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Photoluminescence, open circuit voltage, and photocurrents in Cu(In,Ga)Se2 solar cells with lateral submicron resolution
Thin Solid Films 511 – 512 (2006) 678 – 683 www.elsevier.com/locate/tsf
Photoluminescence, open circuit voltage, and photocurrents in Cu(In,Ga)Se2 solar cells with lateral submicron resolution T. Ju¨rgens, L. Gu¨tay, G.H. Bauer * Institute of Physics, Carl von Ossietzky University Oldenburg, D-26111 Oldenburg, Germany Available online 24 January 2006
Abstract Thin film Cu(In,Ga)Se2 solar cells with average efficiencies in the 14% regime have been analyzed by photoluminescence (PL) with lateral micrometer-resolution. PL has been evaluated according to Planck’s generalized law with respect to splitting of quasi-Fermi levels under open circuit (OC) and short circuit (SC). Lateral patterns of luminescence yields signalize local fluctuations in quasi-Fermi energies in the range of some tens of milli-electron-volt. These patterns spatially extend to some microns and by far exceed the size of individual grains of 1 Am or less. The responses of PL, and thus the local V OC and short circuit currents turn out to be significantly anti-correlated. This behaviour is explained in terms of a two dimensional network coupling illuminated and dark hypothetic micrometer-sized diodes with different characteristic parameters such as series and parallel resistances, as well as reverse saturation currents. D 2005 Elsevier B.V. All rights reserved. Keywords: Cu(In,Ga)Se2; Confocal microscopy; Luminescence; Photocurrents; Lateral inhomogeneities
1. Introduction Chalcopyrites like Cu(In1 x Gax )Se2 (CIGSe) being promising candidates for low-cost high efficiency solar cells exhibit quasi-intrinsic limitations of the performance by their spatial inhomogeneity effects induced by their polycristallinity. Recent publications on inhomogeneity effects in CIGSe have been, e.g. attributed to the granular structure, to lateral variations of element concentration, to different metallurgical phases, locally fluctuating defect densities/minority life times and even to potential fluctuations in the scale of few micrometers [1– 4]. In previous publications we have analyzed the quality of the photo-excited state in CIGSe by PL-analyses with confocal lateral sub-micron resolution. We have shown the existence of lateral patterns in PL-yield, which we interpret as local fluctuations in splitting of the quasi-Fermi levels with geometrical extensions in the few micron regime [4– 6], which by far exceed the grain sizes. Furthermore from spectral studies [7] we have got strong evidence for potential fluctuations and for spatial variations of the optical band gap. In this contribution we concentrate on laterally resolved PL-studies * Corresponding author. E-mail address: [email protected] (G.H. Bauer). 0040-6090/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2005.11.065
of complete CIGSe solar cells with efficiencies in the range of 14% [private communication, IPE] operated in open circuit (OC) and short circuit (SC) and we simultaneously record short circuit currents (I SC) and try to correlate quantitatively ‘‘local open circuit voltages (V OC)’’ with ‘‘local short circuit currents’’. 2. Experimental We have analyzed Mo/Cu(In0.70Ga0.30)Se2/CdS/ZnO/Al solar cells with 2 Am CIGS and a 50 nm CdS-overlayer in a confocal microscope for simultaneous recording of local PL, and optical reflection with sub-micron resolution (see, e.g. Ref. [4]). In addition to our former approach we have simultaneously monitored the output currents of the cell, e.g., in SC, which have been generated by the same micrometer-sized focused excitation as for luminescence and for reflection. For our experiments we apply photon fluxes of about U = 1.5 I 1022 cm 2 s 1 corresponding to 5 I 104 suns equivalent and adjust a temperature of T = 293 K. 3. Experimental results In Fig. 1a,b typical lateral scans ((40 Am)2) of the photoluminescence-yield from a CIGSe cell (Ga-con-
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tent = 30%) in OC and in SC-operation are shown (SC means here: short circuit conditions are met at the metal finger front contact of the cell which is several hundreds of micrometers displaced from the 1 Am sized illuminated excitation spot). In both modes of operation we observe substantial local variations of the PL-yield –in length scales much larger than the grain sizes [4 –7] –which we are used to attribute via Planck’s generalized law to a considerable variation of the splitting of the quasi-Fermi levels which we name ‘‘local open circuit voltages’’ [8– 11]. Due to the extraction of a significant amount of carriers I SC which no longer recombine in the cell in non-radiative and in according radiative transitions, the PL-yields in SC are, of course, lower compared to those emitted in OC mode. Generally we observe in PL-yields in OC- and SC-modes ( Y PL,OC, Y PL,SC) similar patterns where the difference in PL-Yields Y PL,OC Y PL,SC seems to be approximately constant. From the respective ratio of PL-yields n = ( Y PL,max / Y PL,min) (under identical boundary conditions, e.g. OC- or SC-mode) we are able to quantify the lateral fluctuation in quasi-Fermi levels splitting D(( Fn ( Fp) [9,11,12] (here the factors amount to n OC = 7.8 and n SC = 14.6).
