Critical Issues in the Design of Polycrystalline, Thin-film Tandem Solar Cells z

PROGRESS IN PHOTOVOLTAICS: RESEARCH AND APPLICATIONS Prog. Photovolt: Res. Appl. 2003; 11:359 375 (DOI: 10.1002/pip.491) Research Critical Issues in the Design of Polycrystalline, Thin-film Tandem Solar Cells z Timothy J. Coutts,*,y J. Scott Ward, David L. Young, Keith A. Emery, Timothy A. Gessert and Rommel Noufi National Renewable Energy Laboratory, Golden, CO 80401, USA We use an empirical technique for modeling the efficiency of thin-film tandem solar cells and calculate an approximate upper limit on the range of performance of these hypothetical devices. This is shown to be approximately 282%, without losses due to inactive layers at the front of the device, or other parasitic sources. Reduction of the value of the reverse saturation current density by a factor of ten, increases the lossless efficiency by approximately 4% absolute. This change also greatly broadens the range of top and bottom cell bandgaps that would lead to efficiencies greater than 25%, the project goal. These observations emphasize the critical importance of focusing future research on gaining a thorough understanding of recombination losses. We then calculate daily energy density outputs for various direct spectra, computed from meteorological data, and show that the optimum bandgap pairs are relatively insensitive to the detail of the spectral irradiance. We also show that the use of daily energy density output may be a more useful criterion than efficiency in designing tandem thin-film solar cells. We compute contours of equal daily energy density output and show that the range of potentially suitable bandgap pairs is much larger than simple maximization of efficiency implies. The simple parametric approach enables us to investigate the effect of partial loss of photons with energies less than that of the bandgap of the top cell, but greater than that of the bottom cell. These photons are essential to the project goal of 25% efficiency, which emphasizes the need to evaluate the optical properties in this wavelength range very carefully. We also discuss the reduction of the thickness, or the area, of the top cell. When the top subcell generates a greater current than the bottom subcell, either of these parameters may be reduced to enable current-matching, and increased efficiencies, to be achieved. Again, this approach greatly extends the range of bandgaps that could lead to a 25% tandem thin-film cell. Next, we consider the case of concentrated sunlight and show that the optimum bandgap pairs decrease with concentration ratio. This is due to the atmospheric absorption bands. The efficiency increases by approximately 4% absolute per decade increase in concentration ratio. * Correspondence to: T. J. Coutts, National Renewable Energy Laboratory, Golden, CO 80401, USA. y E-mail: tim_coutts@nrel.gov z This article is a U.S. Government work and is in the public domain in the U.S.A. Contract/grant sponsor: US DOE; contract/grant number: DE-AC36-99-GO10337. Received 11 November 2002 Published in 2003 by John Wiley & Sons, Ltd. Accepted 14 February 2003

360 T. J. COUTTS ET AL. Finally, we comment on some of the practical difficulties that can already be anticipated in constructing these devices. Published in 2003 by John Wiley & Sons, Ltd. 1 INTRODUCTION The US Department of Energy added the High Performance Project to the range of activities investigated in its National Photovoltaics Program in the year 2000. 1 The goals of this project are to demonstrate a single-crystal multijunction cell with an efficiency of 40% and a tandem thin-film cell with an efficiency of 25%, in the laboratory, in approximately ten years, by which time it is expected that designs will already be under transition to industry. This paper is concerned with the second of these objectives. The highest efficiency achieved for a thin-film solar cell is 188%, which has taken approximately twenty years of intensive investigation by many groups throughout the international community of photovoltaics researchers to achieve. 2 The goal of 25% is, therefore, extremely ambitious, given the additional severe practical difficulties that can readily be foreseen in making such a device. The NREL team decided, therefore, to spend a significant time at the start of this project evaluating possible routes to this goal, identifying critical issues, and suggesting possible combinations of materials and device designs. 3 In this paper, we shall present findings resulting from our modeling efforts after approximately one year since the start of the project. Most modeling over-simplifies practical difficulties and, in the work presented here, we claim no exception to this criticism. However, the model has enabled us to complete some of our initial tasks. At the PVSEC-12 conference in 2001, we first published data on the influence of the front layers of a tandem device and showed that the goal of 25% was feasible, in principle, but recognized the extent of the probable practical difficulties in achieving this goal. 3 Subsequently, we showed that the use of mild optical concentration (10 20) could be extremely beneficial for both single and tandem junction thin-film devices. 4 The model suggested that a gain of approximately 4% absolute per decade of flux increase may be realized. This was approximately confirmed by Ward et al. 4 who achieved an efficiency of 215% for a single-junction copper indium gallium diselenide (CIGS) device operated at a concentration ratio of about 14. Later, we modeled the performance of thin-film tandem cells on the basis of their daily energy density output for various global reference spectra and demonstrated that the detailed nature of the spectra (cold cloudy, hot cloudy, cold sunny, hot sunny, and cool sunny) had very little influence on the values of the optimum bandgap pairs. 5 In this paper, we shall apply the technique to direct spectra and provide an explanation for this relative insensitivity. Among the practical problems that can be anticipated is the loss of photons with energies less than the bandgap of the top cell. In principle, these should all be used by the bottom cell (at least those with energies greater than the bandgap of the bottom cell), but in practice, there will almost certainly be losses due to reflections (both specular and diffuse) at the many interfaces in the thin-film stack, as well as free-carrier and other nonradiative mechanisms of absorption, and internal collection efficiencies of less than unity. We have extended the modeling to summarize these effects in a single parameter that is not spectrally dependent. Although we fully recognize the spectral dependence of all the above factors, we believe this simple parametric approach is useful and enables us to draw meaningful qualitative conclusions. Finally, we examine the idea of designing devices on the basis of the actual operational conditions they are likely to experience, rather than on the basis of their efficiency measured under laboratory conditions using standard, but possibly unrealistic, spectra. The consumer is unlikely to be concerned with the efficiency of a PV system, but will be much more concerned with the energy output it gives. It makes sense, therefore, to design a system on this basis. Although we have considered this approach previously, it is reexamined here, and we arrive at a conclusion that may be of great relevance to the future of the project. 5 2 THE MODELING APPROACH All the modeling discussed below was performed on an Apple PowerBook G4, operating with system 92. The modeling itself was coded in Mathematica 41. Various spectra were used in the modeling. The standard reference global and direct spectra were used initially. 6 In addition, and as discussed below, global and direct spectra

