Available online at www.sciencedirect.com ScienceDirect Energy Procedia 55 (2014 ) 30 37 4th International Conference on Silicon Photovoltaics, SiliconPV 2014 An alternative one-diode model for illuminated solar cells Otwin Breitenstein* Max Planck Institute of Microstructure Physics, Weinberg 2, 06120 Halle, Germany Abstract A novel one-diode model is proposed for illuminated solar cells, which contains an additional variable resistance describing minority carrier diffusion from the bulk to the p-n junction. This model naturally describes the differences between photo- and electroluminescence imaging without the need for correcting the photoluminescence images by short circuit images. It was successfully applied to the quantitative interpretation of photoluminescence images of an industrial multicrystalline silicon cell, where it provides a realistic prediction of the local short circuit current density. Moreover, the novel model explains a known departure from the superposition principle. 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license 2014 The Author. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the scientific committee of the SiliconPV 2014 conference. Peer-review under responsibility of the scientific committee of the SiliconPV 2014 conference Keywords: One-diode model, quantitative evaluation, photoluminescence imaging, solar cell modeling, superposition principle 1. Introduction The simplest conventional one-diode model for illuminated solar cells consists of a diode to ground, being characterized by a saturation current density J 01 and an ideality factor n, a constant current source in parallel delivering the short circuit current J sc, and a series resistance R s to the terminal, see Fig. 1 (a). In the following we will assume that the properties of the solar cell are dominated by recombination and current generation in the bulk. Then the bulk recombination rate as a function of the pn-junction voltage V pn (expressed as a current density) is given by the dark current density J rec bulk = J d = J 01 exp(v pn /V T ) of the diode in Fig. 1 (a), with V T being the thermal voltage. In this one-diode model, for a given value of V pn, the bulk recombination current is the same in the dark and * Corresponding author. Tel.: +49-345-5582740; fax: +49-345-5511223. E-mail address: breiten@mpi-halle.mpg.de 1876-6102 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the scientific committee of the SiliconPV 2014 conference doi:10.1016/j.egypro.2014.08.024
Otwin Breitenstein / Energy Procedia 55 (2014 ) 30 37 31 under illumination, which is known as the superposition principle. Physically, the bulk recombination rate in a semiconductor diode is governed by the excess carrier lifetime and depends exponentially on the implied voltage V impl, which is the separation of the quasi-fermi level positions of electrons and holes in the bulk, divided by the electron charge q. In this light, for a given value of V pn, the implied voltage in the bulk of the diode in Fig. 1 (a) equals V pn and should be independent on whether the device is illuminated or not. However, it is well known that, for a real solar cell, this is not the case. PC1D simulations have shown that, even at V pn = 0, the implied voltage for AM 1.5 illumination is as large as 0.55 V for a lifetime of 30 μs and approaches V pn only beyond 0.57 V [1]. This is the reason why in the quantitative evaluation of photoluminescence (PL) imaging of solar cells [2-5] always a PL image taken at short circuit (J sc ) condition has to be subtracted from all other PL images for ensuring that the resulting 'net' PL signal under current extraction depends exponentially on V pn [2]. This measure compensates for the diffusion-limited carriers, which are generated by the illumination in the bulk and have to diffuse to the pn-junction. Hence, by subtracting the J sc PL image, the net PL images behave like electroluminescence (EL) images, which are measured in the dark. All previous PL evaluation methods are solely based on evaluating V pn and do not consider V impl at all. In this contribution an alternative one-diode model is introduced, which considers V impl in the bulk and V pn at the pn-junction separately. This model assumes that the excess carrier concentration in the bulk is essentially constant across the depth, except for a small 'drift region' between the pn-junction and the bulk region. PC1D simulations have shown that this drift region has a spatial extension of about 20 μm [1], see also Fig. 2 (a) below. The current flow to the pn-junction is described here as a diffusion process between bulk and pn-junction. The formally introduced effective diffusion coefficient D eff can be expressed by known physical diode parameters. The model leads to an analytic formula for V imp (V pn ), which is its main outcome. In the equivalent circuit model, the voltage drop between V impl and V pn is described by a non-linear (voltage-dependent) 'diffusion' resistance R diff, see Fig. 2 (b). However, this resistance is only introduced here for visualizing the voltage drop between bulk and pn-junction in the equivalent circuit. The most important V imp (V pn )-relation contains only measurable diode parameters and does not contain this resistance explicitly. This alternative diode theory is applied for the interpretation of measured PL images of an industrial multicrystalline solar cell [6]. Moreover, based on this model a known departure from the superposition principle [7] may be explained. 2. The model and its application to PL image evaluation (a) V, J V pn (b) V, J V pn V impl R s J d J sc R s R diff J d J gen Fig. 1. (a) conventional one-diode model; (b) proposed one-diode model. Details to the alternative one-diode model have been described already in [1] and [6]. Therefore here only the starting assumptions and the final results of this model will be reported. In contrast to the conventional one-diode model Fig. 1 (a), the constant current source in the new model (b) describes not only the short circuit current J sc but the total current equivalent to the complete amount of absorbed photons, here called the generation current density J gen. The difference between J gen and J sc is the bulk recombination current density under short circuit condition, here called J rec,0 : Jsc Jrec,0 (1) In the following PL evaluation procedure J gen is assumed to be distributed homogeneous, which is certainly a good approximation as long as the surface texture is homogeneous. Then the spatial inhomogeneity of J rec,0 reflects the inhomogeneity of J sc, which comes out of the final PL evaluation procedure [6].
