advertisement

Tel. Phenomena No.: 0049-0711-6857181 Solid State Vol. 93 (2003) pp 399-404 © (2003) Trans Tech Publications, Switzerland Fax. No.: 0049-0711-6857206 doi:10.4028/www.scientific.net/SSP.93.399 e-mail: kurt.taretto@ipe.uni-stuttgart.de A simple method to extract the diffusion length from the output parameters of solar cells - application to polycrystalline silicon K. Taretto, U. Rau, T. A. Wagner, and J. H. Werner Institut für Physikalische Elektronik, Universität Stuttgart, Pfaffenwaldring 47, 70569 Stuttgart, Germany Keywords: solar cell, diffusion length, internal quantum efficiency, grain boundary recombination, polycrystalline silicon Abstract. This work presents a simple method to obtain the effective diffusion length Leff of a solar cell directly from measured values of open circuit voltage, short circuit current density, and the doping density in the base of the cell. In the second part of this paper, we extract Leff from literature data of polycrystalline silicon cells, with grain sizes from 10-2 to 104 µm, modeling the extracted Leff as a function of the grain size g, and the recombination velocity SGB at the grain boundaries. For g > 1 µm, our model predicts 105 < SGB < 107 cm/s. Cells with g < 1 µm, are understood with 101 < SGB < 103 cm/s. This finding supports the hypothesis that the key to high efficiencies at small grain sizes is the use of {220}-textured films. Introduction The efficiency of solar cells depends strongly on the effective diffusion length Leff of minority carriers. Internal quantum efficiency (IQE) measurements give Leff, but its determination is based on an exact knowledge of the absorption and dispersion of light in the cell. This makes the IQEmethod rather complicated, but up to now, no simple substitute to this method was found. This work presents such a substitute, where Leff is calculated directly from the open circuit voltage, the short circuit current density, and the doping level in the base of the solar cell. In contrast to the IQEmethod, our approach constitutes a fast and simple method to estimate Leff. To prove the validity of our model, we compare a large data set of Leff measured by IQE, and compare these values to the predictions of our model. We show that throughout three orders of magnitude of Leff, our method allows us to determine Leff within an accuracy of 35 %. The efficiency of a solar cell generally increases with increasing grain size g [1]. The explanation for this increase is simple: if g increases, the ratio of grain boundary area to grain volume decreases, reducing the amount of recombination centers, increasing the minority carrier diffusion length. In the second part of this paper, we show the increase of Leff with g, extracting Leff from literature data of polycrystalline silicon cells having 10-2 < g < 104 µm. We model the extracted Leff-data as a function of g and the recombination velocity SGB at the grain boundaries. At g > 1 µm, our model predicts values of SGB between 105 and 107 cm/s, while cells with g < 1 µm are only understood with SGB between 101 and 103 cm/s. This finding supports the hypothesis that the nanocrystalline silicon cells benefit from {220}-textured films introduced in Refs. [2] and [3]. Model The double-diode current(J)/voltage(V) characteristics of the pn cell is given by the equation [4] All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, www.ttp.net. (ID: 130.203.136.75, Pennsylvania State University, University Park, United States of America-04/06/14,22:39:27) 400 Polycrystalline Semiconductors VII J = J 01 exp V V - 1 - J SC , - 1 + J 02 exp Vt 2V t (1) where Vt is the thermal voltage kT/q, J01 and J02 are the saturation current densities, and JSC is the short-circuit current density. The first term of Eq. (1) represents the recombination current in the base, characterized by an ideality factor nid = 1. The second term belongs to the recombination in the space-charge region (SCR), and it is modeled considering Shockley-Read-Hall recombination via a single trap located in the middle of the bandgap, resulting nid = 2 [4]. In a monocrystalline material, J01 is a function of an effective diffusion