SiNx interfaces

JOURNAL OF APPLIED PHYSICS 108, 014506 共2010兲

Interpretation of recombination at c-Si/ SiNx interfaces by surface damage Silke Steingrube,1,a兲 Pietro P. Altermatt,2 Daniel S. Steingrube,3 Jan Schmidt,1 and Rolf Brendel1 1

Institute for Solar Energy Research Hamelin (ISFH), Am Ohrberg 1, 31860 Emmerthal, Germany Department of Solar Energy, Institute of Solid-State Physics, Leibniz University of Hannover, Appelstr. 2, 30167 Hannover, Germany 3 Institute of Quantum Optics, Leibniz University of Hannover, Welfengarten 1, 30167 Hannover, Germany 2

共Received 11 March 2010; accepted 3 May 2010; published online 8 July 2010兲 The measured effective surface recombination velocity Seff at the interface between crystalline p-type silicon 共p-Si兲 and amorphous silicon nitride 共SiNx兲 layers increases with decreasing excess carrier density ⌬n ⬍ 1015 cm−3 at dopant densities below 1017 cm−3. If such an interface is incorporated into Si solar cells, it causes their performance to deteriorate under low-injection conditions. With the present knowledge, this effect can neither be experimentally avoided nor fully understood. In this paper, Seff is theoretically reproduced in both p-type and n-type Si at all relevant ⌬n and all relevant dopant densities. The model incorporates a reduction in the Shockley–Read–Hall lifetime in the Si bulk near the interface, called the surface damage region 共SDR兲. All of the parameters of the model are physically meaningful, and a parametrization is given for numerical device modeling. The model predicts that a ten-fold reduction in the density of defect states within the SDR is sufficient to weaken this undesirable effect to the extent that undiffused surfaces can be incorporated in Si solar cells. This may serve to simplify their fabrication procedures. We further discuss possible causes of the SDR and suggest implications for experiments. © 2010 American Institute of Physics. 关doi:10.1063/1.3437643兴 I. INTRODUCTION

Amorphous silicon nitride 共SiNx兲 layers are deposited on semiconductors for multiple reasons. In the 1960s, such layers were introduced in the microelectronics industry as a barrier against contamination. In 1981, they were first introduced to photovoltaics for the formation of metal–insulator– semiconductor structures1 but were subsequently used mainly as antireflection coatings 共ARC兲. Soon after, it was noted that SiNx layers improve cells made of multicrystalline Si wafers more than the ARC property does,2 but it took until 1989 to verify that SiNx layers reduce the recombination rate of excess carriers at the Si surface.3 This property is called (electronic) passivation, and was intensively investigated in the photovoltaics community in the 1990s,4–6 where it was noted that passivation depends on both the dopant density7 and the density of the excess carriers 共injection density兲.8 In particular, the passivation has to be effective also at low illumination levels in nonfavorable illumination conditions, where excess carrier densities are small 共below 1014 cm−3兲. However, the passivation performance of commonly used SiNx layers is rather weak at low illumination levels if the Si material is p-type with Nacc ⬍ 1017 cm−3.4,6,8,9 This effect has been interpreted in various ways6,10–12 and a detailed understanding of it is still lacking. This has implications on present solar cell design: as most Si solar cells are fabricated on p-type wafers with Nacc ⬍ 1017 cm−3, all their surface parts must usually be diffused with dopants to achieve a good performance under weak illumination conditions. In this paa兲

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per, we use theoretical considerations and semiconductor device modeling to understand the causes of the passivation behavior at low illumination levels, from where suggestions for improvements can be derived. Knowing the recombination mechanisms may assist in simplifying the fabrication procedures for Si solar cells. II. BASIC THEORY

We model the recombination rate Rsurf at the interface between crystalline Si and an amorphous SiNx layer, using the Shockley–Read–Hall 共SRH兲 formalism13,14 Rsurf =

2 psns − ni,eff , 共1/S p兲共ns + n1兲 + 共1/Sn兲共ps + p1兲

共1a兲

where n1 = Nce␤共Ed−Ec兲,

p1 = Nve␤共Ev−Ed兲 .

共1b兲

Rsurf depends strongly on the product ns ps of electron and hole densities at the surface of Si, in excess of their densities 2 , where ni,eff denotes the at thermal equilibrium, n0,s p0,s = ni,eff effective intrinsic carrier density. The recombination properties of the defects at the interface are defined by their surface velocity parameters Sn and S p 共in units of centimeter per second兲 and by Ed, the energy of the defect state with respect to the band edges Ec and Ev with their respective effective densities Nc and Nv. We define ␤ = q / 共kT兲, where q is the unit charge, T is the temperature, and k is Boltzmann’s constant. Instead of using Sn and S p, one may use the capture crosssections ␴n and ␴ p for electron and hole capture, respectively; and the interface defect density Nit,d, where Sn

