28th European Photovoltaic Solar Energy Conference and Exhibition
AN EXAMINATION OF THREE COMMON ASSUMPTIONS USED TO SIMULATE RECOMBINATION IN HEAVILY DOPED SILICON
Keith R. McIntosh,1 Pietro P. Altermatt,2 Tom J. Ratcliff,3 Kean Chern Fong,3 Lachlan E. Black,3 Simeon C. Baker-Finch,1,3 and Malcolm D. Abbott1 1 PV Lighthouse, Coledale, NSW 2515, AUSTRALIA. [email protected]. 2 Leibniz University of Hannover, Inst. of Solid-State Physics, Dep. Solar Energy, Appelstrasse 2, 30167, GERMANY 3 Centre for Sustainable Energy Systems, Australian National University, Canberra, ACT 0200, AUSTRALIA
ABSTRACT: A well-designed silicon solar cell entails a complicated optimisation of its heavily doped surfaces. There exist many analytical and numerical models to assist this optimisation, and each involves a foray of assumptions. In this paper, we examine three of the common assumptions: (i) quasi-neutrality, (ii) the use of effective recombination parameters as a boundary condition at the surface, and (iii) the use of Boltzman rather than the more complicated but more precise Fermi–Dirac statistics. We examine their validity and the computational benefits of their inclusion. We find that (i) the quasi-neutrality assumption is valid over a wide range of conditions, enabling a fast and relatively simple simulation of emitter recombination, (ii) the effective surface recombination velocity depends on doping under most conditions and can give the misleading impression that the density of interface defects depend strongly on surface concentration, and (iii) at carrier concentrations where Boltzmann statistics are invalid (above 1019 cm–3, and especially above 1020 cm–3), it is more difficult to mitigate the error by employing an effective model that combines degeneracy with band-gap narrowing, than to include the more general Fermi–Dirac statistics themselves. Keywords: Recombination, Simulation, Software
1 INTRODUCTION 2 SIMULATION TOOLS
All silicon solar cells contain heavily doped surfaces We employ EDNA 2 [1, 2] to represent a model to ensure low contact resistance, low surface containing the quasi-neutrality assumption and PC1D [3] recombination and low lateral resistance. They are called and Sentaurus [4] to represent models that do not. emitters, diffusions, back-surface fields, front-surface EDNA 2 is an online computer simulation program fields and floating junctions. As the doping increases, that models recombination in a 1D emitter. It is based on contact resistance and lateral resistance decrease (which the freeware Excel model released in 2010 [2]. As well is good), free carrier absorption increases (bad), and as being much faster than its predecessor, EDNA 2 has recombination first decreases (good) and then increases additional features: (bad). Given these competing effects, it is desirable to models for incomplete ionisation of dopant atoms; optimise the dopant profiles at the surfaces, at least to a selection of models for carrier mobility; within the constraints of low-cost manufacturing. the choice of Fermi–Dirac or Boltzmann statistics An optimisation of a cell’s heavily doped surfaces— that is independent of the selected band-gap referred to as emitters from here on—is aided by the narrowing model; ability to accurately simulate their behaviour. This is the ability to sweep input parameters to generate plots more easily said than done. The dopant concentration such as J0E vs surface concentration; varies over many orders of magnitude in a short distance the choice of setting Seff, J0s, and soon, the full set of causing significant variations in the minority carrier interface properties (Dit(E), p(E), n(E), and Q) to concentration, Auger recombination, Shockley–Read– parameterise surface recombination; and Hall recombination, carrier mobility, free-carrier the option to set a J and determine surface absorption, effective masses, and last but not least, the 0E parameters. band gap. Moreover, when the emitter is heavily doped, EDNA 2 runs on a cloud-based server that has high the semiconductor becomes degenerate and the carrier computation speeds. The principle reason, however, that concentrations must be calculated with Fermi–Dirac the computation time is short relative to its alternatives is statistics rather than the simpler Boltzmann statistics. that EDNA 2 assumes the emitter is ‘quasi neutral’. It is More complicated still, the surface of an emitter also specifically designed to rapidly determine an constitutes an interface with a different material, which emitter’s recombination parameters like J and IQE. It necessarily introduces interface defects and the 0E does not solve a complete