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  • Editor: Christian Seassal
  • Vol. 22, Iss. S3 — May. 5, 2014
  • pp: A818–A832
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Time domain simulation of tandem silicon solar cells with optimal textured light trapping enabled by the quadratic complex rational function

H. Chung, K-Y. Jung, X. T. Tee, and P. Bermel  »View Author Affiliations


Optics Express, Vol. 22, Issue S3, pp. A818-A832 (2014)
http://dx.doi.org/10.1364/OE.22.00A818


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Abstract

Amorphous silicon/crystalline silicon (a-Si/c-Si) micromorph tandem cells, with best confirmed efficiency of 12.3%, have yet to fully approach their theoretical performance limits. In this work, we consider a strategy for improving the light trapping and charge collection of a-Si/c-Si micromorph tandem cells using random texturing with adjustable short-range correlations and long-range periodicity. In order to consider the full-spectrum absorption of a-Si and c-Si, a novel dispersion model known as a quadratic complex rational function (QCRF) is applied to photovoltaic materials (e.g., a-Si, c-Si and silver). It has the advantage of accurately modeling experimental semiconductor dielectric values over the entire relevant solar bandwidth from 300—1000 nm in a single simulation. This wide-band dispersion model is then used to model a silicon tandem cell stack (ITO/a-Si:H/c-Si:H/silver), as two parameters are varied: maximum texturing height h and correlation parameter f. Even without any other light trapping methods, our front texturing method demonstrates 12.37% stabilized cell efficiency and 12.79 mA/cm2 in a 2 μm-thick active layer.

© 2014 Optical Society of America

1. Introduction

Sunlight is one of the most promising renewable sources of energy. The amount of solar power incident on the Earth’s surface is 10,000 times greater than the commercial energy used by every human on the planet, even without assuming any improvements in the performance and cost of current solar cell technology [1

1. R. Margolis, ed. SunShot vision study (U.S. Department of Energy, 2012).

]. In recent years, the level of adoption of solar technology has increased enormously, to tens of gigawatts of annual installations worldwide. However, the enormous potential of this resource may not be realized without further improvements in the efficiency of materials usage, and greater reductions in cost of manufacturing, particularly for incumbent technologies based on silicon [2

2. N.S. Lewis, “Toward Cost-Effective Solar Energy Use,” Science 315, 798–801 (2007). [CrossRef] [PubMed]

]. For example, the fluctuating prices of poly-silicon wafers have provided significant motivation for the development and utilization of thin film solar cells (TFSCs). TFSCs have a major advantage over ordinary, wafer-based solar cell technology, i.e., they provide the same power with only a fraction of the materials usage [3

3. P. Bermel, C. Luo, L. Zeng, L. C. Kimerling, and J. D. Joannopoulos, “Improving thin-film crystalline silicon solar cell efficiencies with photonic crystals,” Optics Express 15, 16986–17000 (2007). [CrossRef] [PubMed]

5

5. J. G. Mutitu, S. Shi, C. Chen, T. Creazzo, A. Barnett, C. Honsberg, and D. W. Prather, “Thin film solar cell design based on photonic crystal and diffractive grating structures,” Optics Express 16, 15238–15248 (2008). [CrossRef] [PubMed]

]. However, the maximum efficiencies of some types of TFSCs are lower than those found in wafer-based technology. For example, the best crystalline silicon-based solar cells operate at 25.0% efficiency, whereas nanocrystalline silicon TFSCs only have efficiencies up to 10.7% [6

6. M.A. Green, K. Emery, Y. Hishikawa, W. Warta, and E.D. Dunlop, “Solar cell efficiency tables (version 43),” Prog. Photovolt.: Res. Appl. 21, 1–9 (2013). [CrossRef]

]. The gap in performance between TFSCs and monocrystalline-based cells is believed to arise primarily from differences in optical and electronic design and performance. Because of incomplete light absorption, many photovoltaic cells have lower performance than the theoretical Shockley-Queisser limit associated with their electronic bandgap [7

7. W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” Journal of applied physics 32, 510–519 (1961). [CrossRef]

]. This is particularly a challenge for thin-film materials with low mobilities. Among the variety of commercially manufactured photovoltaic materials, silicon micromorph solar cells have operated far from the theoretical limit of tandem cells: approximately 40% with a-Si (Eg1 = 1.72eV) and c-Si (Eg2 = 1.11eV) [8

8. A. De Vos, “Detailed balance limit of the efficiency of tandem solar cells,” J. Phys. D 13, 839–845 (1980).

,9

9. O. D. Miller, E. Yablonovitch, and S. R. Kurtz, “Strong internal and external luminescence as solar cells approach the shockley-queisser limit,” IEEE J. Photovolt. 2, 303–311 (2012). [CrossRef]

]. On the other hand, thin film silicon micromorph cells have a record efficiency of only 12.3% [6

6. M.A. Green, K. Emery, Y. Hishikawa, W. Warta, and E.D. Dunlop, “Solar cell efficiency tables (version 43),” Prog. Photovolt.: Res. Appl. 21, 1–9 (2013). [CrossRef]

]. Thus, it is very important to identify the best light-trapping structures possible to help maximize the performance and reduce the costs of photovoltaic cells. Light trapping can be achieved by changing the angle of the light as it travels in the solar cell, i.e., by elongating the optical path, and this can be done by using a surface that has a rough texture. Theoretically, a rough-textured surface reduces reflection by increasing the probability that the reflected light will bounce back onto the surface, minimizing its chance of reflecting out of the cell [10

10. M. Berginski, J. Hupkes, M. Schulte, G. Schope, H. Stiebig, B. Rech, and M. Wuttig, “The effect of front zno: Al surface texture and optical transparency on efficient light trapping in silicon thin-film solar cells,” Journal of Applied Physics 101, 074903 (2007). [CrossRef]

]. From early theoretical work on photovoltaic cells [11

11. R. Brendel, M. Hirsch, R. Plieninger, and J. Werner, “Quantum efficiency analysis of thin-layer silicon solar cells with back surface fields and optical confinement,” IEEE Transactions on Electron Devices 43, 1104–1113 (1996). [CrossRef]

, 12

12. T. Tiedje, E. Yablonovitch, G. D. Cody, and B. G. Brooks, “Limiting efficiency of silicon solar cells,” IEEE Transactions on Electron Devices 31, 711–716 (1984). [CrossRef]

], it is known that a perfectly random structure can scatter light at all possible angles inside the active layer, thereby enhancing the absorption (effective path length) up to 4n2 [13

13. E. Yablonovitch, “Statistical ray optics,” JOSA 72, 899–907 (1982). [CrossRef]

]. However, thin-film solar cells, in particular, rarely achieve such high performance [14

14. M. Ghebrebrhan, P. Bermel, Y. Avniel, J. D. Joannopoulos, and S. G. Johnson, “Global optimization of silicon photovoltaic cell front coatings,” Optics express 17, 7505–7518 (2009). [CrossRef] [PubMed]

]. Part of the reason is that analytical approaches do not adequately describe the real structures that have been built experimentally. In particular, the feature size of a randomly-textured surface seems to play an important role for light absorption, based on experimental observations over the last decade [15

15. J. Zhao, A. Wang, M. A. Green, and F. Ferrazza, “19.8% efficient honeycomb textured multicrystalline and 24.4% monocrystalline silicon solar cells,” Applied Physics Letters 73, 1991 (1998). [CrossRef]

, 16

16. R. Dewan, I. Vasilev, V. Jovanov, and D. Knipp, “Optical enhancement and losses of pyramid textured thin-film silicon solar cells,” Journal of Applied Physics 110, 013101 (2011). [CrossRef]

]. Although experimental results must be the ultimate guide, examining and optimizing a very wide range of structures that can potentially be fabricated is extremely expensive and time-consuming. Thus an extremely accurate, simulation-based approach is needed to help guide experimentalists. In this work, we investigate and optimize a broad class of random structures with short-range correlation for experimental realism, and long range periodicity to enhance diffraction into guided modes. As depicted in Fig. 1, the simulation of random texturing is not a trivial undertaking due to the complicated structure and dispersion characteristics of a thin film solar cells.