108, 014506-1

© 2010 American Institute of Physics

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

J. Appl. Phys. 108, 014506 共2010兲

Steingrube et al. 0 .6

(a) 102

0.3 Ωcm

Q

0 .5 N

a c c

< 1 0

1 7

c m

f

= 2 ×1 0

1 2

2

q /c m

-3

B

1.0

B a n d b e n d in g a t th e s u r fa c e

Seff [cm s−1]

s

0.4

[e V ]

014506-2

1.5 10

1

10 4.5

150

10

0

p-type

2

10

Seff [cm s−1]

5 ×1 0

0 .3

1 ×1 0

1 8

0 .2

2 ×1 0

1 8

4 ×1 0

1 8

1 ×1 0

1 9

0 .1

1 ×1 0

0 .0 1 0

n-type

9

1 7

2 0

1 0

1 1

1 0

1 3

1 0

1 5

In je c tio n d e n s ity in th e b u lk

1 0

Ä n

1 7

[c m

1 0 -3

1 9

]

FIG. 2. 共Color online兲 Band bending Bs at the Si surface underneath the SiNx layer as a function of the injection density ⌬n in the quasineutral region of Si, for various acceptor densities Nacc. The fixed charge Q f has a density of 2 ⫻ 1012 unit charges per square centimeter. The regions under 共or above兲 the dashed curve denote accumulation 共or inversion兲 conditions. Calculated using Eq. 共3兲.

0.6Ωcm 101

0 .4

1.5 20 90

100 1012 1013 1014 1015 1016 1017

(b)

Injection density ∆n [cm−3]

FIG. 1. 共Color online兲 Effective recombination velocity Seff in dependence of the excess carrier density ⌬n in the quasineutral region of Si 共symbols兲, measured by Kerr and Cuevas 共Ref. 9兲 in wafers with the denoted resistivity. The calculations 共lines兲 are not a fit to the data, but show the contribution solely from surface recombination in the extended model in Sec. III below using Eqs. 共1兲 and 共3兲 with S p = Sn given by the symbols in Figs. 3共c兲 and 3共d兲.

= Nit,d␴nvth and S p = Nit,d␴ pvth, with vth denoting the thermal velocity of the charge carriers, which is about 107 cm/ s in Si at room temperature. As Rsurf is rather proportional to the injection density ⌬n, it is convenient to asses the recombination properties by means of Rsurf / ⌬n, which is the surface recombination velocity S, and is commonly given in units of centimeter per second. However, the SiNx layers usually contain a high density of positive fixed charges, Q f , typically near 2 ⫻ 1012 q / cm2 共unless additional manipulation is applied15兲. These charges attract the excess carriers to, or repel them from the Si surface, and the excess charge density is different for electrons and holes. Thus, ns and ps are difficult to quantify experimentally, and S is not an easily measurable quantity. One therefore refers to ⌬n at a location zSCR where ⌬n can be easily measured, namely further below the Si surface in the quasineutral region, where the space charge has become negligible and the energy bands are flat. One calls Rsurf = Seff , ⌬n共zSCR兲

共2兲

the effective surface recombination velocity. In the following, ⌬n共zSCR兲 is referred to as ⌬n. The symbols in Fig. 1 show Seff in dependence of ⌬n for various dopant densities, measured by Kerr and Cuevas9 using the quasisteady-state photoconductance method.16 The

Seff values increase with ⌬n at intermediate-injection conditions where ⌬n ⬎ Ndop, where Ndop is the doping density because ns ps increases. However, in p-type material 共with an acceptor density Nacc兲, the Seff values also increase toward a lower ⌬n at low-injection conditions, where ⌬n ⬍ Nacc. This implies that passivation deteriorates at low illumination levels in Si solar cells. In the following, we demonstrate that this feature cannot be explained by means of the SRH formalism at the surface alone. To understand the dynamics of Seff, it is helpful to know that p-Si with Nacc ⬍ 1017 cm−3 is strongly inverted underneath SiNx layers at typical injection conditions 共in pioneering work,10 it was assumed that electron injection under illumination reduces Q f but this was later revoked11兲. In n-type Si, electrons are also attracted by the positive Q f in the nitride layer and are the dominant carrier species at the surface, leading to accumulation conditions. This behavior is most directly understandable in terms of the band bending at the surface Bs, induced by Q f via Poisson’s equation. If Q f is the only charge present at the interface, Bs can be computed by numerically solving the equation17–19 Q2f = p共e−␤Bs + ␤Bs − 1兲 + n共e␤Bs − ␤Bs − 1兲, c