POLYCRYSTALLINE THIN-FILM TANDEM SOLAR CELLS 361 were also extracted from measured meteorological data. In all cases, the variation of flux with wavelength was represented by a linear interpolation function. The device is assumed to be a series-connected tandem and a schematic of the modeled structure is shown in Figure 1. It is assumed that the series and shunt resistance losses are negligible. The current flowing through the two subcells is continuous and passes through both cells. Each subcell has its own values of J sc and J 0, (the calculation of which is discussed below) which govern the voltage developed across it. The ideality factor is assumed to be 15 for both subcells. The diode equation for the individual subcells may be inverted to give voltage in terms of current density, from which the subcell voltages, V 1 and V 2, may be calculated. The voltage developed across the series-connected pair is simply the sum of V 1 and V 2. Using this procedure, the current density voltage characteristic of the tandem device may be calculated. 2.1 The reverse saturation current density Figure 1. Schematic of the tandem cell modeled To calculate the performance of thin-film tandem solar cells, we used the technique originally employed by Fan and, later, by Nell and Barnett, Wanlass et al., and Gee et al. 5,7 10 This assumes that the reverse saturation current density J 0, a critical quantity in device modeling, may be greatly simplified from the full form discussed in the book by Hovel. 11 This equation (eq. (50), p. 51) requires much input information, almost all of which is unavailable; given the wide range of semiconductors and bandgaps modeled. The simplified form is useful, if used cautiously, but must not be taken as providing an upper limit on the possible efficiency. It serves merely as a guide to the likely potential performance under the assumption that it will be possible to fabricate a tandem solar cell having subcells, each of which has a reverse saturation current given by the approach discussed below. Fan et al. 7 showed that J 0 could be expressed as: J 0 ¼ E g ; T T 3 exp E g ð1þ kt In this expression E g is the bandgap of the semiconductor, T is the absolute temperature, and k is Boltzmann s constant. ðe g ; TÞ is determined from world-record devices, measured at NREL under standard conditions or reported in efficiency tables published inprogress in Photovoltaics, in the following way. 12,13 First, we assume that the ideality factor n for thin-film cells 3 may be taken as 15. This is reasonable for good-quality thin-film devices. From the open-circuit voltagev oc (given by Equation 2) of a device of known bandgap, measured at known temperature, and its short-circuit current densityj sc, we may calculate the value of J 0 and, hence, ðe g ; TÞ) from Equation (1). Fan et al. 7 showed that the latter may also be related to the bandgap by Equation (3). V oc ¼ nkt e ln J sc ð2þ J 0

362 T. J. COUTTS ET AL. Figure 2. Plot of log 10 (E g, T) against E g. The symbols beside numbers 1, 2, and 3 refer to CIGS cells. 14 Number 4 refers to a CIS cell. 15 Numbers 5, 6, and 7 refer to CdTe cells. 12,13,16 Numbers 8, and 9 correspond to silicon cells 12,13 BE g E g ; T ¼ A exp kt In Equation (3), A and B are constants that may be determined from a plot of log 10 ðe g ; TÞ against E g. This enabled us to determine J 0 for a semiconductor of arbitrary bandgap. This function is shown in Figure 2 for nine state-of-the-art devices measured under standard conditions. 6 The standard deviation of the intercept is 143, and this is shown approximately by the uppermost and lowermost of the three lines. These lines are drawn 1 with respect to the line of best fit, for convenience. Because the y-axis utilizes logarithms to the base ten, the range of values may be taken as J 0, based on the central line, multiplied and divided by ten for the uppermost and lowermost lines, respectively. For this set of cells, the correlation coefficient was 0886. As will be shown later, this range of values of J 0 leads to rather significant variations in the calculated performance of the devices. The equation derived by Hovel 11 was based on the recombination current mechanism being diffusion. For thin-film devices, it may be necessary to consider several other mechanisms including space-charge recombination, bulk recombination, tunneling, etc., but it is assumed here that these may all be represented functionally by Equations (1) and (3). 2.2 The short-circuit current density The calculation of J sc is simpler than the above procedure, but many assumptions are still implicit. It is assumed that there are no losses of photons incident on the cell, and that all above-bandgap photons are absorbed, and generate minority-carrier pairs, all of which are collected. If the spectral irradiance or photon flux (i.e., number of photons cm 2 s 1 ev 1 ) is known, then it is straightforward to sum their number, multiplied by the electronic charge e, across the relevant spectral range, i.e., from the highest energy for which the flux was measured, to the bandgap of the semiconductor considered, to obtain J sc. We are considering two series-connected subcells and the procedure is to calculate the currents due to each of these and to select the smaller of the two. This quantity is taken as J sc of the tandem circuit. Various spectra were used in this work. One-sun devices have historically been modeled by means of the standard global spectrum. We used direct spectral irradiances when calculating total daily energy density output, but occasionally used the total global irradiance, as indicated. Three direct spectra were used. These were: hot sunny, cold sunny, and nice (now named cool sunny by the IEEE standards group; this term being used subsequently in this paper). 17,18 Each of these spectra was extracted from measured broad-band meteorological parameters (air-temperature, irradiance, time, location, humidity, and pressure) at different geographical ð3þ