Fig. 1 Cross section of a-Si/c-Si tandem solar cell. It is contacted with indium tin oxide on the front, and silver in the back, and encapsulated with glass. The randomly textured front surface is shown from two different perspectives. Note that the same textured surface on the ITO and a-Si is also applied to the top of the c-Si layer. The minimum glass thickness of 1500 nm is used only in simulation. Experimental thicknesses are greater, but Fig. 4 shows that this only has a minor effect on the absorption spectrum.

In this work, we apply a quadratic complex rational function (QCRF) model to accurately capture the dispersion of thin film photovoltaic materials. We employ a FDTD simulation due to its simplicity and accuracy [27

27. S.-G. Ha, J. Cho, J. Choi, H. Kim, and K.-Y. Jung, “FDTD dispersive modeling of human tissues based on quadratic complex rational function,” IEEE Transactions on Antennas and Propagation 61, 996–999 (2013). [CrossRef]

29

29. H. Chung, S.-G. Ha, J. Cho, and K.-Y. Jung, “Accurate FDTD modeling for dispersive media using rational function and particle swarm optimization.” International Journal of Electronics to be published (2014).

]. Specifically, the dielectric function of a-Si is modeled over the wavelength range from 300—1000 nm, where the relevant power-generating absorption of the a-Si active layer mainly occurs. A full-wave optical simulation is performed and the full-spectrum results are compared to an analytic model as well as experimental data. Also, it has been shown that the QCRF model can fit other photovoltaic materials including, but not limited to, c-Si, silver, and CdTe.

In order to trap light optimally, we suggest a statistically correlated random surface texturing algorithm which can reproduce known structures such as perfectly Lambertian surfaces and flat surfaces in the proper limits, as well as yielding physically realistic structures at intermediate values. Combining this proposed texturing with accurate modeling of TFSC materials can then be used to determine the optimum texturing of the front surface texturing of a tandem silicon solar cell (a-Si/c-Si) structure with normal incident light, as shown in Fig. 1. The tandem silicon solar cell has several advantages over a silicon-based single junction solar cell [30

30. K. Yamamoto, A. Nakajima, M. Yoshimi, T. Sawada, S. Fukuda, T. Suezaki, M. Ichikawa, Y. Koi, M. Goto, H. Takata, T. Sasaki, and Y. Tawada, “Novel hybrid thin film silicon solar cell and module,” in Proceedings of 3rd World Conference on Photovoltaic Energy Conversion (IEEE, 2003), vol. 3, pp. 2789–2792.

]. For instance, the tandem silicon cell is more stable than the a-Si:H single junction solar cell, due to the contribution of a bottom μc-Si:H cell, which means that the Staebler-Wronski degradation is smaller. In contrast with the single μc-Si:H single junction solar cell, it can be manufactured as a thinner layer. However, one additional challenge is that unlike a single junction solar cell, a multi-junction solar cell must have the currents generated in each layer matched in order to obtain optimal performance. This requirement further suggests that careful simulation will be necessary for a successful design.

2. Dispersion modeling and validation of its accuracy

Classical modeling methods, such as the Debye, Lorentz and Drude models, have been used extensively for many types of dispersive media [23

23. A. Tavlove and S. C. Hagness, “Computational electrodynamics: the finite-difference time-domain method,” 2, Artech House, (Artech House, 2000).

,31

31. F. L. Teixeira, “Time-domain finite-difference and finite-element methods for maxwell equations in complex media,” IEEE Transactions on Antennas and Propagation 56, 2150–2166 (2008). [CrossRef]

33

33. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the fdtd method,” Computer Physics Communications 181, 687–702 (2010). [CrossRef]

]. However, those models are insufficient for dispersive modeling of some thin film photovoltaic materials such as a-Si, CIGS and CZTS, because in the semiconductor materials, both the conducting term and non-conducting term must be taken into account in their wave equation. The wave equation considering both terms is rather complicated and the solutions are somewhat difficult to interpret [34

34. E. L. Haines and A. B. Whitehead, “Pulse height defect and energy dispersion in semiconductor detectors,” Review of Scientific Instruments 37, 190–194 (1966). [CrossRef]

]. Nevertheless, a qualitative description of many of the optical properties of semiconductors is furnished by classical theory. As a result, there is a promising modeling method called Tauc-Lorentz model [25

25. G. Jellison and F. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Applied Physics Letters 69, 371–373 (1996). [CrossRef]

] which shows very good agreement with measurements of a-Si. However, the Tauc-Lorentz model has an exponential function in its equation, making numerical differentiation very difficult; thus, the Tauc-Lorentz model is not ideal for time domain simulations [26

26. A. Fantoni and P. Pinho, “FDTD simulation of light propagation inside a-si: H structures,” in “ MRS Proceedings (Cambridge University, 2010). [CrossRef]

]. Recently, the quadratic complex rational function (QCRF) model was suggested for dispersive modeling of biological tissues [27

27. S.-G. Ha, J. Cho, J. Choi, H. Kim, and K.-Y. Jung, “FDTD dispersive modeling of human tissues based on quadratic complex rational function,” IEEE Transactions on Antennas and Propagation 61, 996–999 (2013). [CrossRef]

] and concrete materials [28

28. H. Chung, J. Cho, S.-G. Ha, S. Ju, and K.-Y. Jung, “Accurate FDTD dispersive modeling for concrete materials.” ETRI Journal 35, 915–918, (2013). [CrossRef]

], although its potentially usefulness is much more general. The QCRF dielectric function has the following form:
εr,QCRF(ω)=A0+A1(jω)+A2(jω)21+B1(jω)+B2(jω)2,
(1)
where ω is the optical frequency, and A0, A1, A2, B1, and B2 are adjustable parameters.

As shown in Eq. (1), the QCRF dispersion model has an advantage over the Drude, Debye and Lorentz dispersion models in terms of the number of degrees of freedom, which helps make it highly applicable to wide-band dispersive media such as photovoltaic materials. In addition, the coefficients of the QCRF model can be obtained by solving a 5 × 5 matrix inversion analytically — a very computationally efficient procedure [27

27. S.-G. Ha, J. Cho, J. Choi, H. Kim, and K.-Y. Jung, “FDTD dispersive modeling of human tissues based on quadratic complex rational function,” IEEE Transactions on Antennas and Propagation 61, 996–999 (2013). [CrossRef]

29

29. H. Chung, S.-G. Ha, J. Cho, and K.-Y. Jung, “Accurate FDTD modeling for dispersive media using rational function and particle swarm optimization.” International Journal of Electronics to be published (2014).