length Leff = Leff,mono, which is given by Leff,mono = Lnf(W,Sb,Ln), where Ln is the diffusion length of electrons in the p-type base, and f is a function of the base thickness W, the recombination velocity Sb at the back contact, and Ln [5]. The current density J02 depends on Ln, not including contact recombination [4]. In order to simplify the analysis, we make an assumption that allows to use the same diffusion length to calculate J01 and J02: the value of Ln does not depend on the position in the cell (SCR or bulk), and the recombination of carriers at the back contact does not affect strongly Leff. This assumption imposes Ln < W, since, Leff,mono equals Ln within an error smaller than 20 % provided Ln/W < 1. Assuming Ln/W < 1, both current densities J01 and J02 become a function of an unique diffusion length Leff. In the p-type base, the saturation current density J01 is given by [5] qD n n i2 1 J 01 = , (2) N A L eff where Dn is the diffusion constant of electrons, ni the intrinsic carrier concentration, and NA the doping density in the base of the cell. In the SCR, the saturation current density J02 is given by [6] q pD n n i V t 1 . J 02 = (3) Fmax L 2eff If the doping profiles are step-like, the maximum electric field Fmax is given by Fmax =(2qNAVbi/eS)1/2 [6], being Vbi the built-in voltage, and eS the semiconductor’s absolute dielectric constant. Replacing Eqs. (2) and (3) in Eq. (1), the whole J/V-curve is written as a function of Leff. Thus, we express Leff as a function of the cell’s open circuit voltage VOC and JSC. At J = 0, we have V = VOC, and using the definitions of J01, J02, and Fmax, Eq. (1) is rewritten as es qD n n i2 1 V V 1 J SC = exp OC + q pD n n i V t exp OC . (4) 2 N A L eff Vt 2 qN A V bi L eff 2V t Solving this equation for Leff, we obtain z + z 2 + 2 pV t J SC L eff = 2J 2 Dn e S V bi 1/ 2 1/2 z1/2 (5) , SC where z is given by z = qn i D n exp V OC NA - ln Vt ni . (6) Figure 1 shows the increase of Leff with [VOC - Vt ln(NA/ni)] , given by Eqs. (5) and (6) with Dn = 10 cm2/s, and Vbi = 0.8 V. This estimate of Vbi meets commonly found values in silicon cells. Equation (5) yields values of Leff differing less than 1 % for the range 0.5 < Vbi < 1.0 V. Figure 1 indicates that a material with low recombination (high Leff) is required to obtain solar cells with high values of VOC. The plot shows two regions: the region for low Leff, where the recombination in the SCR determines VOC, and nid = 2; and the region of higher Leff, where VOC is limited by bulk recombination, resulting nid = 1. The curves in Figure 1 suggest that, by calculating the ideality factor nid at VOC from the slope of the condition Leff/W < 1. Figure 2 shows that the values of Leff obtained with the present model, agree with the IQE values over three orders of magnitude of Leff. The circles in Figure 2 belong to silicon epitaxial cells prepared with the ion-assisted deposition method [7], while the triangles belong to multicrystalline silicon cells [8]. The solid line in Figure 2 gives the identity Leff (modeled) = Leff (measured by IQE). The dashed lines represent the least-square standard deviation of the data from the identity line, indicating that the present model predicts Leff with an error of 35 % (assuming that the IQE values are exact). Effective diffusion length in polycrystalline cells 401 nid = 1 2 10 2 JSC = 2 mA/cm 1 10 0 10 5 10 nid = 2 20 30 40 -1 10 -0.2 -0.1 0.0 0.1 VOC-Vtln(NA/ni) [V] 0.2 Figure 1: The effective diffusion length Leff obtained by the present model, as a function of the open circuit voltage VOC, and the short circuit current density JSC. At low Leff the curves are described by the recombination in the space-charge region (ideality nid = 2). At high Leff, the recombination in the base (nid = 1) limits VOC. 2 Leff [µm], modeled a measured J/V curve, one can determine where the highest recombination takes place: in the SCR, or in the bulk. Cells with a small diffusion length, will show nid = 2, and will have VOC limited by the recombination in the SCR. By increasing Leff, the generated electron-hole pairs will not recombine in the SCR but mainly in the bulk, showing nid = 