27th European Photovoltaic Solar Energy Conference, Frankfurt, Germany, 24-28 September 2012, 2AV.6.9 MODELING OF MULTICRYSTALLINE SI SOLAR CELLS BASED ON LIFETIME DISTRIBUTIONS H. Wagner1,*, M. Müller2, G. Fischer2 and P.P. Altermatt1 Leibniz University of Hannover, Dep. Solar Energy, Appelstr. 2, 30167 Hannover, Germany 2 SolarWorld Innovations GmbH, Berthelsdorfer Str. 111 A, 09599 Freiberg, Germany 1 ABSTRACT: We present a numerical simulation model for multicrystalline Si (mc-Si) solar cells, which predicts cell behavior from lifetime mappings of mc-Si wafers to a precision of about 3 mV in open-circuit voltage, Voc. From a scientific point of view, the model allows us to answer commonly discussed questions such as: which is a suitable method for averaging lifetimes in the quality control of mc-Si wafers, or the role of lower lifetimes in mc-Si cell performance. It is shown that mathematical averaging procedures of wafer lifetime mappings can predict cell’s Voc performance only within a rather broad range (30 mV for the arithmetic mean value, 7 mV for the geometric and 4 mV for the harmonic mean value). From an industrial point of view, the simulation model enables cell manufacturers to predict cell behavior from lifetime images of mc-Si wafers and hence help in deciding on improvement strategies. Keywords: Electrical properties, Lifetime, Multi-Crystalline, Modeling, Qualification and Testing, Silicon Solar Cell, Simulation 1 INTRODUCTION The influence of lifetime inhomogeneity on mc-Si cell performance has been investigated using various methods referenced e.g. in Refs. [1] and [2]. For a recent study, see for example Ref. [3]. In the present paper, we use the modeling software Sentaurus Device [4] (also called Dessis or TCAD) to predict mc-Si cell performance from lifetime images. The model is tested on mc-Si cells made from various ingots and it is used for investigating various phenomena such as the role of low lifetime regions in cell performance. We critically assess, to which precision mathematical averages of the lifetime images can predict mc-Si cell performance. 2 passivated with Al2O3. Accessible to measurement is only the effective excess carrier lifetime, τeff(x,y), which is related to τbulk(x,y) via [5]: 1 𝜏𝑒𝑓𝑓 = 1 𝜏𝑏𝑢𝑙𝑘 + 2𝑆 𝑊 = 1 𝜏𝑆𝑅𝐻 + 1 𝜏𝐴𝑢 + 1 𝜏𝑟𝑎𝑑 + 2𝑆 , (2) 𝑊 where S is the surface recombination velocity, W is the sample thickness, and τ the Shockley-Read-Hall, Auger, and radiative lifetimes, respectively. Independent measurements yield S ≈ 5 cm/s [6], so we can neglect the term containing S in Eq. (2). τbulk [µs] SIMULATION MODEL FOR MC-SI CELLS To simulate mc-Si solar cells, the following wellknown approach is used. An image of the excess carrier lifetime in the bulk of the cell, τbulk(x,y), is discretized in a lifetime histogram having a number n of lifetime values τsim,i with corresponding areas Ai. The sum of all Ai gives the total cell area. We then simulate a sequence of n monocrystalline Si (c-Si) cells, having the homogeneous lifetime τsim,i, and obtain their I-V curves I(V)mono,i. To calculate the I-V curve of the mc-Si cell I(V)multi, these simulated I(V)mono,i curves are connected in parallel in a numerical spice circuit model, using Ai as area factors. The circuit approximates the lateral current flow within both the silicon and the metallization. Alternatively the I(V)mono,i curves can be analytically calculated using 𝐼(𝑉)𝑚𝑢𝑙𝑡𝑖 = ∑𝑛𝑖=1 𝐴𝑖 ∙ 𝐼(𝑉)𝑚𝑜𝑛𝑜,𝑖 . (1) The resistance due to the front metallization, Rs, is included only as a lumped parameter (0.3 Ωcm2) but can, if required, be simulated with the spice model from finger line resistances. In the following, we establish the model by comparing the simulated with the measured I-V curves. As the excess carrier lifetime in the bulk changes during the fabrication of the cells, lifetime images of finished mc-cells are used. To obtain τbulk(x,y), the metallization, the emitter, and the BSF are etched off prior to measurement, and the newly formed surface is Figure 1: Local excess carrier lifetime in the bulk of industrially fabricated multicrystalline solar cells, measured with the photoconductance-calibrated photoluminescence technique. The emitter and the BSF were etched off and the newly formed surface was passivated with Al2O3 prior to the measurement. We measure τeff(x,y) with photoconductancecalibrated photoluminescence images [7-9] using the LIS-R1 system from BT Imaging [10], which contains a Sinton Instruments photoconductance set-up [11]. The optical reflectivity, required by the LIS-R1 system, is derived using a PerkinElmer UV/VIS spectrometer [12], while the wafer thickness is measured with a Käfer digital-dial gauge [13]. An example of a τbulk(x,y) image is shown in Fig. 1. 