]. In this paper, the QCRF model is applied to thin film photovoltaic materials.

2.1. Dispersion modeling

Despite the simplicity and accuracy of the QCRF method, the conventional QCRF model does not always precisely fit measurements of all materials, especially when the imaginary part of epsilon exponentially approaches zero. Also, obtaining the coefficients of the QCRF model through matrix inversion does not consider the Kramers-Kronig (K-K) relations explicitly. However, an optimization method respecting K-K can be used to overcome this limitation [29

29. H. Chung, S.-G. Ha, J. Cho, and K.-Y. Jung, “Accurate FDTD modeling for dispersive media using rational function and particle swarm optimization.” International Journal of Electronics to be published (2014).

].

As shown in Fig. 2, the QCRF model fits fairly well with measurement data for photovoltaic materials. Since the light absorbed by a dispersive material is directly proportional to an exponential of ε″, the imaginary part of the dielectric function of a-Si, the imaginary part of the dielectric function of which varies sharply, is plotted as a log scale. Also, in this work, silver is treated as a non-absorbing material, so only the real part of permittivity is considered. Although it is possible to fit lossy silver with the QCRF model, if we were to treat silver as a lossy material explicitly, it would result in a predictive error, in which parasitic loss would be incorrectly counted as absorption contribution to open-circuit voltage. This means that our estimates of short-circuit current enhancement may slightly underestimate the relative enhancement associated with our light-trapping approach. The optical constants of the photovoltaic materials considered in this manuscript were acquired from the literature [35

35. R. Collins, A. Ferlauto, G. Ferreira, C. Chen, J. Koh, R. Koval, Y. Lee, J. Pearce, and C. Wronski, “Evolution of microstructure and phase in amorphous, protocrystalline, and microcrystalline silicon studied by real time spectroscopic ellipsometry,” Solar Energy Materials and Solar Cells 78, 143–180 (2003). [CrossRef]

, 36

36. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1998).

].

Fig. 2 Dispersion curve fittings of photovoltaic materials using the QCRF model. The solid lines and symbols indicate the results of the QCRF model and the experimental data of dispersive material, respectively: (a) Real part of relative permittivity of a-Si. (b) Imaginary part of relative permittivity of a-Si. (c) Real part of relative permittivity of c-Si. (d) Imaginary part of relative permittivity of c-Si. (e) Real part of relative permittivity of silver.

2.2. Theoretical absorption and simulation result

In this section, the accuracy of the QCRF model is verified by comparison to analytical predictions for a dielectric slab. In the 300 nm thickness of a single a-Si dielectric slab, the 3-D FDTD simulation results are compared to analytic absorption derived by a theoretical calculation [37

37. S. J. Orfanidis, Electromagnetic waves and antennas (Rutgers University, 2002).

]. The total reflection and transmission coefficients for the electric fields can be calculated analytically using multiple reflections, and then summed exactly to yield:
r(λ)=ρ1+n=1τ1τ1(ρ1)n1ρ2nejωt=ρ1+τ1τ1ρ21ejωtρ1
(2)
t(λ)=τ1τ2n=0(ρ2ρ1)nejωt=τ1τ21ρ2ρ1ejωt,
(3)
where ρ1 is the electric field reflection coefficient at the left boundary of the dielectric slab when traveling to the right, ρ′1 is the same coefficient when traveling in the reverse direction, τ1 is the transmission at the left boundary when traveling to the right, τ′1 is the same coefficient traveling in the reverse direction, ρ2 is the reflection coefficient at the right boundary of the dielectric slab when traveling to the right, ω = 2πc/λ is the optical frequency, and t is time required for light travel through the certain thickness of dielectric material. Note that τk = 1 + ρk and τ′k = 1 + ρ′k for all integer k, because of phase shifts. Using Eqs. (2) and (3), reflected power R(λ) = |r(λ)|2 and transmitted power T (λ) = |t(λ)|2 can also be obtained. The light absorption spectrum A(λ) is then given simply by A(λ) = 1 − T (λ) − R(λ).

Using this approach, the light absorption spectrum for a 300 nm dielectric slab of a-Si is obtained. Also, a 3-D QCRF based FDTD simulation is performed with 300 nm thickness of a-Si material. The dispersive QCRF material is located at the center of 3-D space which has 100 × 100 × 900 cells. The x and y boundaries are connected periodically; perfectly matched layers are implemented near the z boundaries (at the top and bottom of the simulation geometry). The Yee lattice spacing is set to 3.86 nm, resulting in the minimum resolution of 20 cells per optical wavelength within all non-metallic materials. The rest of the simulation region is set to be free space.

As shown in Fig. 3, the dispersive FDTD simulation, which is performed only once, predicts the absorption of the a-Si material very accurately, including both the material dispersion and the Fabry-Perot oscillations that occur in the dielectric slab. Comparing the two data sets, we find that the root mean square error between them is 3.97%.

Fig. 3 The theoretical and simulated absorption rates of 300 nm thick a-Si, the former being obtained from Eq. (3) combined with literature data from ref. [36], and the latter being obtained from our QCRF model. The root mean square error from comparing the two data sets is 3.97%.

2.3. Experimental and simulated absorption result in the solar cell structure

This section presents the results of the 3-D FDTD simulation of a c-Si single junction solar cell structure found in a recent experiment. In this experimental study, the absorption of the solar cell was measured over a broad range of wavelengths both with and without light trapping. The thickness of c-Si layer is 1500 nm and ITO is considered as a charge transport and anti-reflection coating layer [38

38. L. T. Varghese, Y. Xuan, B. Niu, L. Fan, P. Bermel, and M. Qi, “Enhanced photon management of thin-film silicon solar cells using inverse opal photonic crystals with 3d photonic bandgaps,” Advanced Optical Materials 1, 692–698 (2013). [CrossRef]

]. A 3-D QCRF-FDTD simulation is performed on the same geometries in order to establish its accuracy.

The experimentally measured absorption spectra of c-Si solar cells, shown on the left hand side of Fig. 4, can be predicted accurately using our simulation technique. More specifically, the absorption spectrum measured for the flat structure is very similar to our simulation results, except for the Fabry-Perot oscillations in the short wavelength range. This difference is mainly because the experiment had glass that is more than 100 μm thick deposited on the top of the solar cell, whereas, in order to save simulation time, it is assumed that the thickness of the glass is less than a few microns. Even so, the overall absorption curve for the flat case matches very well with the experimental data. For the textured structure, the simulation predicted a slightly lower absorption than observed in experiment, particularly for wavelengths around 800 nm. This is mainly because different texturing methods are used in the simulation and the experiment.

Fig. 4 The left figure indicates the experimental absorption rate for a 1500 nm thick c-Si solar cell. It is adapted from recently published research [38]. The right figure indicates the absorption rate obtained by the simulation.