Fig. 2. Lateral scans (20 Am)2 of (a) PL-yield (293 K) and (b) photo current measured at same location and (c) plotted linescans (40 Am) of PL-yield and photocurrent (I SC) at the same position, of a polycrystalline Cu(In0.7Ga0.3)Se2 on Mo substrate with a 50 nm CdS overlayer and ZnO front electrode (photon flux for excitation 5 1022 cm 2 s 1).
Fig. 1. Lateral scans (40 Am)2 of PL-yield (293 K) of polycrystalline Cu(In0.70Ga0.30)Se2 on Mo substrate with a 50 nm CdS overlayer and ZnO front electrode in OC-mode (a) and in SC-mode (b) (photon flux for excitation 5 1022 cm 2 s 1).
In Fig. 2a and b we present a luminescence scans ((20 Am)2) in OC-mode ( Y PL,OC) and the according photocurrent scans (I SC) recorded at identical positions (I SC = (69.5 T 5) AA). Fig. 2c displays a line scan of Y PL representing the local V OC and a line scan of I SC which evidently are counter-related to one another.
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4. Interpretation of experimental results and modeling The luminescence yield of photoexcited matter Y PL generally can be explained in terms of local splitting of quasi-Fermi levels [8,9,11,12] and hence in laterally inhomogeneous matter the variations in Y PL can be translated into local fluctuation such as D(( Fn ( Fp). For OC-operation we assume the PLyield to reflect the local quality of the photo-excited state. Because of an eventual transfer of minorities generated in the illuminated regime to the dark environment, we only see the minimum local variation in quality [5,6,13]. In SC-mode the PL-yield is further reduced by the component of charge carrier extraction from the illuminated cell area to the contacts and consequently the interpretation of Y PL,SC gets much more complicated and less visually (see modeling approach below). (For the presented sample we get D(( Fn ( Fp)OC = 52 meV and D(( Fn ( Fp)SC = 68 meV). From the approximately constant difference in PL-yield between the two operation points we must not extrapolate strict-forwardly towards constant output currents (I SC), since the luminescence yield consists of the spatial integral of the product of minority and majority density and their respective coefficients for radiative transitions. Yet carrier transport bases on gradients of Fermi levels, related carrier mobilities and the necessity of current continuity in steady state. Without further careful experimental studies on the exact differences ( Y PL,OC Y PL,SC) and the appropriate correlation with the local variation of I SC, which lies in the range of ten percent we so far hesitate to discuss this observation in detail. In Fig. 2c comparatively high ‘‘voltages’’ are correlated with low ‘‘photocurrents’’ and vice versa which signalize a remarkable anti-correlation that has been quantified for the 2 dimensional scans of Y PL,OC and I SC. For each lateral position both magnitudes Y PL,OC(x, y) and I SC(x, y) have been combined to a local correlation coefficient c(x, y) by the relation YPL;OC ð x; yÞ bYPL;OC IðISC ð x; yÞ bISC Þ cð x; yÞ ¼ ð1Þ r YPL;OC IrðISC Þ (terms in angular brackets designate mean values and r the standard deviation). The resulting correlation pattern is outlined in Fig. 3; the mean value of the entire pattern represents the correlation coefficient r, which by definition in the complete data set is normalized with 1 r 1 [14], even if local correlation coefficients may exceed this regime. We observe from Fig. 3 a decent average degree of anti-correlation of Y PL,OC and I SC in terms of mostly small or considerably negative local correlation coefficient (integrated: r = 0.5). On the basis of this anti-correlation of Y PL,OC with I SC we propose a model based upon a mechanism to prevent comparatively high excitation states from easy dispersion to their neighborhood which is realized in an equivalent circuit diagram of substantial higher degree of complexity and non-linearity than some traditional ones [15,16]. Our model circuit consists of micron-sized diodes, one of them illuminated representing the illuminated area in the confocal setup and a non-illuminated dark