27th European Photovoltaic Solar Energy Conference, Frankfurt, Germany, 24-28 September 2012, 2AV.6.9 For validating the model, six mc-Si cells were taken from an industrial production line; they were made from six different ingots. Two groups of three cells were produced separately, what means that in each case three of them have an almost identical emitter phosphorus diffusion profile, Al-induced back-surface-field (BSF), geometry, etc. Their I-V parameters were measured with a flash I-V tester from h.a.l.m. [14] such as the shortcircuit current density Jsc, the open-circuit voltage Voc, fill factor FF and efficiency η. After this, they underwent the above described procedure so τbulk(x,y) could be extracted. The two groups were numerically simulated in two dimensions using the software Sentaurus Device [4] from Synopsys to solve the complete set of semiconductor equations [15]. The physical models of Refs. [16] – [18] are applied. All dopant profiles were independently measured with the electrochemical capacitance-voltage technique (ECV) using the CVP21 profiler [19] and the procedure described in Ref. [20]. The front surface recombination velocity Sfront is adjusted such that the measured saturation current-density J0e of the emitter is reproduced in the model. J0e is measured with the procedure of Kan and Swanson [21] on independently fabricated, textured samples. It turns out that Sfront, necessary to reproduce J0e, is compatible with the data on planar wafers in Ref. [18] when taking Ref. [22] for extrapolating from planar to textured wafers. The τsim,i values, extracted from the histogram of the measured τbulk(x,y), are inserted into the simulation as τSRH in Eq. (2), concretely by choosing equal lifetime parameters τn and τp in the well-known SRH equation [23,24] and the defect level energy at midgap (τAu and τrad are chosen as given in Ref. [16], Table 3 therein). The only free parameter is the SRH lifetime in the Al-induced BSF, which seems to be limited by the density of Al-O complexes [17] and hence depends on processing conditions. We adjusted it in a separate study which is compatible with Ref. [17]. With all these parameters independently measured, our model contains no remaining free parameters. As described at the beginning of this Section, we include the I-V curves, simulated with the homogeneous τsim,i values, into a circuit simulation to obtain e.g. the Voc of the multicrystalline cells. A comparison between the experimental and simulated IV parameters is shown in Fig 2. Overall, the simulation model reproduces the measurements very well. For example, the model predicts Voc with a precision of about 3 mV. The fill factor is reproduced with the least precision because the resistive losses due to the metallization are included in the model merely as a lumped value. 3 Experimental Data Simulated Data 16.8 η [%] 16.5 16.2 78.9 FF [%] 78.6 78.3 34.4 Jsc [mA/cm2] 34.0 33.6 621 618 Voc [mV] 615 612 1 2 3 1. Cell Process 4 5 6 2. Cell Process Figure 2: Comparison between experimental and simulated I-V parameters of multicrystalline Si solar cells from an industrial production line made from six different ingots and applying two different fabrication processes. The histograms used in Sec. 2 are obtained from lifetime images, containing a certain number of τbulk(x,y) values, which are binned into n bins. In this section, we compute the histograms synthetically, while we do not care about how the τbulk(x,y) values could be distributed in the synthetic lifetime image. The boundary conditions for our synthetic histograms are as follows. The number of τbulk values is kept constant at 106, which is comparable to the number of pixels in the experimental lifetime images of Sec. 2. Each histogram has an adjustable lowest lifetime value τmin, as indicated in Fig. 3. Each bin has the same area A0, except a selectable area A1 ≥ A0 with lifetime values between selectable limits τ1 and τ2. Additionally, the histograms’ mean lifetime M is INFLUENCE OF LOW LIFETIME REGIONS ON CELL PERFORMANCE It is widely known that the low lifetime regions in mc-Si limit the cell’s performance. If two mc-Si cells have equal mean lifetimes but different lifetime distributions, the wafer having smaller regions of low lifetimes yields a better cell performance than the other. In this Section, we synthetically vary the lifetime distribution while keeping the mean lifetime constant, and show that Voc may vary up to 30 mV because Voc is influenced by the varying areas of low lifetimes. Figure 3: Schematic illustration of synthetic lifetime distributions. The values τmin, τ1, τ2 and A1 are set as input parameters. A selectable mean value M is achieved by variation of τmax and A0. 27th European Photovoltaic Solar Energy Conference, Frankfurt, Germany, 24-28 September 2012, 2AV.6.9 (ii) The slope of the Voc lines is influenced by τ2. At the line with the strongest decrease (lowest curve of blue squares) the area set by the parameters τ1 and τ2 contains in this case only the lowest lifetimes. By increasing the parameter τ2, higher lifetimes become more abundant, resulting in a smaller slope of Voc. This shows that the lowest lifetimes limit the Voc values the most. (iii) The second set of synthetic lifetime distributions (green circles) starts at higher lifetime, τmin = 10 µs, which shifts all Voc values up. This shows again that the lowest lifetimes limit the maximum achievable Voc. (iv) Note that the spread in Voc is about 22 mV, causing a spread of cell efficiency η of 1.5 % absolute as is apparent in Fig. 5. This demonstrates that the arithmetic mean of wafer lifetimes (which remains constant in Figs. 4 and 5) is a poor indicator for cell performance. 