共3兲

where c = 2⑀0⑀ / 共q␤兲. The electron and hole densities, n = n0 + ⌬n and p = p0 + ⌬p, respectively, are referred to in the quasineutral region 共at the location zSCR兲 where ⌬n = ⌬p, while n0 and p0 denote the carrier densities in the quasineutral region at thermal equilibrium. Equation 共3兲 holds only if both quasi-Fermi levels, EFn and EFp, are constant in the depletion region 共between zSCR and the surface of Si兲. This is the case except for very high Rsurf values, where a large number of carriers must be supplied from the quasineutral region, which causes a considerable gradient in the quasiFermi levels. In this case, Rsurf may be limited by the supply of carriers instead of by Sn and S p and is called diffusionlimited recombination. Bs, obtained from Eq. 共3兲, is shown in Fig. 2 as a func-

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tion of ⌬n. Note that Bs is almost independent of Nacc up to Nacc ⬇ 1 ⫻ 1017 cm−3. This is due to the fact that there are a sufficient number of electrons available to compensate for the positive Q f of the SiNx layer. Because Q f is independent of both Nacc and ⌬n, so is ns, implying that Bs follows Ec − EFn with increasing ⌬n. Hence, ns depends solely on Q f . This explains the linear decrease in Bs as a function of ⌬n in Fig. 2. Thus, analytical expressions can be applied to quantify the band parameters in a situation of strong inversion. Regarding accumulation conditions, Bs prevails below the dashed line in Fig. 2. Here, the Q f are compensated mainly by the depletion of holes, whose density does not significantly change with ⌬n. Therefore, Bs is insensitive to ⌬n. Between accumulation and strong inversion, weak inversion prevails, were Q f is compensated to a similar extent by the accumulation of electrons and by the depletion of holes. In the case of strong inversion in p-type Si, and in the case of accumulation in n-type Si, Eq. 共3兲 is dominated by the term n exp共␤Bs兲 on the right-hand side. Hence it holds that Bs =

1 Q2f ln . ␤ cn

共4兲

By inserting this equation into these well known relationships ns = Nce␤共EFn−Ec+Bs兲,

ps = Nve␤共Ev−EFp−Bs兲 ,

共5兲

both ps and ns can be written in the simple form ps = c

pn , Q2f

ns =

Q2f . c

共6兲

These two expressions can be inserted into Eq. 共1兲, so Seff can be explicitly computed as a function of ⌬n, as is shown by the lines in Fig. 1. Note that the parameters for the computed Seff curves are not fitted to the measurements. Instead, they are chosen from the extended recombination model described in Sec. III to show the surface contributions in the extended model. It is apparent that for all Ndon levels shown in Fig. 1, the Seff values stay independent of ⌬n at low-injection conditions. This constancy is due to the fixed charges. It can be expressed analytically by inserting Eq. 共6兲 into Eq. 共1兲 and approximating the latter for low-injection conditions Seff = S p

c Ndop , Q2f

共7兲

with Ndop = Nacc in p-Si and Ndop = Ndon in n-Si.20 Hence, SRH formalism alone cannot explain the increase of Seff in p-type Si toward low ⌬n. A straightforward extension21,22 of SRH formalism can account for the distribution of defect states D共Ed兲 within the band gap of Si at the Si/ SiNx interface. Measured defect distributions were reported in Refs. 6 and 23. The overall density of the distribution seems to vary with the quality of the nitride layer such as: both the defect density and the Seff values in the former reference are about ten times larger than in the latter reference. However, the qualitative shape of D共Ed兲 does not seem to vary greatly, although the relative

proportion between the observed peaks in the density distribution may vary to some extent. Taking D共Ed兲 into account cannot explain the increase in Seff with decreasing ⌬n, for the following reasons. It is a general feature of SRH recombination that only defects within a limited range of Ed in the band gap can effectively contribute to Rsurf. This range becomes smaller with decreasing ⌬n, implying that Seff would decrease 共not increase兲 with decreasing ⌬n, regardless of the Ed dependencies of the capture cross-sections ␴n and ␴ p. This contradiction in measurements necessitated a further extension of the hypothesis such as:10 to distinguish between acceptorlike and donorlike defects, and to include their charge occupation Qit in the electrostatic condition at the interface. A comparison of Eq. 共7兲 and 共3兲 with the experimental values in Fig. 1 reveals that a negative Qit of at least 1.4⫻ 1012 q / cm2 共or 2 ⫻ 1012 q / cm2兲 in 1 ⍀ cm 共or 10 ⍀ cm兲 material would be required to compensate Q f = 2.3⫻ 1012 q / cm2 and to increase Seff sufficiently strongly. The total defect density in Ref. 23 is definitely too small to host this large amount of charge, ruling it out as a cause of the increasing Seff values. A further way to obtain the desired Seff values is to exploit the fact that a smaller Q f can be more easily compensated by Qit. In pioneering work,10 the behavior of Seff was indeed interpreted with significantly lower Q f values 共⬇1 ⫻ 1011 q / cm2兲, causing accumulation conditions in lowinjection. However, such low Q f values were revoked later when detailed measurements became available.11 Other authors6 reported very high defect densities, even higher than those measured in the pioneering work.3 Assuming that Q f is rather low 共1 ⫻ 1012 q / cm2兲 and that all defect states below the midgap are donorlike, and that states above the midgap are acceptorlike, the Qit can indeed influence Seff in the desired way. However, the samples shown in Fig. 1 have low values of Seff, and are thus well passivated. This is most probably due to a higher Q f 共ⲏ2 ⫻ 1012 q / cm2兲 and a low density of interface states similar to that reported in Ref. 23 rather than that reported in Ref. 6. Such low defect densities cannot compensate Q f sufficiently, as will be shown quantitatively in Sec. III. Hence the behavior of Seff cannot be mainly caused by the defect charges alone. Furthermore, the model for field-effect passivation on c-Si/ SiO2 interfaces24 was adapted to c-Si/ SiNx interfaces.11 This model suggests that SRH recombination in the space charge region 共SCR兲 causes the behavior of Seff at lowinjection condition. Hence, a diode equation was added to Eq. 共1兲 and fitted to the observed Seff values by choosing a saturation current density J02 = 800 pA/ cm2.11 While such a model and data fit is plausible, it does not reveal the underlying causes, making it difficult to draw conclusions for improvements in surface passivation. Therefore, we give a more detailed analysis of recombination in the SCR in Sec. IV.25 III. RECOMBINATION NEAR THE INTERFACE