environment (see Fig. 4a). Each of the micron-diodes has an identical area and is ‘‘equipped’’ with a parallel R pA and a series resistor R sA, as well as with a reverse saturation current I 0A, each of these values in micrometer-scales, and are connected via series resistors; the one illuminated diode, of course, is outfitted by a respective photocurrent source Iphoto. We have downscaled R pA and I 0A from their square-centimeter-values which have been deduced in the traditional procedures for extraction of solar cell parameters. As in our approach R s in the centimeter-scale is non-linearly composed of a sequence of series and parallel micrometer-scale resistors and diodes we determined the range of R sA by measurements of luminescence yields and I SC (see below). In a simple diode model like ours R p represents the recombination losses, say, the non-radiative part whereas the current across the ideal diode (I Diode) is linearly related to the rate of radiative recombination which equals the PL-yield. By our highly resolved luminescence measurement we get experimental access to the current through the micron-sized illuminated ideal diode. According to the generalized Planck’s Law the PL-yield Y PL is given by Z V dx YPL ” x2 ð2Þ h x l ¯ eG exp 1 kT where x is the angular frequency of the emitted light; l, the chemical potential of the electron-hole ensemble; ( G, the gap energy; k, Boltzmann’s constant; and T, absolute temperature. For an at least decently illuminated solar cell we approximate ¯hxl kT H1, hence neglect the ‘‘ 1’’ in the denominator and solve the above integral analytically and finally get: l YPL ¼ C ðeG ; T ÞIexp : ð3Þ kT (C is a constant that depends only on temperature and gap energy; l, equals the splitting of quasi-Fermi-levels, which is an upper limit for the voltage V in the illuminated solar-cell.) The excess luminescence DY PL which consists of the total PL-yield except the portion attributed to thermal
Fig. 3. (20 Am)2 calculated correlation pattern of PL-scan and I SC-scan from Fig. 2a and b.
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different values of R sA. On the ordinate the intercept represents the photo-current induced by the illumination (estimated from the light flux, I photo = 80 AA), at large ring numbers, n, the
Fig. 4. 2-dimensional network. (a) Topological setup: illuminated cell in the middle, dark cells radial symmetrically around; every cell has the same area (1 Am2) and the same distance d to its neighbours (b) 2-dimensional equivalent circuit diagram network with illuminated cell in the middle, cells are coupled via serial resistors R sA.
equilibrium radiation (l = 0) gets the form identical with a diode-equation qV ð4Þ DYPL ¼ ½YPL YPL ðl ¼ 0Þ” exp 1 IDiode : kT In a first step for the estimation of the so far unknown value of R sA we simplify our model network (Fig. 4b) by adjusting each of the elements R pA, R sA, I 0A identical and scale down the 0.5 cm2-numbers (I 0 = 3.25 10 10 A, R p = 20.1 kV) to 1 Am2 values (I 0A = 6.5 10 18 A, R pA = 1.0 1012 V) and rearrange the circuitry with higher degree of symmetry by summarizing equipotential cells to rings. Now the numerical problem for the determination of currents and voltages at different locations in the network is reduced to n nonlinear Kirchhoff-equations, where n is the number of the respective, equipotential rings around the illuminated area (Fig. 4a) and the number of variables as well (currents from one ring to the next outer one). The numerical solution of this set of equations found by Newton’s gradient method is displayed in Fig. 5a which shows the distribution of currents versus ring number in SC-mode for
Fig. 5. (a) and (b) Currents in homogeneous simplified network, (a) from one ring to the next outer one versus ring number in short circuit mode, and (b) across Diode1 versus serial resistance R sA in open-circuit and short-circuit-mode. (c) and (d) Current through Diode1(OC) and I sc in inhomogeneous simplified network, (c) versus serial resistance R sA(1), (d) versus reverse saturation current I 0A(1).