device but solely the emitter, possibility of surface charge. which introduces error when recombination in or near the There are many analytical and numerical models for depletion region is significant. simulating emitters. In this paper, we discuss three of the Sentaurus is the most complete and accurate assumptions that are employed in many of those models: approach to modelling semiconductor devices. For (1) quasi-neutrality, (2) the effective recombination simulating emitters, it has practically all of the velocity as a boundary condition, and (3) Boltzmann functionality of EDNA 2 and far more besides. It does statistics. We describe conditions under which they are not assume quasi-neutrality, it allows 2D and 3D devices, valid and invalid. and it simulates complete and sophisticated devices [4]. PC1D is the most commonly used program for
simulating solar cells. It is limited to Boltzmann and a
single model for carrier mobility, band-gap narrowing
and Auger recombination. It does not assume quasi-
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Figure 1: The upper graphs plot J0E determined from PC1D and EDNA 2 as a function of (a) surface recombination 19 –3 velocity Seff with surface dopant concentration NS = 3.05 10 cm and (b) Ns with Seff = 3000 cm/s; the lower graphs plot the relative difference in J0E determined by EDNA 2 compared to the J0E determined by PC1D. neutrality, it simulates complete 1D devices, and it is fast, band-gap narrowing model. Figure 2 plots the results of free and open source [3]. the comparison between EDNA 2 and Sentaurus. This second comparison contains more sophisticated models, including Fermi–Dirac statistics, Klaassen’s mobility 3 ASSUMPTION 1: QUASI-NEUTRALITY model, and Schenk’s band-gap narrowing. Appendix A describes the simulation inputs in labourious detail. The quasi-neutrality (QN) assumption states that the The upper graphs in Figures 1 and 2 plot the J0E excess hole concentration p equals the excess electron determined from the programs as a function of (a) surface concentration n. This assumption is contained in many recombination velocity and (b) surface dopant recent solar cell programs [1, 2, 5–7] and in many concentration Ns. The figures indicates a close agreement analytical emitter models [e.g, 8–10]. The Shockley between EDNA 2 and PC1D for the simpler comparison, diode equation is also limited to the QN regions of the and between EDNA 2 and Sentaurus for the more cell. sophisticated comparison. When the QN assumption is applied, the carrier The lower graphs in Figures 1 and 2 plot the relative concentrations can be determined without recourse to the difference between the simulations; they show that J0E electric field. That is, the three coupled differential determined by EDNA 2 tends to be a little less than that equations that relate the electron concentration n, the hole determined by PC1D or Sentaurus, but by no more than concentration p, and the electric field (or potential ) can 4% over the tested range of inputs (except for the most be simplified into one differential equation, making it heavily doped diffusion in the comparison wit Sentaurus, much simpler and faster to solve. for which it is 6% smaller). The source of these small The QN assumption is valid in regions where the discrepancies was not determined. electric field is small. It becomes increasingly invalid as It is concluded from this study that, over a wide the gradient of the excess carriers increases, and it is variety of emitters, and with and without sophisticated entirely invalid in a depletion region (where the electric semiconductor models, the QN assumption can be field is large). The QN assumption is also invalid near imposed with little loss in accuracy. The error introduced the surface when surface charge is significant (i.e., when into the emitter’s J0E by the QN assumption is within the the surface is in accumulation, depletion or inversion). range +0.3% to –6%. In this section, we examine the applicability of the QN assumption for emitter modelling. It is conducted by comparing the results of EDNA 2 with PC1D and Sentaurus. Figure 1 plots the results of the comparison between EDNA 2 and PC1D. This first comparison contains simple physical models, such as Boltmann statistics, constant carrier mobilities, and an exponential apparent
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19 – Figure 2: The upper graphs plot J0E determined from Sentaurs and EDNA 2 as a function of (a) Seff with NS = 1 10 cm 3 and (b) Ns with Seff = 1000 cm/s; the lower graphs plot the relative difference in J0E determined by EDNA 2 compared to J0E determined by Sentaurus.
