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depends on the photogenerated current Iph(τ) and the open circuit voltage Voc(τ). A PC1D [12] model determines both Iph(τ) as well as Voc(τ) depending on the ...

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Energy Procedia 27 (2012) 129 – 134

SiliconPV: April 03-05, 2012, Leuven, Belgium

Absolute quantification of gettering efficiency from luminescence images L. Stoicescua*, M. Reuterb and J. H. Wernera,b a

Institut für Photovoltaik, Universität Stuttgart, Pfaffenwaldring 47, 70569 Stuttgart, Germany b Steinbeis Center Photovoltaics, Rotebühlstr. 145, 70197 Stuttgart, Germany

Abstract A multidiode simulation determines the absolute solar cell efficiency gain due to gettering. The simulation predicts current/voltage characteristics of lateral inhomogeneous industrial solar cells from luminescence images measured before and after gettering or any other processing step. Improved lifetime distributions lead to a predicted increase in solar cell efficiency 'ηabs = +0.46 % for a POCl3 diffusion on multicrystalline silicon wafers. The simulative approach determines the gettering efficiency of any process step on any wafer material without preselection.

© andand peer-review under responsibility of the c committee of © 2012 2012Published Publishedby byElsevier ElsevierLtd. Ltd.Selection Selection peer-review under responsibility ofscientifi the scientific committee of 2012 the SiliconPV 2012 conference the SiliconPV conference. Keywords: Solar cells; SPICE; Simulation; Luminescence; Gettering

1. Introduction So far gettering efficiency is quantified either by processing sister wafers [1], or by measuring lifetimes before and after defect engineering [1, 2]. The first approach compares solar cell efficiencies of sister wafers with almost identical lifetime distribution which undergo different defect engineering steps. Only a careful preselection of wafer material leads to meaningful results. Nevertheless, only the difference between the applied recipes is determined, while the actual efficiency gain due to gettering remains unknown. Lifetime imaging [3-5] has improved the situation. Measurements before and after any defect engineering step are able to quantify gettering effects locally [1, 2]. Area-weighted lifetime averaging [6] and harmonic averaging [7] have been used to predict solar cell efficiency from spatially resolved

* Corresponding author. Tel: + 49 711 685 67245; fax: + 49 711 685 67143 E-mail address: [email protected]

1876-6102 © 2012 Published by Elsevier Ltd. Selection and peer-review under responsibility of the scientific committee of the SiliconPV 2012 conference. doi:10.1016/j.egypro.2012.07.040

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measurements of the bulk material quality. Both methods used SPICE circuit simulations [8] to legitimate the employed averaging schemes. However, so far it has not been possible to directly incorporate lifetime images with high spatial resolution into SPICE circuit simulations due large circuit sizes and associated long simulation durations. The current contribution applies a multidiode simulation of spatially inhomogeneous solar cells in order to determine the absolute gettering efficiency. The simulation predicts the efficiency of the solar cell from spatially resolved luminescence measurements of the wafer. A smart image decomposition & merging algorithm drastically reduces the circuit size resulting in an effective simulation duration of less than 2 h for a full area industrial solar cell. Comparing absolute solar cell efficiencies calculated from lifetime distributions before and after the defect engineering step yields its absolute gettering efficiency. 2. Simulation Use of the Berkeley SPICE circuit simulator [9] elucidates the effect of inhomogeneously distributed bulk lifetimes on solar cell efficiency. An 8 core 2.27 GHz Intel Xenon workstation with 32 GB RAM evaluates the current - voltage characteristic of a multidiode solar cell circuit with N nodes in ܶ ൌ ͳǤͲʹ ൈ ͳͲି଺ ܰ ଶǤ଴ଷ seconds. Ideally, each pixel of a spatially resolved lifetime image would be modeled as an elementary solar cell in the circuit. This procedure leads at a resolution of ͳͷ͸ ൈ ͳͷ͸ μm2 per pixel.to ܰ ൌ  ͳͲ଺ circuit nodes and an estimated 20 days simulation time. Our smart image decomposition & merging algorithm segments the image into elementary cells of different sizes according to local features. The decomposition enables us to reduce the size of the resulting network to N = 105 nodes without compromising fine image details. The aforementioned workstation is able to simultaneously process 4 simulations each with 105 nodes within 5 hours, thus reducing the simulation time to less than 2 h. 2.1. Image decomposition and merging algorithm The image decomposition and merging algorithm segments the full area lifetime image into small elementary solar cells. The elementary cells have a small size at grain boundaries, front side finger grid and contact points to the cell. The size of the cells increases with a 2:1 ratio towards homogeneous regions. Consequently the algorithm reduces the network complexity while preserving high resolution, where it is necessary. The algorithm operates on feature images which, layered on top of each other, split the solar cell into multiple regions. We distinguish 4 features, each described in one feature image: x grain boundaries detected via the canny edge detection algorithm [10] x front side finger grid x busbars x contact points connected to the cell A decomposition algorithm derived from Eberdt [11] further segments those regions to create a network of elementary cells of size ‫ݏ‬௫ ൈ ‫ݏ‬௬ according to following rules: x Fill the image with cells of size ʹ ൈ ʹ, ʹ ൈ ͳ, ͳ ൈ ʹ or ͳ ൈ ͳ depending on the constraints: o Bricking: a cell may only be placed with its upper left corner in a position ሺ‫ݔ‬ǡ ‫ݕ‬ሻ if ݉‫݀݋‬ሺ‫ ݔ‬െ  ܿ௫ ሻ ൌ ݉‫݀݋‬ሺ‫ ݕ‬െ  ܿ௬ ሻ ൌ Ͳ. Symmetry points like the image boundaries or the repeating solar cell front grid finger structure are effective definitions for the limits cx, cy. o Size: Employ the largest possible cell size, without overlapping an already existing cell. o Containment: A cell must be fully contained in one region. Consequently a rectangular cell may not be part grain boundary or part finger. x Merge all placed cells according to following rules: o A cell may only have two neighbors on each side.

