband momentum matrix between Bloch-cell parts |Si, |Xi, |Y i, |Zi are equal to zero, and since p~ (φ) is very small the one-band model will underestimate the intraband transition strength. However, for intraband transitions within the one-band model the dipole matrix can be calculated directly by: Z φ φ(n)∗~rφ(m) dV, (4.50) ~rnm = V

as this expression is well-defined due to the boundness of the envelope functions.

82


Chapter 5

The GaN/AlN wurtzite quantum dot The relevance of GaN/AlN quantum dot structures for optoelectronic devices has been widely shown [89]. For example, by using GaN quantum dots, intense room-temperature visible luminescence can be obtained on Si 111 substrate, and the emission energy can be continuously tuned from blue to orange by simply controlling the quantum dot size. Moreover, the mixing of properly chosen quantum dot sizes leads to white light emission [88]. An important aspect in this context is the influence of electromechanicalfield interactions and their combined effects on the bandstructure and eigenstates of the quantum dots. The governing equations for the electromechanical fields of wurtzite structures are axisymmetric, hence all electric- and mechanical-field solutions are axisymmetric as well and the original three-dimensional problem can be solved as a two-dimensional mathematical-model problem [41]. So it has been first developed a two-dimensional continuum model for computing the electromechanical fields of a cylindrical GaN/AlN quantum dot showing the importance of piezoelectric effects, spontaneous polarization, and lattice mismatch with quantitative comparisons. These results have been compared with a simpler one-dimensional case. Then, in order to also check the validity of the two-dimensional model, a complete three-dimensional fullycoupled continuum model is presented, so the results for these two models can be compared. Finally, a more realistic quantum dot shape, an hexagonal pyramid has been studied. Since the linear continuum mechanics models do not contain the full crystal symmetry, and therefore show a higher degree of symmetry than atomistic models [91], in order to quantitatively address the importance of strain, a comparison of the three-dimensional continuum mechanics and the atomistic valence-force field strain models has been made. The implications for the electronic state energies for this latter structure have been 83

84

CHAPTER 5. THE GAN/ALN WURTZITE QUANTUM DOT z (nm)

bc 1

z AlN

1.5

bc 2

GaN bc 4 r(nm)

y −1.5

φ

x

r

6

bc 3

Figure 5.1: Geometry of the system considered (left) and of the twodimensional model (right) analyzed.

5.1 5.1.1

GaN/AlN wurtzite cylindrical quantum dot Two-dimensional rotational invariant continuum model

The original three-dimensional problem of a cylindrical wurtzite quantum dot has been solved with a two-dimensional mathematical-model using equations (2.70) in a system of cylindrical coordinates (r, z, ϕ). As explained in section (2.5.2), these equations are invariant with respect to rotations around the z axis [42], hence solutions can be separated into a (r, z) part and a ϕ part. Figure 5.1 plots the three-dimensional geometry of the cylindrical quantum that has been considered (left) and the (r, z) representation (right). Because of the geometry, Dirichlet boundary conditions have been im∂V z posed along bc: 1, 2, and 3 while ur = 0, ∂u ∂r = 0 and ∂r = 0 at bc 4. The axisymmetry of the system (equations, geometry and boundary conditions) implies no angular dependence, therefore uφ = 0, and the electromechanical fields of the system are calculated by equations (2.70). Results for a cylindrical GaN quantum dot with a radius of 6 nm and an height of 3 nm embedded in a matrix of AlN are presented, with a comparison of the four following cases: • Case 1: With lattice mismatch, piezoelectric effect, and spontaneous polarization. • Case 2: With lattice mismatch and piezoelectric effect.

5.1. GAN/ALN WURTZITE CYLINDRICAL QUANTUM DOT

85

• Case 3: With piezoelectric effect and spontaneous polarization. • Case 4: Lattice mismatch only. In Figure 5.2, is plotted ur (r) for z = 0 (top) and uz (z) for r = 0 (bottom). By comparing cases 1, 2, and 4, seen to be that lattice mismatch is the main driving force for the strain-distribution respect to piezoelectricity and spontaneous polarization, as it also appears from the source term [Equation (2.72)]. A similar result for a one-dimensional GaN quantum well surrounded by AlN layers was shown in [50]. However, for an AlN quantum well surrounded by GaN, piezoelectric effect and spontaneous polarization contribute to strain more than 30% in absolute values [50], and the analysis confirms that contributions from piezoelectric and spontaneous polarization amount to approximately 30% of the lattice mismatch. Since spontaneous polarization and piezoelectricity destroy the inversion symmetry, the solutions to the set of partial-differential equations in general do not satisfy parity. This is indeed found in cases 1-3 as the electric field (displacement vector) is even (odd) with respect to inversion (see also the figures 4 and 5). In the inversion symmetry case (case 4), the electric field is zero and the displacement vector is odd with respect to inversion. In Figure 5.3 the strain tensors εrr and εzz at r = 0 and at z = 0 are plotted for the various cases, while in Figure 5.4 two-dimensional plots of the strains εrr (top), εzz (center), and εrz (bottom) are indicated. First it is apparent that the dot material is compressed by the barrier material and the barrier material close to the dot is compressed by the dot material. This is in accordance with what is observed in other systems and can be understood from the conceptual idea of preparing the system with a shrink fit procedure [8]. Even if the εrz component is divergent at the corners, the solution away from the corners are still expected to be reasonable [85]. Furthermore, the reason why the most relevant difference between case 1 and cases 2 and 4 is along the z-direction is that the z-components Pz of the polarization [see equation (2.43)] are the strongest. This is due both to the spontaneous polarization, which in a wurtzite structure is directed along the z-axis, and to the fact that the diagonal components of the strain tensor, on which Pz depends, are stronger than the off-diagonal which govern the planar components of the polarization. For completeness the z and r components of the electric field E are plotted in Figure 5.5 .

5.1.2

Comparison with a one-dimensional case

It is interesting to make a comparison between the results recorded for the three-dimensional system from the previous section and some analytical results for a simpler one-dimensional case [50]. Results are compared not only for the previous system: a GaN quantum dot embedded in a matrix of AlN

86

CHAPTER 5. THE GAN/ALN WURTZITE QUANTUM DOT

Figure 5.2: Displacement vectors ur (r) for z = 0 (top) and uz (z) for r = 0 (bottom). (GaN/AlN), but also for an AlN quantum embedded in a matrix of GaN (AlN/GaN). The analytical expression for the stress tensor in one dimension can be written as a sum of three terms: an uncoupled term due to mechanical fields only, a coupled piezoelectric term combining electrical and mechanical fields, and finally a term due to couplings between spontaneous polarization and piezoelectricity. If we identify the quantum dot as layer (1) and the matrix as layer (2) it can be written as [50]:


87

Figure 5.3: Strain tensors εrr (top left) and εzz (top right) at r = 0, and εrr (bottom left) and εzz (bottom right) at z = 0.