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These simulated values are in good agreement with experimental observation of the local variations in PL and LBICpatterns (Fig. 2a and b). Another option for the implementation of local inhomogeneities consists in the defined variation of the reverse saturation current, e.g., of the illuminated cell (I 0A(1)). Fig. 5d shows the effect of this variation on I Diode1,OC and I SC; evidently, we see for both kinds of introducing inhomogeneities into the network, a significant anti-correlation of I SC with Y PL,OC. At variance with the former results, an anti-correlation as a consequence of the variation of R pA(1) could not be achieved with our model and with an appropriate choice of RpA(1). By a more careful inspection of the comparison of simulated data from Fig. 5c and d summarized in Fig. 6a, with the experimental values in Fig. 6b, which have been composed of the data of Fig. 2a and 2b, the anti-correlation can be reproduced qualitatively. Assuming these two contributions to be independent from one another the negative curvature for the simulated behaviour in Fig. 6a may be recognized in an ensemble of lots of experimental data points on the left hand side in Fig. 6b, and the linear relation due to I 0A(1)-variation may be associated with the less numerous data points on the right hand side in Fig. 6b. 5. Conclusions
Fig. 6. (a). Anti-correlation of short-circuit-current in units of I SC of homogeneous network and current through Diode1,OC because of inhomogeneities in the network due to variations of R sA(1)(negative curvature) and I 0A(1)(linear), same data as in Fig. 5c and d. (b). Experimental anti-correlation of short-circuit-current and photoluminescence yield OC taken from scans of Mo/Cu(In0.70Ga0.30)Se2/ CdS/ZnO/Al solar cells, T = 293 K, / exc = 1.5I 1022 s 1 cm 2.
calculated current has to match the experimental value I SC at the contact of the diode. In this configuration almost the complete current flows across the series resistors to the contact and only a small fraction, depending on R sA, recombines in the area of the illuminated cell; the contribution of the photocurrent to the dark diodes in the range in which we varied R sA seems to be negligible. Via the ratio of Y PL,OC / Y PL,SC we have estimated the regime of R sA when fitting the numerical ratio to the experimental one Y Y which amounts to YPL;OC V2 and correlates directly to YDiode1;OC V2. PL;SC Diode1;SC In agreement with the functions in Fig. 5b we find the lower limit of R sA of about 13 kV and consequently adjust it to the regime of (10 –20) kV. In a second step of our modeling approach we have introduced local inhomogeneities, firstly by varying the quality of connection of the illuminated area to its dark environment, which means a variation of the first series resistor (R sA(1)). In the range 0.2 R sA R sA(1) 1.6 R sA (see Fig. 5c) the model yields a current through the first diode under open circuit conditions (I Diode1,OC) representing the PLsignal in V OC which increases by a factor of 7 while the simulated short circuit current I SC is decreasing by 10%.
The laterally resolved luminescence of polycrystalline Mo/ Cu(In0.70Ga0.30)Se2/CdS/ZnO/Al cells with submicron lateral resolution shows variations that correspond to variations in splitting of quasi-Fermi levels in the few-micron length scale. These local fluctuations are accompanied by significant variations in photocurrent which are considerably anticorrelated. From these results we get strong evidence for the influence of transport properties on local inhomogeneity effects. For the description of this interplay a two-dimensional network of equivalent circuit diagrams of diodes coupled by serial resistors has been introduced. Defined variations of series resistors and variations of reverse saturation currents in our strongly coupled non-linear model network qualitatively and quantitatively reproduce our experimental observations. Yet, from our experimental data so far we have not been able to discuss in detail the quantitative difference in PLyields of open circuit with that in short circuit. In our simulation approach, so far, a statistical variation of elements of the equivalent circuit such as series and parallel resistances, or reverse saturation currents in order to fit the laterally coupled effects of inhomogeneities experimentally observed has not been performed.
Acknowledgements The authors like to thank U. Rau, IPE University Stuttgart for sample preparation, R. Bru¨ggemann for scientific and P. Pargmann for technical support and gratefully acknowledge funds from former BMBF contract. No. 01SF0115.
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References [1] D. Eich, U. Herber, U. Groh, U. Stahl, C. Heske, M. Marsi, M. Kiskinova, W. Riedl, R. Fink, E. Umbach, Thin Solid Films 361 – 362 (2000) 258. [2] J.R. Sites, J.E. Granata, J.F. Hiltner, Sol. Energy Mater. Sol. Cells 55 (1998) 43. [3] N.G. Dhere, S.R. Ghonadi, M.B. Pandit, A.A. Kadam, A.H. Jahagirdar, V.S. Gade, in: J. Benner (Ed.), Conf. Rec. 29 IEEE Photovolt. Spec. Conf., IEEE, Piscataway N.J., USA, 2002, p. 876. [4] K. Bothe, G.H. Bauer, T. Unold, Thin Solid Films 403 (2002) 453. [5] G.H. Bauer, K. Bothe, T. Unold, in: J. Benner (Ed.), Conf. Rec. 29 IEEE Photov. Spec. Conf., IEEE, Piscataway N.J., USA, 2002, p. 700. [6] G.H. Bauer, L. Gu¨tay, R. Kniese, Thin Solid Films 480 – 481 (2005) 259.