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A cell may have a maximal aspect ratio ‫ šƒܣ‬ൌ ݉ܽ‫ݔ‬൫‫ݏ‬௫ ǡ ‫ݏ‬௬ ൯Ȁ݉݅݊ሺ‫ݏ‬௫ ǡ ‫ݏ‬௬ ሻ of ‫ šƒܣ‬ൌ Ͷ if it is below a front side grid finger or ‫ šƒܣ‬ൌ ʹ elsewhere. o Merge only if the containment and bricking constraints above are fulfilled. The aspect ratio ‫ šƒܣ‬ൌ Ͷ for finger cells avoids clogging the space between fingers with small cells. Figure 1 shows the decomposed lifetime image of an ͳͷ͸ ൈ ͳͷ͸ mm2 multicrystalline solar cell. The algorithm reduces the image with 106 pixel resolution to 105 segments. The amount of segments depends strongly on the quantity and density of grain boundaries detected, as well as on the number of fingers and their spacing. The close up views in Figure 1 show details of the feature images and decomposition results: a) shows the detected grain boundaries layered on top of the original image, b) shows the front side fingers and busbar, c) shows the individual regions decomposed into cells, d) is a closeup of the original image, e) layers the cell boundaries on top of the original image, f) averages the lifetime values over the area of each elementary cell into the local average employed by the simulation. Fine structures are still well preserved, while the amount of data is effectively reduced. o

2.2. Network model Figure 2 shows the network creation. Each cell defined during the decomposition process represents an elementary solar cell. The elementary solar cells are described by the one-diode-model which contains a photogenerated current source Iph(τ), a dark diode and a shunt resistor Rp. The dark diode has the saturation current density I0 and the ideality factor n. The dark saturation current density ‫ܫ‬଴ ሺ߬ሻ ൌ ‫ܫ‬୮୦ ሺ߬ሻ݁ ି

௤௏౥ౙ ሺఛሻ ௡௞் 

(1)

depends on the photogenerated current Iph(τ) and the open circuit voltage Voc(τ). A PC1D [12] model determines both Iph(τ) as well as Voc(τ) depending on the averaged local lifetime τ of the elementary cell. The aluminium rear contact grounds all elementary solar cells, while the emitter connects them with each other. The interconnecting emitter resistor Re:x,y is directional and depends on the local emitter sheet

Fig. 1. Decomposed lifetime image of an ͳͷ͸ ൈ ͳͷ͸ mm2 multicrystalline solar cell. The algorithm reduces the image from 106 pixels to 105 segments. a) detected grain boundaries layered on top of the original, b) front side fingers and busbar, c) individual regions decomposed into elementary cells, d) original image closeup view, e) cell boundaries on top of the original, f) local average lifetime of each elementary cell for simulation: Fine structures are well preserved, while the amount of data is reduced

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resistance Re,sh as well as on the size of the element in x and y direction according to ‫ݏ‬௬ ‫ݏ‬௫ ܴ‡ǣš ൌ ܴ‡ǡ•Š andܴ‡ǣ› ൌ ܴ‡ǡ•Š Ǥ ʹ‫ݏ‬௬ ʹ‫ݏ‬௫