1. Equation (5) is extremely useful for the experimentalist who wants to estimate Leff, because JSC, VOC, and NA are easy to measure. The standard technique to determine Leff is much more complicated, since it is based on internal quantumefficiency (IQE) measurements, which require an exact knowledge of the absorption constant of the material [5], making a determination of Leff rather intricate. Now we prove that Eq. (5) gives the correct value of Leff, using literature data of silicon solar cells where Leff was obtained from IQE measurements, and compare them to the values of Leff predicted by Eq. (5). The selected data satisfies effective diff. length Leff [µm] Solid State Phenomena Vol. 93 10 1 10 0 10 0 1 2 10 10 10 Leff [µm], from IQE measurements Figure 2: The effective diffusion length Leff of silicon solar cells determined by our model agrees with the values of Leff obtained by internal quantum efficiency measurements, over three orders of magnitude of Leff. The circles belong to silicon epitaxial cells [7], while the triangles correspond to polycrystalline silicon cells [8]. In a polycrystalline material, we must consider a diffusion length Leff,poly, which contains the recombination SGB velocity at the grain boundaries (GBs). In Ref. [10], the diffusion length Leff,poly was calculated considering the diffusion and recombination of minority carriers in the base of a three-dimensional np cell, with no GBs perpendicular to carrier flow. With the recombination of carriers inside the grains described by Leff,mono, the diffusion length Leff,poly is given by the equation L eff ,mono L eff , poly = , (7) 2 S GB L 2eff , mono 1+ Dn g 402 Polycrystalline Semiconductors VII Table 1: Experimental polycrystalline silicon solar cell parameters extracted from the literature. The geometrical quantities given are the area A, the cell thickness W, and the grain size g. The doping density NA corresponds to the p-type base of the cells. The electrical parameters, given under AM1.5 illumination conditions, are the efficiency h, the open circuit voltage VOC, the fill factor FF, and the short circuit current density JSC. Cells denoted as pnn-type have low doped and n-type middle-layers. cell cell Ref. A W g NA VOC FF JSC h 2 -3 2 type [cm ] [mm] [cm ] [%] [mV] [%] [mA/cm ] [mm] a np [11] 4 60 104 2x1016 (a) 16.5 608 77 35.1 3 b np [12] 1 100 10 2x1016 (a) 16.6 608 82 33.5 3 16 (b) c np [13] 1 72 10 2x10 9.3 567 76 21.6 d np [13] 1 30 103 2x1016 (b) 11 570 76 25.6 16 (b) e np [15] 1 300 500 2x10 9.95 517 7.2 27.1 f np [15] 1 300 500 2x1016 (b) 11.1 538 72.4 28.5 16 (b) g np [14] 1 500 250 2x10 10.7 527 69 31.1 h np [17] 1 49 200 2x1017 8.2 525 66 23.8 16 i np [16] 1.3 30 150 3x10 8.3 561 74 20.1 j np [18] ? 330 20 2x1016 (b) 4.3 430 64 16.7 k np [19] 0.01 4.2 10 4.3x1017 6.5 480 53 25.5 17 l np [21] 1 15 7 1x10 5.2 461 64 17.5 m np [17] 1 15 5 2x1017 2.8 368 59 12.8 17 n np [20] 0.17 15 1 1x10 5.3 400 58 23 o np [22] 1 20 1-3 2x1017 2.0 340 59 10.1 77 24.35 p [23] ?1 2 2x1016 (b) 10.1 539 pnn »0.5 16 (b) q [24] 1 5.2 1 2x10 9.2 553 66 25 pnn 16 (b) r pin [26] 0.7 2 0.05 2x10 7.5 499 68.7 22 s pin [25] 0.25 2.1 0.042 2x1016 (b) 9.5 500 68 28 16 (b) t pin [27] 0.25 2.5 8.6 500 66 26.2 »0.01 [28] 2x10 16 (b) u pin [29,30] 0.33 2 2x10 8.5 531 70 22.9 »0.01 a) this value an estimate that corresponds to commonly utilized doping levels. b) value estimated from the resistivity values between 1-2 Wcm (p-type material), given in the paper corresponding to each cell. which shows that at high grain sizes g, Leff,poly approaches the limit given by the monocrystalline value Leff,mono. Figure 3 shows the increase of Leff with g, where the values of Leff were obtained using the data of Table 1 using Eq. (5). All the circles belong to np-type cells, and all triangles to pin- or pnn- cells. The solid lines give Leff,poly from Eq.