3. Statistical random surface texturing model

A 2-D simulation of the solar cell is performed using a FDTD tool known as MEEP [33

33. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the fdtd method,” Computer Physics Communications 181, 687–702 (2010). [CrossRef]

] to obtain the optimum texturing height and correlation factor. The algorithm for generating the random surface texturing is based on the double correlation equation above, Eq. (5). In this 2-D simulation, a TM-polarized wave is incident from the normal direction. The structure defined in the 2-D simulation, as shown in Fig. 5, consists of a metal back reflector, a silicon (absorbing material) block with a fixed thickness, and a non-absorbing random surface texturing. The transmission spectrum T (λ) and the reflection spectrum R(λ) are computed and used to compute the absorption spectra, A(λ) = 1 − T (λ) − R(λ) over a specified range of wavelengths. The absorption is then fed into the short-circuit current density (Jsc) which serves as a proxy for the efficiency of the photovoltaic cells.

Fig. 5 (a) Contour plot showing calculated short-circuit current density as a function of maximum texturing height and correlation factor for 2D solar cells, using TM-polarized light incident at normal incidence. Note that the optimal performance is expected to occur at f = 0.975, ln(1 − f) = −3.689 and h = 1000 nm in 2-D structure. (b) The optimized 2D geometry used to generate our contour plot.

As depicted in Fig. 5, the highest Jsc of 7.466 mA/cm2 is obtained at the optimum texturing height of 1000 nm and the correlation factor of 0.975. Each Jsc value is collected from multiple simulations of runs (5 runs) and averaged out for plotting the contour plot. This provides a proof of concept that correlated random structures can provide additional light absorption.

Fig. 6 Random surface texturing algorithm represented in terms of correlation factor (f).

Due to the differing resolutions employed in our 2-D and 3-D solar cell simulations, the optimum correlation factor obtained in each simulation also differs. In order to compare them properly, the correlation factor from the 3-D simulation result is exponentiated by a factor given by the product of the block width N times the ratio of the y-grid values in absolute units (e.g., in nm) as shown in the equation below:
f2D=f3D(NΔy2D/Δy3D),
(9)
where Δy2D and Δy3D are the grid values used in the 2-D and 3-D simulations, respectively. Given that the optimum correlation factor for 2-D simulation is 0.975, its normalized 3-D correlation factor is equal to 0.9998.

It can be useful to compare these results with recent experimental work on thin-film textures for light-trapping. The specific random texturing methods discussed in a recent comprehensive study [40

40. M. J. Keevers, T. L. Young, U. Schubert, and M. A. Green, “10% efficient CSG minimodules,” Proceedings of the 22nd European Photovoltaic Solar Energy Conference and Exhibition, Milan (2007).

] gave rise to several local peaks, with widths of approximately 50 — 100 nm. Comparing the local peak width of the experimental geometry with our random surface texturing statistical model, the sample with f = 0.99 matches well with the structure introduced in the reference paper [40

40. M. J. Keevers, T. L. Young, U. Schubert, and M. A. Green, “10% efficient CSG minimodules,” Proceedings of the 22nd European Photovoltaic Solar Energy Conference and Exhibition, Milan (2007).

]. Looking at the surfaces shown in Fig. 6, one hardly can distinguish which one will be the best light trapping structure among a variety of randomly-textured surfaces. It will become more apparent when these randomly-textured surfaces are applied to the front of the solar cell structure.

4. Enhanced light trapping in a tandem cell application

The efficiency as a function of the thicknesses of a-Si:H, c-Si:H and ITO layers in a flat tandem cell structure is shown in Fig. 7. In order to find the best thicknesses, we first run 3-D FDTD simulations after varying all the geometric parameters. We then project these results onto a single axis at a time, while look for a cluster of points with the highest calculated efficiencies. Using this procedure, we find that the optimum thicknesses of a-Si:H and c-Si:H are 205 nm and 1795 nm, respectively; ITO had its best performance when its thickness is 60 nm.

Fig. 7 Efficiency versus thickness of (left) indium-tin oxide and (right) a-Si. Each red dot corresponds to a single 3-D FDTD simulation and is projected from a higher-dimension manifold of design space onto the axes displayed, in order to identify the optimal values for these individual parameters. Note that each simulation is performed in a flat solar cell structure without texturing.

Starting with the best parameters observed in Fig. 7, the statistical texturing algorithm is applied to determine the optimum surface texturing of the tandem cell structure. As shown in Fig. 1, the same randomly textured surface is applied to the ITO, a-Si and c-Si layers. Because a maximum texturing height is also considered as an important factor in random texturing, we introduce a feature which controls the maximum texturing height of the random texturing algorithm. In order to calculate the overall cell efficiency of the cell, the internal quantum efficiency of a-Si and c-Si are calculated from a semiclassical drift-diffusion simulation tool capable of calculating recombination losses known as ADEPT 2.0 [43

43. J. L. Gray, X. Wang, X. Sun, and J. R. Wilcox, “Adept 2.0,” (2011).

]. In the combination of recombination and optical losses, the overall cell efficiencies of the random texturing model are plotted as shown in Fig. 8. The best cell efficiency is 12.37%; the associated short circuit currents are 12.79 mA/cm2 at the a-Si layer and 12.88 mA/cm2, while the open circuit voltages are 875.9 mV for the a-Si layer and 520.0 mV for the c-Si layer. The optimum texturing height is 1158 nm and the optimum correlation factor is 0.999. The efficiency of cell tends to plateau after it reaches a high enough correlation factor of 0.999 or more. The reason is that our algorithm re-scales the height of random structures in order to find the optimum texturing height, so that random surfaces with higher correlation factors tend to create structures very similar to those obtained with lower correlation factors. Thus we can feel confident that we have found a global optimum with respect to these two key parameter values.

Fig. 8 Contour plot showing silicon tandem cell efficiency versus texturing height and the correlation factor. Note that the optimal performance is predicted to occur when f = 0.999 and h = 1158 nm, as explained in the text.

Light absorption of the a-Si and c-Si layers is shown in Fig. 9. We compare the best-performing structure from Fig. 8 with flat (f = 1) and totally random (f = 0) structures at normal incidence. In the a-Si layer, the optimized structure shows enhanced light trapping over the entire range of wavelengths. The absorption of c-Si shown in Fig. 9 (b) should not be directly compared to each structure, because the amount of light arriving at the c-Si layer is different due to a filtering effect caused by absorption in the a-Si layer. Thus, light absorption in the c-Si layer is re-normalized by including that effect. Fig. 9 (c) shows that the normalized absorption in the c-Si layer is also enhanced, compared to both the flat and random structures. Excepting the Fresnel reflection associated with the air-SiO2 boundary (0.0349 from analytical calculation), the optimized tandem cell has almost full absorption for wavelengths from 300 nm to 550 nm; after that, it decreases as shown in Fig. 9 (d). Front texturing itself with the statistical algorithm shows promising light absorption enhancement for normal incidence; however, it would not be expected to retain the same advantage at all angles. In future work, this shortcoming should be addressed and studied over all angles by adding complementary light trapping methods, such as photonic crystal [44