4 ASSUMPTION 2: THE EFFECTIVE SURFACE interfaces, it has recently be concluded that Sn0 and Q are RECOMBINATION VELOCITY relatively constant over a wide range of Ns [17]. In this section, we describe the conditions under 4.1 Introduction to Seff which Seff is independent of Ns. We also describe an A well-known parameter that quantifies surface alternative surface parameter J0s that is independent of Ns passivation is the effective surface recombination when the surface charge is large [18]. (The J0s is velocity Seff. It is employed as a boundary condition in analogous to the popular emitter saturation current the current versions of Quokka [5], CoBoGUI [11, 12] density J0E, as described in Section 4.3). The purpose of and EDNA [1, 2], as well as a host of analytical equations employing these quasi-parameters is that (i) they [8–10]. represent a single metric for the quality of a surface The merit in using Seff is that it incorporates many passivation scheme, and (ii) they can readily measured by aspects of surface recombination into a single parameter. conventional lifetime measurements on undiffused Specifically, it combines the interface state density silicon. Dit(E), the capture cross section of electrons n(E) and holes p(E), and the net surface charge Q whether it be in 4.2 The dependence of Seff on Ns an insulator or at an insulator–semiconductor interface. The effective surface recombination velocity Seff is The drawback in using Seff is that by lumping various defined as physical phenomena together, it is difficult to recognise J = q U = q S n (1) the physical reasons behind its behaviour. Under some rec s eff d conditions, for example, the application of Seff can be where Jrec is the current density that flows into the misleading, making it easy to use erroneous values of Seff surface to recombine, q is the charge of an electron, Us is –2 –1 during emitter simulations. the recombination rate at the surface in cm s , and nd When Q is significant, Seff becomes dependent on the is the excess carrier concentration ‘near’ the surface. The surface dopant concentration Ns, even when the actual subscript d refers to the distance from the surface at SRH surface recombination velocity parameters, Sn0 and which the carrier concentrations are no longer affected by Sp0, are independent of Ns [13–15]. Plots of Seff against surface charge. This is illustrated in Figure 3. Usually Ns invariably exhibit a strong increase of Seff with Ns, and this distance is sufficiently close to the surface s that it has subsequently been construed that the surface defect there is negligible difference between the doping density must increase with Ns when in fact there might be concentration at s and d. The recombination and no such evidence. This is not to say that the defect generation rates between s and d tend to be negligible density cannot increase with Ns, only that it might not compared to Us. necessarily increase with Ns. In fact, for Al2O3–Si
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We now derive Seff for an n-type emitter. In this case, positive (because minority carrier holes are repelled from electrons are the majority carriers and holes are the the surface, making the bands bend upwards and s minority carriers. More detailed derivations are given in negative); and Seff is greater than Sp0 when Q is negative [13–18]. (s is positive). Due to the high dopant concentration at d, Equation 1 If s were dependent solely on Q, then Seff would can be rewritten depend only on the surface defects and Q, making it a S = U / p , (2) convenient and uncomplicated metric to quantify the eff s d surface passivation provided by a charged dielectric. But where pd is the hole concentration at d and equal to nd s also depends on Ns and nd [13–21]. This means that (since the excess carrier concentration is the same for when Q is significant, Seff is not just related to the surface electrons and holes when quasi-neutrality is valid). defects, but also to the doping and the excess carrier Now, at s, the hole concentration is far smaller than concentration. Being unaware of this dependence makes the electron concentration, so Shockley–Read–Hall it easy to misinterpret a plot of Seff against Ns and to (SRH) recombination at the surface can be given by the conclude that Sp0 increases with doping when it might simple relation, not! Figure 