(7), assuming Dn = 10 cm2/s, Leff,mono = 102 mm, and Vbi = 0.8 V. Associating the data points to the solid lines in Figure , we distinguish two groups of data with different ranges of SGB: i) cells with g > 1 mm, show values of SGB in the range 105 < SGB < 107 cm/s, ii) cells made from nano- and microcrystalline material, where g < 1 mm, are only understood with 101 < SGB < 103 cm/s. Despite the fact that the present model assumes np junctions and not pin junctions, the difference of SGB between the two regions is large. Are the low SGB values found for the pin cells misleading because the model does not strictly apply to them? The answer to this question is given by the numerical simulations of pin cells given in Ref. [32]. The simulations show that SGB indeed must have values between 300 to 1100 cm/s (at grain sizes around g = 1 mm) in those pin cells. Solid State Phenomena Vol. 93 403 diffusion length Leff,poly [µm] An explanation for such low grain 2 boundary recombination velocities is given in 1 limit Leff,mono=10 µm S = 10 cm/s GB 2 Refs. [2] and [3], where it is argued that the 10 B A low SGB comes from structural differences D 3 C between the cells with g < 1 mm and cells P 10 I 1 Q F 10 G showing g > 1 mm. The pattern found to make 5 U E 10 R that estimation is that all cells with g < 1 mm S H T L 0 J of Table 1, were reported to have a {220} 7 10 10 N M K surface texture. With that information, the O low SGB is explained as follows: “The -1 10 measured {220}-texture implies a (110)oriented surface for most of the grains. A -2 -1 0 1 2 3 4 large number of the columnar grains must 10 10 10 10 10 10 10 therefore be separated by [110] tilt grain grain size g [µm] boundaries. Symmetrical grain boundaries of this type are electrically inactive because they Figure 3: Data points give the diffusion length Leff extracted from the data of Table 1 using Eq. (5). contain no broken bonds” [3]. Circles correspond to np cells, and triangles to pin cells. The overall increase of Leff with the grain Discussion and Conclusions size g, indicates that the recombination velocity SGB at the grain boundaries determines Leff The In the first part of this contribution we solid lines model L considering a recombination eff present a method to extract the effective velocity S at the grain boundaries. GB diffusion length in pn-type solar cells from I/V measurements. Our model considers recombination in the bulk as well as in the space-charge region of the cell. We prove that over three orders of magnitude of the effective diffusion length, our method agrees very well with values obtained from internal quantum-efficiency measurements. The method to extract the effective diffusion length presented here requires only the short-circuit current of the cell, the open-circuit voltage, and the doping density in the base of the solar cell, making it simple compared to quantum-efficiency measurements. The second part of this contribution extracts the diffusion length of polycrystalline silicon solar cells with grain sizes between 10-2 and 104 mm. The increase of diffusion length with grain size is then modeled with an equation that considers the recombination velocity at the grain boundaries. We find that the reported nanocrystalline and microcrystalline cells reaching efficiencies up to 10 % are only explained with low grain boundary recombination velocities between 101 and 103 cm/s. Such low recombination velocities support earlier predictions ascribing low defect densities to the GBs in those small grained cells. Literature [1] See for example R. B. Bergmann, Appl. Phys. A 69, 187 (1999) [2] J. H. Werner, in Techn. Dig. 13th Sunshine workshop on Thin Film Solar Cells, ed. by M. Konagai (NE-DO, Tokyo, Japan, 2000), p. 41 [3] J. H. Werner, K. Taretto, and U. Rau, Solid State Phenomena 80-81, 299 (2001) [4] A. Goetzberger, B. Voß, and J. Knobloch, Sonnenenergie: Photovoltaik (Teubner, Stuttgart, 1997), p. 94 [5] See, for example, P. Basore, in Proc. 23rd IEEE-Photovoltaic Specialists Conf. (IEEE, New York, 1993), p. 149 [6] N. Jensen, U. Rau, R. M. Hausner, S. Uppal, L. Oberbeck, R. B. Bergmann, and J. H. Werner, J. Appl. Phys. 87, 2640 (2000) [7] L. Oberbeck, Ionenassistierte Deposition von Siliciumschichten, Dissertation thesis, (University 404 Polycrystalline Semiconductors VII Stuttgart 2001) [8] F. Duerinckx, J. Szlufcik, Solar Energy Materials & Solar Cells 72, 231 (2002) [9] A. Poruba, A. Fejfar, Z. Remes, J. Springer, M. Vanecek, J. Kocka, J. Meier, P. Torres, and