(1)

ε(1) zz =

2C13

(1)

C33

(1)

a−

(1)

(1)

(1)

(1)

2e33 [e33 C13 − e13 C33 ] (1)

(1)

(1)

(1)

C33 [ˆ ε33 C33 + (e33 )2 ]

(1)

a+

(2)

(1)

e33 (Psp − Psp ) (1)

(1)

(1)

εˆ33 C33 + (e33 )2

εzz = εuncoupled + εcoupled + εsp zz zz zz .

,

(5.1)

(5.2)

In an analogous way the z component of the electric field for the coupled and the uncoupled cases [49] (setting eij = 0) is given by: (2)

Ezuncoupled,1 =

(1)

Ezcoupled,1 =

(1)

(1)

(1)

2[e33 C13 − e13 C33 ] (1)

(1)

(1)

(1)

Psp − Psp

εˆ33 C33 + (e33 )2

(1)

εˆ33

(1)

a + C33

,

(5.3)

(2)

(1)

Psp − Psp

(1) (1)

(1)

C33 εˆ33 + (e33 )2

.

(5.4)

88


Figure 5.4: Strains εrr (left), εzz (center), and εrz (right) in the r, z plane. All the effects have been included.

In Table 5.1, we can see results for the strain tensor for the one-dimensional case (1D) and the model presented in the previous section (3D) correspond-


89

Figure 5.5: The z (top) and r (bottom) components of the electric field in the r, z plane. ing to r = 0 and z = 0 introducing: ∆ε =

εzz (3D) − εzz (1D) , εzz (1D)

(5.5)

and in Table 5.2 the electric field for the uncoupled and the coupled cases (1D) and the 3D result (evaluated at r = 0 and z = 0) are listed in addition

90


to the relative difference ∆E defined in a similar manner as ∆ε .

GaN/AlN AlN/GaN

εuncoupled zz 0.0123 −0.0143

εcoupled zz −0.0008 0.0026

εsp zz −0.0014 0.0026

εzz (1D) 0.0101 −0.0091

εzz (3D) −0.0050 0.0068

|∆ε | 150% 175%

Table 5.1: The εzz strain values in the different (one-dimensional) cases, the total 1D value, the corresponding 3D value at r = 0 and z = 0 (3D), and finally the difference in percentage between a 1D and a 3D analysis (|∆ε |).

GaN/AlN AlN/GaN

Ezuncoupled(GV /m) −0.608 0.716

Ezcoupled (GV /m) −0.980 1.309

Ez (3D)(GV /m) −0.613 0.765

|∆E | 37% 42%

Table 5.2: Electric-field z-component for the uncoupled and coupled cases (1D) compared to the present 2D results evaluated at r = 0 and z = 0. What is apparent is the relatively big impact of employing a 3D analysis as compared to a 1D analysis on both strains and electric-field values (∆ε is numerically larger than 100 %). It should also be noted that the piezoelectric and spontaneous-polarization coupling effects play an important part on the strain results in AlN/GaN system. It is instructive to observe that the latter two coupling contributions compensate, albeit slightly, to the big difference between the 1D and 3D results. Furthermore, the strain fields is compressive in the 3D case whereas in the 1D case it is tensile for the GaN/AlN system. In the 1D case the expansion in the z-direction is due to the Poisson effect (the substrate compresses the material perpendicular to the z axis resulting in an expansion in z-direction) [26]. In the 3D case the dot is not free to expand in the z-direction due to the presence of matrix material around the dot (and the opposite applies to the AlN/GaN system). This is the main reason behind the pronounced difference observed. From Table 5.2 it can be seen that the electric field in the 1D and the 3D case has the same direction but the magnitude is smaller in the 3D case. The reason for this is twofold. Firstly, as was observable in Table 5.1 there is a substantial difference in the strain between the 1D and the 3D cases. This will also change the electric field due to the coupling in the constitutive relation for the electric displacement (last three equations of (2.68)). Using the constitutive relation this difference is estimated to be around 0.13 GV/m. Secondly, a reduction in the electric field exists due to the geometry induced fringe field.

5.2. THE GAN/ALN WURTZITE HEXAGONAL PYRAMID QUANTUM DOT91

5.1.3

The three-dimensional fully-coupled continuum model

In order to check the validity of the two-dimensional rotational-invariant model of section (2.5.2), the complete three-dimensional fully coupled model given by equations (2.48) for the same cylindrical wurtzite GaN/AlN quantum dot as in Figure 5.1 (r = 6 nm, h = 3 nm) has been solved. The strain tensor εzz calculated by the two-dimensional model (top) and the three-dimensional model (bottom) is plotted in Figure 5.6. The plot for the three-dimensional model shows a slide of the quantum dot in the xy plane, correspondent to the double of the surfaces in the r, z plane given by the rotational-invariant model in two dimensions. Not only is an almost perfect qualitative agreement of the strain field in the two plots observable, but there is also an excellent agreement of the maximum and minimum values of εzz inside of the dot and in the matrix. With the three-dimensional model we can also verify the cylindrical symmetry of the electric field E for a wurtzite cylindrical quantum dot in the xy plane, showing in Figure 5.7 the absolute value of E on this plane.

5.2

The GaN/AlN wurtzite hexagonal pyramid quantum dot

In this section three different hexagonal quantum dots with wetting layer (refer to Figure 5.8 for parameter meanings and geometry) are studied. The dimensions of the dots are given in Table 5.3. The first two dots (Dot 1 and 2) are narrow dots with a bottom diameter of 4.936 nm and the last (Dot 3) is wide having a bottom diameter of 8.638 nm.