POLYCRYSTALLINE THIN-FILM TANDEM SOLAR CELLS 363 locations and dates in the USA at hourly intervals, for the period of daylight at each location. 17,19 To model the efficiency of concentrator devices, we used the direct component of the noon, cool sunny spectrum. After reading into the program an irradiance file (each of which consisted of 221 flux/wavelength points, in the case of the hourly spectra), a linear interpolation formula was calculated to represent the data. In this way, the irradiance could be calculated for arbitrary intervals of wavelength (or energy). Nevertheless, with a data set consisting of only 221 points, distributed over a relatively wide range of wavelengths, it is unavoidable that some of the more detailed features in the spectra may be overlooked. 2.3 The open-circuit voltage and fill factor From the values of J 0, J sc, and n, the current voltage equation is fully defined, within its limitations. The procedure for calculating the output of the tandem cell was described earlier. The open-circuit voltage V oc of a single cell is approximately given by Equation (2). The power density is defined by the product of the current density and the total voltage across the two series-connected subcells. The power density was calculated as a function of voltage, for 50 equally spaced points, and an interpolation function of third degree was derived to fit this set of data. This function was then differentiated and equated to zero to find the coordinates J mp and V mp of the maximum power point. These quantities, together with V oc and J sc, were then used to calculate the fill factor and efficiency. 2.4 Identification of the optimum bandgap pair Modeling was generally performed with a resolution of 001 or 0005 ev in the bandgaps of both the top and bottom cells and the ranges of bandgaps were typically, but not always, 13 2 ev for the top cell and 05 13 ev for the bottom cell. The data were presented as contours of equal efficiency for all permutations of bandgap in these ranges. A typical set of contours is shown in Figure 3, which was calculated on the basis of the global reference spectrum and a temperature of 300 K. The maximum predicted efficiency is 282% for top and bottom cell bandgaps of 174 and 114 ev, respectively. For this calculation, it was assumed that the entire photon flux reaches the top absorber (i.e., that there are no front-layer losses) and all the photons that have energies less than the bandgap of the top cell, but greater than that of the bottom cell, are absorbed in the latter. The quantum efficiency of both the top and bottom cells was taken as unity up to their bandgaps. This set of contours is based on the central correlation line shown in Figure 2. It is very instructive to study the influence of J 0 on the optimum device design. From the lowermost of the three lines in Figure 2, we derive the contours shown in Figure 4. This figure is identical to Figure 3 with the Figure 3. Contours of equal efficiency for a monolithic series-connected tandem thin-film cell (300 K, global). 6 It is assumed that there are no optical losses associated with the inactive front layers of the device and that both cells have unity quantum efficiency

364 T. J. COUTTS ET AL. Figure 4. Contours of equal efficiency for the global reference spectrum, for the lossless case. The calculations are identical to those in Figure 3 with the exception that the lowermost of the three correlation lines from Figure 2 has been used exception that J 0 has been decreased by a factor of ten. The important features of this figure are that the maximum efficiency increases by approximately 4% absolute to 321%, and that the optimum bandgaps decrease to 161 and 096 ev for the top and bottom cells, respectively. The range for which efficiencies greater than 30% may be achieved is also very broad, further emphasizing the importance of J 0. The tenfold decrease in J 0 means that recombination has been greatly reduced, which makes the point that this a critical topic for future investigation. We shall return to this later. When modeling the total daily energy density output, the power density for each of the three reference days, for which direct spectra had been derived, was calculated at each hourly interval, again for all permutations of bandgaps in the above ranges. This led to a matrix of data with the bandgaps of the top and bottom cells in the first two columns, respectively, and with the power densities corresponding to each pair of these, and to the particular hour of the day, in the following columns. For a range of 13 2 ev and 05 12 ev in the top and bottom cell bandgaps, respectively, and a resolution of 001 ev in each, there were 5751 (81 71) calculations of power density for each hour of each reference day (depending, of course, on the range and resolution of bandgaps) and it took approximately 70 s to calculate the data set for each hour. The resolution could easily be increased, but at the cost of time and memory required. The dimensions were converted from power density (W cm 2 ) to energy density (W h cm 2 ) for each hourly interval, and the total daily energy density output was calculated by summing all the energy density data for each bandgap pair at each hour of the day. This was the final column of the matrix (again consisting of 5751 results) and the maximum daily energy density output as well as the corresponding bandgap pair could readily be identified by inspection. For the example of the cool sunny reference day, irradiances were measured at fourteen hourly intervals (from 6 am to 7 pm). These data, and the ambient temperatures, were measured at the same time as the meteorological data. The cell temperature and the ambient temperature are connected by the empirical relationship: T ¼ T air þ NOCT E tot =800 ð4þ that was used to calculate the former. 20 The direct component of the total global irradiance was used for the work on concentrator devices. 3 MODELING OF DAILY ENERGY DENSITY OUTPUT We previously showed that the daily energy density output may be found for the global spectra by using irradiances and temperatures, measured hourly throughout various standard days, and full details are given elsewhere. 5 An algorithm exists that enables the total global component to be calculated from the direct data, which