A. Shah, J. Appl. Phys. 88, 148 (2000) [10] R. Brendel and U. Rau, Solid State Phenomena 67-68, 81 (1999) [11] A. Takami, S. Arimoto, H. Morikawa, S. Hamamoto, T. Ishihara, H. Kumabe and T. Murotani, in Proc. 12th Europ. Photovoltaic Solar Energy Conf. (H. S. Stephens & Assoc., Bedford, 1994), p. 59 [12] Y. Bai, D. H. Ford, J. A. Rand, R. B. Hall and A. M. Barnett, in Proc. 26th IEEE Photovoltaic Specialists Conf. (IEEE, Piscataway, 1997), p. 35 [13] C. Hebling, S. W. Glunz, J. O. Schumacher, and J. Knobloch, in Proc. 14th Europ. Photovoltaic Solar Energy Conf., (H. S. Stephens & Assoc., Bedford, 1997), p. 2318 [14] T. Mishima, S. Itoh, G. Matuda, M. Yamamoto, K. Yamamoto, H. Kiyama, and T. Yokoyama, in Tech. Digest 9th Int. Photovolt. Solar Energy Conf. (Dept. Electrical and Eletronic Engin., Tokyo, 1996), p. 243 [15] M. Spiegel, C. Zechner, B. Bitnar, G. Hahn, W. Jooss, P. Fath, G. Willeke, E. Bucher, H.-U. Höfs, and C. Häßler, Solar Energy Materials & Solar Cells 55, 331 (1998) [16] R. Auer, J. Zettner, J. Krinke, G. Polisski, T. Hierl, R. Hezel, M. Schulz, H. P. Strunk, F. Koch, D. Nikl, H. v. Campe, in Proc. 26th IEEE Photovoltaic Specialists Conf. (IEEE, Piscataway, 1997), p. 739 [17] A. Slaoui, S. Bourdais, G. Beaucarne, J. Poortmans, and S. Reber, Solar Eergy Materials & Solar Cells 71, 245 (2002) [18] F. Tamura, Y. Okayasu, K. Kumagai, in Techn. Digest 7th Intern. Photovolt. Sci. Eng. Conf. (Dept. Electric. and Computer Eng., Nagoya, 1993), p. 237 [19] R. Shimokawa, K. Ishii, H. Nishikawa, T. Takahashi, Y. Hayashi, I. Saito, F. Nagamina, S. Igari, Solar Energy Materials & Solar Cells 34, 277 (1994), the value of the doping concentration in the base was given in K. Ishii, H. Nishikawa, T. Takahashi, and Y. Hayashi, Jpn. J. Appl. Phys. 32, L 770 (1993) [20] H. S. Reehal, M. J. Thwaites, T. M. Bruton, phys. stat. sol. (a) 154, 623 (1996) [21] G. Beaucarne, M. Caymax, I. Peytier, and J. Poortmans, Solid State Phenomena 80-81, 269 (2001) [22] R. Brendel, R. B. Bergmann, B. Fischer, J. Krinke, R. Plieninger, U. Rau, J. Reiß, H. P. Strunk, H. Wanka, and J. H. Werner, in Conf. Rec. 26th IEEE Photovolt. Spec. Conf. (IEEE, New York, 1997), p. 635 [23] K. Yamamoto, M. Yoshimi, Y. Tawada, Y. Okamoto, A. Nakajima, S. Igari, Applied Physics A 69, 179 (1999) [24] T. Matsuyama, N. Terada, T. Baba, T. Sawada, S. Tsuge, K. Wakisaka, S. Tsuda, J. Non-Cryst. Solids 198-200, 940 (1996) [25] K. Saito, M. Sano, K. Matzuda, T, Jibdim N, Higasikawa, and T. Kariya, in Techn. Digest 11th Intern. Photovolt. Sci. Eng. Conf., ed. T. Saitoh (Tokyo Univ. A&T, Tokyo, 1999), p.229 [26] O. Vetterl, F. Finger, R. Carius, P. Hapke, L. Houben, O. Kluth, A. Lambertz, A. Mück, B. Rech, and H. Wagner, Solar Energy Materials & Solar Cells 62, 97 (2000) [27] M. Kondo and A. Matsuda, Thin Solid Films 383, 1 (2000) [28] private communication of J. H. Werner with Michio Kondo, 21 July 2000 [29] N. Wyrsch, P. Torres, M. Goerlitzer, E. Vallat, U. Kroll, A. Shah, A. Poruba, M. Vanecek, Solid State Phenomena 67-68, p. 89 [30] H. Keppner, J. Meier, P. Torres, D. Fischer, A. Shah, Applied Physics A 69, 169 (1999) [31] M. A. Green, Solar Cells (Prentice-Hall, Englewood Cliffs, 1982), p. 97 [32] K. Taretto, U. Rau, and J. H. Werner, Solid State Phenomena 80-81, 311 (2001) Polycrystalline Semiconductors VII 10.4028/www.scientific.net/SSP.93 A Simple Method to Extract the Diffusion Length from the Output Parameters of Solar Cells Application to Polycrystalline Silicon 10.4028/www.scientific.net/SSP.93.399 DOI References [3] J. H. Werner, K. Taretto, and U. Rau, Solid State Phenomena 80-81, 299 (2001) doi:10.4028/www.scientific.net/SSP.80-81.299 [10] R. Brendel and U. Rau, Solid State Phenomena 67-68, 81 (1999) doi:10.4028/www.scientific.net/SSP.67-68.81 [15] M. Spiegel, C. Zechner, B. Bitnar, G. Hahn, W. Jooss, P. Fath, G. Willeke, E. Bucher, H.-U. Hfs, and C. Hler, Solar Energy Materials & Solar Cells 55, 331 (1998) doi:10.1002/(SICI)1099-159X(199805/06)6:3<163::AID-PIP219>3.0.CO;2-H [32] K. Taretto, U. Rau, and J. H. Werner, Solid State Phenomena 80-81, 311 (2001) doi:10.4028/www.scientific.net/SSP.80-81.311