Development of a three-dimensional numerical model of grain ...

JOURNAL OF APPLIED PHYSICS

VOLUME 91, NUMBER 7

1 APRIL 2002

Development of a three-dimensional numerical model of grain boundaries in highly doped polycrystalline silicon and applications to solar cells Pietro P. Altermatta) and Gernot Heiserb)

Centre for PV Engineering, University of New South Wales, Sydney 2052, Australia

共Received 26 November 2001; accepted for publication 14 January 2002兲 We have developed a three-dimensional numerical model of grain boundaries to simulate the electrical properties of polycrystalline silicon with doping densities larger than approximately 5 ⫻1017 cm⫺3 . We show that three-dimensional effects play an important role in quantifying the minority-carrier properties of polycrystalline silicon. Our simulations reproduce the open-circuit voltage of a wide range of published experiments on thin-film silicon p-n junction solar cells, choosing a velocity parameter for recombination at the grain boundaries, S, in the order of 105 – 106 cm/s. The simulations indicate that, although S has been reduced by one order of magnitude over the last two decades, improvements in the open-circuit voltage have mainly been achieved by increasing the grain size. A few options are proposed to further reduce S. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1456962兴 I. INTRODUCTION

has a peak. The trapped charges are supplied by the adjacent grains, which may suffer depletion and hence need to be simulated with appropriate modifications to the Poisson equation. However, the situation becomes simpler at N dop ⲏ5⫻1017 cm⫺3 , for two reasons.

Polycrystalline silicon is used for many thin-film applications, such as gate transistors, nonvolatile and dynamic access memories, passive resistors, bipolar transistor diodes, and solar cells. The electrical behavior of these polycrystalline devices is rather complex to model. On the one hand, the amount of trapped charge at the grain boundaries depends on the position of the quasi-Fermi levels, causing complicated electrostatic conditions. On the other hand, important material properties 共such as the lifetime of excess carriers兲 do not solely depend on the material quality within the grains, but are strongly influenced by the grain boundaries as well. To reduce this complexity, many models ‘‘lump’’ the influence of various physical mechanisms into ‘‘effective’’ parameters. However, we aim to model polycrystalline devices on the basis of the underlying physical effects, avoiding lumped input parameters as far as possible. This makes it necessary to use numerical techniques to solve the coupled set of semiconductor differential equations self-consistently. To our knowledge, multi- and polycrystalline silicon solar cells have been simulated numerically in two dimensions only.1 Three-dimensional 共3D兲 models have been restricted to analytical formulations, as for example in Refs. 2– 4. Because computer technology has improved, we are able to develop more sophisticated models, as is shown in the following.

共1兲 Q t is rather insensitive to the shape of the DOS because most traps are filled 共in n type兲 or empty 共in p type兲. Consequently, Q t depends only weakly on bias and illumination levels. 共2兲 Q t is small compared to the number of majority carriers in the adjacent grains, and therefore the grains are not considerably depleted. Under these circumstances, we can avoid modifying the Poisson equation, and we approximate the grain boundary by a boundary condition, i.e., as a charged interface with charge Qt .1 We choose Q t and the recombination model as follows. Grain boundaries in moderately and highly doped silicon are usually surrounded by a depletion region,5 hence Q t is negative in n type and positive in p type. It is therefore commonly believed that silicon grain boundaries contain acceptor- and donor-like traps, and their DOS is shaped in such a way that Q t ⫽0 when the Fermi level is near midgap.6 We mentioned that most traps in highly doped material are either filled 共in n type兲 or empty 共in p type兲 and that no substantial pinning of the Fermi level occurs. Hence, to describe the electrostatic conditions, we may integrate the DOS over the entire band gap to obtain the total density of traps N t . 1,7 It then follows from the Shockley–Read–Hall 共SRH兲 theory8,9 that