POLYCRYSTALLINE THIN-FILM TANDEM SOLAR CELLS 365 Figure 5. Contours of daily energy density output. The contours were calculated using the direct cool sunny hourly reference spectra. The maximum daily output is 0185 W h cm 2 for bandgaps of 171 and 114 ev for the top and bottom cells, respectively we shall use in this section. 18,19 One of the striking features to emerge from the previous diurnal modeling exercise was the relative invariance of the optimum bandgap pairs for the wide range of standard days. These varied from cold cloudy to hot sunny with corresponding variations in absolute daily irradiance levels. We shall demonstrate the same striking result for the direct component but, as well as the optimum pairs being invariant from one reference day to the next, the observation will also be shown to hold for global and direct measurements. However, we shall use only the three reference days for which the direct component was the major fraction of the total irradiance. The reason is that we shall use these later for modeling of concentrator devices for which the cloudy reference days are inappropriate. We first calculated the contour sets for each hour of each reference day and added all the power density outputs for the hours of the day for which measurements were taken. This gives a good approximation to the total daily energy density output for any bandgap combination. The maximum output and the optimum bandgap pair may be identified by inspection of the data. A set of contours of the energy density output, for the direct cool sunny reference spectra, is shown in Figure 5 and the position of the optimum bandgaps is shown by the small cross within the central contour. A summary of the optimum bandgaps, for each of the direct spectra, is shown in Table I. The invariance in the optimum bandgap pairs, previously reported for global reference days, is again found. 5 We shall discuss the reasons for this later. In Figure 5, the maximum energy density output is shown by the cross within the innermost contour. However, values within 5% of this may be obtained for a wide range of bandgap combinations. In fact, contours normalized to the maximum energy density output show that this can be achieved for 15 < E g1 < 185 ev and 085 < E g2 < 13 ev. Of course, it is assumed that E g1 > E g2. If we regard the 95% contour roughly as an ellipse, then the major axis may be represented by the line E g1 E g2 þ 06. This algorithm may be used to select bandgap pairs, within the limits of the 95% contour, with little loss of energy density. In practice, it may be difficult to manufacture tandem devices with bandgaps sufficiently well controlled to obtain the absolute maximum pair. Relaxing the design in this way greatly widens the range of potentially suitable bandgaps that may be used to make efficient tandem cells. In reality, it may be Table I. Optimum bandgap pairs for direct spectra derived from several reference days Reference spectra Optimum top cell (ev ) Optimum bottom cell (ev) Hot sunny 172 116 Cold sunny 170 114 Cool sunny (nice) 171 114

366 T. J. COUTTS ET AL. acceptable to select the materials on the basis of those that may be readily manufactured, which also contain plentiful, nontoxic materials, and which may be fabricated at modest deposition temperatures. 4 OPTICAL LOSSES AND THEIR MITIGATION 4.1 Losses in the range E g1 < E < E g2 In this section we consider the effect of losses of photons due to the front layers. This is done by assuming that optical losses due to absorption and reflection may be taken as spectrally independent and lumped under a single parameter, T1, which is the flux entering the top absorber layer. We use a similarly spectrally independent parameter, T 2, to represent the loss of photons with energies in the range E g1 < E < E g2. In the former case, the losses are likely to be quite small; perhaps of the order of 5 10%. In the latter case, several mechanisms can cause the loss of photons with energies less than E g1, including free-carrier absorption and nonradiative subbandgap absorption via impurities or defects. 21 In general, these are dependent on the energies of the various species, but we shall neglect these and simply obtain a qualitative sense of their effects. Figure 6 shows the short-circuit currents of the two subcells. The global reference spectrum is used in these calculations. In this figure, we have taken T1 ¼ 095. T 2 is varied between 01 and 1 and the bandgap of the bottom cell is treated parametrically. The current of the top cell is, of course, independent of T 2 and the horizontal line shows the current for E g1 ¼ 173 ev, which is approximately equal to the values of the top cell bandgaps listed in Table I. Notice that if E g2 ¼ 11 ev, the currents of the subcells are matched when T 2 ¼ 1, which is as expected from the above calculations. However, when T 2 is reduced, it is necessary to reduce E g2 to maintain current-matching, as expected intuitively. Figure 7 shows an alternative way of looking at the problem. In this, we show lines of current-matching for various values of T 2. Each line shows the value of E g1 required to match a corresponding value of E g2. As the value of T 2 is reduced, it is necessary to increase E g1 to maintain current-matching. Figures 6 and 7 provide an interesting insight. Clearly it is possible for both cells to change their bandgaps, depending on the value of T 2. To assess the combined effect of T1 and T 2, it is necessary to calculate the contours of equal efficiency and, to emphasize this point, we show the contours for T 2 ¼ 05 in Figure 8. The maximum efficiency has decreased to 206% for bandgaps of 19 and 094 ev, for the top and bottom cells, respectively. From this, we conclude that the transmittance of photons, in the range of energies E g1 < E < E g2, to the bottom cell is critical, if the ultimate goal of 25% is ever to be achieved. At 206%, it Figure 6. Variation of the short-circuit current density of the two subcells with transmittance T 2 into the bottom cell. In this figure T1 ¼ 095. The values of E g2 increase in steps of 01 ev from top to bottom