4 shows the conditions under which is U = S p (3) s s p0 s sufficiently small that there is no doping dependence where Sp0 is the hole recombination velocity, which under accumulation or inversion. This region is coloured depends solely on the surface defects, i.e. S = N v . yellow. It indicates that for typical undiffused wafers p0 d th p 16 –3 In fact, Equation 3 is valid only under the provisos that (Ns < 10 cm ), Seff is probably never constant with (i) the principle defect is within the quasi-Fermi levels of doping, since it is unlikely that Q would ever be smaller 9 –2 trapped charges; (ii) ns/p >> ps/n, which is valid in an than 3 10 cm . Consequently, using Seff to represent n-type emitter except when there is a large negative surface passivation has no merit without also stating Ns. charge at the surface; and (iii) the surface concentrations Thus, for undiffused wafers, one cannot determine Seff at 2 are not near their equilibrium values (i.e. psns >> ni ), some Ns and apply it at another Ns, even if Sn0 and Q are which is valid in all conditions except in the dark with independent of Ns. little or no applied bias. That may sound like a lot of Figure 4 also shows that for dopant-diffused wafers 18 –3 provisos, but they are regularly justifiable for an emitter. (Ns > 10 cm ), Seff is only constant when there is small When Q is too small to affect the carrier to moderate Q. Has Q ever been measured for diffused concentrations, ps = pd, and Equations 2 and 3 combine to silicon? Is it possible that the apparent dependence of Seff give, on Ns previously observed for Ph-doped SiO2–Si S = S . (4) eff p0 1E+14
This states that in this case Seff depends solely on the )
2 - surface defects and is independent of Q of Ns. This 1E+13 equality is regularly assumed (despite often being
unjustified!). (cm /q 1E+12 Si When Q is sufficiently large that it alters the carrier 1E+11 concentrations near the surface, there is ps pd. This is represented by the band-bending depicted in Figure 3. In such case, and when the quasi-Fermi levels are constant 1E+10 between s and d, ps = pd·exp(s/VT), where s is the 1E+9 band-bending in Volts and VT is the thermal voltage. Q Accumulation Hence, 1E+8
( ). (5) 1E+8
Equation 5 states that S is less than S when Q is 1E+9
eff p0 )
2 -
1E+10 /q (cm /q Si 1E+11
1E+12
Inversion Q Inversion 1E+13
1E+14 1E+12 1E+14 1E+16 1E+18 1E+20 Dopant concentration at surface (cm-3)
Figure 4: Contours of Q vs Ns at which one can assume J0s and Seff are independent of Ns in equilibrium. The contours represent a 10% error in the calculation of Figure 3: Band diagram at the surface of an illuminated surface carrier concentration and therefore Us, for low n-type semiconductor coated with a positively charged injection, c-Si, and 300 K; valid for either p- or n-type. insulator.
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interfaces (e.g., [22–26]) is in fact due to a large Q? For the minority carrier concentration at the surface. Three 2 some surface treatments, Seff of B-doped silicon was of the four contours are governed by the ratio Q /Ns, 17 -3 2 7 found to vary little over Ns = 3–300 10 cm [27]; and where Q /Ns = 1.50 × 10 cm for the green line in in a recent study of Al2O3–Si, it was concluded that accumulation, 1900 cm for the yellow line in neither Q nor Sp0 varied over a wide range of B-doped Ns accumulation, and 1600 cm for the yellow line in [17]. inversion. The green line in inversion does not have an 1.85 6 We take this opportunity to re-evaluate previously explicit solution but is approximately Q /Ns = 1.5 × 10 published data for SiNx-passivated Ph-diffused silicon cm. [26]. In that study, Seff was extracted from simulations of The definition of J0s is in analogy to the saturation J0e measurements. Zero surface charge Q was assumed, current in the Shockley diode equation. It is given by and the blue line in Fig. 5 was obtained [26], indicating that Seff increases strongly with the phosphorus dopant [ ]. (6) density at the surface Ns. We repeated those simulations, 12 2 for a constant Q of +2 and +3 10 C/cm , as is typically When the quasi-Fermi levels are flat between s and d, this measured at low dopant densities for SiNx. We then plot is equivalent to S (not S because electrostatics are now considered) p0 eff against N as the green and red lines in Fig. 5. By [ ]. (7) s including this