32 Otwin Breitenstein / Energy Procedia 55 ( 2014 ) 30 37 As mentioned above, in this model the current under illumination is proportional to the difference between the excess carrier concentration in the bulk n bulk and that at the pn-junction n pn and is formally described by an effective diffusion coefficient D eff (n i = intrinsic carrier concentration, N A = bulk net acceptor concentration, V T = thermal voltage): 2 qd effn i Vimpl Vpn nbulk npn exp exp N A VT VT J qdeff (2) In contrast to the usual definition, this "effective diffusion coefficient" D eff has the dimensions of cm/s. In analogy to a recombination velocity it could also be called an extraction velocity, which is zero under open-circuit condition. On the other hand, the extracted cell current is the difference between J gen and the bias-dependent recombination rate in the bulk, which may be expressed by the saturation current density of the bulk recombination J 01 : Vimpl J J01exp V T (3) From the short circuit condition an expression for D eff can be derived: JscJ01N D A eff 2 q ni Jrec,0 (4) If (4) is inserted into (2), this leads together with (3) to the major result of this theory, which expresses V impl by V pn under illumination: J rec,0 J Vpn V sc impl( Vpn ) VT ln exp (5) J01 VT Here the material parameters n i and N A cancel and only measurable diode parameters remain. The diffusion resistance R diff between the bulk and the pn-junction in Fig. 1 (b) can also be expressed analytically as a function of V pn, see [1]. For low and medium voltages it closely approximates a constant current source for J sc, as expected, and for V pn approaching V oc, as well as for the un-illuminated case, it approaches zero. Since (5) is the most important outcome of this theory, the value of R diff is not important. If (5) is inserted into (3) for calculating the illuminated current density, the result is basically the same as for the conventional diode model: J Vpn J ( V sc pn ) Jsc J01exp (6) VT The factor (J sc /J gen ) in (6) is due to the fact that, in the new model, the open circuit condition is established by balancing J gen to the dark current, instead of J sc in the conventional diode model. Thus, the alternative one-diode model does not change the solar cell equations significantly, it only provides a deeper understanding of the pnjunction physics and provides a simple way to express V impl by V pn analytically. All previous luminescence evaluation methods for solar cells (e.g. [2-5]) start from this basic equation:
Otwin Breitenstein / Energy Procedia 55 (2014 ) 30 37 33 Vpn, i i Ci exp V T (7) Here C i is the so-called luminescence scaling parameter, which depends e.g. on the local surface conditions and recombination properties, and i is the position index. However, as mentioned in the introduction, (7) can be applied directly only to electroluminescence (EL) data. For PL data, only a 'net' PL image generated by subtracting a J sc PL image from that measured at a certain voltage V can be described by (7). In the frame of the model described above, the PL signal must be described by: Vimpl, i i Ci exp VT (8) Under short circuit condition the implied voltage V impl 0 establishes in the bulk. Since the newly introduced diode parameter J rec,0 describes the bulk recombination rate at short circuit condition, it can be described by: 0 Vimpl, i J rec, 0, i J01 exp V T (9) Together with (8) this leads to: Jrec0, i J01, i ( sc) i Ci (10) Hence, the PL signal measured under short circuit i (sc) provides information on the new parameter J rec,0 and thus via (1) on J sc. Here the influence of the light shadowing by the busbars and grid lines has to be considered, otherwise the simulated local short circuit current density becomes too low. Note that I sc of the cell always refers to the whole cell area, but locally (between the grid lines) the averaged J sc is larger than I sc /A (A = cell area). This influence is considered here by dividing the generated current density by a factor of (1 - f met ), with f met being the fraction of metallized area. Since the global current of the cell is the sum of all pixel currents, the global generation current is the global sum of I sc and of I rec,0 of all pixels, see (1). Expressed as current densities, this leads to: Isc A 1 J rec0, i X * Y i 1 fmet (11) This equation allows to calculate the value of J