POLYCRYSTALLINE THIN-FILM TANDEM SOLAR CELLS 367 Figure 7. Lines of current-matching for various values of the parameter T 2. The values of T 2 increase in steps of 01 from top to bottom would be questionable whether the effort to develop tandem devices is worthwhile, given that the efficiency of single-junction devices is already 188%. 2 Hence, the issue of loss of photons in this energy range requires careful evaluation during the experimental phase of this work. 4.2 Possible means of achieving current-matching Current-mismatch may be reduced, assuming that the top subcell generates a greater photocurrent than the bottom subcell, by thinning the top cell and transmitting some of the photons with E 5 E g1, or by making the area of the top cell less than that of the bottom cell. Both approaches could be used to achieve currentmatching. The latter possibility was considered by Fan et al. 7 who also considered the influence of a varying air mass number during the day. Although this is, in part, taken into account in Section 3 by the use of practical data, the associated scattering effects that may or may not occur at the various interfaces, is not. This technique permits the tandem device to have a higher efficiency than would be the case if both subcells were optically thick and the modeling of this approach was demonstrated by Kurtz et al. 22 for single-crystal Figure 8. Contours of equal efficiency for T 2 ¼ 05. This calculation used the global reference spectrum

368 T. J. COUTTS ET AL. Figure 9. Variation of J sc1,2 with the fraction of photons with E5E g1 that are absorbed in the top cell. The values of T 2 decrease in steps of 01 from 1, uppermost line, to 05, lowermost line III V tandem solar cells. This approach has also been investigated here, once again assuming that there is no spectral dependence. The objective is to broaden the range of bandgaps and materials for which current matching may be obtained. At NREL, the majority of effort in recent years, in thin-film devices, has been on CdTe (E g 15 ev) and CIS (CIGS) (E g 116 ev) cells, and it may be advantageous to construct a tandem device based on these materials. There is not, at present, an alternative to CdTe as a demonstrated high-efficiency top cell and, given that the top cell of a tandem generates at least two-thirds of the total power, it makes good sense to use it. However, the bandgaps of these materials are not current-matched. The CdTe top subcell is expected to generate more current than the CIS subcell under the filtered global spectrum, but this may be overcome by thinning, or reduction of area. Figure 9 shows the values of J sc1,2 versus the fraction of photons a, with E 5 E g1 actually absorbed by the top cell. Here, it is implicitly assumed that the top cell absorber has either been thinned or has deliberately been grown thin. The isolated lower line shows J sc1, which increases linearly with a. The family of lines shows the values of J sc2 for several values of T 2. The lines show that the condition for current-matching now depends on both T 2 and on a. Figure 10 extends this approach, and shows the efficiency as a function of the parameter a. The sharp feature at the intersections can readily be explained by referring to Figure 8. In the lower range of values of a, J sc1 is less Figure 10. Variation of efficiency with the parameter a

POLYCRYSTALLINE THIN-FILM TANDEM SOLAR CELLS 369 than J sc2, which means that the former is selected as the short-circuit current density J sc of the tandem circuit. Of course, J sc1 is a linear function of a. For values of a beyond the intersection with each of the J sc2 lines, J sc2 is less than J sc1 and is selected as J sc of the tandem circuit. The variation of efficiency with a therefore follows the curves shown in Figure 9. Bearing in mind that these calculations were performed for a CdTe/CIS tandem cell, we now see that the efficiency for an optically thin top cell with 09 < a < 065, depending on T 2, is greater than that for an optically thick top cell. In fact, the maximum efficiency is about 26%, for a 09 and T2 1, for these assumptions. These calculations were performed with the standard global reference spectrum. 5 THIN-FILM CONCENTRATOR CELLS When thin-film cells are considered, it is sometimes argued that the use of optical concentration is not viable, economically. The argument is generally based on the belief that the cost of the concentrator optics, tracking equipment, etc., negates the economic advantage of thin-film cells. This is not necessarily the case and the argument depends on the successful development of low-cost concentrator optics and low-cost, high-efficiency solar cells. In this section, we shall show the results of modeling the performance of tandem thin-film cells for a wide range of concentration ratios. We have calculated sets of daily energy density contours for the direct, cool sunny spectra for several concentration ratios. The optimum bandgap pairs at the maximum daily energy density outputs are shown in Table II, along with the concentration ratios. Similar sets of results can be obtained using each of the other direct spectra shown in Table I. The striking features are that the optimum bandgaps are reduced for a concentration ratio of 3 and then remain constant up to 100. Between 100 1000, the optimum bandgap pair decreases again. We shall discuss these features in the next section. This is illustrated in Figure 11, which shows several important items of information. The curve of more regular slope shows the current-matched condition, as in Figure 6, and this refers to the left-hand y-axis and the lower x-axis. On this curve we have superimposed the coordinates of the bandgap pairs for each of the concentration ratios in Table II, corresponding to maximum daily energy density outputs. These correspond to the filled circles. The curve of more variable slope shows the cool sunny day, noon flux density and it refers to the right-hand y-axis and the upper x-axis. On this we have superimposed the optimum bandgaps of single-junction concentrator cells, shown by the star symbols. Notice that the coordinates of the lower bandgap cell of the tandem cells, and the bandgaps of the optimum single-junction cells all lie at approximately the same energies, although not for the same concentration ratios. These bandgaps all lie at the high-energy end of an atmospheric absorption band. The interpretation is that, at lower energies, J 0, which is independent of photon flux, increases with decreasing bandgap energy more rapidly than J sc, which, of course, depends on photon flux. This is true for some ranges of concentration, but not for others. Hence, the atmospheric absorption bands determine the bandgaps of the bottom cells and the need for current-matching determines the bandgap of the top cells, as supported by the fact that the coordinates of the optimum bandgap pairs all lie on the current-matched curve in Figure 11. These observations explain all the features we have seen in this work. Table II. Variation of bandgap pairs giving maximum daily energy density output for direct, hourly, cool sunny reference spectra and various concentration ratios Concentration ratio Bandgap of optimum Bandgap of optimum top cell (ev) bottom cell (ev) 1 171 114 2 170 112 3 157 094 10 156 094 20 156 094 100 156 094 1000 143 070 10000 143 070