High-quality float zone 共FZ兲 wafers were used for the measurements shown in Fig. 1. Their measured bulk excess carrier lifetimes, ␶bulk, are high and cannot cause a sufficient

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amount of recombination near the interface. Consequently, we assume that ␶bulk is degraded near the Si surface. This is in agreement with the observation that hydrogenation, e.g., during SiNx processing, induces electrically active defects near the middle of the Si band gap up to 0.8 ␮m away from the interface.26 The origin of this surface damage region 共SDR兲 is not yet clear, and will be discussed in Sec. IV. As a motivation for our model, we outline only one hypothesis: an excessive density of hydrogen. It was shown that during the deposition and annealing of the SiNx layer, up to 2 ⫻ 1022 hydrogen atoms per cubic centimeter are released into the Si.27 This density comes close to the density of Si atoms 共4.9 ⫻ 1022 cm−3兲 and may weaken the covalent bonds of Si to the extent that it causes a considerable number of lattice defects, and hence a reduction in ␶bulk near the interface. An experimental validation of H-induced deep-level defects was reported in Ref. 26. In the present work, we examine highquality FZ wafers with a low inherent defect density; hence, hydrogen may interact mainly with dopant impurities by forming stationary defects in the form of complexes.28 Deeper in the Si, the hydrogen density drops and is well known to improve ␶bulk, e.g., in multicrystalline Si materials.2,29 Measurements of the hydrogen-density profile near the interface by means of secondary ion mass spectroscopy indicate that there is a very high hydrogen density present to a depth of z ⬇ 0.1, . . . , 0.2 ␮m from the interface.30 We model recombination in the SDR using the Shockley–Read–Hall 共SRH兲 formalism is the bulk13,14 RSRH =

2 pn − ni,eff , ␶ p共n + n1兲 + ␶n共p + p1兲

共8兲

where n1 and p1 are defined in Eq. 共1b兲, and ␶n and ␶ p are called lifetime parameters for electrons and holes, respectively. Consequently, ␶bulk = ⌬n / RSRH. We introduce a reduction in ␶bulk near the Si/ SiNx interface with a depth-profile of ␶n共z兲 and ␶ p共z兲. This is equivalent to a profile of the volume density of defects Nvol,d because, first, 1 / ␶n = Nvol,d␴nvth and 1 / ␶ p = Nvol,d␴ pvth and, second, we assume that the defect properties 共␴n and ␴ p兲 do not depend on depth. This lifetime reduction enhances Seff, described in Eq. 共2兲, by the following additive term: Sdeg =

兰RSRH共z兲dz . ⌬n

共9兲

Because most of the measured30 hydrogen profiles can be approximated by an exponential function, we choose

␶n共z兲 = min共␶it,nez/zdeg, ␶0兲,

共10a兲

␶ p共z兲 = min共␶it,pez/zdeg, ␶0兲.

共10b兲

where ␶0 is equal to ␶n = ␶ p of the undegraded bulk, and zdeg is the slope of the exponential increase of ␶bulk, as shown in the middle graph of Fig. 4 共please note that Eq. 共10兲 appears in this double-logarithmic representation as if it had a plateau down to a depth of zdeg兲. In the course of our work, we realized that the exact shape of this profile is rather irrel-