Dot 1 Dot 2 Dot 3

Db [nm] 4.936 4.936 8.638

Dt [nm] 4.319 2.468 8.021

H [nm] 1.011 2.526 1.011

W [nm] 0.505 0.505 0.505

Table 5.3: The dimensions of the quantum dots. In the first three rows of Figure 5.10 the strain components εxx and εzz and the electric potential V for the three dots along the center of the structures are shown. There is a clear qualitative agreement for the εxx and εzz strain components between the three models: VFF, fully-coupled, and semi-coupled continuum. However, quantitative differences exist and locally up to approximately 25 % between VFF and continuum models. Further, these results demonstrate good quantitative agreement between semi-coupled and fully-coupled continuum data (locally up to maximum 5 %). Similarly for

92


Figure 5.6: The strain tensor εzz given by the two-dimensional model in the r, z plane (top) and by the three-dimensional model in the y, z plane (bottom).

the electric potential a notable difference between the VFF and the continuum results is observed. The differences in the electric field are a direct


Figure 5.7: The absolute value of the electric field E in the x, y plane for a cylindrical GaN/AlN wurtzite quantum dot. H W

Dt

Db

Figure 5.8: The geometry and the parameters of the hexagonal pyramid quantum dots. consequence of the differences in the strain fields. Since VFF parameters are computed using non-piezoelectric corrected stiffness coefficients, better agreement between semi-coupled continuum and VFF vs. fully-coupled continuum and VFF was expected. This is indeed confirmed by the results in Figure 5.10. Deviations between VFF and continuum results for the electric potential are mainly due to discrepancies in the off-diagonal strain components εij (i 6= j) (see Figure 5.9).

94


Figure 5.9: The εxz strain component for Dot 1. In Table 5.4 the electronic groundstate energies for the three dots found using the effective mass approximation for the conduction band are shown. The effect of strain and electric field has been included via the effective potential [40]: Vef f = a(εxx + εyy ) + b(εzz ) + eV + Eedge . Here a and b are deformation potentials, e is the electronic charge, and Eedge is the bulk band edge. The effective potential for the three dots is shown in row four of Figure 5.10. Firstly, a difference between the VFF and the continuum results of up to 100 meV is observed. Secondly, a smaller difference between the fully-coupled and the semi-coupled models for dots 1 − 3 of up to 15 meV is found. The differences in the effective potential are responsible for the variations in groundstate energies observed in Table 5.4.

Fully-coupled model (meV) Semi-coupled model (meV) Valence force field model (meV)

Dot 1 489 474 401

Dot 2 251 237 121

Dot 3 390 371 308

Table 5.4: The groundstate energies of the quantum dots.


Figure 5.10: Solid black, dashed blue, and dashed red line codings correspond to fully-coupled continuum, semi-coupled continuum, and VFF data, respectively. The first, second, and third columns show results for Dot 1, 2, and 3, respectively.

96


Chapter 6

The InAs/GaAs zincblende quantum dot The huge impact the InAs/GaAs quantum dot structures could have in the field of optoelectronic devices has recently been shown [89]. They have been proposed as components in devices for controlling the emission pattern of phased array antennas [94], as ultrafast optical amplifiers or in all–optical switches [95], as active media in buffers based on slow-light phenomena, utilizing either electromagnetically induced transparency (EIT) [20] or population oscillation [96]. Important aspects in this context are to accurately model the optical properties of the quantum dots and to consider the influence of the strain field and its effects on the bandstructure and eigenstates. In order to calculate the strain field, due to computational considerations and using the isotropic assumption (2.65) a two-dimensional model in cylindrical coordinates has been developed, and results for the strain fields with a fully coupled three-dimensional continuum model compared, in order also to verify the validity of our assumption. We know that optoelectronic properties of the InGaAs zincblende quantum dot with varying shape and size based on ~k · p~ theory have already been studied by Schliwa et al. [97] and Veprek et al. [98]. However, so far, only a small selection of all possible transitions have been studied. Here the derived eight-band model as presented in section (3.3.3) is applied to zincblende InAs/GaAs conical quantum dots and the impact of different size and shape is studied, with a focus on the most relevant interband transitions [see section (4.2)]. Furthermore, there is an investigation of the effect of strain and band-mixing between the conduction band and valence bands by comparing four different ~k · ~ p model. Although the previous studies do include these effects their impact on dipole moments have as yet not been investigated. 97

98

CHAPTER 6. THE INAS/GAAS ZINCBLENDE QUANTUM DOT

6.1

Three-dimensional strain distributions due to anisotropic effects

It has been shown in section (2.5.1) that zincblende materials are not axisymmetric. This led to the isotropic assumption 2C44 + C12 = C11 (2.65) used to write equations (2.67) in cylindrical coordinates (r, z, ϕ), which have been solved for the InAs cylindrical quantum dot as shown in Figure 6.1, embedded in a GaAs matrix. Because of the geometry, we have imposed z (nm)

bc 1

15

z

GaAs

1.5

bc 2

InAs bc 4 r(nm)

y x

−1.5

φ

6

r

−15

bc 3

60

Figure 6.1: Geometry of the system considered (left) and of the twodimensional model (right) z Dirichlet boundary conditions along bc: 1, 2, and 3 while ur = 0, ∂u ∂r = 0 and ∂V ∂r = 0 at bc 4. Using these conditions it is also possible to remove the angular dependence. Table 6.1 shows the real values for the elastic tensors and the values given by the assumption for both materials, in order to evaluate the difference for these parameters.

InAs GaAs

C11 (GPa) 83.29 122.1

C12 (GPa) 45.26 56.6

C44 (GPa) 39.59 60.0

C12 + 2C44 (GPa) 124.44 176.60

Table 6.1: The real values and in the last column the values given by the isotropic assumption (2.65) for the elastic tensors. To verify the validity of the assumption for the calculations of the strain fields, the three-dimensional equations (2.47) for the cylindrical quantum dot of Figure 6.1, with radius r = 6 nm and height h = 3 nm, have also been solved.

6.1. THREE-DIMENSIONAL STRAIN DISTRIBUTIONS DUE TO ANISOTROPIC EFFECTS99

Figure 6.2: Strain fields εzz (top) and εyy (bottom) at x = 0, y = 0 (left) and x = 0, z = 0 (right), respectively (red line Anisotropic, blue line Isotropic). Figure 6.2 plots the strain fields εzz (top) and εyy (bottom) at x = 0, y = 0 (left) and x = 0, z = 0 (right), respectively, and compares the results of the two-dimensional model with the isotropic assumption (blue line) and of the three-dimensional anisotropic model (red line). The first thing to notice is that there is a qualitatively different behavior of the strain fields in the middle of the dot along the z direction: there is a minimum for the isotropic model while the real anisotropic model shows a maximum. In addition, a remarkable quantitative difference both inside the dot and in the matrix close to the dot, with an error of almost 100% for εzz in the middle of the dot can be observed. Therefore, the first choice for the isotropic assumption does not seem to be very reliable. However, looking at the plot of the strain fields εzz and (εxx + εyy ) given by the anisotropic three-dimensional model as shown in Figure 6.3, it is still apparent that they show a sort of cylindrical symmetry. In order to satisfy the isotropic assumption (2.65), the real values for C12