370 T. J. COUTTS ET AL. Figure 11. The current-matched bandgaps (smoother) for top and bottom cells of a tandem device at various concentration ratios. The curve of varying slope shows the direct, noon, cool sunny photon flux (right-hand y-axis) versus photon energy. The ordinates, and the star symbols, refer to the bandgaps of single-junction cells, optimized for maximum daily energy density output, at various concentration ratios The efficiency (for the direct, noon, cool sunny day spectrum), calculated for the optimum bandgap pairs shown in Table I, is plotted in Figure 12 against the logarithm to the base ten of the concentration ratio. The term efficiency is usually defined with respect to a specific reference spectrum and temperature. Here, it is defined with respect to the direct, noon, cool sunny spectrum. The optimum bandgap pairs shown in Table I give maximum daily energy density output, inclusive of variations in irradiance throughout the day. These pairs were used in calculating the data in Figure 11. If one of the standard reference spectra (for example, the direct spectrum) is used, higher efficiencies result. On the other hand, the efficiency could also be defined, as the total daily energy density output divided by the total daily irradiance. This result in similar efficiencies to the second of these definitions. Figure 12 shows that the efficiency increases by about 4% absolute per decade increase in concentration ratio. Of course, we do not regard concentration ratios of 10 4 seriously, but we have performed the calculations to illustrate the points made in connection with Figures 10. Figure 12. Variation of the efficiency (direct, noon, cool sunny reference day) of thin-film tandem cells under optical concentration

POLYCRYSTALLINE THIN-FILM TANDEM SOLAR CELLS 371 6 SUMMARY AND INTERPRETATION OF MODELING RESULTS The water vapor absorption bands, which are present at the same wavelengths (energies) in all reference spectra used, are responsible for the invariance of the bandgap pairs for the daily energy density outputs, irrespective of whether global or direct spectra are used, and irrespective of the detailed nature of the spectra. These results are illustrated in Tables I and II. The absorption bands are also responsible for the invariance of the bandgaps, whether or not front layers are included in the design, because their effect is greater than that of the front layers. Modeling of the device under optical concentration, also reveals the dominant influence of the absorption bands. The reduction of E g2 (as the concentration ratio is initially increased) is due to the fact that the increased J sc, resulting from optical concentration and lower E g2, more than offsets the increased J 0, resulting from lower E g2. However, at some point, E g2 lies at the high-energy end of an absorption band and a further reduction in energy does not lead to a further increase in J sc, but does lead to an increase in J 0. Consequently, the optimum pair does not change over the range of energies corresponding to the absorption band. However, when the concentration ratio is sufficiently high, E g2 decreases to the high-energy end of the next-lowest energy band. In addition, the absorption bands are responsible for the behavior of the optimum bandgaps when the transmittance, T 2 of photons with E g1 < E < E g2 is reduced. When T 2 is significantly less than unity, the optimization code reduces E g2 to increase J sc2 thereby maintaining a current-match. However, as seen in Figure 7, at a sufficiently small value of T2, E g2 is reduced to the point at which it lies at the high-energy end of an absorption band and the response of the software is to increase E g1 to maintain current-matching, because this cannot be achieved by reducing E g2 any further. This is similar to the effect of optical concentration, as described above. As illustrated in Figures 9 and 10, the current mismatch caused by the reduction in J sc2 may be overcome by reducing the thickness of the top cell to reduce J sc1 accordingly. Of course, a further assumption in this latter calculation is that T2 does not affect the transmittance of photons with E > E g1. This may or may not be true, but the essential features of the model are likely to remain valid because even large changes, such as increased concentration ratio, do not radically influence the optimum bandgap pairs. Using an optically-thin or reduced-area top cell extends the range of bandgap pairs that could be considered suitable, and permits the possibility of achieving the target efficiency of 25% with subcells of CdTe and CuInSe 2, both of which are technologically well developed although not necessarily compatible. This goal of 25% would become even more realistic if techniques for reducing J 0 were developed. 7 PRACTICAL CONSIDERATIONS We have emphasized several times that the goal of 25% is very challenging although there does not appear to be any fundamental limitation to achieving it. In this section, we shall briefly discuss some of the practical difficulties that can be anticipated, even at this early point in the project. Doubtless, time will reveal additional problems. 7.1 Temperature compatibility Most successful thin-film solar cells involve a high-temperature step at some stage in their construction. For CdTe and CIS or CIGS, for example, the deposition temperature may be in excess of 600 C. If we consider a structure grown with the bottom cell first on the substrate, then it is important that the deposition of the second device does not disrupt the delicate interfaces and compositional gradients that may have been carefully built into the bottom cell. Even if the top cell were grown first on a transparent substrate, it would still be essential to deposit the second cell at a temperature that the top cell would survive. Disrupt implies chemical interactions or interdiffusion. Eventually, this may necessitate learning how to optimize the growth of the second cell at significantly lower temperatures that will not disrupt the first cell. Clearly, this is a challenging objective, although there are several possible approaches.