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[7] L. Gu¨tay, G.H. Bauer, Thin Solid Films 487 (2005) 8. [8] G.H. Bauer, R. Bru¨ggemann, S. Tardon, S. Vignoli, R. Kniese, Thin Solid Films 480 – 481 (2005) 410. [9] K. Schick, E. Daub, S. Finkbeiner, P. Wu¨rfel, Appl. Phys. A54 (1992) 109. [10] G. Smestad, H. Ries, Sol. Energy Mater. Sol. Cells 25 (1992) 51. [11] W. van Rooosbroeck, W. Shockley, Phys. Rev. 94 (1954) 1558. [12] P. Wu¨rfel, Physics of Solar Cells, Wiley-VCH, Weinheim (F.R.G.), 2005. [13] A.A. McDaniel, J.W.P. Hsu, Appl. Phys. Lett. 70 (26) (1997). [14] R. Schlittgen, B. Streitberg, Zeitreihenanalyse, Oldenbourg, 2001. [15] U. Rau, P.O. Grabitz, J.H. Werner, Appl. Phys. Lett. 85 (24) (2004) 6010. [16] V.G. Karpov, A.D. Compaan, D. Shvydka, Phys. Rev., B 69 (2004) 045325.
Photoluminescence, open circuit voltage, and photocurrents in Cu(In,Ga)Se2 solar cells with lateral submicron resolution T. Ju¨rgens, L. Gu¨tay, G.H. Bauer * Institute of Physics, Carl von Ossietzky University Oldenburg, D-26111 Oldenburg, Germany Available online 24 January 2006
Abstract Thin film Cu(In,Ga)Se2 solar cells with average efficiencies in the 14% regime have been analyzed by photoluminescence (PL) with lateral micrometer-resolution. PL has been evaluated according to Planck’s generalized law with respect to splitting of quasi-Fermi levels under open circuit (OC) and short circuit (SC). Lateral patterns of luminescence yields signalize local fluctuations in quasi-Fermi energies in the range of some tens of milli-electron-volt. These patterns spatially extend to some microns and by far exceed the size of individual grains of 1 Am or less. The responses of PL, and thus the local V OC and short circuit currents turn out to be significantly anti-correlated. This behaviour is explained in terms of a two dimensional network coupling illuminated and dark hypothetic micrometer-sized diodes with different characteristic parameters such as series and parallel resistances, as well as reverse saturation currents. D 2005 Elsevier B.V. All rights reserved. Keywords: Cu(In,Ga)Se2; Confocal microscopy; Luminescence; Photocurrents; Lateral inhomogeneities
1. Introduction Chalcopyrites like Cu(In1 x Gax )Se2 (CIGSe) being promising candidates for low-cost high efficiency solar cells exhibit quasi-intrinsic limitations of the performance by their spatial inhomogeneity effects induced by their polycristallinity. Recent publications on inhomogeneity effects in CIGSe have been, e.g. attributed to the granular structure, to lateral variations of element concentration, to different metallurgical phases, locally fluctuating defect densities/minority life times and even to potential fluctuations in the scale of few micrometers [1– 4]. In previous publications we have analyzed the quality of the photo-excited state in CIGSe by PL-analyses with confocal lateral sub-micron resolution. We have shown the existence of lateral patterns in PL-yield, which we interpret as local fluctuations in splitting of the quasi-Fermi levels with geometrical extensions in the few micron regime [4– 6], which by far exceed the grain sizes. Furthermore from spectral studies [7] we have got strong evidence for potential fluctuations and for spatial variations of the optical band gap. In this contribution we concentrate on laterally resolved PL-studies * Corresponding author. E-mail address: [email protected] (G.H. Bauer). 0040-6090/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2005.11.065
of complete CIGSe solar cells with efficiencies in the range of 14% [private communication, IPE] operated in open circuit (OC) and short circuit (SC) and we simultaneously record short circuit currents (I SC) and try to correlate quantitatively ‘‘local open circuit voltages (V OC)’’ with ‘‘local short circuit currents’’. 2. Experimental We have analyzed Mo/Cu(In0.70Ga0.30)Se2/CdS/ZnO/Al solar cells with 2 Am CIGS and a 50 nm CdS-overlayer in a confocal microscope for simultaneous recording of local PL, and optical reflection with sub-micron resolution (see, e.g. Ref. [4]). In addition to our former approach we have simultaneously monitored the output currents of the cell, e.g., in SC, which have been generated by the same micrometer-sized focused excitation as for luminescence and for reflection. For our experiments we apply photon fluxes of about U = 1.5 I 1022 cm 2 s 1 corresponding to 5 I 104 suns equivalent and adjust a temperature of T = 293 K. 3. Experimental results In Fig. 1a,b typical lateral scans ((40 Am)2) of the photoluminescence-yield from a CIGSe cell (Ga-con-