100 CHAPTER 6. THE INAS/GAAS ZINCBLENDE QUANTUM DOT

Figure 6.3: εzz in the (x, y) (top left) and (y, z) (top right) plane, respectively, and (εxx + εyy ) in the (x, y) plane (bottom). and C44 and the value given by the assumption for C11 have been chosen. But obviously there are different ways to satisfy this assumption, and so another one was tried. This second choice has been derived from an analytical strain distribution around a spherical inclusion [99], and is also supported by the experimental data for the Poisson ratio ν of isotropic semiconductors, which approximately equals one third [99]. The values of the Poisson ratio and of the elastic tensors are related [26], so by imposing ν = 13 it is possible to derive the following relations: C12 =

C11 2

C44 =

C11 . 4

(6.1)

In other words, only C11 was fixed and C12 and C44 were calculated using (6.1). Table 6.2 shows the real values for the elastic tensors, together with the values given by both the choices for the assumption. It can be seen that

InAs GaAs

C11 83.29 122.1

C12 45.26 56.6

C44 39.59 60.0

C12 + 2C44 124.44 176.60

C11 /2 41.64 61.05

C11 /4 20.82 30.52

Table 6.2: The real values and the values given by the two possible choices for the isotropic assumption for the elastic tensors (all values in GPa). with the first choice C11 is affected by an error of about 50%, while in the

6.1. THREE-DIMENSIONAL STRAIN DISTRIBUTIONS DUE TO ANISOTROPIC EFFECTS101 second case there is an error of about 100% for C44 , while C12 is almost correct, and that the elastic tensors have a different weight on the strain fields is known [see again equations (2.67)]. Thus, the comparison made in Figure 6.2 is also repeated in Figure 6.4, using the same legend, i.e., blue line for the isotropic and red line for anisotropic case, respectively, but this time using the values for the elastic tensors given by (6.1).

Figure 6.4: Strain fields εzz (top) and εyy (bottom) at x = 0, y = 0 (left) and x = 0, z = 0 (right), respectively (Red line Anisotropic, blue line Isotropic). It is possible to notice that, although there is still a qualitatively different slope of the strain fields in the middle of the dot, a considerable reduction of the errors is achieved. This entails that if the three-dimensional problem of calculating the strain fields for a zincblende quantum dot is reduced to a twodimensional model, satisfying the isotropic assumption (2.65) by the (6.1) we arrive at the most correct values. This can be observed in Figure 6.5, where the strain field εzz at x = 0, y = 0 for the three-dimensional model (red line) and the isotropic two-dimensional model with the first (blue line)

102 CHAPTER 6. THE INAS/GAAS ZINCBLENDE QUANTUM DOT and the second (green line) choice for the values of the elastic tensors are plotted together.

Figure 6.5: Strain fields εzz at (x = 0, y = 0) given by the three-dimensional model (red line), and by the two-dimensional isotropic model with the first (blue) and second (green) choice for the values of the elastic tensors. It is important to mention in this context that solving a three-dimensional fully-coupled or semi-coupled model gives almost the same results, i.e., a solution to equations (2.47) including or excluding the piezoelectric effect affects the strain fields very slightly. This is mainly due to the fact that in a zincblende quantum dot the polarization is given by the off-diagonal components of the strain fields, as can be seen from equations (2.40). This leads to a weak electric field almost all outside of the dot, as we can see from Figure 6.6, where it is shown the absolute value of the electric field for this cylindrical quantum dot, given by the fully-coupled three-dimensional model. It is interesting to compare this last result with the one given in the previous chapter in Figure for a GaN/AlN wurtzite quantum dot with the same shape and dimensions. Not only the electric fields present a completely different symmetry, because of the different components which generate the piezoelectrical fields [see equations (2.40) and (2.43)], but the maximum value in the zincblende case is almost 30 time smaller as well. A similar result has already been observed in [40].

6.2 6.2.1

Optical properties of strained quantum dots Theory and system

The dipole moments presented in the following section are determined using eight-band ~k · p~ theory including strain effects. There is an investigation of the effects of both strain and band mixing. The effect of band mixing is determined by comparing the eight-band results with one-band model results for the conduction band and the heavy holes, models which were presented in

6.2. OPTICAL PROPERTIES OF STRAINED QUANTUM DOTS

103

Figure 6.6: The absolute value of the electric field in the (x, y) plane. details with regards zincblende quantum dots is sections (3.3.1) and (3.3.3). Due to the lattice mismatch present in the systems under consideration the materials will be strained. The strain fields are found by minimizing the elastic strain energy, as shown in Chapter 2. In the eight-band model the wavefunctions are given by a linear combination of the eight Bloch states weighted by an envelope function, ψn =

8 X

(n)

φi ui ,

(6.2)

i=1

(n)

where φi are the envelope functions and ui are the Bloch states. The eight-band Hamiltonian is used as described in section (3.3.3), with the strain-dependent part given in section (3.3.6). As already shown in section (4.3) the evaluation of the momentum matrix element p~nm is meaningful while the dipole matrix elements are ill-defined in crystals involving (infinitely-extended) Bloch states. Thus, first it is necessary to calculate p~nm and then use the relation between the momentum and the electric dipole matrix element given by [87]: p~nm =

im0 ωnm ~µnm , e

(6.3)

where ¯hωnm = En − Em , m0 is the free electron mass and e is the electronic charge. Due to computational considerations only cylindrical symmetric quantum dots are investigated. Specifically the study is centred on the conical quantum dots shown in Fig. 6.7, since nearly conical-shaped InAs/GaAs


Figure 6.7: The shape of the quantum dots under consideration.

quantum dots have been grown [90]. The dot material is InAs and the barrier material is GaAs. Both InAs and GaAs are zincblende materials so in order to reduce the problem to a two dimensional model the isotropic assumption (6.1) is made. This entails that we disregard phenomena such as piezoelectricity and atomistic anisotropic effects [40, 100]. The atomistic anisotropic effects have been investigated by Bester and Zunger [91] showing that this leads amongst other things to a splitting of states which in model in question are degenerate. This splitting however is less pronounced for cylindrical shaped quantum dots as studied here. Recently it has been shown that second order piezoelectric terms effectively cancel out the linear piezoelectric effects for cylindrical shaped quantum dots [101, 102, 97]. The isotropic assumption has been checked and a maximum error for the strain fields of 8% along the z axis identified, going rapidly to zero at the edges of the dot. For the material parameters used in calculations the reader is referred to Ref. [103].