evant, but with higher-order exponentials like a Gaussian function or with the extreme case of an abrupt step-function we were unable to obtain satisfactory fits to the experimental Seff values of Fig. 1 in the whole range of Ndop and ⌬n. Before proceeding to adjust the parameters in Eq. 共10兲, we would like to comment on the procedure Kerr and Cuevas9 used to obtain their Seff-data. They measured the effective lifetime ␶eff and decomposed the contributions from the bulk and the surface. For the contribution from bulk SRH recombination, they chose an injection-independent but doping-dependent lifetime by using an equation proposed by Kendall.9 Here, we adapt the interpretation of their measured ␶eff-values by using Eq. 共8兲 instead of the Kendall expression to take account of the injection dependence of ␶bulk, and because ␶SRH is independent of the dopant density in modern Si wafer material. The resulting Seff-curves are rather similar to the original publication, but are more consistent, as shown in Fig. 3. Our aim is to fit the experimental Seff values of Fig. 3 with a set of parameters in Eqs. 共1兲, 共8兲, and 共10兲. An important prerequisite for interpreting experiments based on a parametric model is that the model parameters are physically meaningful. Only then can the model provide indications to experimenters as to how to optimize the passivation of SiNx layers. Considering the large set of free parameters, namely, ␶it,n, ␶it,p, zdeg, ␶0, Sn, S p, and Q f , the most precise data fitting may be obtained when all parameters are allowed to vary with dopant density. In this case, however, the fitted parameters are not necessarily physically meaningful. Therefore, we reduce the number of free parameters as much as possible and emphasize the behavior of the few parameters that remain free. To the best of our knowledge, it is reasonable to assume that Q f , ␶it,n, and ␶it,p depend on the deposition process rather than on the wafer doping. The magnitude of ␶0 is obtained from the extraction of Seff from the ␶eff data 共e.g., a too-high ␶0-value would cause too-low Seff values in the intermediate ⌬n range兲. Hence, all these parameters are optimized globally for all samples of Fig. 1, i.e., we treat them as independent of Ndop. In contrast, it is established7,31,32 that Sn and S p depend on the dopant type and dopant density. Moreover, Rsurf is limited by holes in the investigated samples due to the large ns. Hence, Sn has a negligible influence on data fitting and we choose Sn = S p. Assuming that hydrogen is a possible cause of the SDR, the penetration behavior of hydrogen during the deposition process and hence zdeg are likely to depend on the dopant density and on dopant type in the Si wafer because the diffusivity of hydrogen was found to depend strongly on various parameters, such as material quality, process conditions, annealing, firing, etc.28,30 Also, the hydrogen solubility depends on the dopant species and Ndop due to the formation of B–H or P–H complexes.28 Therefore, we choose S p and zdeg as the sole parameters that are allowed to vary with Ndop. For data fitting, we use a genetic algorithm based on the least-square method. The fitting results are listed in Tab. I and displayed in Figs. 3共a兲 and 3共b兲 as lines. Indeed, the magnitude of the global parameters 共␶0, Q f , ␶it,n, and ␶it,p兲 make sense physically, as so does the behavior of the two

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(a) 102

p-type

n-type

Seff [cm s−1]

0.3 Ωcm 1.0 10

0.4 0.6Ωcm

1.5

1

1.5

10

100

1013

FIG. 3. 共Color online兲 Fits to measurements 共Ref. 9兲 including recombination in the SDR. 关共a兲 and 共b兲兴 Effective recombination velocity Seff 共symbols兲 as obtained using a slightly revised procedure from the measurements by Kerr and Cuevas 共Ref. 9兲 共compare to Fig. 1兲. The lines are calculated by means of Eqs. 共1兲–共3兲, 共8兲, 共9兲, 共10a兲, and 共10b兲, with the parameter values given in Table I and by the symbols in the lower two panels. 关共c兲 and 共d兲兴 S p of Eq. 共1兲 and the degradation depth zdeg of Eq. 共10兲 dependent on the dopant density for n-type and p-type substrates. The dependencies are fitted 共lines兲 according to Eqs. 共11a兲 and 共11b兲, and Table I.

20

4.5 150

90 1014

1015

1016

1013

Injection density ∆n [cm−3]

1014

1015

1016 −3

Injection density ∆n [cm ]

4

10

p-type

10

n-type

−4

Sp [cm s−1]

Sp zdeg [cm]

zdeg Sp

10−5

zdeg 3

10

(c)

10

14

10

15

16

10

10

17

Acceptor density Nacc [cm−3]

10

(d)

14

10

15

Donor density Ndon [cm−3]

free parameters 共S p and zdeg兲: Fig. 3 shows that S p increases with Ndop as is observed at high Ndop,7,31,32 while zdeg seems to vary slightly but monotonically with Ndop. This monotonous behavior may be statistically insignificant if a larger number of experiments are fitted. In order to arrive at a global parametrization for numerical device simulators, we now parameterize the observed dependence of S p and zdeg 共symbols in Fig. 3兲 using the equations

冋 冉 冊册 冋 冉 冊册

S p,i = Sref,i + ␣i ⫻ 1 −

zdeg,i = zref,i + ␤i ⫻ 1 −

Ndop Nref

Ndop Nref

␥i

共11a兲

␦i

.

