Theoretical Efficiency of Intermediate Band Solar Cells with Overlapping Absorption Coefficients for Various Combinations of Band Gaps Rune Strandberg
Teknova AS, Gimlemoen 19, 4630 Kristiansand, Norway,
[email protected]
Abstract-The most commonly used criteria for evaluation
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of the suitability of different intermediate band materials for use in intermediate band solar cells are their band gaps. One often sees that such an evaluation is made based on theoretical efficiency limits with non-overlapping absorption coefficients. In this work the theoretical efficiency limits for various degrees of overlap are calculated for relevant combinations of band gaps.
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Fig. 2. The theoretical efficiency at 1 sun, calculated as in ref. [2], for IBSCs with no overlap between the absorption coefficients.
of the work is to serve as a quick reference for researchers looking for materials with good combinations of band gaps for use in IB SCs. The results to be presented will give an indication of the sensitivity of the efficiency on the overlap for various band gap combinations. II.
MODELING DETAILS
Efficiencies have been calculated by the model developed by Cuadra et al. [3], to which we refer the reader for mathematical details. The model assumes that the IB has no energetical width, that only radiative recombination occur and that all principally avoidable losses, like resistive and reflective losses, are zero. The optical path length in the cell is assumed to be twice the cell thickness due to a perfect reflector located at the rear side of the cell. The absorption coefficients have been set to 4 . 104 cm-1 in the intervals where they are non-zero (see figure 1). The 6000 K black body spectrum is used and the cell temperature is set to 300 K. With overlapping absorption coefficients the cell thickness or, equivalently, the absorbance of the cell is a parameter to optimize [3]. It has recently been shown that efficiency vs. thickness plots can have two peaks [4]. In the referred work the second peak appeared for very thick cells. In this work only efficiencies that can be achieved by cells with a reasonable thickness are considered to be of interest. Therefore, for all cases to be presented, an optimizing algorithm have been used that identifies the maximum efficiency in an absorbance range from 0 to 80. This upper limit is sufficiently high for practically all incoming photons to be absorbed. If the maximum value is found on the upper absorbance limit the range is extended until a peak is identified. The theoretical efficiency of some combinations of band gaps has been shown to increase if spectrally selective layers (layers that are reflective for some photon energies and trans parent for others) are applied to the cells [5]. In this work such layers are not taken into account.
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Fig. 3. The theoretical efficiency at 1000 suns, calculated as in ref. [2], for IBSCs with no overlap between the absorption coefficients.
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Fig. 4. Efficiency map of the theoretical efficiency for various combinations of band gaps with an overlap of 0.1 eV with a light concentration of 1000 suns.
III.
RESULTS
For reference, plots of the efficiency limits of cells with non-overlapping absorption coefficients are shown in figures 2 and 3 for light intensities of 1 and 1000 suns, calculated with a resolution of 0.01 eV. Similar efficiency maps for various combinations of band gaps are presented for 6 different cases with overlapping absorption coefficients in the following. Due to high computation time, the resolution of these maps is 0.02 eV. When peak efficiency and optimal band gaps are mentioned later in this article it is meant 'peak' and 'optimal' within the points of this grid. Slightly higher values might be found between the data points. The first three maps are shown in figures 4, 5 and 6 and are computed with a light intensity corresponding to 1000 suns. The overlap is 0.1, 0.5 and 1.0 eV, for the three plots. For 1000 suns and an overlap of 0.1 eV, the highest efficiency, 54.6 %, is found for sub-band gaps of 0.8 and 1.28 eV. For overlaps of 0.5 and 1.0 eV the highest efficien-
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Fig. 5. Efficiency map of the theoretical efficiency for various combinations of band gaps with an overlap of 0.5 eV with a light concentration of 1000 suns.
Fig. 6. Efficiency map of the theoretical efficiency for various combinations of band gaps with an overlap of 1 eV with a light concentration of 1000 suns.
cies are 49.6 % and 45.2 % for the band gap combinations 0.88 eV /1.18 eV and 0.98 eV /1.1 eV, respectively. Note that when the sub-band gaps are approximately equal, that is, when the IE is close to the middle of Eg, the efficiency is higher with some overlap than without it. When the sub-band gaps are of almost the same size, the energy interval from EL to EH is narrow and contains a small number of incoming photons. Since the net generation via the IE is limited by the smallest of the generation rates involving this band, the net generation will be small without an overlap. An overlap between the absorption coefficients can increase the number of photons exciting electrons over EL and thus increase the net generation via the IE and the efficiency of the cell. Also, when EL � EH, the overlap-related loss of energy is small when a photon energetically allowed to excite an electron over EH excites an electron over EL. As a consequence of the reduced or even positive impact of
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Fig. 7. Efficiency map of the theoretical efficiency for various combinations of band gaps with an overlap of 0.1 eV at 1 sun.