6.2.2

Spurious solutions

In section (3.3.4) and (3.3.5) it has been shown how it is possible to circumvent the problem of spurious solutions by changing the Kane energy Ep as suggested by Foreman [74] or by the cut-off method. In this short section the results given by applying these methods for a conical quantum dot, as shown in Figure 6.7 with r = 18nm and h = 9nm using eight band k · p theory including strain, are presented. √ Besides the isotropic assumption (6.1), it is assumed that γ2 = γ3 and 3b = d (displacement potentials) as this ensures that the resulting model is cylindrical symmetric [see section (3.3.7) for the reasons of this choice].


105

So in order to use the plane wave cut-off approach for a cylindrical symmetric system Bessel functions should be used as the radial expansion basis [Sections (3.3.4), (3.3.5) and (3.3.7)]. Table 6.3 lists the first four conduction-band energies found using the two approaches. Changed parameters Cut-off Difference

0.481 0.473 0.008

0.517 0.506 0.011

0.517 0.507 0.010

0.544 0.531 0.013

Table 6.3: The first four conduction-band energies found using the planewave cut-off method and the change of parameters approach and the difference between them. All energies are in eV. The choice of approach for the elimination of spurious solutions influences the energy spectrum resulting in a difference of around 10 meV for the conical dot under consideration here. The following results have been obtained using the changed parameters approach.

6.2.3

Results: bandstructure and dipole moments

The first results presented are related to a set of conical quantum dots where the aspect ratio between the radius r and the height h of the dot has been fixed so that r = 2h. Focus has been centred on the first twelve bound states for both bands. Since all the states are at least double degenerated (spin degeneracy), six energy levels (labeled from 1 to 6) are connsidered for both bands. In the one-band model, due to the conical quantum-dot symmetry and isotropy assumption (giving a inversion symmetric model), level 2 and level 3 are degenerate for both the conduction and the valence band, level 4 and level 5 are degenerate for the valence band, and level 5 and level 6 are degenerate for the conduction band. In the upper part of Fig. 6.8 and Fig. 6.9 are shown the levels and the most relevant interband dipole moments corresponding to a dot with h = 7.5 nm for four different models: • one-band model without strain • one-band model with strain • eight-band model without strain • eight-band model with strain. With µij the interband dipole moment between energy level i in the conduction band and energy level j in the valence band is denoted. It has been decided to consider only right-handed circularly polarized light.


Eight band without Strain

One band without Strain

6 5 4 3 2 1

5, 6 4 2, 3 1 µ µ

µ

16 41 46

µ

22

µ

µ

11

4, 5 6

8

8

6

6

µ (Åe)

10

µ (Åe)

10

2 0

µ

µ

11

1 2 3 4 5 6

1 2, 3

4

µ µ

16 41 46 22

4 2

µ33 µ11 µ55 µ22 µ64 µ46 µ41 µ16

0

µ33 µ11 µ55 µ22 µ64 µ46 µ41 µ16

Figure 6.8: Energy levels and the most relevant dipole moments (top) and values of the eight strongest interband dipole moments (bottom) for oneband (left) and eight-band (right) model without strain for a dot with h = 7.5 nm.

In Figs. 6.8 and 6.9 the thickness of lines indicating the transitions are proportional to the corresponding dipole moments. Obviously, an eightband model calculation leads to a higher valence-band density-of-states as compared to a one-band calculation. Hence, also more interband transitions result, in a given energy range, when using an eight-band model. Only the strongest eight dipole-matrix elements for both one-band and eight-band models are shown in the bottom part of Fig. 6.8 and Fig. 6.9. First, it can be observed that there is a qualitative agreement between dipole-moments results for the one-band and eight-band models with strain (see Figure 6.9). This is due to the fact that in the eight-band model the biaxial strain component of Eq. (3.37) shifts heavy-holes (light-holes) to higher (lower) energies. As a consequence the valence-band groundstate


Eight band with Strain

One band with Strain

6 5 4 3 2 1

5, 6 4 2, 3 1 µ µ µ µ µ 16 41 46 22 11

µ µ µ

8

8

6

6

µ (Åe)

10

µ (Åe)

10

0

µ

11

1 2 3 4 5 6

6

2

µ

16 41 46 22

1 2, 3 4, 5

4

107

4 2

µ33 µ11 µ55 µ22 µ64 µ46 µ41 µ16

0

µ33 µ11 µ55 µ22 µ64 µ46 µ41 µ16

Figure 6.9: Energy levels and the most relevant dipole moments (top) and values of the eight strongest interband dipole moments (bottom) for oneband (left) and eight-band (right) model with strain for a dot with h = 7.5 nm.

and the first excited states are predominantly heavy holes-like giving rise to a general better agreement between the one-band model and the eight-band model. Second, the inclusion of strain reduces the strength of the dipole moments significantly. This is because there is a non-trivial influence of strain on dipole moments. The conduction-band states are only affected by the hydrostatic strain component of Eq. (3.36) giving rise more or less to a constant shift in the effective potential inside the dot while the valence band states, in addition to the hydrostatic strain, are also affected by the biaxial strain component [see Eq. (3.37)]. The latter component is highly inhomogeneous inside the dot. In the eight-band model there is a third contribution from ǫxz and ǫyz strain components [75] [see also section (3.3.6)] but this term is not as significant as the biaxial of Eq. (3.37).

108 CHAPTER 6. THE INAS/GAAS ZINCBLENDE QUANTUM DOT In order to understand the influence of strain on the dipole moment we compare in Fig. 6.10 the valence-band groundstate probability density |ψ|2 (eight-band model) with and without the influence of the strain field for a quantum dot with height h = 11.5 nm and radius r = 23 nm, namely the bigger dot which has been investigated.

Figure 6.10: Probability density |ψ|2 of the valence-band groundstate for the eight-band model without (up) and with (down) strain. The dimensions of the quantum dot are h = 11.5 nm and r = 23 nm.

While in those cases without strain the groundstate shows a s-like shape, the biaxial-strain component modifies the hole wave function into a toroidal shape moving it away from the center of the dot where the potential is stronger. This drastically reduces the overlap between the envelope functions φi of the conduction and valence band and consequently the corresponding dipole moments. The reduction of the dipole moment due to strain is evident in Fig. 6.11 where the interband dipole moment µ11 between the conduction- and the valence-band ground states for the four different models is plotted as a function of h. In the models with strain we have a maximum around h = 5 nm while the dipole moments for the two models without strain grow monotonically with increasing height and eventually reach a plateau value corresponding to the bulk value of 16.8 ˚ Ae. The monotonic increase of the dipole moments can be understood based on Eq. (6.3). Without strain the momentum matrix elements p~ [mainly determined by p~ (u) , see equations (4.20)] remains constant with increasing height whereas the energy difference ωnm decreased.