共11b兲

where i = n-type, p-type, and all parameters are specified in Table I. This parametrization is shown as lines in Figs. 3共c兲 and 3共d兲. In the following, we discuss the recombination dynamics. As mentioned in Sec. II, the interface of SiNx-passivated p-type Si is strongly inverted for typical injection conditions as a consequence of noncompensated positive Q f . Figure 4 shows the spatially resolved n and p at three different ⌬n 共top panel兲. With increasing ⌬n, the position where n is equal to p moves toward the interface. This is indicated by the dashed-dotted line and the background color. In general, the SRH recombination rate is maximal when ␶ p共z兲n共z兲 / ␶n共z兲p共z兲 = 1. Apart from the ratio n / p, the recombination rate depends on ␶n共z兲 and ␶ p共z兲 shown in the middle panel of Fig. 4. For the regions very close to the interface, ␶ p共z兲n共z兲 / ␶n共z兲p共z兲 ⬎ 1 and recombination is limited by the density of holes. Deeper within the wafer, ␶ p共z兲n共z兲 / ␶n共z兲p共z兲 ⬍ 1 and recombination is limited by the

10

16

density of electrons. The corresponding recombination rates for the three values of ⌬n are shown in the bottom panel of Fig. 4. With increasing zdeg, the maximum of RSRH is shifted deeper into the wafer and, accordingly, Sdeg and Seff increase. Hence, the fit to the experiment depends rather sensitively on zdeg, ␶it,n, and ␶it,p. A dependence of zdeg on Ndop according to Eq. 共11b兲 can cause Seff to be higher in lower-doped samples than in higher-doped samples. Comparing for example Seff of the samples doped with 150 and 10 ⍀ cm in Fig. 3共a兲 shows that recombination in the degraded bulk region contributes significantly to Seff. The lines in Fig. 1 show the contribution from the surface recombination alone. They are situated significantly below the measured Seff values except at high injection densities in n-type Si. This implies that recombination in the surface damaged region is also important in n-type Si materials, although it does not lead to an increasing Seff toward low ⌬n. So far, we have modeled the interface recombination without considering a distribution of interface states D共Ed兲 within the band gap, and without the interface charge Qit that results from the occupation of D共Ed兲 by electrons and holes. In the following, we show how sensitive the modeled Seff is to D共Ed兲 and Qit by replacing Sn and S p with D共Ed兲, ␴n, and ␴ p. We keep all the other parameters the same as above. The contribution to Qit of acceptorlike and donorlike defects is Qit,A = −

冕

Ec

f共Ed兲DA共Ed兲dE,

共12a兲

Ev

Qit,D =

冕

Ec

共1 − f共Ed兲兲DD共Ed兲dE,

共12b兲

Ev

where Ev and Ec are the energy levels of the valence band and the conduction band, respectively. Assuming the thermal

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TABLE I. Fundamental and global optimized simulation parameters for the fitted curves in Fig. 3. Abbreviation

Value

Description

Nc Nv ␧R Eg

Fundamental parameters 2.86⫻ 1019 cm−3 Conduction band effective density of states 3.10⫻ 1019 cm−3 Valence band effective density of states 11.9 Relative dielectric constant 1.12 eV c-Si energy band gap

␶bulk ␶it,n ␶it,n / ␶it,p Sn / S p Qf

Global fit parameters 3.5⫻ 10−2 s Bulk lifetime parameter 5 ⫻ 10−7 s Electron lifetime parameter at interface 6.4⫻ 10−2 Ratio of ␶it between electrons and holes 1 Ratio of SRV between electrons and holes 2.21⫻ 1012 q cm−2 Fixed positive charge density

Nref

Parameterization of fit results 关Eq. 共11兲兴 1 ⫻ 1014 cm−3 Reference dopant density p-type

Sref,p ␣p ␥p zref,p ␤p

2.536⫻ 103 cm s−1 3.7013 cm s−1 −1.004⫻ 10−2 4.069⫻ 10−5 cm −3.479−5 cm −3.569⫻ 10−1

Sref,n ␣n ␥n zref,n ␤n

4.375.98⫻ 103 cm s−1 3.645⫻ 103 cm s−1 −1.852⫻ 10−1 3.672⫻ 10−5 cm 3.299−3 cm −1.071⫻ 10−3

␦p

SRV of holes at Nref,p Prefactor for increase in S p with doping Exponent for increase in S p with doping Degradation depth of holes at Nref,p Prefactor for increase in zdeg with doping Exponent for decrease in zdeg with doping n-type

␦n

velocity of electrons and holes to be equal, the occupation fraction f共Ed兲 is calculated as f共Ed兲 =