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tent = 30%) in OC and in SC-operation are shown (SC means here: short circuit conditions are met at the metal finger front contact of the cell which is several hundreds of micrometers displaced from the 1 Am sized illuminated excitation spot). In both modes of operation we observe substantial local variations of the PL-yield –in length scales much larger than the grain sizes [4 –7] –which we are used to attribute via Planck’s generalized law to a considerable variation of the splitting of the quasi-Fermi levels which we name ‘‘local open circuit voltages’’ [8– 11]. Due to the extraction of a significant amount of carriers I SC which no longer recombine in the cell in non-radiative and in according radiative transitions, the PL-yields in SC are, of course, lower compared to those emitted in OC mode. Generally we observe in PL-yields in OC- and SC-modes ( Y PL,OC, Y PL,SC) similar patterns where the difference in PL-Yields Y PL,OC Y PL,SC seems to be approximately constant. From the respective ratio of PL-yields n = ( Y PL,max / Y PL,min) (under identical boundary conditions, e.g. OC- or SC-mode) we are able to quantify the lateral fluctuation in quasi-Fermi levels splitting D(( Fn ( Fp) [9,11,12] (here the factors amount to n OC = 7.8 and n SC = 14.6).
Fig. 2. Lateral scans (20 Am)2 of (a) PL-yield (293 K) and (b) photo current measured at same location and (c) plotted linescans (40 Am) of PL-yield and photocurrent (I SC) at the same position, of a polycrystalline Cu(In0.7Ga0.3)Se2 on Mo substrate with a 50 nm CdS overlayer and ZnO front electrode (photon flux for excitation 5 1022 cm 2 s 1).
Fig. 1. Lateral scans (40 Am)2 of PL-yield (293 K) of polycrystalline Cu(In0.70Ga0.30)Se2 on Mo substrate with a 50 nm CdS overlayer and ZnO front electrode in OC-mode (a) and in SC-mode (b) (photon flux for excitation 5 1022 cm 2 s 1).
In Fig. 2a and b we present a luminescence scans ((20 Am)2) in OC-mode ( Y PL,OC) and the according photocurrent scans (I SC) recorded at identical positions (I SC = (69.5 T 5) AA). Fig. 2c displays a line scan of Y PL representing the local V OC and a line scan of I SC which evidently are counter-related to one another.
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4. Interpretation of experimental results and modeling The luminescence yield of photoexcited matter Y PL generally can be explained in terms of local splitting of quasi-Fermi levels [8,9,11,12] and hence in laterally inhomogeneous matter the variations in Y PL can be translated into local fluctuation such as D(( Fn ( Fp). For OC-operation we assume the PLyield to reflect the local quality of the photo-excited state. Because of an eventual transfer of minorities generated in the illuminated regime to the dark environment, we only see the minimum local variation in quality [5,6,13]. In SC-mode the PL-yield is further reduced by the component of charge carrier extraction from the illuminated cell area to the contacts and consequently the interpretation of Y PL,SC gets much more complicated and less visually (see modeling approach below). (For the presented sample we get D(( Fn ( Fp)OC = 52 meV and D(( Fn ( Fp)SC = 68 meV). From the approximately constant difference in PL-yield between the two operation points we must not extrapolate strict-forwardly towards constant output currents (I SC), since the luminescence yield consists of the spatial integral of the product of minority and majority density and their respective coefficients for radiative transitions. Yet carrier transport bases on gradients of Fermi levels, related carrier mobilities and the necessity of current continuity in steady state. Without further careful experimental studies on the exact differences ( Y PL,OC Y PL,SC) and the appropriate correlation with the local variation of I SC, which lies in the range of ten percent we so far hesitate to discuss this observation in detail. In Fig. 2c comparatively high ‘‘voltages’’ are correlated with low ‘‘photocurrents’’ and vice versa which signalize a remarkable anti-correlation that has been quantified for the 2 dimensional scans of Y PL,OC and I SC. For each lateral position both magnitudes Y PL,OC(x, y) and I SC(x, y) have been combined to a local correlation coefficient c(x, y) by the relation YPL;OC ð x; yÞ bYPL;OC IðISC ð x; yÞ bISC Þ cð x; yÞ ¼ ð1Þ r YPL;OC IrðISC Þ (terms in angular brackets designate mean values and r the standard deviation). The resulting correlation pattern is outlined in Fig. 3; the mean value of the entire pattern represents the correlation coefficient r, which by definition in the complete data set is normalized with 1 r 1 [14], even if local correlation coefficients may exceed this regime. We observe from Fig. 3 a decent average degree of anti-correlation of Y PL,OC and I SC in terms of mostly small or considerably negative local correlation coefficient (integrated: r = 0.5). On the basis of this anti-correlation of Y PL,OC with I SC we propose a model based upon a mechanism to prevent comparatively high excitation states from easy dispersion to their neighborhood which is realized in an equivalent circuit diagram of substantial higher degree of complexity and non-linearity than some traditional ones [15,16]. Our model circuit consists of micron-sized diodes, one of them illuminated representing the illuminated area in the confocal setup and a non-illuminated dark