6.2. OPTICAL PROPERTIES OF STRAINED QUANTUM DOTS 10

109

One band without Strain One band with Strain Eight band without Strain Eight band with Strain

9 8

µ11 (Åe)

7 6 5 4 3 2

4

5

6

7

8

9

10

11

h (nm) Figure 6.11: Interband µ11 dipole moments as a function of h for the four different models.

The presence of strain reduces the overlap of electron and hole distributions as a result of the increased displacement of the hole wave functions away from the center leading to the observed decrease in the dipole moments. In Fig. 6.12 the energies of the first six levels in the conduction (top) and valence band (bottom) are plotted as a function of h. The wave functions of the confined states are characterized (in the eight-band model) by eight envelopes weighted differently depending on the state. As mentioned above, the biaxial strain is inhomogeneous and this, combined with the differently spatially distributed envelope functions, leads to a higher sensitivity against strain as compared to a one-band model. This effect grows with volume. These coupled strain-band mixing effects lead to energy crossings in the valence band. Also plotted is where the wetting layer continuum starts (WL), considering a 0.5 nm wetting layer which has not been included in the bandstructure calculations. Is is observable that only the last three considered conduction-band levels for the smallest quantum dot lay above the lower bound of the wetting layer continuum states. The wetting layer continuum has been calculated with a Ben Daniel-Duke model for a onedimensional quantum well. The most relevant dipole moments are plotted for the eight-band model in the upper part of Fig. 6.13 as a function of h. In the bottom part, the related energy differences ∆E are indicated. Due to strain it is possible to

E (eV)

0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5

c

E1 c E2 c E3 c E4 c E5 c E6 WL

−0.35 v

E (eV)

−0.4

E1 v E2 v E3 v E4 v E5 v E6 WL

−0.45 −0.5

−0.55 −0.6

4

5

6

7

8

9

10

11

h (nm) Figure 6.12: Energies of the six considered levels in conduction (top) and valence (bottom) band as a function of h for eight-band model with strain. The onset of continuous wetting layer states (WL) is also indicated (dashedhorizontal line).

detect an almost general reduction in dipole-moment strengths with increasing volume as a result of the decreasing wave function overlap (similar to what was found for µ11 ). This geometric effect is indeed mainly a function of the dot volume as Fig. 6.14 shows. Here, a second set of quantum dots with different aspect ratios (Asp = hr ) having the same volume (V = 226.19 (nm)3 ) are referred to. The interband dipole moments (top) and relative energy differences (bottom) are depicted as a function of Asp and evidently results attained are rather insensitive to the aspect ratio.

110

µ (Åe)

7 6 5 4 3 2 1 0

µ11 µ22 µ46 µ41 µ16

∆E (eV)

1.3 1.2 1.1 1 0.9 0.8

4

5

6

7

8

h (nm)

9

10

11

Figure 6.13: Interband dipole moments µ (top) and relative band energy difference ∆E (bottom) as a function of the height h with fixed aspect ratio r = 2h.

7

µ (Åe)

6 5 4 3 2 1

∆E (eV)

0 1.4 1.3 1.2 1.1

µ11 µ22 µ46 µ41 µ16

1 0.9 1

1.5

2

2.5

3

3.5

4

Aspect ratio Figure 6.14: Interband dipole moments µ (top) and relative band energy difference ∆E (bottom) as a function of the aspect ratio at constant volume.

111

112

Conclusions The development of mathematical models for the electromechanical, electronic and optical properties of quantum dot heterostructures, in order to present realistic and reliable numerical results was the main goal of this thesis. I focused on two different structures, due to the relevance they have shown for new optoelectronic devices: GaN/AlN wurtzite quantum dots and InAs/GaAs zincblende quantum dots. For the analysis of the electromechanical fields different appropriate versions of the continuum mechanical model have mainly been implemented, while the calculations of the bandstructures have been achieved using multiband ~k · ~p theory. A first set of results is given for the strain and electric field distributions, emphasizing the coupled influence of lattice mismatch, piezoelectricity, and spontaneous polarization in an axisymmetrical cylindrical GaN wurtzite quantum dot embedded in an AlN matrix, using a two-dimensional rotational-invariant model in cylindrical coordinates. Results indicate that lattice mismatch dominates the resulting field-distribution values, however, in cases with piezoelectricity and spontaneous polarization, a violation of inversion symmetry is demonstrated. It is important to emphasize that the approach undertaken is general and can be easily applied to any geometry, material combination, and crystal structure. These results have been compared with the electromechanical fields given by a complete three-dimensional model for the same cylindrical structure, showing both a qualitative and quantitative agreement for these two models for a wurtzite structure, which is due to the rotational invariance of the governing equations. The cylindrical symmetry of the electric field generated by the piezoelectrical effect has also been demonstrated. As a further step, a wurtzite hexagonal quantum-dot structure has been studied, and there has been a comparison undertaken of the results for the electromechanical fields given by three different models: the atomistic valence force field and the continuum semi- and fully-coupled models, respectively. In addition, the impact of the electromechanical fields on the electronic bandstructure of these dots has been shown. A general qualitative agreement between atomistic strain calculations and continuum elastic models for this wurtzite hexagonal quantum-dot structure has been shown, while quantitative discrepancies were observed for all dots taken into account. Less 113

114 CHAPTER 6. THE INAS/GAAS ZINCBLENDE QUANTUM DOT significant differences were found between semi- and fully-coupled continuum models. Due to computational reasons, it was developed a two-dimensional continuum model for a cylindrical InAs zincblende quantum dot as well, although the governing equations for this structure are not axisymmetric. It has been shown that it is possible to obtain results comparable to the real three-dimensional case, with an appropriate choice for the material parameters in the isotropic assumption adopted. The same isotropic assumption has been used to include strain effect in an eight-band ~k · p~ model used for the analysis of the optical properties of conical InAS/GaAs quantum dot. For the system under investigation it has been shown that the strain field generally reduces optical transition strengths as a function of the volume of the quantum dot for a fixed aspect ratio. This is due to a decreasing overlap of the involved wave functions. This geometric effect has been shown to be an effect of volume rather than of shape. Moreover, the combined influence of band mixing and strain entails state crossings in the valence band, and a separation of the heavy and light holes. The latter leads to a qualitative agreement between the lower conduction and upper most valence-band states computed using the one-band model and the eight-band model, with discrepancies which could be relevant in optical applications.