gen, assumed here to be homogeneous between the grid lines, if I sc is known and J rec,0,i is measured after (10). Regarding a local series resistance R s,i to the position i, the local pnjunction voltage for an applied voltage V calculates by regarding (6): J sc, i Vpn, i V pn, i V Rs, ij V Rs, i J sc, i J01, i exp (12) V T The actual PL evaluation procedure based on this theory, which is described in more detail in [6], is based on the measurement of four PL images, one at reduced illumination intensity at open circuit, one at full illumination intensity at short circuit, and two at full illumination intensities at two different loading conditions. From the J sc PL
34 Otwin Breitenstein / Energy Procedia 55 ( 2014 ) 30 37 image the value of J gen and the J sc,i distribution are obtained via (10) and (11). The other PL images are converted into V impl via (9), then converted to V pn via (5) and are then fitted to (12), leading to the parameters C i, J 01,i, and R s,i. A software package called "PL-imp" has been developed, which applies an iteration procedure for simultaneously fitting the local diode parameters to the PL image data by evaluating the implied voltage as described above. 3. Results 3.1. Simulation results, comparison to PC1D In the following the theoretical predictions of Sect. 2 are applied to a model solar cell and compared to PC1D simulations. The model solar cell is based on the "Pvcell_15%.prm" example in [8] describing a standard industrial cell at AM 1.5. For getting rid of all series resistance effects, the contact and base circuit resistances have been chosen close to zero. The parallel resistance was disabled, the cell thickness was reduced to 200 μm, and the bulk lifetime was reduced to 30 μs for coming closer to the multicrystalline case. Fig. 1 (a) shows the PC1D simulated electron concentration in the bulk at AM 1.5 and V = 0.5 V (close to the maximum power point), together with the assumption of the model as a dashed line. The magnitudes n pn and n bulk and the assumed extension of the drift region are indicated. Fig. 2 (b) shows the dependence of V impl on V pn under AM 1.5 illumination as simulated after (5) and, measured in the middle of the bulk, after PC1D simulations (square symbols). The PC1D simulations can nicely be fitted by (5) by assuming J sc = 32.9 ma/cm 2, J 01 = 1.3*10-12 A/cm 2 (both the same as for the PC1D analysis) and J rec,0 = 3.65 ma/cm 2 [1]. It is visible that, for V pn < 0.45 V, the implied voltage remains virtually constant at about 0.55 V and only beyond slowly approaches V pn. In the dark, V impl in the middle of the bulk is constantly a few mv lower than V pn, which is due to the finite diffusion length in this device. In the dark there is no additional diffusion resistance. (a) n [cm -3 ] 1.5x10 13 n bulk 1.0x10 13 5.0x10 12 PC1D model n pn drift region (b) V impl [V] 0.6 0.5 Vi_ill_PC1D Vi_dark_PC1D V_pn V_impl 0.0 0 50 100 150 200 z [μm] 0.4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 V pn [V] Fig. 2. (a) Depth-dependent electron concentration at AM 1.5 and V pn = 0.5 V according to PC1D simulation (full line) and to the proposed model (dashed line); (b) dependence of V impl on V pn according to PC1D simulation (symbols) and the proposed model (line). 3.2. Application to PL imaging results The following results are used to illustrate the application of the new model to the quantitative interpretation of PL images, for more details see [6]. For this analysis four PL images of an industrial multicrystalline silicon solar cell were used, which are (1) one image taken under open circuit at a reduced intensity of about 0.1 sun, (2) one image taken under short circuit condition at 1 sun intensity, and (3) and (4) two images taken at 1 sun intensity for two different electric load conditions. In contrast to earlier PL evaluation methods, here the PL images are directly evaluated instead of producing 'net' PL images by subtracting the J sc PL image from the other ones, see [2-5]. In Fig. 3 typical results of the PL evaluation procedure described in Sect. 2 are shown. In the image (a) of the calibration