plausible level of surface charge in the calculations, we find that Sp0 increases only slowly over and when Boltzmann statistics apply, 19 -3 the range Ns = 1 to ~7 10 cm . At higher Ns, Sp0 [ ( ) ], (8) might be overestimated because we neglect phosphorus precipitation, which often occurs at high Ns and where V is the separation of quasi-Fermi levels. The introduces additional SRH recombination. Sp0 is a direct measure of interface quality effects. That it appears definition of J0s is examined in more detail in Ref. [18]. When Q is sufficiently large that J0s is constant with rather insensitive to Ns might be explained by (i) the P atoms have 5 instead of 4 valence bonds, which may give Ns (whether in accumulation or inversion), J0s can be an additional degree of freedom to the lattice for relaxing succinctly written in terms of Sp0 and Q [13, 18]: the twisted, stretched and dangling bonds at the interface, . (9) or (ii) dopant states may simply introduce very few defect states at the interface. 4.4 Summary We presented the conditions under which Seff and J0s are independent of Ns and showed how they relate to Q and Sp0 (or Sn0). When neither Seff nor J0s can be justifiably applied, emitters cannot be accurately modelled without accounting for Q and Ns, and solutions can be found in the manner of Refs. [28, 29]. Preferably, this is performed with inputs for the energy dependent density of states of each donor and acceptor defect, their associated capture cross sections, and insulator charge. We also note that Sp0 and Q for Al2O3–Si interface have recently been found to be constant over a wide range of Ns [17] for several different deposition and annealing conditions. In passivation schemes where this is the case, emitter modelling far simpler than is currently appreciated, since one can then measure Sp0 and Q on an undiffused sample and apply those values at any other Ns. Over certain ranges of Ns (see Figure 4), one can also combine S and Q into a single parameter (either S p0 eff from Equation 4 or J0s from Equation 9), making emitter Figure 5: The surface recombination velocity parameter simulation simpler and faster. Sp0 of SiNx-passivated surfaces and its dependence on the phosphorus dopant concentration at the surface, assuming the indicated interface charge Q. Sp0 is extracted from 5 ASSUMPTION 3: BOLTZMANN STATISTICS. simulations of J0E measurements as in Ref. [26]. In non-degenerate semiconductors, the number of free
carriers is small relative to the number of states that those
carriers can occupy. The conentration of electrons and 4.3 The surface saturation current density J 0s holes can therefore be described by ideal-gas theory. An alternative to S is the “surface recombination eff When this assumption is valid, Boltzmann (B) statistics current density” J , which is introduced in Ref. [18]. 0s are applicable and the concentration of free electrons Contrary to S , it is independent of N when Q is large eff s depends exponentially on the Fermi energy, but proportional to Ns when Q is small. Thus, it is preferable to use J when Q is large, and S when Q is 0s eff ( ). (10) small. The green region in Figure 4 is where J0s is An equivalent equation exists for holes. independent of Ns. To be more specific, the contours in B statistics do not accurately determine the majority Figure 4 show the points where there is a 10% error in carrier concentration when carrier–carrier interactions are
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significant. In c-Si at 300 K, this occurs when the doping little error. Having evaluated various options, we exceeds 1019 cm–3 [26]. In such case, the more general implemented the algorithm described in [34] into EDNA Fermi–Dirac (FD) statistics are required, by which 2. Compared to a fine-resolution numerical solution of the integral, the approximation gives a 100-fold reduction ( ). (11) in computation time at a maximum cost of 0.002% relative error for Fermi levels within 100kT of the band Figure 6 shows how the application of (10) rather edges. We also rely on an analytical approximation to than (11) introduces error at high phosphorus dopant facilitate rapid evaluation of the inverse FD function concentrations. In Figure 6(a), the equilibrium hole [35]; this algorithm introduces less than 0.003% relative concentration p0 (i.e., the minority carrier concentration) error