␴n共Ed兲ns + ␴ p共Ed兲p1共Ed兲 , 共13兲 ␴n共Ed兲关ns + n1共Ed兲兴 + ␴ p共Ed兲关ps + p1共Ed兲兴

where n1共Ed兲 and p1共Ed兲 are given by Eq. 共1b兲. To obtain the band bending Bs, Q f in Eq. 共3兲 is replaced by Q f + Qit. In Ref. 23, D共Ed兲 was measured in samples prepared by direct plasma-enhanced chemical vapor deposition 共PECVD兲 at low-frequency. The resulting D共Ed兲 has three distinct peaks labeled A, B, and C. However, Kerr and Cuevas9 prepared their SiNx films at high-frequency, so we must neglect the peak C.10 The remaining two peaks are shown in Fig. 5共a兲. Their ␴n共Ed兲 and ␴ p共Ed兲 dependencies were measured in the same samples using small-pulse deep-level transient spectroscopy 共DLTS兲, shown as symbols in Fig. 5共b兲. With this technique, one can only determine ␴ p in the lower half of the band gap and ␴n in the upper half of the band gap. Therefore, the measured values must be extrapolated to the other unknown half of the band gap, which we do either by means of a Gaussian function 共scenario “Gauss”兲 or as a sigmoidal function 共scenario “sigmoid”兲, similar to Ref. 10. These two functions are exemplarily shown in Fig. 5共b兲 for ␴A,n 共lines兲. Note that an exponential extrapolation of the measured ␴共Ed兲 dependence would lead to Seff values that

SRV of electrons at Nref,n Prefactor for decrease in S p with doping Exponent for increase in S p with doping Degradation depth of electrons at Nref,n Prefactor for increase in zdeg with doping Exponent for decrease in zdeg with doping

exceed the measured values tremendously. We assign donorlike 共acceptorlike兲 properties to these defect states having ␴n ⬎ ␴ p 共␴n ⬍ ␴ p兲, based on the assumption that the Coulomb interaction dominates the ratio between ␴n and ␴ p. The calculated Seff values are shown in Fig. 5共c兲 with and without surface damage for both scenarios. Since smaller Q f can be more easily compensated by Qit, we also show calculations made using Q f = 1 ⫻ 1012 qcm−2. It is apparent that Qit has no noticeable influence on Seff; consequently, the rise in Seff toward low ⌬n can only be achieved with local degradation. We do not perform a fit of the entire data with this extended model, because the measured samples23 had an intentionally high D共Ed兲 in order to obtain sufficiently large DLTS signals.33 Hence, a reduction in D共Ed兲 by about one order of magnitude may be assumed for well-passivated wafers. In such samples, the influence of Qit is definitely negligible, and we see no need to include Eqs. 共12兲 and 共13兲 for a device model. IV. IMPLICATIONS ON EXPERIMENTS

In the previous sections, we have shown that a SDR characterized by a strongly reduced carrier lifetime leads to the pronounced Seff共⌬n兲 dependence which is experimentally observed on SiNx-passivated p-type Si wafers. An important practical consequence of the pronounced Seff共⌬n兲 depen-

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J. Appl. Phys. 108, 014506 共2010兲

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Seff [cm s−1]

0.3Ωcm 10

1

10

0

it,n

-7

= 5x10 s -6

5x10

-3

-4

10

10 Ωcm

-5

5x10

5x10

5x10

−1

10

12

10

13

10

14

10

15

p-type 10

16

10

17

Injection density ∆n [cm−3] FIG. 6. 共Color online兲 Calculated Seff in the p-type Si samples with 10 and 0.3 ⍀ cm resistivity. Both ␶it,p and ␶it,n of Eq. 共10兲 are multiplied by up to several orders of magnitude. This shows that Seff does not increase significantly toward low ⌬n if the reduced lifetime parameters are increased by only one order of magnitude.

lifetime in the degraded volume is necessary to maintain the good performance of a Si solar cell also at low illumination levels, even though it contains undiffused surfaces passivated with SiNx. Figure 6 shows the Seff values of the 10 ⍀ cm sample, calculated using ␶it,p and ␶it,n multiplied by up to several orders of magnitude. Surprisingly, an improvement by a factor of 10 would already lead to the desired amelioration. A detailed understanding of the origin of the deteriorated region may help to achieve this improvement. One main cause for the formation of such a SDR might be the extremely high hydrogen content in the SiNx layer deposited by PECVD, typically 10–15 at. %, diffusing into the Si substrate during deposition. At typical SiNx deposition temperatures around 400 ° C the effective diffusion coefficient Deff of hydrogen in Si was experimentally determined by Pearton34 to be Deff ⬇ 10−11 cm2 s−1. In most experiments published in the literature the deposition time is in the order of t = 2 – 10 min, leading to a roughly estimated penetration depth of 冑Defft = 0.3– 0.8 ␮m which is in the same order of magnitude as the depth of the SDR determined in this work. Very high densities of hydrogen in Si are known to lead to the formation of platelets as well as recombination-active deep-level defects.26,35,36 An additional effect which could