Otwin Breitenstein / Energy Procedia 55 (2014 ) 30 37 35 constant C i some grains seem to be homogeneously darker than the others. This was also visible in the PL images, but not at the visual inspection. We believe that here the acidic "iso texture" is not as isotropic anymore at the PL wavelength of about 1.1 μm. Note that the J rec,0 and the J sc images in Fig. 3 (d) and (e) cannot result from any of the previous PL evaluation methods. Qualitatively, J rec,0 looks very similar to J 01, but it is not identical, see eq. (8). The comparison of J sc simulated from PL (Fig. 3 e) with a slightly blurred LBIC image (f) shows that the predicted J sc image (e) is quite realistic. The blurring of the LBIC image was useful for a better comparison, since the directly measured LBIC image showed a distinctly better spatial resolution than the PL-based J sc image (e) [6]. This is due to the lateral spreading of the local voltage in a solar cell due to the horizontal balancing currents in the emitter and the grid. Note that this PL evaluation procedure is based on the model of independent diodes, which neglects any lateral interaction between neighboring pixels. In reality, the series resistance is a distributed one, which is not considered here. We believe that the remaining differences between the simulated J sc image and the blurred LBIC image may come e.g. from the very simple physical model, assuming independent diodes, a constant carrier concentration throughout the bulk, and the very formal effective diffusion coefficient. They also may stem partly from the blurring procedure, which not only blurs the dark LBIC lines but also the grid lines, which remain sharp in the PL evaluation. max (a) C i 0-10 a.u. (b) J 01 0-3*10-12 A/cm 2 (c) R s 0-1.5 cm 2 min 2 cm (d) J rec,0 0-3 ma/cm 2 (e) J sc from PL 38-41 ma/cm 2 (f) J sc from LBIC, blurred 34-41 ma/cm 2 Fig. 3. PL-based images of the (a) luminescence calibration factor C; (b) saturation current density J 01, (c) series resistance R s, (d) bulk short circuit recombination current density J rec,0, (e) PL-simulated short circuit current density J sc, (f) artificially blurred LBIC image of J sc. 3.3. Explanation of a known departure from the superposition principle In 1994, based on PC1D simulations, Robinson et al. [7] have described two different departures from the principle of superposition occurring in silicon solar cells. "Departure 1" leads to J sc -shifted illuminated currents below that of the dark characteristic and "departure 2" to shifted illuminated currents above the dark characteristic. While departure 1 was clearly attributed to an excitation intensity-dependent lifetime of the dielectric backside
36 Otwin Breitenstein / Energy Procedia 55 ( 2014 ) 30 37 passivation, the physical explanation of departure 2 was less clear. It is shown here that the alternative one-diode model provides a simple explanation of departure 2. In Fig. 4 (a) results of PC1D simulations of a dark and some illuminated characteristics (J sc -shifted) of the model diode for several parameters are shown [1]. It is visible that the departure only regards the low-voltage part of the characteristic and that it increases with increasing illumination intensity and decreasing bulk lifetime. Fig. 4 (b) shows the low voltage part of these characteristics in linear scale, together with the characteristic of a 58 kcm 2 resistor. This shows that the departure essentially leads to a weak ohmic contribution to the illuminated characteristic, which is not present in the dark characteristic. It is proposed here to explain this departure by the bias-dependent shift of the edge of the depletion region, which leads to a weak bias-dependent shift of the thickness of the recombination region of the bulk. Note that, according to Fig. 2 (a), the excess carrier concentration close to the pn-junction is considerably reduced. This means that, under these conditions (illumination with current drain), only the region deeper in the bulk contributes to the bulk recombination, where the excess carrier concentration is highest. If the edge of the depletion region shifts, also the extension of the recombination-active part of the bulk shifts. This can be modeled by a slightly bias-dependent value of J 01 [1] (W = depletion region width, d = bulk thickness, d 0 = width of drift region with reduced excess carrier concentration): W d d0 V J V J dark V 01 ( ) 01 (13) d (a) A/cm 2 0.01 1E-3 1E-4 1E-5 Jred_30 μs_1 sun Jred_30 μs_0.1 sun Jred_300 μs_1 sun Jdark_30 μs (b) 6.0x10-6 A/cm 2 5.0x10-6 4.0x10-6 3.0x10-6 Jred_30 μs_1 sun Jred_30 μs_0.1 sun Jred_300 μs_1 sun