in the carrier concentrations within the is plotted as a function of the ionised dopant aforementioned range of Fermi levels (100kT from the concentration ND when one applies B (green) and FD band edge). (orange) statistics. The dashed lines show p0 when band- Thus, with fast and accurate application of FD gap narrowing (BGN) is omitted and the solid lines show statistics, we conclude that for heavily doped silicon it is p0 when BGN is included using Schenk’s model for BGN preferable to account for degeneracy correctly, and to [30, 2]. The figure shows a marked deviation between B 19 –3 apply a separate model for BGN. Yan and Cuevas give and FD statistics at ND > 10 cm . such an option determined from experimental data for ND Figure 6(b) plots the relative error in p0 when B < 1020 cm–3: statistics are used instead of FD. (The error is the same with or without BGN). The figure indicates that the 19 –3 ( ), (13) relative error in p0 is 10% at 10 cm and over 100% at 20 –3 10 cm . Note that p0 is overestimated by B statistics –5 14 3 20 where = 4.2 10 [ln(ND/10 )] for ND < 10 and, consequently, the calculated recombination is also -3 20 –3 cm [36]. overestimated, especially at ND > 10 cm where Pauli- We emphasise, however, that the simulation of very blocking becomes significant. This ultimately leads to an heavily doped emitters is problematic irrespective of overestimation of J0E when modelling heavily doped whether FD statistics are implemented. The main reason emitters with (10). The amount to which J0E is is that phosphorus precipitates readily in silicon and overestimated has a complicated dependence on the causes, in many cases, more SRH recombination than dopant profile and surface recombination. there is Auger recombination [37]. The precipitates are In addition to degeneracy, BGN also causes a stable under usual processing conditions, e.g., during deviation in p0 at high doping [26]. As evident in drive-in, and therefore may affect a whole series of Figure 5, the deviation due to BGN occurs at a lower ND experiments consistently (potentially in experimental 17 3 (~ 10 cm ), and the effect on p0 is the opposite to that studies designed to determine BGN). This consistency caused by degeneracy. may give the impression that the SRH recombination can The opposing trends introduced by degeneracy and BGN make p0(ND) reasonably ‘exponential’ until ND > 20 –3 1E+6 10 cm (see the solid orange line in Figure 5). This has (a) been observed and exploited in a number of studies (e.g., 1E+5 [22–25, 27, 38, 39]). They find that the influence of 1E+4
degeneracy and BGN can be adequately combined into a 1E+3
) 3
20 –3 - simple exponential for ND < 10 cm , and thus the 1E+2
equations for B statistics can still be used by introducing (cm
0 1E+1
‘apparent BGN’ that combines degeneracy and BGN. p B stats, with BGN (Schenk) A recent example from Yan and Cuevas is plotted in 1E+0 FD stats, with BGN(Schenk) Fig. 6(a) as the dotted black line [36]. Its equation is 1E-1 B stats, no BGN 1E-2 FD stats, no BGN
( ), (12) Yan-Cuevas (Eq 12) 1E-3 1E+14 1E+15 1E+16 1E+17 1E+18 1E+19 1E+20 1E+21 where = 13.0 10–3[ln(N /1017)] in meV for N 1000 D D N (cm-3) > 1017 cm–3 and zero otherwise, E 0 is the equilibrium (b) D c (%) conduction band energy. A similar equation is employed 0 100 in PC1D [3]. Although there are various inconsistencies by 10 applying this approach [31, 32], the inclusion of an exponential apparent BGN describes p0(ND) very well for 1 20 –3 ND < 10 cm . At higher ND, Pauli-blocking becomes strong, and the apparent BGN would need to decrease p in error Relative 0.1 strongly to account for the increasing influence of degeneracy. To our knowledge, there is no apparent 0.01 model that describes this “turning point”, and thus, 1E+14 1E+15 1E+16 1E+17 1E+18 1E+19 1E+20 1E+21 industrial phosphorus-diffused emitters, which typically -3 20 –3 ND (cm ) have ND > 10 cm , cannot yet be accurately modelled by apparent BGN models. Figure 6: (a) Equilibrium minority carrier concentration FD statistics are often avoided because F1/2 cannot be rigorously expressed analytically. But there exist many p0 as a function of the ionised phosphorus concentration explicit approximations [33–35] to the FD integral that ND, and (b) relative error in p0 by applying B instead of significantly reduce computation time while introducing FD stats.