44. A. Bielawny, J. Üpping, P. T. Miclea, R. B. Wehrspohn, C. Rockstuhl, F. Lederer, M. Peters, L. Steidl, R. Zentel, S.-M. Lee, M. Knez, A. Lambertz, and R. Carius, “3d photonic crystal intermediate reflector for micromorph thin-film tandem solar cell,” physica status solidi (a) 205, 2796–2810 (2008). [CrossRef]

, 45

45. J. Üpping, A. Bielawny, R. B. Wehrspohn, T. Beckers, R. Carius, U. Rau, S. Fahr, C. Rockstuhl, F. Lederer, M. Kroll, T. Pertsch, L. Steidl, and R. Zentel, “Three-dimensional photonic crystal intermediate reflectors for enhanced light-trapping in tandem solar cells,” Advanced Materials 23, 3896–3900 (2011). [CrossRef] [PubMed]

], back grating [46

46. D. Madzharov, R. Dewan, and D. Knipp, “Influence of front and back grating on light trapping in microcrystalline thin-film silicon solar cells,” Optics express 19, A95–A107 (2011). [CrossRef] [PubMed]

, 47

47. H. Sai, H. Fujiwara, M. Kondo, and Y. Kanamori, “Enhancement of light trapping in thin-film hydrogenated microcrystalline si solar cells using back reflectors with self-ordered dimple pattern,” Applied Physics Letters 93, 143501 (2008). [CrossRef]

] and intermediate layers [48

48. K. Yamamoto, A. Nakajima, M. Yoshimi, T. Sawada, S. Fukuda, T. Suezaki, M. Ichikawa, Y. Koi, M. Goto, T. Meguro, T. Matsuda, M. Kondo, T. Sasaki, and Y. Tawada, “A high efficiency thin film silicon solar cell and module,” Solar Energy 77, 939–949 (2004). [CrossRef]

].

Fig. 9 Light absorption rate of the optimized tandem silicon solar cell with two reference absorption curves that are obtained from a flat structure and a totally random structure for normal incidence. (a) Light absorption in the a-Si layer. (b) Light absorption in the a-Si layer. (c) Normalized light absorption in the c-Si layer with rest of light filtered by the a-Si layer and by subtraction of the first reflected light at the SiO2 layer. (d) Total light absorption in both layers.

The random textured surface of the best-performing cell is shown in Fig. 10. It has a large structure close to the x-axis, which is connected to the opposite side via periodic boundary conditions, and also has a number of small random structures on its surface. It seems like that a combination of one large structure and several small structures ensures that incident light will be scattered in all directions, so that enhanced absorption can be achieved. It is shown that an enhanced light trapping structure can be obtained by adjusting the correlation factor and the texturing height in our random texturing algorithm.

Fig. 10 Optimal random surface texturing in a tandem cell application shown from two different perspectives.

Also, it should be emphasized that each simulation in Fig. 8 takes approximately 75 hours on a single core computer; entire ensembles of simulations are performed on our computational cluster, Conte, on a dedicated queue that has 64 cores with a memory capacity of 4GB/core. Without an accurate dispersion model, simulating the full bandwidth at an acceptable frequency resolution (as shown in Fig. 9) would not be viable with such a computational resource. In this work, the QCRF dispersion model enables us to reduce a potentially large number of simulations to a single calculation over the entire relevant portion of the solar spectrum for a given geometry.

5. Conclusion

In conclusion, we have investigated a randomly textured surface in a-Si/c-Si micromorph tandem cells using correlated random texturing with QCRF to calculate the entire solar spectrum in a single calculation. The QCRF model satisfies the Kramers-Kronig relation for real materials, is numerically stable, and can be used to achieve accurate curve fitting to experimental semiconductor material dispersion data (e.g., a-Si, c-Si and silver) for wavelengths ranging from 300—1000 nm. Its accuracy is verified in two ways: first, by comparing the results it produced with experimental results acquired for photovoltaic materials; and second, in comparing this simulation technique with analytical results, where the root mean squared error is observed to be 3.967%. The QCRF model is applied to a 3-D FDTD simulation; used properly, it reduces a potentially large number of simulations required for full solar spectrum analysis to a single simulation run. Taking advantage of this capability, a range of correlated random textures for light trapping are examined to find the optimal parameter values. Constraining ourselves to a 2 μm-thick active material combination of a-Si/c-Si, we have found that the best-performing micromorph tandem cell structure has 1158 nm of maximum texturing height and a relatively high correlation factor (f = 0.999). It is predicted to have 12.37% stabilized cell efficiency, 12.79 mA/cm2 of short-circuit current and 1395.9 mV of open-circuit voltage. In short, a randomly textured tandem cell with optimized parameters shows meaningful enhancement of light absorption at normal incidence. In future work, alternative and complementary designs with more general applicability will be considered, including but not limited to photonic crystal structures [44

44. A. Bielawny, J. Üpping, P. T. Miclea, R. B. Wehrspohn, C. Rockstuhl, F. Lederer, M. Peters, L. Steidl, R. Zentel, S.-M. Lee, M. Knez, A. Lambertz, and R. Carius, “3d photonic crystal intermediate reflector for micromorph thin-film tandem solar cell,” physica status solidi (a) 205, 2796–2810 (2008). [CrossRef]

,45

45. J. Üpping, A. Bielawny, R. B. Wehrspohn, T. Beckers, R. Carius, U. Rau, S. Fahr, C. Rockstuhl, F. Lederer, M. Kroll, T. Pertsch, L. Steidl, and R. Zentel, “Three-dimensional photonic crystal intermediate reflectors for enhanced light-trapping in tandem solar cells,” Advanced Materials 23, 3896–3900 (2011). [CrossRef] [PubMed]

], back grating [46

46. D. Madzharov, R. Dewan, and D. Knipp, “Influence of front and back grating on light trapping in microcrystalline thin-film silicon solar cells,” Optics express 19, A95–A107 (2011). [CrossRef] [PubMed]

,47

47. H. Sai, H. Fujiwara, M. Kondo, and Y. Kanamori, “Enhancement of light trapping in thin-film hydrogenated microcrystalline si solar cells using back reflectors with self-ordered dimple pattern,” Applied Physics Letters 93, 143501 (2008). [CrossRef]

] and intermediate layer [48

48. K. Yamamoto, A. Nakajima, M. Yoshimi, T. Sawada, S. Fukuda, T. Suezaki, M. Ichikawa, Y. Koi, M. Goto, T. Meguro, T. Matsuda, M. Kondo, T. Sasaki, and Y. Tawada, “A high efficiency thin film silicon solar cell and module,” Solar Energy 77, 939–949 (2004). [CrossRef]

]. In all these cases, as well as for other photovoltaic absorber or window layer materials, the QCRF and statistical texturing models can play a critical role in enabling accurate single simulations encompassing the entire solar spectrum.

Acknowledgments

The authors thank Muhammad Ashraful Alam and David Janes for valuable discussions. Support was provided by the Department of Energy, under DOE Cooperative Agreement No. DEEE0004946 (PVMI Bay Area PV Consortium), as well as the Semiconductor Research Corporation, under Research Task No. 2110.006 (Network for Photovoltaic Technologies) and the Basic Research Program through National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (No. 2012R1A1A1015159).

References and links

1.