372 T. J. COUTTS ET AL. 7.2 Subcell configuration issues In addition to producing high-performance top and bottom devices at mutually compatible processing temperatures, it is equally important that the devices be made in compatible configurations. Both subcells must be superstrate devices, or they must both be substrate devices. Presently, there are no leading pairs of candidates that meet this criterion. For example, high-efficiency top subcells based on optically thin CdTe (or optically thick films of its wider bandgap alloys) may be attractive because some of the processes and understanding that already exists may be applicable to tandem cells. In this regard, it is noted that high-efficiency CdTe thin-film devices have been made only in the superstrate configuration (light enters through a glass superstrate). However, the leading candidates for the bottom cell are based on CuInSe 2 and related alloys, which have only been made in the substrate configuration (light enters from air side ). To date, neither of these devices has been explored extensively in the opposite configuration, and thus the fabrication of a monolithic structure similar to that identified in Figure 1 will be challenging. 7.3 Interface roughness All the optical modeling described here assumes that the interfaces are totally specular. However, given the predominant role of the water vapor absorption bands in the solar spectrum, it seems less likely that the values of the optimum bandgaps would change greatly, if diffuse scattering were present. It is unclear, however, whether scattering would lead to an increase or a decrease in the performance of the devices. Roughness may increase the reverse saturation current simply because of a large effective area, but there may be a corresponding increase in J sc because of longer optical path lengths. This issue requires careful evaluation although there may be lessons to be learned from single-junction devices. Roughness is also a factor of potential importance to the analysis of optically thin cells. Although the analysis is valid for calculation of efficiency with respect to a specific standard reference spectrum, if an analysis of the total daily energy output were performed, the results may well be influenced by interface roughness as the Sun moves through a range of angles during the day and the path-length through the stack of films changes. 7 If coherence is retained, then the daily energy density output may change by interference effects. On the other hand, if the films are optically rough, then the optical effects may be minimal, other than those due to variations in the spectral irradiance. This issue requires further clarification. 7.4 High-bandgap subcell The early modeling on this device implied that the bandgap of the optimum top subcell would be about 17 ev. As seen in Sections 3 and 4, this does not seem to be essential. Nevertheless, we identified several materials of approximately this bandgap, including CuGaSe 2, Cu(In x Al 1 x )Se 2, Cd x Zn 1 x Te, Cd x Mn 1 x Te and CdSe. Work is underway on all these materials within the High Performance Project. However, the construction of an efficient, wide bandgap solar cell is clearly a major challenge. Very-high-efficiency wide-bandgap single-crystal devices have been made and the well-known GaInP/GaAs tandem is one such example. However, we have not yet considered this combination in polycrystalline thin-film form. The top cell of a two-subcell tandem contributes at least two-thirds of the total power generated and it is essential, therefore, to maximize its efficiency. To achieve an efficiency of 25% for the tandem implies that the top cell must contribute almost 17% of this. Given that approximately 20 years were required to develop the CIGS cell to efficiencies of this level, the goal of achieving 25% within 10 years is ambitious, even though we expect to build on experience gained during the development of single-junction devices. 7.5 Tunnel junction issues The two layers of the p þ /n þ tunnel junction are assumed to be approximately 20 nm each. The interface roughness discussed in Section 7.3, may make it difficult to deposit films that are continuous and which will survive the overall cell fabrication, i.e., deposition of the second cell.

POLYCRYSTALLINE THIN-FILM TANDEM SOLAR CELLS 373 The model previously discussed considered the materials of the tunnel junction to be TCOs for two reasons. 3 First, their bandgaps are so large (3 ev or greater) that they would have no impact on the performance of the bottom cell. Second, it is straightforward to calculate their optical constants from the Drude free-electron theory. In reality, there is no need for the tunnel junction to be made of TCOs and it may be difficult or impossible to make p-type material with sufficiently high hole concentrations to reach degeneracy, as required for a tunnel junction. The bandgap of the interconnect layers need only be equal to, or greater than, that of the top cell to avoid optical losses. However, given that the layers are very thin, lower bandgaps could, perhaps, be used without incurring excessive optical losses. This may also reduce the electrical losses associated with the tunnel junction. In addition, it may be possible to develop an interconnection that does not use a tunnel junctions, such as that used in multi-junction amorphous silicon cells. 7.6 The reverse saturation current density We have emphasized several times in this paper the need to focus research in this field on the nature, and reduction, of J 0. The calculations of the maximum efficiency are based on the performance of the world-record cells of today and, with future development, one might expect the correlation line shown in Figure 2, to move downwards. To achieve this, would be extremely beneficial and would lead to higher efficiencies as well as a wider range of materials from which to construct the tandem device. For example, if the lowermost of the three lines in Figure 2 is used to calculate the efficiency contours shown in Figure 3, the maximum efficiency increases from about 28% (no front-surface losses) to greater than 32% (global reference spectrum) and the optimum bandgaps decrease to 161 and 096 ev, for the top and bottom cells, respectively. It is also found that this reduction in J 0 would enable the project goal of 25% efficiency to be achieved over a wide range of bandgap pairs. With the record thin-film cells of today, the recombination mechanisms are only now starting to be understood, even though high efficiencies have been achieved for many years. We intend to devote considerable attention to this issue early in the project, given the incentives described above. In fact, it may be prudent to begin investigation of lower bandgap pairs early in the project, in anticipation of future improvements in J 0. 7.7 Optically thin top cell Optically thin top cells have been used for many years in the GaInP/GaAs tandem device. The appeal of this approach is that the two absorbers discussed in Section 4.2 (CdTe and CIS) are both very-well-developed. In fact, CdTe, which is used as a superstrate device, may be considered to be the only available top cell that exhibits the level of efficiency needed for the tandem. 16 However, it will be necessary to learn how to make the device much thinner than usual, which may be challenging. 7.8 Possibility of a thin-film silicon bottom cell Initial modeling showed that the bandgap of the optimum bottom cell corresponded to that of silicon. This is an interesting possibility because of the availability of reasonably efficient thin-film devices, their temperature stability, the fact that a tunnel junction could be fabricated in the silicon, and that it has an indirect bandgap, which would minimize optical losses in the tunnel junction. The disadvantage is that the selection of silicon defines the bandgap of the current-matched top cell as approximately 17 ev, and there would once again be the difficulty of developing a suitable material to a high level of performance, as required for a high-efficiency device. 8 CONCLUSIONS We have presented a simple qualitative model to calculate the likely achievable performance of thin-film tandem cells. We conclude that the project goal of 25% is extremely challenging, but not unrealistic. The model is based on the assumption that the performances of the individual subcells of the tandem device will be similar to those of record-efficiency single-junction cells of today. We used the model to show that the efficiency will