(2)

in y direction. Boundaries where 3 cells connect have to be resolved via the star to delta impedance transformation to eliminate the unnecessary intersection node. The elementary solar cells below the finger and busbar structures are connected to the grid above via the contact resistor Rc. These cells are furthermore considered as not illuminated and have a photocurrent Iph =0. The front side grid sheet resistance Rg,sh may be defined depending on measured grid resistivity and print height. Consequently, grid elements are interconnected via the grid resistor Rg:x,y, which is calculated in the same way as Re:x,y above. 3. Experimental Figures 3a and 3b show the local bulk lifetimes from photoluminescence images captured before and after the phosphorous gettering process. Figure 3a was measured on an as-grown wafer passivated with plasma enhanced chemical vapor deposition (PECVD) SiNx. Figure 3b was measured on an equally passivated sister wafer after emitter formation and subsequent emitter etch back. Both images are simulated with the presented SPICE multidiode simulation. PC1D simulations are performed for comparison. The first PC1D simulation uses the arithmetic lifetime average over the whole image, while the second simulation uses the area-weighted lifetime average [6]. Both PC1D simulations use the model employed for the calculation of Iph(τ) and Voc(τ), however, the model now contains a dark diode, to better describe the cell area shaded below the front side grid. Furthermore, the series and shunt resistance determined from the SPICE simulation are inserted into the PC1D model. Both the SPICE as well as the PC1D simulation employ the same ideality factor n = 1. 4. Results and Discussion The smart image decomposition & merging algorithm produces an error of less than 0.015 %, as determined from simulations of a 5 × 5 cm2 crop from Fig. 3a with N1 = 90 000 nodes at full resolution and only N2 = 18 000 nodes after decomposition & merging.

Fig. 2. Network model: Each image segment represents an elementary solar cell. Each elementary solar cell consists of a photogenerated current source Iph, a dark diode and a shunt resistor Rp. The dark diode is modelled with the saturation current I0 and the ideality factor n. Each elementary solar cell is connected to its neighbours via the emitter sheet resistor Re and the aluminium rear contact, which serves as common ground. Elements below the front side grid are not illuminated (Iph = 0) and connected to the grid via the contact resistor Rc. The front side grid consists of interconnected grid resistors Rg

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Fig. 3. Bulk lifetimes before and after the gettering step. a) SiNx passivated as-grown wafer b) Sister wafer after POCl3 diffusion, emitter etch back and subsequent SiNx passivation

Table 1 displays the solar cell parameters simulated from the data in Figure 3a,b comparing PC1D simulations of the arithmetic and area-weighted lifetime averages to the full area SPICE simulation. After the gettering process the average lifetime τavg, the short circuit current density Jsc as well as the open circuit voltage Voc increase. The increased lifetimes lead to an increased fill factor FF and efficiency η. PC1D has no straightforward way to simulate a device with an ideality factor n other than n = 1. For comparison all simulations use n = 1, which leads to the unrealistic high FF. Differing average lifetimes τavg are the only cause for the different fill factors of the PC1D simulations. However, the full area SPICE simulation incorporates the effect of distributed series resistance into the calculated fill factor. Arithmetic averaging of lifetimes underestimates the effect of low lifetime regions and thus overestimates Jsc, Voc and η [6] for the as-grown wafer. The calculated absolute efficiency increase due to gettering 'ηabs=+0.38 % is too low. Area-weighted lifetime averaging calculates τavg according to its impact on the short circuit current density Jsc. The employed class model yields good results as long as Jsc(τ) and Voc(τ) show similar lifetime dependence. In the hypothetical case of a thin, well passivated sample Jsc(τ) quickly saturates, while Voc(τ) changes over a long range of lifetimes τ; at this, area-weighted lifetime averaging fails to determine the correct Voc as opposed to a SPICE simulation. The calculated efficiencies from the data in Figure 3a,b are very close to the full area SPICE results, while 'ηabs=+0.51 % is slightly higher. The full area SPICE simulation calculates an absolute efficiency increase due to gettering of 'ηabs=+0.46 %. The SPICE simulation considers lateral currents in the emitter and front side grid [7, 13], which have a strong influence the open circuit voltage Voc and a weak influence on the fill factor. Table 1. Simulation results