FIG. 4. 共Color online兲 Electron and hole densities 共top graph兲, lifetime parameters 共middle graph兲, and recombination rates 共bottom graph兲 as a function of depth from the Si/ SiNx interface in 10 ⍀ cm p-type Si material, for three different excess carrier densities ⌬n in the quasineutral region, and for two different values of the degradation depth zdeg of Eq. 共10兲. If not otherwise specified, the parameters are chosen according to Table I.

dence is that at low-injection density, which is typical for solar cell operation conditions, the surface passivation quality is strongly deteriorated compared to higher injection densities. It is worthwhile to study which improvement of the (b)

(c)

10−11

Qf =1.0x10¹² qcm−2 Qf =2.2x10¹² qcm−2 without degradation

2

Capture cross sections [cm ]

1012

1011 Defect A

1010

0.1

0.3 0.5 0.7 0.9 Energy Ed−Ev [eV]

1.1

sigmoidal

−15

10

−17

10

−19

σn,B

σn,A

σp,B σp,A

10

0.1 0.3 0.5 0.7

0.9 1.1

Energy Ed −Ev [eV]

−1

10−13 Gaussian

Seff [cm s ]

Defect B

−2

−1

Interface DOS D(E) [cm eV ]

(a)

10

2

10

1

sigmoidal

Gaussian

1012

1013

1014

1015

1016

−3 Injection density ∆n [cm ]

FIG. 5. 共Color online兲 Application of the extended SRH formalism to a continuous distribution of defect states. 共a兲 Measured 共Ref. 23兲 densities of interface defect states and 共b兲 measured 共Ref. 23兲 capture cross-sections for the electron capture ␴n and the hole capture ␴ p at the Si/ SiNx interface, as a function of energy Ed from the valence band edge Ev within the Si band gap 共symbols兲 with an extrapolation 共lines兲 using either a Gaussian or a sigmoidal function. 共c兲 Resulting calculated Seff values without bulk degradation 共triangles兲 and with bulk degradation 共lines兲 at two different fixed charge densities Q f . These graphs show that the charge stored in the defects has no significant influence on Seff.

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also be envisaged as contributing to the SDR might be the chemical etching of the Si surface prior to the SiNx deposition. Wet chemical etching leads to a microscopic roughening of the surface and also introduces hydrogen in high concentrations into the Si wafer. This has been experimentally proven via acceptor deactivation in a near-surface layer after wet chemical etching.36 From these considerations, we may derive some rules of thumb for experimenters to further improve the passivation quality of SiNx on p-type Si surfaces at low-injection densities. One obvious direction for avoiding the SDR would be to strongly reduce the hydrogen concentration in the SiNx film. In fact, there are deposition techniques, such as low-pressure CVD 共LPCVD兲, that lead to a strongly reduced hydrogen content compared to PECVD films. However, these hydrogen-lean films do not provide as good passivation properties as PECVD-SiNx does, due to the lack of hydrogen at the Si/ SiNx interface, where the hydrogen passivation of interface states effectively reduces interface recombination. Hence, a reduction in the hydrogen content might not be the best approach for avoiding the SDR, as the interface passivation is deteriorated at the same time. Another, more promising idea would be to invert the positive fixed charge density Q f within the SiNx. This would reduce the impact of the SDR on Seff, and could thereby significantly improve the passivation quality of SiNx on p-type Si at low-injection densities. It is well known from the literature that the Q f in SiNx can be manipulated by charge injection.37 This property of SiNx is used in memory devices.38 It has recently been experimentally demonstrated that the Q f in SiNx films deposited by LPCVD can be permanently switched from large positive to large negative values, leading to a pronounced improvement in the surface passivation of p-type Si.15 Hence, the development of negatively charged SiNx may be a promising approach for improving SiNx passivation on p-type Si surfaces. It might also be worthwhile to study in more detail the impact of the wet chemical etching prior to the SiNx deposition in more detail. Very little data exists concerning the impact of the chemical pretreatment of the Si surface on the passivation quality after SiNx deposition. The effect of the chemical etchant and the duration of the chemical treatment should be studied with regard to the formation of a SDR. V. CONCLUSION

Recombination at c-Si/ SiNx interfaces is quantified by the SRH formalism with the inclusion of lifetime reduction in the bulk of Si near the surface, called the SDR. The model explains the measured effective surface recombination velocity Seff to a high precision level in a wide range of excess carrier densities ⌬n and dopant densities, in both p-type and n-type Si. All model parameters are physically meaningful, and predict that a ten-fold improvement of the SRH lifetime in the SDR, corresponding to a ten-fold reduction in the density of states, eliminates the strong increase of Seff toward low ⌬n in p-type Si. With an extension of the model to include measured distributions of the interface states and energy-dependent capture cross-sections for electron and

hole capture, respectively, it is confirmed that, at typical values for the fixed charge density Q f , the charge stored at the interface defects has a negligible influence on Seff. A parametrization suitable for numerical device modeling is given by Eqs. 共1兲, 共8兲, 共10兲, and 共11兲, and in Table I. ACKNOWLEDGMENTS

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