photon exchange between the sub-band gaps when EL � EH, the peak in the contour plots move towards such combinations of band gaps as the overlap becomes larger. There is still a loss, however, when a photon is emitted by de-excitation of an electron from the CB to the VB and reabsorbed by one of the sub-band gaps. Therefore the efficiency decrease with the overlap also for these combinations of band gaps, after an initial increase for a small overlap. The optimal thickness of the cells are between 0.5 and 4/Jm for all investigated cases with X = 1000. The largest values are found when EL � EH, since this allows transfer of more photons from EH to EL which is beneficial for such band gap combinations. Of course the numbers for the optimal thickness are only valid for the particular values of the absorption coefficients that have been used here. It is more general to refer to the optimal absorbance, that is, the value of the absorption coefficient multiplied with the optical path length in a cell with the optimal thickness. Here, the optical path length is twice the cell thickness, which gives an optimal absorbance between 2 and 16. For 1 sun illumination, shown in figures 7, 8 and 9, the qualitative findings are similar to the findings for 1000 suns. The peak moves towards the upper left corner of the contour plots with increasing overlap. Note that the color scale is different in the plots for 1 sun and 1000 suns. The peak efficiency is 44.5 %, 40.4 % and 36.8 % for overlaps of 0.1, 0.5 and 1.0 eV. The peaks are found for the EL /Ewpairs 0.96/1.46, 1.0411.36 and 1.1411.28 electron volts. For this light intensity the optimal cell thickness is between 0.5 and 29/Jm for all investigated cases. This correspond to absorbances between 2 and 116. The optimal ratio between EL and EH is approximately the same for 1 sun and 1000 suns for the degrees of overlap investigated here. It grows from 0.6 without overlap to ap proximately 0.75 with an overlap of 0.5 eV, and 0.9 with an overlap of 1 eV.
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the intention of this work is only to serve as a quick reference for a fast and dirty evaluation of IE-materials. As mentioned in the introduction, a better analysis should be performed based the actual shape of the material's absorption coefficients, for example as in ref [4].
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Fig. 8. Efficiency map of the theoretical efficiency for various combinations of band gaps with an overlap of 0.5 eV at I sun.
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CONCLUSIONS
Overlap between the absorption coefficients are found to extend the usable range of band gap combinations for IESCs. With some overlap the theoretical efficiency can get well above the Shockley-Queisser limit for some band gap combinations that are below this limit without overlap. For other band gap combinations the overlap reduces the theoretical efficiency. Without overlapping absorption coefficients, the highest efficiency is found when ELi EH � 0.6 for both 1 sun and 1000 suns. In general one sees that band gap combinations with a ratio larger than this is less sensitive to overlapping absorption coefficients than combinations with a smaller ratio. As the overlap grows, the highest efficiency is found for larger ratios of EL to EH. When searching for materials that can be used to implement the IESC-concept it is important to be aware of the fact that overlapping absorption coefficients increases the range of potentially usable combinations of band gaps. ACKNOWLEDGEMENTS
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35
The author would like to thank S0rlandets Kompetansefond for supporting this work. REFERENCES
Fig. 9. Efficiency map of the theoretical efficiency for various combinations of band gaps with an overlap of 1 eV at I sun.
The band gap combinations that are optimal without overlap are far from optimal with an overlap of 1 eV. For both 1 sun and 1000 suns the theoretical efficiency for these combinations of band gaps are then only slightly above the Shockley Queisser limit [6], and several percentage points lower than the best band gap combination. (The Shockley-Queisser limit is 31.0 % for 1 sun and 37.1 % for 1000 suns.) Real materials have absorption coefficients with rather ir regular shapes and the degree of overlap between EL and EH is likely to be different from the overlap between EH and Eg. In addition, the IE in real materials will have an energetical width larger than zero, which will also influence the theoretical efficiency [7]. As mentioned in the introduction,
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[I] G. L. Araujo and A. Marti, " Absolute limiting efficiencies for photovoltaic energy conversion," Solar energy materials and solar cells, vol. 33, no. 2, pp. 213-240, 1994. [Online]. Available: http://caLinisLfrl?aModele=afficheN&cpsidt=4275134 [2] A. Luque and A. Marti, "Increasing the Efficiency of Ideal Solar Cells by Photon Induced Transitions at Intermediate Levels," Physical Review Letters, vol. 78, no. 26, pp. 5014-5017, Jun. 1997. [Online]. Available: http://prl.aps. org/abstract/PRL/v781i26/p5014_1 [3] L. Cuadra, A. Marti, and A. Luque, "Influence of the Overlap Between the Absorption Coefficients on the Efficiency of the Intermediate Band Solar Cell," IEEE Transactions on Electron Devices, vol. 51, no. 6, pp. 1002-1007, Jun. 2004. [Online]. Available: http://ieeexplore. ieee. org/xpl/freeabs_all.jsp?arnumber=1300837 [4] R. Strandberg and 1. Aguilera, "Evaluation of vanadium substituted In2S3 as a material for intermediate band solar cells," Solar Energy Materials and Solar Cells, Nov. 2011. [Online]. Available: http://linkinghub.elsevier.com/retrieve/pii/S0927024811005733 [5] R. Strandberg and T. W. Reenaas, "Limiting efficiency of intermediate band solar cells with spectrally selective reflectors," Applied Physics Letters, vol. 97, no. 3, p. 031910, 2010. [Online]. Available: http://Iink.aip.orgllinkldoi/l 0.1063/1.3466269lhtml [6] W. Shockley and H. J. Queisser, "Detailed Balance Limit of Efficiency of p-n Junction Solar Cells," Journal of Applied Physics, vol. 32, no. 3, pp. 510-519, Mar. 1961. [Online]. Available: http://link.aip.orgllinkl?JAP/32/510/1 [7] M. Levy and C. Honsberg, "Solar cell with an intermediate band of finite width," Physical Review B, vol. 78, no. 16, OcL 2008. [Online]. Available: http://prb.aps. org/abstract/PRB/v78/i16/e165122
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