environment (see Fig. 4a). Each of the micron-diodes has an identical area and is ‘‘equipped’’ with a parallel R pA and a series resistor R sA, as well as with a reverse saturation current I 0A, each of these values in micrometer-scales, and are connected via series resistors; the one illuminated diode, of course, is outfitted by a respective photocurrent source Iphoto. We have downscaled R pA and I 0A from their square-centimeter-values which have been deduced in the traditional procedures for extraction of solar cell parameters. As in our approach R s in the centimeter-scale is non-linearly composed of a sequence of series and parallel micrometer-scale resistors and diodes we determined the range of R sA by measurements of luminescence yields and I SC (see below). In a simple diode model like ours R p represents the recombination losses, say, the non-radiative part whereas the current across the ideal diode (I Diode) is linearly related to the rate of radiative recombination which equals the PL-yield. By our highly resolved luminescence measurement we get experimental access to the current through the micron-sized illuminated ideal diode. According to the generalized Planck’s Law the PL-yield Y PL is given by Z V dx YPL ” x2 ð2Þ h x l ¯ eG exp 1 kT where x is the angular frequency of the emitted light; l, the chemical potential of the electron-hole ensemble; ( G, the gap energy; k, Boltzmann’s constant; and T, absolute temperature. For an at least decently illuminated solar cell we approximate ¯hxl kT H1, hence neglect the ‘‘ 1’’ in the denominator and solve the above integral analytically and finally get: l YPL ¼ C ðeG ; T ÞIexp : ð3Þ kT (C is a constant that depends only on temperature and gap energy; l, equals the splitting of quasi-Fermi-levels, which is an upper limit for the voltage V in the illuminated solar-cell.) The excess luminescence DY PL which consists of the total PL-yield except the portion attributed to thermal
Fig. 3. (20 Am)2 calculated correlation pattern of PL-scan and I SC-scan from Fig. 2a and b.
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different values of R sA. On the ordinate the intercept represents the photo-current induced by the illumination (estimated from the light flux, I photo = 80 AA), at large ring numbers, n, the
Fig. 4. 2-dimensional network. (a) Topological setup: illuminated cell in the middle, dark cells radial symmetrically around; every cell has the same area (1 Am2) and the same distance d to its neighbours (b) 2-dimensional equivalent circuit diagram network with illuminated cell in the middle, cells are coupled via serial resistors R sA.
equilibrium radiation (l = 0) gets the form identical with a diode-equation qV ð4Þ DYPL ¼ ½YPL YPL ðl ¼ 0Þ” exp 1 IDiode : kT In a first step for the estimation of the so far unknown value of R sA we simplify our model network (Fig. 4b) by adjusting each of the elements R pA, R sA, I 0A identical and scale down the 0.5 cm2-numbers (I 0 = 3.25 10 10 A, R p = 20.1 kV) to 1 Am2 values (I 0A = 6.5 10 18 A, R pA = 1.0 1012 V) and rearrange the circuitry with higher degree of symmetry by summarizing equipotential cells to rings. Now the numerical problem for the determination of currents and voltages at different locations in the network is reduced to n nonlinear Kirchhoff-equations, where n is the number of the respective, equipotential rings around the illuminated area (Fig. 4a) and the number of variables as well (currents from one ring to the next outer one). The numerical solution of this set of equations found by Newton’s gradient method is displayed in Fig. 5a which shows the distribution of currents versus ring number in SC-mode for
Fig. 5. (a) and (b) Currents in homogeneous simplified network, (a) from one ring to the next outer one versus ring number in short circuit mode, and (b) across Diode1 versus serial resistance R sA in open-circuit and short-circuit-mode. (c) and (d) Current through Diode1(OC) and I sc in inhomogeneous simplified network, (c) versus serial resistance R sA(1), (d) versus reverse saturation current I 0A(1).