R. Margolis, ed. SunShot vision study (U.S. Department of Energy, 2012).

2.

N.S. Lewis, “Toward Cost-Effective Solar Energy Use,” Science 315, 798–801 (2007). [CrossRef] [PubMed]

3.

P. Bermel, C. Luo, L. Zeng, L. C. Kimerling, and J. D. Joannopoulos, “Improving thin-film crystalline silicon solar cell efficiencies with photonic crystals,” Optics Express 15, 16986–17000 (2007). [CrossRef] [PubMed]

4.

A. G. Aberle, “Thin-film solar cells,” Thin Solid Films 517, 4706–4710 (2009). [CrossRef]

5.

J. G. Mutitu, S. Shi, C. Chen, T. Creazzo, A. Barnett, C. Honsberg, and D. W. Prather, “Thin film solar cell design based on photonic crystal and diffractive grating structures,” Optics Express 16, 15238–15248 (2008). [CrossRef] [PubMed]

6.

M.A. Green, K. Emery, Y. Hishikawa, W. Warta, and E.D. Dunlop, “Solar cell efficiency tables (version 43),” Prog. Photovolt.: Res. Appl. 21, 1–9 (2013). [CrossRef]

7.

W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” Journal of applied physics 32, 510–519 (1961). [CrossRef]

8.

A. De Vos, “Detailed balance limit of the efficiency of tandem solar cells,” J. Phys. D 13, 839–845 (1980).

9.

O. D. Miller, E. Yablonovitch, and S. R. Kurtz, “Strong internal and external luminescence as solar cells approach the shockley-queisser limit,” IEEE J. Photovolt. 2, 303–311 (2012). [CrossRef]

10.

M. Berginski, J. Hupkes, M. Schulte, G. Schope, H. Stiebig, B. Rech, and M. Wuttig, “The effect of front zno: Al surface texture and optical transparency on efficient light trapping in silicon thin-film solar cells,” Journal of Applied Physics 101, 074903 (2007). [CrossRef]

11.

R. Brendel, M. Hirsch, R. Plieninger, and J. Werner, “Quantum efficiency analysis of thin-layer silicon solar cells with back surface fields and optical confinement,” IEEE Transactions on Electron Devices 43, 1104–1113 (1996). [CrossRef]

12.

T. Tiedje, E. Yablonovitch, G. D. Cody, and B. G. Brooks, “Limiting efficiency of silicon solar cells,” IEEE Transactions on Electron Devices 31, 711–716 (1984). [CrossRef]

13.

E. Yablonovitch, “Statistical ray optics,” JOSA 72, 899–907 (1982). [CrossRef]

14.

M. Ghebrebrhan, P. Bermel, Y. Avniel, J. D. Joannopoulos, and S. G. Johnson, “Global optimization of silicon photovoltaic cell front coatings,” Optics express 17, 7505–7518 (2009). [CrossRef] [PubMed]

15.

J. Zhao, A. Wang, M. A. Green, and F. Ferrazza, “19.8% efficient honeycomb textured multicrystalline and 24.4% monocrystalline silicon solar cells,” Applied Physics Letters 73, 1991 (1998). [CrossRef]

16.

R. Dewan, I. Vasilev, V. Jovanov, and D. Knipp, “Optical enhancement and losses of pyramid textured thin-film silicon solar cells,” Journal of Applied Physics 110, 013101 (2011). [CrossRef]

17.

C. L. Tan, A. Karar, K. Alameh, and Y. T. Lee, “Optical absorption enhancement of hybrid-plasmonic-based metal-semiconductor-metal photodetector incorporating metal nanogratings and embedded metal nanoparticles,” Optics express 21, 1713–1725 (2013). [CrossRef] [PubMed]

18.

C. Rockstuhl, S. Fahr, K. Bittkau, T. Beckers, R. Carius, F.-J. Haug, T. Söderström, C. Ballif, and F. Lederer, “Comparison and optimization of randomly textured surfaces in thin-film solar cells,” Optics express 18, A335–A341 (2010). [CrossRef] [PubMed]

19.

S.-S. Lo, C.-C. Chen, F. Garwe, and T. Pertch, “Broad-band anti-reflection coupler for a: Si thin-film solar cell,” Journal of Physics D: Applied Physics 40, 754 (2007). [CrossRef]

20.

J. Lacombe, O. Sergeev, K. Chakanga, K. von Maydell, and C. Agert, “Three dimensional optical modeling of amorphous silicon thin film solar cells using the finite-difference time-domain method including real randomly surface topographies,” Journal of Applied Physics 110, 023102 (2011). [CrossRef]

21.

Y.-C. Tsao, C. Fisker, and T. Garm Pedersen, “Optical absorption of amorphous silicon on anodized aluminum substrates for solar cell applications,” Optics Communications 315, 17–25 (2014). [CrossRef]

22.

V. Jovanov, U. Palanchoke, P. Magnus, H. Stiebig, J. Hüpkes, P. Sichanugrist, M. Konagai, S. Wiesendanger, C. Rockstuhl, and D. Knipp, “Light trapping in periodically textured amorphous silicon thin film solar cells using realistic interface morphologies,” Optics Express 21, A595–A606 (2013). [CrossRef] [PubMed]

23.

A. Tavlove and S. C. Hagness, “Computational electrodynamics: the finite-difference time-domain method,” 2, Artech House, (Artech House, 2000).

24.

W. C. Chew, Waves and fields in inhomogenous media (Van Nostrand Reinhold, 1990).

25.

G. Jellison and F. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Applied Physics Letters 69, 371–373 (1996). [CrossRef]

26.

A. Fantoni and P. Pinho, “FDTD simulation of light propagation inside a-si: H structures,” in “ MRS Proceedings (Cambridge University, 2010). [CrossRef]

27.

S.-G. Ha, J. Cho, J. Choi, H. Kim, and K.-Y. Jung, “FDTD dispersive modeling of human tissues based on quadratic complex rational function,” IEEE Transactions on Antennas and Propagation 61, 996–999 (2013). [CrossRef]

28.

H. Chung, J. Cho, S.-G. Ha, S. Ju, and K.-Y. Jung, “Accurate FDTD dispersive modeling for concrete materials.” ETRI Journal 35, 915–918, (2013). [CrossRef]

29.

H. Chung, S.-G. Ha, J. Cho, and K.-Y. Jung, “Accurate FDTD modeling for dispersive media using rational function and particle swarm optimization.” International Journal of Electronics to be published (2014).

30.

K. Yamamoto, A. Nakajima, M. Yoshimi, T. Sawada, S. Fukuda, T. Suezaki, M. Ichikawa, Y. Koi, M. Goto, H. Takata, T. Sasaki, and Y. Tawada, “Novel hybrid thin film silicon solar cell and module,” in Proceedings of 3rd World Conference on Photovoltaic Energy Conversion (IEEE, 2003), vol. 3, pp. 2789–2792.

31.

F. L. Teixeira, “Time-domain finite-difference and finite-element methods for maxwell equations in complex media,” IEEE Transactions on Antennas and Propagation 56, 2150–2166 (2008). [CrossRef]

32.