Arithmetic PC1D

as-grown gettered

τavg [μs] 41,0 148.6

Area Weighted PC1D

as-grown gettered

28,8 117,0

Full Area SPICE

as-grown gettered

2 Jsc [mA/cm ] -33.96 -34.27

Voc [mV] 606.98 612.76

FF [%] 79.94 80.32

η [%] 16.48 16.86

-33.79 -34.23

604.56 612.17

79.89 80.31

16.32 16.83

+0.51

-33.79 -34.23

604.33 611.57

80.05 80.29

16.35 16.81

+0.46

'ηabs [%] +0.38

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5. Conclusion In conclusion, we have introduced an elegant tool to quantify the absolute gettering efficiency of any solar cell processing step. Our approach is especially efficient for material, where defect engineering cannot be evaluated by the sister wafer method. The full area SPICE simulation considers the effect of lateral currents and exemplarily calculates 'ηabs=+0.46 % efficiency gain due to phosphorous gettering on multicrystalline silicon wafers. The advantage of a multidiode SPICE simulation over averaging schemes is an easy incorporation of effects like laterally inhomogeneous ideality factors [14, 15] and edge recombination [16]. The presented simulation delivers a decision tool to determine a priori the necessity of defect engineering at a certain stage in the manufacturing process. References [1] Schultz O, Glunz SW, Riepe S, Willeke GP. High-Efficiency Solar Cells on Phosphorus Gettered Multicrystalline Silicon Substrates. Progress in Photovoltaics: Research and Applications 2006;14:711-9. [2] Tan J, Cuevas A, Macdonald D, Trupke T, Bardos R, Roth K. On the Electronic Improvement of Multi-Crystalline Silicon via Gettering and Hydrogenation. Progress in Photovoltaics: Research and Applications 2008;16:129-134. [3] Ramspeck K, Bothe K, Schmidt J, Brendel R. Minority charge carrier lifetime mapping of crystalline silicon wafers by timeresolved photoluminescence imaging. Journal of Materials Science: Materials in Electronics 2008;19:S4-S8. [4] Kiliani D, Micard G, Steuer B, Raabe B, Herguth A, Hahn G. Correlation between spatially resolved solar cell efficiency and carrier lifetime of multicrystalline silicon. Journal of Applied Physics 2011;110:054508. [5] Giesecke JA, Michl B, Schindler F, Schubert MC, Warta W. Minority carrier lifetime of silicon solar cells from quasisteady-state photoluminescence. Solar Energy Materials & Solar Cells 2011;95:1979–1982. [6] Isenberg J, Dicker J, Warta W. Averaging of laterally inhomogeneous lifetimes for one-dimensional modeling of solar cells. Journal of Applied Physics 2003;94:4122-4130. [7] Michl B, Rüdiger M, Giesecke JA, Hermle M, Warta W, Schubert M. Efficiency limiting bulk recombination in multicrystalline silicon solar cells. Solar Energy Materials & Solar Cells 2012;98:441–447. [8] Nagel LW. SPICE2: A computer program to simulate semiconductor circuits. Electronics Research Laboratory, Univ. California, Technical Report No. UCB/ERL M520 1975. [9] Grabitz PO, Rau U, Werner JH. A Multi-diode Model for Spatially Inhomogeneous Solar Cells. Thin Solid Films 2005;487 (1-2):14-8. [10] Canny J. A Computational Approach To Edge Detection. IEEE Trans. Pattern Analysis and Machine Intelligence 1986:8(6):679–698. [11] Eberdt M, Brown PK, Lazzi G. Two-Dimensional SPICE-Linked Multiresolution Impedance Method for Low-Frequency Electromagnetic Interactions. IEEE Transactions on Biomedical Engineering 2003;50:881-9. [12] Clugston DA., Basore PA. PC1D version 5: 32-bit solar cell modeling on personal computers. In: Conference Record of the 26th IEEE Photovoltaic Specialists Conference, New York: IEEE; 1997, p. 207 – 210. [13] Stoicescu L, Gedeon P, Gläser GC. Full Area Simulation of Multicrystalline Silicon Solar Cells with High Spatial Resolution. In: Conference Record of the 37th IEEE Photovoltaic Specialists Conference; Seattle, USA, 2011, p. 003495 – 003497. [14] Breitenstein O. Nondestructive local analysis of current–voltage characteristics of solar cells by lock-in thermography. Solar Energy Materials & Solar Cells 2011;95:2933–2936. [15] Stoicescu L, Glaeser GC, Reuter M, Rau U, Werner JH. Ideality factor extraction from photoluminescence images. In: Proceedings of the 25th European Photovoltaic Solar Energy Conference,Valencia,Spain,2010, p 29 – 32. [16] Kessler M, Ohrdes T, Altermatt PP, Brendel R. The effect of sample edge recombination on the averaged injectiondependent carrier lifetime in silicon. Journal of Applied Physics 2012;111:054508.

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