620 Voc [mV] 615 610 τmin τ2 605 τ2 600 595 First set of synthetic lifetime distributions Second set of synthetic lifetime distributions 0 10 20 A1 [%] 30 40 Figure 4: Modeled Voc values from 720 synthetic lifetime histograms having a constant arithmetic mean value of 64 µs but different amounts of lower and higher lifetime values. The variation in Voc is 22 mV. 16.5 16.2 τmin τ2 15.9 η [%] kept constant. To achieve this selected M value, the parameter τmax must be allowed to be free. And to keep the number of τbulk values at 106, the value A0 must be free as well. This procedure allows us to synthetize different histograms having equal mean values M by using τmin, τ1, τ 2, and A1 as input parameters. In the following, we generate histograms that have a constant arithmetic mean lifetime Mari of 64 µm, as observed in the sample 5 shown in Fig. 2. The first set of distributions starts at a very low lifetime τmin = τ1 = 1 µs with a variation of τ2 from 6 µs to 30 µs in steps of 3 µs. For all of these distributions, the area A1 of lower lifetimes between τ1 and τ2 is varied from 1 to 40% of the total area. With the device model of Sec. 2, Voc is simulated and shown as blue squares in Fig. 4. A second set of distributions is generated with a higher minimal value of τmin = τ1 = 10 µs plus variations in τ2 from 15 µs to 39 µs in steps of 3 µs and A1 from 1 to 40%. The resulting Voc values are indicated by the green circles in Fig. 4. A clear tendency of Voc is observable: (i) With increasing A1 the number of low lifetime values set between τ1 and τ2 increases. To keep the mean value of the distributions constant, the upper limit of high lifetimes, τmax, increases in these histograms. As a result, Voc continuously decreases because the higher lifetimes cannot compensate the influence of the lower lifetimes. 15.6 τ2 15.3 15.0 First set of synthetic lifetime distributions Second set of synthetic lifetime distributions 0 10 20 A [%] 30 40 1 Figure 5: Modeled η values from the same simulations as shown in Fig. 4. The maximum variation in η is 1.5 % absolute. More synthetic distributions can be produced by moving the area between τ1 and τ2 from the lowest lifetimes up to the highest, while holding the mean value constant at 64 µs. The maximum variation in Voc then becomes 30 mV. Besides the arithmetic mean value, the harmonic and geometric mean values have been analyzed in the same way. A maximum range in Voc of 7 mV for the geometric and 4 mV for the harmonic mean is observable. The arithmetic mean value has the highest variation in Voc because it weights the lowest lifetimes most, the harmonic mean weights the lowest lifetimes least and is most stable by changing the synthetic distributions. The geometric mean weights all lifetime values equal. 4 MEAN VALUES FOR PREDICTING CELL PERFORMANCE In the following, we show that no global mean value exists for predicting cell performance from lifetime images with certainty. We broaden our scope with the generalized theory of mean values, as was done in Ref. [2]. The generalized mean value of x1,…,xn positive real numbers is defined as: 𝑝 1/𝑝 𝑀𝑝 (𝑥1 , … , 𝑥𝑛 ) = �𝑛1 ∑𝑛𝑖=1 𝑥𝑖 � , (3) where the exponent p is a real number and characterizes the mode of mean. For example, p = 1 is the arithmetic mean, p = –1 the harmonic mean and p = 0 the geometric mean. In the following, we show that there is no global p value for predicting cell performance (with a given fabrication process). Or in other words, each wafer has its individual p value for predicting cell performance, which can only be found by physical modeling, not by pure mathematical means. Each simulated or experimental Voc value of a mc-Si cell can be reproduced with the simulation model described in Sec. 2 using a c-Si cell and a single, welldefined τsim,i value; it acts as a kind of average value, which we now call τVoc,av. Hence, to each τVoc,av value corresponds a p value in Eq. (3) with τVoc,av = Mp. For example, the sample 3 shown in Fig. 2 has p = –0.69 and 27th European Photovoltaic Solar Energy Conference, Frankfurt, Germany, 24-28 September 2012, 2AV.6.9 τVoc,av = 54.97 µs, whereas sample 5 in Fig. 2 has p = –0.58 and τVoc,av = 43.70 µs. We have investigated a [8] number of mc-Si cells and found that each waver has its individual p value which, on top of this, may depend on the fabrication process. Hence, whatever kind of mathematical averaging procedure we use for assessing the quality of mc-Si wafers, we cannot predict cell performance with certainty. This makes sense because, if a mathematical averaging procedure existed to predict cell performance, no physical models would be necessary to describe solar cells – it could be done on purely mathematical grounds. [12] 5 EXCLUSION PROCEDURE FOR LOW-QUALITY WAFERS BEFRORE FABRICATION [13] This still leaves the question of which averaging procedure predicts cell performance most precisely so it can be used to exclude low-quality wafers from fabrication. A possible way for obtaining a procedure to exclude low-quality mc-Si wafers from cell fabrication may be as follows: [14] 1. Measure τeff(x,y) images of wafers before production and compute their mean values. 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