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Acknowledgements This thesis is the end of three years of work. A long journey which started in September 2006 when I decided to move away from my hometown Rome, away from my entire world to move to the small, and at that time completely unknown city of Sønderborg. This would never have been possible without all the people teaching me, supporting me, helping me and loving me all along my way. First of all, I want to thank my supervisors, Prof. Morten Willatzen and Dr. Benny Lassen, who have been my irreplaceable guides to my PhD degree. They really showed me in details what it means to do research. Not only have they taught me this and helped me to improve my capacities, they have as well been an example of the sobriety and responsibility that a researcher should always have in mind. Last but not least, they gave me their dear and sincere friendship. I want to thank all the other people involved in the QUEST project with whom I have worked together: Prof. Jesper Mørk, Prof. Antti-Pekka Jauho, Prof. Torben Nielsen and finally Jakob Houmark. I also thank Prof. Roderick Melnick and Prof. Lok C. Lew Yan Voon for the work and discussions that we have shared and the precious suggestions that they have always given me. I want to thank Professor Stephen Parsons for his wonderful job of checking my English. Any remaining errors are mine, of course. My sincere thanks also go to all my colleagues at the Mads Clausen Institute. I thank especially Jerome and Krzysztof for all the nice talks that we have had, and the other PhD students with whom I shared the PhD room: Lars, Fei, Feng, Yu Go and all the others. Working is always less hard if you are in good company. I thank all the colleagues from the administrative staff for the fundamental service that they always provide. In particular, I want to thank my friend Charlotte Bolding Andersen, since she not only managed to solve all my problems - even the most complicated ones, and always with a smile - but also because our “caffé” breaks have been precious moments of rest in even the busiest working days. Moreover, because she really helped me to understand the Danish everyday life. And now to something more personal. I have dedicated this thesis to two of the most important persons in my life, my mother and my brother. My 121

mum has always been my angel; from the day that I was born to the recent days while I was finishing this thesis, and she took care of the rest of my life which I was too busy to think about. Without her my physicist career would never have started. My brother is my brother, and this says everything about what we have shared and will share for the rest of our lives. I also want to thank the other members of my family who have supported me along these years: my aunt Anna, my uncle Renato, Federica and finally Ilaria. I thank my father, wherever his smile is now. I thank Silvia for the sunniest evenings, the warmest winter, the most talkative silences, the happiest words I have ever had. I thank all my friends who, here and there, now and yesterday, supported me with their love. I thank “the Italians” Francesco and Sabina, Luciano, Luigi, Vanessa, Giorgio, Gianni, Betta, Raffa, Fra, Mapi, Pierpaolo and family, Paolo and Silvia, Tommaso, Carlo, Daniele, Jean Michel, the geese & Co, my swimmers, Frappi and “the ferox” in particular. Mainly, I thank Grazia for her complete friendship in the good as in the bad times; Lisa for her generosity and her dearness void of any egoism; and Simona, my port among the storms of life. Finally, I also want to thank all the people that I have met in Sønderborg and who have helped me to start a new life in this small, but also surprising city. I thank Roberto, Santino, Giovanni, Nico and all the rest of his staff; Brian and his family; Dragan and his family; Jesper, Henrik, Tim; Nils and his family, Zuhair and Linea, Daimi and all the “gang” at the Agurketid. Especially, I thank Giorgia for all the laughs, dinners, blues and happiness that we have shared in this far place. Tilkendegivelser Denne afhandling afslutter tre ˚ ars arbejde. En lang rejse, der startede i september 2006, da jeg besluttede mig for at rejse væk fra min hjemby Rom; væk fra hele min verden for at flytte til den lille og dengang endnu helt ukendte by Sønderborg. Dette ville aldrig have kunnet lade sig gøre uden de mennesker, som har undervist mig, støttet mig, hjulpet mig og elsket mig hele vejen. For det første vil jeg takke mine vejledere, Professor Morten Willatzen og Adjunkt Benny Lassen, som har været mine uvurderlige vejvisere til min Ph.d. grad. De har virkeligt i detaljer vist mig hvad det vil sige at udføre forskning. De har ikke alene lært mig dette og hjulpet mig til at forbedre mine evner; de har ogs˚ aværet et eksempel p˚ aden seriøsitet og ansvarlighed, som en forsker altid skal udvise. Sidst men ikke mindst har de givet mig deres varme og oprigtige venskab. Jeg ønsker at takke alle de øvrige personer involveret i QUEST projektet, og som jeg har arbejdet sammen med: Prof. Jesper Mørk, Prof. Antti-Pekka Jauho, Prof. Torben Nielsen og endeligt Jakob Houmark. Jeg takker ogs˚ a Prof. Roderick Melnick og Prof. Lok C. Lew Yan Voon 122

for samarbejdet og diskussionerne, som vi har haft med hinanden; og for de værdifulde forslag de altid er kommet med. Jeg takker Professor Stephen Parsons for hans fantastiske arbejde med at tjekke mit engelske sprog. Enhver resterende fejl er min egen, naturligvis. Min varme tak g˚ ar ogs˚ atil kollegerne p˚ a Mads Clausen Instituttet. Jeg takker særligt Jerome and Krzysztof for alle de gode samtaler, vi har haft, og de øvrige Ph.d. studerende, som jeg har delt lokale med: Lars, Fei, Feng, Yu Go og alle de andre. Arbejdet g˚ ar altid lettere, n˚ ar man er i godt selskab. Jeg takker det administrative personale for den generelle service, som de altid udviser. Især vil jeg takke min ven Charlotte Bolding Andersen, da hun ikke kun har været i stand til at løse alle min problemer - endda de mest indviklede, og altid med et smil - men ogs˚ afordi vores “caffé” pauser har været værdifulde øjeblikke, hvor der blev tid til et afbræk p˚ aselv de travleste dage. Endvidere har hun virkeligt hjulpet mig til at forst˚ adanskerne og deres m˚ ade at leve p˚ a. Og nu til noget mere personligt. Jeg har tilegnet denne afhandling til de 2 vigtigste personer i mit liv; min mor og min bror. Min mor har altid været min engel; lige fra den dag, hvor jeg blev født til de seneste dage, hvor jeg har arbejdet p˚ amin afhandling, medens hun tog sig af resten af mit liv, som jeg var for presset til at tænke p˚ a. Uden hende var min fysikerkarriere aldrig startet. Min bror er min bror, og det siger alt om, hvad vi har delt igennem livet og vil dele resten af vores liv. Jeg vil ogs˚ atakke de øvrige medlemmer af min familie, som har støttet mig igennem de seneste ˚ ar: Min tante Anna, min onkel Renato, Federica og endeligt Ilaria. Jeg takker min far, hvorfra han end smiler lige nu. Jeg takker Silvia for de mest solrige aftener, for den varmeste vinter, for de mest ordrige tavse øjeblikke og for de lykkeligste samtaler i mit liv. Jeg takker alle mine venner som her og der, nu og i g˚ ar har støttet mig med deres kærlighed. Jeg takker “italienerne” Francesco og Sabina, Luciano, Luigi, Vanessa, Giorgio, Gianni, Betta, Raffa, Fra, Mapi, Pierpaolo og famile, Paolo e Silvia, Tommaso, Carlo, Daniele, Jean Michel; “Gæssene & Co.”, mine svømmere; særligt Frappi og “the ferox”. Hovedsaligt takker jeg Grazia for hendes trofaste venskab i s˚ avel gode som d˚ arlige tider; Lisa for hendes gavmildhed og uvurderlige mangel p˚ aenhver form for egoisme, og Simona for at være min havn i stormfulde perioder i mit liv. Endeligt, ønsker jeg ogs˚ aat takke alle de mennesker, som jeg har mødt i Sønderborg, og som har hjulpet mig med at starte et nyt liv i denne lille, omend overraskende by. Jeg takker Roberto, Santino, Giovanni; Nico og hans personale, Brian og hans familie; Dragan og hans familie; Jesper, Henrik, Tim; Nils og hans familie, Zuhair og Linea, Daimi og hele slænget fra Agurketid. I særdeleshed takker jeg Giorgia for de mange grin, middage, triste s˚ avel som glade stunder, vi har haft sammen p˚ adenne afsides plet.