II. THE MODEL

In general, the amount of trapped charge Q t at a grain boundary depends sensitively on the density-of-states 共DOS兲 within the forbidden band gap, and considerable pinning of the quasi-Fermi levels may occur at energies where the DOS

冉 冊

Q t ⫽qN t f ⫺

共1兲

where q is the electron charge and f is the occupancy probability of a defect at energy E t within the band gap, given as

a兲

Also with Inianga Consulting, 92/125 Oxford Street, Bondi Junction NSW 2022, Australia. b兲 Also with School of Computer Science and Engineering, University of New South Wales, Sydney 2052, Australia; Electronic mail: [email protected] 0021-8979/2002/91(7)/4271/4/$19.00

1 , 2

f⫽ 4271

S no n g ⫹S po p 1 . S no 共 n g ⫹n 1 兲 ⫹S po 共 p g ⫹ p 1 兲

共2兲

© 2002 American Institute of Physics

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The parameters, 2 /n 1 , n 1 ⫽n i,eff e 共 E t ⫹E i 兲 /kT and p 1 ⫽n i,eff

共3兲

relate E t to the intrinsic energy level E i . The subscript g denotes the grain boundary, p and n are the hole and electron densities, and n i,eff is the effective intrinsic density of the semiconductor. S no and S po are the recombination velocity parameters at the grain boundary 共in units of cm/s兲. In highly doped material, the SRH recombination rate is insensitive to E t as long as E t is below 共above兲 the quasiFermi level for electrons 共holes兲. Hence, we can again neglect the shape of the DOS and ‘‘squeeze’’ all recombination through a single defect level near midgap instead of considering a distribution of defect levels. The recombination rate at the grain boundary is then given by the usual expression for SRH recombination 共in units of cm⫺2 s⫺1兲: U g⫽

2 p g n g ⫺n i,eff

1 1 共 n ⫹n 1 兲 ⫹ 共 p ⫹p 1 兲 S po g S no g

.

共4兲

Since it is commonly assumed that recombination occurs predominantly via the charged and not via the neutral traps, we choose S no ⫽S po for both n- and p-type material.1 We should mention that Eqs. 共1兲–共3兲 would also describe the trap occupancy at the Si–SiO2 and Si–SiNx interfaces, but these charges are usually neglected since they are dominated by the fixed positive charges within the oxide or nitride layer.10 Nowadays, computers with 1 GB random access memory and 1 GHz frequency are affordable and make numerical simulations of grain boundaries in three dimensions feasible. We use the device simulator DESSIS11 which—in contrast to the analytical models—solves the fully coupled set of semiconductor differential equations numerically and self-consistently; no further assumptions need be made as to the quasi-Fermi levels, band bending, etc. Apart from the grain boundaries, we use the same models as in Refs. 12 and 13. For example, the recombination in the grains is quantified by using the analogous expression to Eq. 共4兲, U⫽

2 pn⫺n i,eff

␶ po 共 n⫹n 1 兲 ⫹ ␶ no 共 p⫹ p 1 兲

,

共5兲

which has units of cm⫺3 s⫺1, and where ␶ no and ␶ po are the capture-time constants for electrons and holes, respectively.14 In contrast to Ref. 1, the present model is fully 3D. III. COMPARISON WITH PUBLISHED EXPERIMENTS

To compare our simulations with published experiments, we must restrict ourselves to p-n junction devices where the quasineutral regions extend over the majority of the volume of the device 共a p-i-n structure would contain lowly doped device regions, where the aforementioned approximations would be too coarse兲. We simulate V oc since this parameter is sensitive to the minority-carrier properties, i.e., to the amount of excess carrier recombination. Moreover, V oc is only weakly effected by the various light trapping schemes applied by different groups, and no resistive losses need to be accounted for. We model 30 ␮m 共or 5 ␮m兲 thick n ⫹ p p ⫹