Jdark_30 μs 58 kcm 2 1E-6 2.0x10-6 1E-7 1.0x10-6 1E-8 0.1 0.2 0.3 0.4 0.5 0.6 V loc [V] 0.0 0.0 0.1 0.2 0.3 V loc [V] Fig. 4. (a) PC1D simulations of dark and illuminated characteristics (reduced by J sc) of silicon solar cells in semi-logarithmic drawing; (b) the same data in linear drawing. Fig. 5 shows the bias-dependent shift of the edge position of the depletion layer simulated by PC1D and according to a simple linear model based on the total depletion approximation [1]. In the low voltage range the results nicely coincide. If this slope is used in (13), the illuminated current is calculated after (2), and a more accurate calculation of V pn is used [1], the result shown in Fig. 5 (b) appears. It is visible that this "advanced model" nicely explains the departure 2 from the superposition principle. For the interpretation of PL images, on the other hand, this advanced model does not provide any advantages, since it only influences the results in the low voltage range (< 0.4 V), where no PL measurements are performed. 4. Discussion and conclusions If the same PL images are evaluated by one of the previous methods, like the "coupled voltage calibration" (C- VC) method [4], the results do not differ significantly from that shown in Sect. 3.2. [6]. This can be expected, since the underlying physics is basically the same, and the bulk recombination current density J rec,0 additionally regarded
Otwin Breitenstein / Energy Procedia 55 (2014 ) 30 37 37 here accounts for less than 10 % of J sc. It had been shown theoretically by Glatthaar et al. [6] that, as long as the excess carrier lifetime is independent of the excitation intensity, the technique of subtracting a J sc PL image indeed works correct. However, this does not hold anymore if the lifetime depends on the excitation intensity, hence on the value of V impl, since this magnitude does not appear in any of the previous methods at all. Therefore the main value of the method proposed here is that it can easily be extended to regard an excitation intensity-dependent lifetime. Moreover, this new method is slightly more correct than the old ones, since it regards and even predicts the inhomogeneity of J sc, which is obvious for multicrystalline solar cells. Finally, in an advanced version, this model delivers an explanation of a well-known departure from the superposition principle. (a) edge position shift [μm] 0.60 0.58 0.56 0.54 0.52 0.50 0.48 x[μm]_pc1d analyt. model (b) 0.01 1E-3 1E-4 1E-5 1E-6 J - J sc PC1D simulation Jred adv. model 0.46 0.0 0.1 0.2 0.3 0.4 0.5 0.6 bias [V] 1E-7 0.1 0.2 0.3 0.4 0.5 0.6 bias [V] Fig. 5. (a) PC1D simulation (dots) and analytic linear model of the shift of the depletion region edge position; (b) PV1D simulation (dots) and analytic simulation (line) of the J sc-reduced illuminated characteristic of the model diode Acknowledgements The author thanks H. Höffler and J. Haunschild from Fraunhofer Institute of Solar Energy Systems (ISE) in Freiburg, Germany, for performing the PL measurements and writing the "PL-imp" software package used for evaluating the PL images [6]. References [1] Breitenstein O. An alternative one-diode model for illuminated solar cells. IEEE J. Photovolt. 2014; 4:899-905. [2] Trupke T, Pink E, Bardos RA, Abbott MD. Spatially resolved series resistance of silicon solar cells obtained from luminescence imaging. Appl. Phys. Lett. 2007;90:093506 [3] Glatthaar M, Haunschild J, Kasemann M, Giesecke J, Warta W. Spatially resolved determination of dark saturation current and series resistance of silicon solar cells. Phys. Stat. Sol. RRL 2010;4:13-15. [4] Glatthaar M, Haunschild J, Zeidler R, Demant M, Greulich J, Michl B, Warta W, Rein S, Preu R. Evaluating luminescence based voltage images of solar cells. J. Appl. Phys. 2010;108:014501. [5] Shen C, Green MA, Breitenstein O, Trupke T, Zhang M, Kampwerth H. Improved local efficiency imaging via photoluminescence for silicon solar cells. Solar En. Mat. & Solar Cells 2014;123:41-46. [6] Breitenstein O, Höffler H, Haunschild J. Photoluminescence image evaluation of solar cells based on implied voltage distribution. Solar En. Mat. & Solar Cells, in print. [7] Robinson SR, Aberle AG, Green MA. Departures from the principle of superposition in silicon solar cells. J. Appl. Phys. 1994;76:7820-7930. [8] http://www.pv.unsw.edu.au/info-about/our-school/products-services/pc1d