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be included as an apparent BGN expression. However, [14] J. Brody and A. Rohatgi, “Analytical such apparent BGN models only work for certain approximation of effective surface recombination amounts of precipitation and do not have general velocity of dielectric-passivated p-type silicon,” applicability. Moreover, inactive phosphorus usually Solid-State Electronics 45, pp. 1549 –1557, 2001. exists in various forms to various relative amounts, from [15] S. Steingrube, P.P. Altermatt, D.S. Steingrube J. interstitial phosphorus atoms, inactive phosphorus Schmidt, R. Brendel, “Interpretation of clusters, fine silicon phosphide (SiP) precipitates, to rod- recombination at c-Si/SiNx interfaces by surface like precipitate structures. Hence, we do not expect that damage,” Journal of Applied Physics 108, 013506, such an apparent BGN expression can ever be universally 2010. applicable. [17] L.E. Black, T. Allen, K.R. McIntosh and A. Apart from precipitates, there are additional Cuevas, “Surface passivation of crystalline silicon uncertainties in simulating very heavily doped emitters: by Al2O3: dependence on surface boron (i) the minority carrier mobility is not very well known in concentration,” In preparation, 2013. heavily doped silicon; (ii) Auger coefficients are not well [18] K.R. McIntosh and L.E. Black, “On the analytical established for high dopant concentrations; and (iii) the equations for field-effect passivation,” In effective electron mass starts to increase above ND > preparation, 2013. 1020 cm–3, which is neglected in Schenk’s BGN model. [19] R.H. Kingston and S.F. Neustadter, J. Appl. Phys. Because POCl3 furnaces are a very cheap means of 26, p. 718, 1955. forming an emitter, it can be expected that highly doped [20] C.E. Young, J. Appl. Phys. 32, p. 329,1961. emitters will continue to be used in the commercial [21] A.S. Grove and D.J. Fitzgerald, Solid-State production of silicon solar cells. Thus, the development Electron. 9, p. 783, 1966. 20 –3 of accurate simulation models for Ns > 10 cm is not [22] A. Cuevas, P.A. Basore, G. Giroult–Matlakowski just enjoyable, it has industrial relevance as well. and C Dubois, “Surface recombination velocity of highly doped n-type silicon,” Journal of Applied Physics 80, pp. 3370–3375, 1996. 6 REFERENCES [23] S. W. Glunz, S. Sterk, R. Steeman, W. Warta, J. Knobloch, and W. Wettling, 13th EU PVSEC, [1] EDNA 2 available at www.pvlighthouse.com.au. Nice, pp. 409–412,1995. [2] K.R. McIntosh and P.P. Altermatt, “A freeware 1D [24] M.J. Kerr, J. Schmidt, A. Cuevas and J.H. Bultman, emitter model for silicon solar cells,” Proc. 35th “Surface recombination velocity of phosphorus- IEEE PVSC, Honolulu, pp. 2188–2193, 2010. diffused silicon solar cell emitters passivated with [3] P. A. Basore and D. A. 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Kuhlmann, Charakterisierung und Physcis 84, pp. 3684–3695, 1998. mehrdimensionale Simulation von MIS- [31] H. Kroemer, “Quasi-electric and quasi-magnetic Inversionsschichtsolarzellen. PhD Thesis, Univ. fields in nonuniform semiconductors”, RCA Rev. Hannover, Germany, 1998. 28, 332 (1957).