374 T. J. COUTTS ET AL. increase substantially if the reverse saturation current density can be decreased. Although the projected increase in efficiency is a self-evident conclusion, the reduction in reverse saturation current density would also greatly increase the range of potentially suitable materials for the tandem device. Consequently, we believe that it is important to focus effort on the reduction of this quantity early in the project and to gain a thorough understanding of its origins. We believe that it may be prudent to begin work on the development of lower bandgap materials in anticipation of future improvements in the materials and junctions, and the associated reductions in reverse saturation current density. The effect of decreased reverse saturation current density is qualitatively similar to that of optical concentration, which also leads to a reduction in the optimum bandgap pairs. These are fixed by the absorption bands, and the need for current-matching of the subcells. Devices operated at relatively low concentration ratios increase rapidly in efficiency and their optimum bandgaps decrease substantially from the 1-sun values. Efficiencies of greater than 30% could, in principle, be achieved with concentration ratios of about 10. The absorption bands also play a significant role in the behavior of devices with an optically thin top cell. This, or one with reduced area, may be a suitable way to achieve current-matching for cells that are not currentmatched in the optically thick or equal area case. We also calculated the daily energy density output for a variety of direct spectra that were based on measured meteorological data. As with a previous study that used diurnal global spectra, we demonstrated that the optimum bandgap pairs are dominated by the presence of the atmospheric absorption bands. In general, we concluded that the practical range of acceptable bandgaps may be wider than previously recognized. Acknowledgements This work was supported by the US Department of Energy under Contract DE-AC36-99-GO10337. The authors are grateful to their NREL colleagues, Dr Kannan Ramanathan, Dr Ramesh Dhere, and Dr Miguel Contreras for their helpful discussions and collaborations. We are also grateful to Dr Martha Symko-Davies, for her continuing support for our work. Finally, we are particularly indebted to John P. Benner for his critical review of the manuscript. His profound familiarity with the literature enabled significant improvements to be made to the paper. REFERENCES 1. Symko-Davies M, Zweibel K, Benner J, Sheldon P, Noufi R, Kurtz SR, Coutts TJ, Hulstrom R. High Performance Photovoltaic Project: Identifying Critical Paths, Report NREL /CP-520-31030, 2001. Relevant website is http:// www.nrel.gov/highperformancepv/ 2. Contreras MA, Egaas B, Ramanathan K, Hitner J, Swartzlander A, Hasoon F, Noufi R. Progress toward 20% efficiency in Cu(In, Ga)Se 2 polycrystalline thin-film solar cells. Progress in Photovoltaics: Research and Applications 1999; 7: 311 316. 3. Coutts TJ, Ward JS, Young DL, Gessert TA, Noufi R. In The Search for and Potential Impact of Improved Transparent Conducting Oxides on Thin-film Solar Cells. 12th International Photovoltaic Science and Engineering Conference, Kyung: Hee, Seoul, 2001; 277 280. 4. Ward JS, Ramanathan K, Hasoon F, Coutts TJ, Keane J, Contreras MA, Moriarty T, Noufi R. A 215% efficient Cu(In, Ga)Se 2 thin-film concentrato solar cell. Progress in Photovoltaics: Research and Applications 2002; 10: 41 46. 5. Coutts TJ, Emery KA, Ward JS. Modeled performance of polycrystalline thin-film tandem solar cells. Progress in Photovoltaics 2002; 10: 195 203. 6. Standard Tables for Solar Spectral Irradiance at Air Mass 15: Direct Normal and Hemispherical for a 37 Tilted Surface, Conshohocken, PA, ASTM Standard G159-98, 1995. 7. Fan JCC, Tsaur B-R, Palm BJ. Proceedings of the IEEE Conference on Optimal Design of High-efficiency Tandem Cells, 1982; 692 701. 8. Nell ME, Barnett AM. The spectral p-n junction model for tandem solar-cell design. IEEE Transactions on Electron Devices 1987; ED-24: 257 266.