C. Herzinger, B. Johs, W. McGahan, J. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” Journal of Applied Physics 83, 3323–3336 (1998). [CrossRef]

33.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the fdtd method,” Computer Physics Communications 181, 687–702 (2010). [CrossRef]

34.

E. L. Haines and A. B. Whitehead, “Pulse height defect and energy dispersion in semiconductor detectors,” Review of Scientific Instruments 37, 190–194 (1966). [CrossRef]

35.

R. Collins, A. Ferlauto, G. Ferreira, C. Chen, J. Koh, R. Koval, Y. Lee, J. Pearce, and C. Wronski, “Evolution of microstructure and phase in amorphous, protocrystalline, and microcrystalline silicon studied by real time spectroscopic ellipsometry,” Solar Energy Materials and Solar Cells 78, 143–180 (2003). [CrossRef]

36.

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1998).

37.

S. J. Orfanidis, Electromagnetic waves and antennas (Rutgers University, 2002).

38.

L. T. Varghese, Y. Xuan, B. Niu, L. Fan, P. Bermel, and M. Qi, “Enhanced photon management of thin-film silicon solar cells using inverse opal photonic crystals with 3d photonic bandgaps,” Advanced Optical Materials 1, 692–698 (2013). [CrossRef]

39.

S. Wiesendanger, M. Zilk, T. Pertsch, F. Lederer, and C. Rockstuhl, “A path to implement optimized randomly textured surfaces for solar cells,” Applied Physics Letters 103, 131115 (2013). [CrossRef]

40.

M. J. Keevers, T. L. Young, U. Schubert, and M. A. Green, “10% efficient CSG minimodules,” Proceedings of the 22nd European Photovoltaic Solar Energy Conference and Exhibition, Milan (2007).

41.

Z. Yu, A. Raman, and S. Fan, “Nanophotonic light-trapping theory for solar cells,” Applied Physics A 105, 329–339 (2011). [CrossRef]

42.

Z. Yu, A. Raman, and S. Fan, “Thermodynamic upper bound on broadband light coupling with photonic structures,” Physical review letters 109, 173901 (2012). [CrossRef] [PubMed]

43.

J. L. Gray, X. Wang, X. Sun, and J. R. Wilcox, “Adept 2.0,” (2011).

44.

A. Bielawny, J. Üpping, P. T. Miclea, R. B. Wehrspohn, C. Rockstuhl, F. Lederer, M. Peters, L. Steidl, R. Zentel, S.-M. Lee, M. Knez, A. Lambertz, and R. Carius, “3d photonic crystal intermediate reflector for micromorph thin-film tandem solar cell,” physica status solidi (a) 205, 2796–2810 (2008). [CrossRef]

45.

J. Üpping, A. Bielawny, R. B. Wehrspohn, T. Beckers, R. Carius, U. Rau, S. Fahr, C. Rockstuhl, F. Lederer, M. Kroll, T. Pertsch, L. Steidl, and R. Zentel, “Three-dimensional photonic crystal intermediate reflectors for enhanced light-trapping in tandem solar cells,” Advanced Materials 23, 3896–3900 (2011). [CrossRef] [PubMed]

46.

D. Madzharov, R. Dewan, and D. Knipp, “Influence of front and back grating on light trapping in microcrystalline thin-film silicon solar cells,” Optics express 19, A95–A107 (2011). [CrossRef] [PubMed]

47.

H. Sai, H. Fujiwara, M. Kondo, and Y. Kanamori, “Enhancement of light trapping in thin-film hydrogenated microcrystalline si solar cells using back reflectors with self-ordered dimple pattern,” Applied Physics Letters 93, 143501 (2008). [CrossRef]

48.

K. Yamamoto, A. Nakajima, M. Yoshimi, T. Sawada, S. Fukuda, T. Suezaki, M. Ichikawa, Y. Koi, M. Goto, T. Meguro, T. Matsuda, M. Kondo, T. Sasaki, and Y. Tawada, “A high efficiency thin film silicon solar cell and module,” Solar Energy 77, 939–949 (2004). [CrossRef]

OCIS Codes
(350.6050) Other areas of optics : Solar energy
(350.4238) Other areas of optics : Nanophotonics and photonic crystals

ToC Category:
Light Trapping for Photovoltaics

History
Original Manuscript: March 4, 2014
Revised Manuscript: March 25, 2014
Manuscript Accepted: March 25, 2014
Published: April 10, 2014

Citation
H. Chung, K-Y. Jung, X. T. Tee, and P. Bermel, "Time domain simulation of tandem silicon solar cells with optimal textured light trapping enabled by the quadratic complex rational function," Opt. Express 22, A818-A832 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-S3-A818