FIG. 1. Bold lines show 3D simulations of the open-circuit voltage of 30 ␮m thick n ⫹ pp ⫹ polycrystalline silicon solar cells as a function of grain size 共at 1-sun illumination and 300 K兲. Variations are made to the recombination velocity parameter at the grain boundaries, S, and to the capture-time constant of excess carriers in the grains, ␶. The symbols represent thin p-n junction devices as reviewed by Gosh et al. 共see Ref. 21兲 and Bergmann and Werner 共see Ref. 22兲. 2D simulations 共thin lines兲 do not describe the experiments sufficiently 共shown are ␶ ⫽10⫺6 s with S⫽104 and 105 , respectively兲.

solar cells. The n ⫹ emitter is 1 ␮m deep and has N dop ⫽1019 cm⫺3 . The p ⫹ region is 5 ␮m 共or 1 ␮m兲 deep and has N dop⫽5⫻1019 cm⫺3 . The p base has N dop⫽1018 cm⫺3 . The grains are assumed to grow in columnar rectangular shapes; cubic grains cause only a slight additional reduction in V oc . Our simulated V oc of the 30 ␮m thick cells is shown in Fig. 1 as lines. The curves at grain sizes g⬎1 mm are estimated with continuity arguments, using the simulations at smaller g and at g⫽⬁. Since the V oc -limiting region of the simulated cells is p type, we choose a positive charge; its magnitude is N t ⫽1012 cm⫺2 , in accordance with a broad range of experiments.15–20 Moreover, we choose E t ⫽E i in all simulations presented here, for the reasons previously explained. Varying either S no ⫽S po ⫽S or ␶ no ⫽ ␶ po ⫽ ␶ reveals whether either the recombination at the grain boundaries or within the grains dominates the overall recombination losses. Which of the two losses is dominant depends on the combination of S, ␶, and g. For example, if the grain boundaries are not passivated (S⫽106 cm/s), they dominate the recombination losses up to g⫽1 mm. If they are well passivated (S ⫽104 cm/s), they dominate the recombination losses only up to g⫽50 ␮ m if ␶ ⫽10⫺8 s 共or up to g⫽200 ␮ m if ␶ ⫽10⫺7 s兲. We should keep in mind that varying S would imply some variation in N t . However, we can not quantify N t as a function of S in the present model; this approximation will be removed in a forthcoming article, where we will extend the present model to all relevant doping and injection levels. The symbols represent experiments on polycrystalline thin-film silicon p-n junction solar cells as reviewed by Ghosh et al.21 and Bergmann and Werner.22 It is quite remarkable that most of these published experiments can be described by our model with S⬇105 – 106 cm/s, regardless

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FIG. 2. Thin lines are the same 3D simulations of 30 ␮m thick cells as in Figs. 1 and 3. Bold lines: simulations of 5 ␮m thin cells.

of grain size and deposition method. Various authors measured a similar S with the light-beam induced current technique.23–27 Comparing our simulations with the experiments indicates that S has been reduced by one order of magnitude over the last two decades. If we simulate only in two dimensions, we obtain V oc values that are considerably higher 共thin lines in Fig. 1兲. We would need to increase S well beyond 106 cm/s to describe the experiments, and V oc would be less sensitive to g than in the 3D model. This is so because in two dimensions, all grain boundaries lie parallel, and the excess carriers can only flow towards a single boundary to recombine there. In 3D, however, the flow of carriers is determined by at least two adjacent boundaries. The two-dimensional 共2D兲 simulations would falsely imply that, in the experiments shown, S improved with increasing grain size. This is not the case in the more realistic 3D model, hence 3D effects are very important in the process of interpreting the experiments, especially at g⬍1 mm. We also simulated cells that are 5 ␮m instead of 30 ␮m thick, shown as bold lines in Fig. 2. Thinning the cells improves V oc by about 50 meV at small g and about 20–30 meV at larger grain sizes. The contrary happens in the case of the smallest recombination losses simulated here 共␶ ⫽10⫺6 s and S⫽103 cm/s兲: V oc decreases by thinning at g ⬎100 ␮ m because fewer photogenerated carriers are injected accross the p-n junction. Such effects depend strongly on the applied light trapping scheme. Our simulated devices are planar and have a 110 nm thick front oxide as an antireflection layer, and the internal reflectivity is 0.95 and 0.65 at the rear and front surface, respectively. In Fig. 3, we tracked the progress of a few laboratories.28 – 47 We only consider publications where both V oc and g were reported 共many more publications exist that report on only one of these two parameters兲. Although Fig. 1 indicates that S has been reduced by one order of magnitude over the last two decades, Fig. 3 shows that most laboratories made progress mainly by increasing the grain size. An exception to this may be IMEC.46,47 However, the question