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Ringraziamenti Questa tesi é la conclusione di tre anni di lavoro, un lungo viaggio cominciato nel Settembre 2006 quando decisi di partire dalla mia citt´ a natale, Roma, e da tutto il mio mondo, per trasferirmi nella piccola, ed a quel tempo sconosciuta, citt´ a di Sønderborg. Questo lavoro non sarebbe mai stato possibile senza tutte quelle persone che mi hanno insegnato, che mi hanno sostenuto, che mi hanno aiutato e che mi hanno amato lungo il mio percorso. Per primi voglio quindi ringraziare i miei supervisori, il Professor Morten Willatzen e il Dottor Benny Lassen, che sono stati la mia insostituibile guida per il mio PhD. Mi hanno mostrato fin nei dettagli che cosa significa fare ricerca, non solo insegnandomi il mio lavoro ed aiutandomi a migliorare le mie capacit´ a, ma sono anche stati un esempio di quella seriet´ a e responsabilit´ a che un ricercatore dovrebbe avere sempre presente nella sua mente, e ad ultimo, ma non meno importante, mi hanno dato la loro cara e sincera amicizia. Voglio ringraziare tutte le altre persone coinvolte nel progetto QUEST con cui io ho proficuamente lavorato: il Professor Jesper Mørk, il Professor Antti-Pekka Jauho, il Professor Torben Nielsen e Jakob Houmark. Ringrazio anche il Professor Roderick Melnick e il Professor Lok C. Lew Yan Voon, per il lavoro e le discussioni che abbiamo condiviso, ed i preziosi suggerimenti che mi hanno sempre dato. Ringrazio il Professor Stephen Parsons per il magnifico lavoro svolto nel correggere il mio Inglese. Tutti gli errori che restano sono solo miei, ovviamente. Il mio sincero grazie va anche a tutti i miei colleghi del Mads Clausen Institute che ho conosciuto ed incontrato durante questi tre anni, ringrazio specialmente Jerome e Krzysztof per le nostre chiacchierate, e gli altri sudenti di dottorato con cui ho diviso l’ufficio: Lars, Fei, Feng, Yu Go e tutti gli altri. Il lavoro é sempre un po’ pi´ u leggero quando dividi la stanza con gente in gamba. Ringrazio tutti i colleghi dell’amministrazione per il fondamentale aiuto che hanno sempre fornito. In particolare voglio ringraziare la mia amica Charlotte Bolding Andersen, perché é riuscita sempre non solo a rispondere a tutte le mie domande, anche le pi´ u complicate, e sempre con un sorriso, ma anche perché le nostre pause caffé sono sempre state preziosi momenti di riposo anche nei giorni in cui eravamo sommersi di lavoro, e soprattutto perché lei mi ha veramente aiutato a comprendere la vita danese di tutti i giorni. E ora qualcosa di pi´ u personale. Ho dedicato questa tesi a due fra le persone pi´ u importanti della mia vita, mia madre e mio fratello. Mia mamma é sempre stata il mio angelo, dal giorno in cui sono nato fino a questi ultimi giorni in cui stavo terminando questa tesi, e lei si é presa cura di tutto il resto della mia vita, visto che io ero troppo occupato per poterci anche solo pensare. Senza di lei la mia carriera di fisico non sarebbe mai 124

cominciata. Mio fratello é mio fratello, e la parola dice tutto, di quello che abbiamo condiviso e di quello che condivideremo per il resto della nostra vita. Voglio anche ringraziare gli altri membri della mia famiglia che mi hanno sostenuto durante questi anni, mia zia Anna, mio zio Renato, Federica ed Ilaria. Ringrazio mio padre, ovunque sia ora il suo sorriso. Ringrazio Silvia, per le sere pi´ u solari, l’inverno pi´ u caldo, i silenzi pi´ u loquaci, le parole pi´ u felici che io abbia mia avuto. Ringrazio tutti i miei amici che ovunque, ora come ieri, mi hanno sostenuto con il loro affetto. Ringrazio gli “italiani” Francesco e Sabina, Luciano, Luigi, Vanessa, Giorgio, Gianni, Betta, Raffa, Fra, Pierpaolo e famiglia, Paolo and Silvia, Tommaso, Carlo, Daniele, Jean Michel, il gruppo delle oche con tutti gli affiliati, i miei nuotatori, in particolare Frappi e “the ferox”. E pi´ u di tutti ringrazio Grazia, per la sua piena amicizia nel buono come nel cattivo tempo, Lisa, per la sua generosit´ a ed il suo affetto privo di ogni egoismo, e Simona, il mio porto sicuro fra le tempeste della vita. E ringrazio anche tutte le persone che ho incontrato a Sønderborg e che mi hanno aiutato a cominciare una nuova vita in questa piccola, ma anche sorprendente citt´ a. Ringrazio Roberto, Santino, Giovanni, Nico e tutto lo staff, Brian e la sua famiglia, Dragan e la sua famiglia, Jesper, Henrik, Tim, Nils e la sua famiglia, Zuahair e Linea, Daimi e tutta la “banda” dell’Agurketid. Soprattutto ringrazio Giorgia, per tutte le risate, cene, tristezze e gioie che abbiamo condiviso in questo posto lontano.

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