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References

  1. R. Margolis, ed. SunShot vision study (U.S. Department of Energy, 2012).
  2. N.S. Lewis, “Toward Cost-Effective Solar Energy Use,” Science315, 798–801 (2007). [CrossRef] [PubMed]
  3. P. Bermel, C. Luo, L. Zeng, L. C. Kimerling, and J. D. Joannopoulos, “Improving thin-film crystalline silicon solar cell efficiencies with photonic crystals,” Optics Express15, 16986–17000 (2007). [CrossRef] [PubMed]
  4. A. G. Aberle, “Thin-film solar cells,” Thin Solid Films517, 4706–4710 (2009). [CrossRef]
  5. J. G. Mutitu, S. Shi, C. Chen, T. Creazzo, A. Barnett, C. Honsberg, and D. W. Prather, “Thin film solar cell design based on photonic crystal and diffractive grating structures,” Optics Express16, 15238–15248 (2008). [CrossRef] [PubMed]
  6. M.A. Green, K. Emery, Y. Hishikawa, W. Warta, and E.D. Dunlop, “Solar cell efficiency tables (version 43),” Prog. Photovolt.: Res. Appl.21, 1–9 (2013). [CrossRef]
  7. W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” Journal of applied physics32, 510–519 (1961). [CrossRef]
  8. A. De Vos, “Detailed balance limit of the efficiency of tandem solar cells,” J. Phys. D13, 839–845 (1980).
  9. O. D. Miller, E. Yablonovitch, and S. R. Kurtz, “Strong internal and external luminescence as solar cells approach the shockley-queisser limit,” IEEE J. Photovolt.2, 303–311 (2012). [CrossRef]
  10. M. Berginski, J. Hupkes, M. Schulte, G. Schope, H. Stiebig, B. Rech, and M. Wuttig, “The effect of front zno: Al surface texture and optical transparency on efficient light trapping in silicon thin-film solar cells,” Journal of Applied Physics101, 074903 (2007). [CrossRef]
  11. R. Brendel, M. Hirsch, R. Plieninger, and J. Werner, “Quantum efficiency analysis of thin-layer silicon solar cells with back surface fields and optical confinement,” IEEE Transactions on Electron Devices43, 1104–1113 (1996). [CrossRef]
  12. T. Tiedje, E. Yablonovitch, G. D. Cody, and B. G. Brooks, “Limiting efficiency of silicon solar cells,” IEEE Transactions on Electron Devices31, 711–716 (1984). [CrossRef]
  13. E. Yablonovitch, “Statistical ray optics,” JOSA72, 899–907 (1982). [CrossRef]
  14. M. Ghebrebrhan, P. Bermel, Y. Avniel, J. D. Joannopoulos, and S. G. Johnson, “Global optimization of silicon photovoltaic cell front coatings,” Optics express17, 7505–7518 (2009). [CrossRef] [PubMed]
  15. J. Zhao, A. Wang, M. A. Green, and F. Ferrazza, “19.8% efficient honeycomb textured multicrystalline and 24.4% monocrystalline silicon solar cells,” Applied Physics Letters73, 1991 (1998). [CrossRef]
  16. R. Dewan, I. Vasilev, V. Jovanov, and D. Knipp, “Optical enhancement and losses of pyramid textured thin-film silicon solar cells,” Journal of Applied Physics110, 013101 (2011). [CrossRef]
  17. C. L. Tan, A. Karar, K. Alameh, and Y. T. Lee, “Optical absorption enhancement of hybrid-plasmonic-based metal-semiconductor-metal photodetector incorporating metal nanogratings and embedded metal nanoparticles,” Optics express21, 1713–1725 (2013). [CrossRef] [PubMed]
  18. C. Rockstuhl, S. Fahr, K. Bittkau, T. Beckers, R. Carius, F.-J. Haug, T. Söderström, C. Ballif, and F. Lederer, “Comparison and optimization of randomly textured surfaces in thin-film solar cells,” Optics express18, A335–A341 (2010). [CrossRef] [PubMed]
  19. S.-S. Lo, C.-C. Chen, F. Garwe, and T. Pertch, “Broad-band anti-reflection coupler for a: Si thin-film solar cell,” Journal of Physics D: Applied Physics40, 754 (2007). [CrossRef]
  20. J. Lacombe, O. Sergeev, K. Chakanga, K. von Maydell, and C. Agert, “Three dimensional optical modeling of amorphous silicon thin film solar cells using the finite-difference time-domain method including real randomly surface topographies,” Journal of Applied Physics110, 023102 (2011). [CrossRef]
  21. Y.-C. Tsao, C. Fisker, and T. Garm Pedersen, “Optical absorption of amorphous silicon on anodized aluminum substrates for solar cell applications,” Optics Communications315, 17–25 (2014). [CrossRef]
  22. V. Jovanov, U. Palanchoke, P. Magnus, H. Stiebig, J. Hüpkes, P. Sichanugrist, M. Konagai, S. Wiesendanger, C. Rockstuhl, and D. Knipp, “Light trapping in periodically textured amorphous silicon thin film solar cells using realistic interface morphologies,” Optics Express21, A595–A606 (2013). [CrossRef] [PubMed]
  23. A. Tavlove and S. C. Hagness, “Computational electrodynamics: the finite-difference time-domain method,” 2, Artech House, (Artech House, 2000).
  24. W. C. Chew, Waves and fields in inhomogenous media (Van Nostrand Reinhold, 1990).
  25. G. Jellison and F. Modine, “Parameterization of the optical functions of amorphous materials in the interband region,” Applied Physics Letters69, 371–373 (1996). [CrossRef]
  26. A. Fantoni and P. Pinho, “FDTD simulation of light propagation inside a-si: H structures,” in “ MRS Proceedings (Cambridge University, 2010). [CrossRef]
  27. S.-G. Ha, J. Cho, J. Choi, H. Kim, and K.-Y. Jung, “FDTD dispersive modeling of human tissues based on quadratic complex rational function,” IEEE Transactions on Antennas and Propagation61, 996–999 (2013). [CrossRef]
  28. H. Chung, J. Cho, S.-G. Ha, S. Ju, and K.-Y. Jung, “Accurate FDTD dispersive modeling for concrete materials.” ETRI Journal35, 915–918, (2013). [CrossRef]
  29. H. Chung, S.-G. Ha, J. Cho, and K.-Y. Jung, “Accurate FDTD modeling for dispersive media using rational function and particle swarm optimization.” International Journal of Electronics to be published (2014).
  30. K. Yamamoto, A. Nakajima, M. Yoshimi, T. Sawada, S. Fukuda, T. Suezaki, M. Ichikawa, Y. Koi, M. Goto, H. Takata, T. Sasaki, and Y. Tawada, “Novel hybrid thin film silicon solar cell and module,” in Proceedings of 3rd World Conference on Photovoltaic Energy Conversion (IEEE, 2003), vol. 3, pp. 2789–2792.
  31. F. L. Teixeira, “Time-domain finite-difference and finite-element methods for maxwell equations in complex media,” IEEE Transactions on Antennas and Propagation56, 2150–2166 (2008). [CrossRef]
  32. C. Herzinger, B. Johs, W. McGahan, J. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” Journal of Applied Physics83, 3323–3336 (1998). [CrossRef]
  33. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the fdtd method,” Computer Physics Communications181, 687–702 (2010). [CrossRef]
  34. E. L. Haines and A. B. Whitehead, “Pulse height defect and energy dispersion in semiconductor detectors,” Review of Scientific Instruments37, 190–194 (1966). [CrossRef]
  35. R. Collins, A. Ferlauto, G. Ferreira, C. Chen, J. Koh, R. Koval, Y. Lee, J. Pearce, and C. Wronski, “Evolution of microstructure and phase in amorphous, protocrystalline, and microcrystalline silicon studied by real time spectroscopic ellipsometry,” Solar Energy Materials and Solar Cells78, 143–180 (2003). [CrossRef]
  36. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1998).
  37. S. J. Orfanidis, Electromagnetic waves and antennas (Rutgers University, 2002).
  38. L. T. Varghese, Y. Xuan, B. Niu, L. Fan, P. Bermel, and M. Qi, “Enhanced photon management of thin-film silicon solar cells using inverse opal photonic crystals with 3d photonic bandgaps,” Advanced Optical Materials1, 692–698 (2013). [CrossRef]
  39. S. Wiesendanger, M. Zilk, T. Pertsch, F. Lederer, and C. Rockstuhl, “A path to implement optimized randomly textured surfaces for solar cells,” Applied Physics Letters103, 131115 (2013). [CrossRef]
  40. M. J. Keevers, T. L. Young, U. Schubert, and M. A. Green, “10% efficient CSG minimodules,” Proceedings of the 22nd European Photovoltaic Solar Energy Conference and Exhibition, Milan (2007).
  41. Z. Yu, A. Raman, and S. Fan, “Nanophotonic light-trapping theory for solar cells,” Applied Physics A105, 329–339 (2011). [CrossRef]
  42. Z. Yu, A. Raman, and S. Fan, “Thermodynamic upper bound on broadband light coupling with photonic structures,” Physical review letters109, 173901 (2012). [CrossRef] [PubMed]
  43. J. L. Gray, X. Wang, X. Sun, and J. R. Wilcox, “Adept 2.0,” (2011).
  44. A. Bielawny, J. Üpping, P. T. Miclea, R. B. Wehrspohn, C. Rockstuhl, F. Lederer, M. Peters, L. Steidl, R. Zentel, S.-M. Lee, M. Knez, A. Lambertz, and R. Carius, “3d photonic crystal intermediate reflector for micromorph thin-film tandem solar cell,” physica status solidi (a)205, 2796–2810 (2008). [CrossRef]
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