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These simulated values are in good agreement with experimental observation of the local variations in PL and LBICpatterns (Fig. 2a and b). Another option for the implementation of local inhomogeneities consists in the defined variation of the reverse saturation current, e.g., of the illuminated cell (I 0A(1)). Fig. 5d shows the effect of this variation on I Diode1,OC and I SC; evidently, we see for both kinds of introducing inhomogeneities into the network, a significant anti-correlation of I SC with Y PL,OC. At variance with the former results, an anti-correlation as a consequence of the variation of R pA(1) could not be achieved with our model and with an appropriate choice of RpA(1). By a more careful inspection of the comparison of simulated data from Fig. 5c and d summarized in Fig. 6a, with the experimental values in Fig. 6b, which have been composed of the data of Fig. 2a and 2b, the anti-correlation can be reproduced qualitatively. Assuming these two contributions to be independent from one another the negative curvature for the simulated behaviour in Fig. 6a may be recognized in an ensemble of lots of experimental data points on the left hand side in Fig. 6b, and the linear relation due to I 0A(1)-variation may be associated with the less numerous data points on the right hand side in Fig. 6b. 5. Conclusions
Fig. 6. (a). Anti-correlation of short-circuit-current in units of I SC of homogeneous network and current through Diode1,OC because of inhomogeneities in the network due to variations of R sA(1)(negative curvature) and I 0A(1)(linear), same data as in Fig. 5c and d. (b). Experimental anti-correlation of short-circuit-current and photoluminescence yield OC taken from scans of Mo/Cu(In0.70Ga0.30)Se2/ CdS/ZnO/Al solar cells, T = 293 K, / exc = 1.5I 1022 s 1 cm 2.
calculated current has to match the experimental value I SC at the contact of the diode. In this configuration almost the complete current flows across the series resistors to the contact and only a small fraction, depending on R sA, recombines in the area of the illuminated cell; the contribution of the photocurrent to the dark diodes in the range in which we varied R sA seems to be negligible. Via the ratio of Y PL,OC / Y PL,SC we have estimated the regime of R sA when fitting the numerical ratio to the experimental one Y Y which amounts to YPL;OC V2 and correlates directly to YDiode1;OC V2. PL;SC Diode1;SC In agreement with the functions in Fig. 5b we find the lower limit of R sA of about 13 kV and consequently adjust it to the regime of (10 –20) kV. In a second step of our modeling approach we have introduced local inhomogeneities, firstly by varying the quality of connection of the illuminated area to its dark environment, which means a variation of the first series resistor (R sA(1)). In the range 0.2 R sA R sA(1) 1.6 R sA (see Fig. 5c) the model yields a current through the first diode under open circuit conditions (I Diode1,OC) representing the PLsignal in V OC which increases by a factor of 7 while the simulated short circuit current I SC is decreasing by 10%.
The laterally resolved luminescence of polycrystalline Mo/ Cu(In0.70Ga0.30)Se2/CdS/ZnO/Al cells with submicron lateral resolution shows variations that correspond to variations in splitting of quasi-Fermi levels in the few-micron length scale. These local fluctuations are accompanied by significant variations in photocurrent which are considerably anticorrelated. From these results we get strong evidence for the influence of transport properties on local inhomogeneity effects. For the description of this interplay a two-dimensional network of equivalent circuit diagrams of diodes coupled by serial resistors has been introduced. Defined variations of series resistors and variations of reverse saturation currents in our strongly coupled non-linear model network qualitatively and quantitatively reproduce our experimental observations. Yet, from our experimental data so far we have not been able to discuss in detail the quantitative difference in PLyields of open circuit with that in short circuit. In our simulation approach, so far, a statistical variation of elements of the equivalent circuit such as series and parallel resistances, or reverse saturation currents in order to fit the laterally coupled effects of inhomogeneities experimentally observed has not been performed.
Acknowledgements The authors like to thank U. Rau, IPE University Stuttgart for sample preparation, R. Bru¨ggemann for scientific and P. Pargmann for technical support and gratefully acknowledge funds from former BMBF contract. No. 01SF0115.
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