FIG. 3. Symbols represent published experiments tracked over the years. We consider only polycrystalline silicon p-n homojunction solar cells that have no substantial i layer, are thinner than 100 ␮m, and have a grain size larger than 1 ␮m. The simulations 共lines兲 are the same as in Fig. 1.

arises of how the grain size should be measured to compare the experiments with our simulation. Ideally, a scanning technique that is sensitive to the amount of recombination should be applied for measuring g—such as the electronbeam induced current 共EBIC兲 technique, where electrically active grain boundaries appear as dips in the output graph. In our simulations, all grain boundaries are active recombination centres, so the g of the model should be compared to the average length between the dips in EBIC measurements. At IMEC however, the grain size was determined by means of transmission electron micrographs 共TEM’s兲, where no distinction is made between twin boundaries and high-energy grain boundaries. Undecorated twins are not electrically active, so TEM’s may therefore lead to an ‘‘underestimation’’ of g in the context of our simulations. It is commonly reported that the usual hydrogen plasma anneal at 400 °C passivates about 9/10th of all initial traps.17,48,49 If all the traps are equally efficient recombination centers, S decreases by one order of magnitude during this anneal 共in this case, ␶ would change by the same amount兲. This may be part of the reason why the recent experiments in Fig. 1 共filled circles兲 are modeled with S about ten times smaller than the older experiments 共open circles兲. However, it would be wrong to conclude from these findings that S can not be reduced further. It only means that more specific techniques for passivating the grain boundaries should be found. This may be achieved in a few ways: 共i兲 hydrogenation may be made more efficient by shifting the chemical equilibrium between hydrogen passivation and defect formation;50 共ii兲 the crystal orientation of adjacent grains may be manipulated during growth so that it correlates with

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the coincidence-site lattice theory,51–54 and 共iii兲 dopant segregation may be used more intentionally to surround the grain boundaries with a highly doped region.47 Please note that our collection of experimental values does not aim at completeness, and that we restrict ourselves to thin p-n junction devices with g⬎1 ␮ m and without a substantial i layer. Also, we do not consider devices with a heterostructure 共e.g., with an amorphous emitter兲. IV. CONCLUSIONS

We developed a 3D grain boundary model, suitable for the simulation of the electronic properties of polycrystalline silicon with a doping level of N dopⲏ5⫻1017 cm⫺3 . It was shown that 3D effects play an important role in materials with a grain size of g⬍1 mm. Comparing our simulations with published experiments on thin p-n junction silicon devices indicates that in general, the velocity parameter S at the grain boundaries has been reduced by one order of magnitude over the last two decades. Most experiments can be simulated with S⫽105 – 106 cm/s, regardless of grain size and deposition method. Our simulations indicate that, over the last two decades, improvements in open-circuit voltage have been made mainly by increasing the grain size, rather than by reducing S. We made a few suggestions to decrease S beyond presently estimated values. ACKNOWLEDGMENTS

The authors are grateful for the support of the Australian Research Council 共ARC兲. One of the authors 共P.P.A.兲 is funded by an ARC Postdoctoral Fellowship. 1

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