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[32] M. S. Lundstrom, R. J. Schwartz, and J. L. Gray, n were the carrier concentrations at 497 μm. By this “Transport equations for the analysis of heavily method, J0E was found to be constant over a wide range doped semiconductor devices”, Solid-State of n, increasing from about n > 1015 cm–3. Figure 1 15 –3 Electron. 24, 195 (1981). shows the J0E for n < 10 cm . In EDNA 2, J0E was [33] J. S. Blakemore, “Approximations for Fermi-Dirac determined at a junction voltage of 0.55 V. integrals, especially the function F1/2() used to describe electron density in a semiconductor”, Solid-State Electronics 25, 1067-1076 (1982). A.2 Comparison of EDNA 2 to Sentaurus [34] Van Halen, P and Pulfrey DL. "Accurate short series The temperature was 300 K. The bulk was p-type 14 –3 approximations to Fermi–Dirac integrals of order with NA = 1 10 cm . The diffusion was n-type with a −1/2, 1/2, 1, 3/2, 2, 5/2, 3, and 7/2," Journal of Gaussian profile defined by a depth factor of 0.5 μm and Applied Physics, 57(12), 1985; pp. 5271-5274. a variable surface concentration. Identical models were [35] Antia, HM. "Rational function approximations for applied for determining the intrinsic band gap (Sentaurus Fermi-Dirac Integrals," The Astrophysical Journal with a band gap multiplier of 1.00547), the density of Supplement Series, 84, 1993; pp. 101-108. states (Sentaurus Formula 1), band gap narrowing [36] Di. Yan and A. Cuevas, “Empirical determination (Schenk), carrier mobility (Klaassen), Auger of the energy band gap narrowing in highly doped recombination (Altermatt), and radiative recombination n+ silicon,” Journal of Applied Physics 114, (B = 4.73 10–15 cm3/s). Bulk SRH recombination was 044508, 2013. included with n0 = p0 = 100 μs and Et = Ei. The effective [37] P. Ostoja, S. Guerri, P. Negrini, S. Solmi, “The surface recombination velocity was varied. Fermi–Dirac effects of phosphorus precipitation on the open- statistics and 100% ionisation were instituted. J0E was circuit voltage in n+/p silicon solar cells”, Solar determined at a junction voltage of 0.65 V. Cells 11, 1-12 (1984) The device that was modelled in Sentaurus had a [38] J. del Alamo, S. Swirhun, and R. M. Swanson, width in the x-dimension of 200 μm and contained four “Simulataneous measurement of hole lifetime, hole contacts. Two contacts were located on the front surface mobility and band-gap narrowing in heavily doped and two on the rear. On both front and rear, the contacts n-type silicon, IEEE PVSC, 1985. were 2 μm wide and located at the outer edges (leaving [39] J. Del Alamo, S. Swirhun and R.M. Swanson, 196 μm between the contacts). The contacts were "Measure and modelling minority carrier transport sufficiently small that contact recombination contributed in heavily doped silicon,” Solid-State Electronics negligibly to the total recombination in the diffusion, and 28, pp. 47–54, 1985. the lateral variation in the J0E was also negligible. The J was calculated from pn at 0.65 V, p n at 0 V, and 0E 0 0 the hole current density Jp, at a position very close to the APPENDIX A: INPUTS IN SIMULATIONS metallurgical junction, using the equation
Jp = J0E( pn / p0n0). A.1 Comparison of EDNA 2 to PC1D The temperature was 300 K. The bulk was p-type 15 –3 with NA = 1.335 10 cm . The diffusion was p-type with an ERFC profile defined by a depth factor of 0.5 μm and a variable surface concentration. The intrinsic carrier concentration was 1010 cm–3 (where in EDNA 2 this was attained with the Passler model for the intrinsic band gap, Sentaurus Formula 2 for the density of states, and a band gap multiplier of 1.00385). PC1D defaults were used for band gap narrowing (i.e., del Alamo model with an onset of 1.4 1010 cm–3 and a slope of 0.014 eV), Auger –31 6 recombination (Cn LLI = 2.2 10 cm /s, Cp LLI = 9.9 –32 6 –30 6 10 cm /s, CHLI = 1.66 10 cm /s), and radiative recombination (B = 9.5 10–15 cm3/s). The mobility of electrons and holes was fixed at 1360 and 470 cm2V–1s-1, respectively. Bulk SRH recombination was 12 made negligible by setting n0 = p0 = 10 μs and Et = Ei. The surface recombination velocity was varied (where in PC1D, the electron and hole recombination velocities were equal, Sn = Sp). Boltzmann statistics and 100% ionisation were instituted. To determine J0E in PC1D, identical diffusion were introduced to the front and rear surfaces and the width was 500 μm. J0E was determined for a range of excess carriers by setting the front illumination at 1190 nm and varying the intensity from 10,000 to 0.0001 W/cm2 in logarithmic steps. This gave very close to uniform generation and therefore a symmetrical carrier profile. At each intensity, the J0E was determined by taking the net recombination, subtracting the bulk Auger and radiative 2 recombination, and dividing by 2pn/(qni ) where p and
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