## Time domain simulation of tandem silicon solar cells with optimal textured light trapping enabled by the quadratic complex rational function |

Optics Express, Vol. 22, Issue S3, pp. A818-A832 (2014)

http://dx.doi.org/10.1364/OE.22.00A818

Acrobat PDF (8297 KB)

### 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/cm^{2} in a 2 *μ*m-thick active layer.

© 2014 Optical Society of America

## 1. Introduction

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]

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]

_{1}= 1.72eV) and c-Si (Eg

_{2}= 1.11eV) [8,9

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]

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]

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]

*n*

^{2}[13

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]

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]

## 2. Dispersion modeling and validation of its accuracy

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. 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]

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]

*ω*is the optical frequency, and

*A*

_{0},

*A*

_{1},

*A*

_{2},

*B*

_{1}, and

*B*

_{2}are adjustable parameters.

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]

### 2.1. Dispersion modeling

*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]

### 2.2. Theoretical absorption and simulation result

*ρ*

_{1}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,

*τ*

_{1}is the transmission at the left boundary when traveling to the right,

*τ′*

_{1}is the same coefficient traveling in the reverse direction,

*ρ*

_{2}is the reflection coefficient at the right boundary of the dielectric slab when traveling to the right,

*ω*= 2

*πc/λ*is the optical frequency, and

*t*is time required for light travel through the certain thickness of dielectric material. Note that

*τ*= 1 +

_{k}*ρ*and

_{k}*τ′*= 1 +

_{k}*ρ′*for all integer

_{k}*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*).

*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.

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

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]

*μ*m 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.

## 3. Statistical random surface texturing model

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]

*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 (

*J*) which serves as a proxy for the efficiency of the photovoltaic cells.

_{sc}*i*represents the

*x*index,

*j*represents the y index,

*N*is the maximum index of

_{i}*i*,

*N*is the maximum index of

_{j}*j*and

*w*() denotes a 2-D weighting function: The textured features of the 2-D surface are created in a similar fashion to those in the original 1-D texturing algorithm. As depicted in Fig. 6, Eq. (7) introduces correlations across the 2-D surface, which reflects the limited aspect ratios associated with random texturing methods. The maximum texturing height of the random surfaces is controlled simply by rescaling the standard deviation (

*σ*) of the Gaussian distribution. The application of long-range periodicity enhances diffraction into guided modes and saves a tremendous amount of computational cost, since simulating a single segment of a periodic structure can closely approximate the entire architecture of a large 3-D solar cell.

*N*times the ratio of the y-grid values in absolute units (e.g., in nm) as shown in the equation below: where Δ

*y*2

*D*and Δ

*y*3

*D*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.

*f*= 0.99 matches well with the structure introduced in the reference paper [40]. 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

^{2}at the a-Si layer and 12.88 mA/cm

^{2}, 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.

*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-SiO

_{2}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. 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]

*single*calculation over the

*entire*relevant portion of the solar spectrum for a given geometry.

## 5. Conclusion

## Acknowledgments

## References and links

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2. | N.S. Lewis, “Toward Cost-Effective Solar Energy Use,” Science |

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 |

4. | A. G. Aberle, “Thin-film solar cells,” Thin Solid Films |

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 |

6. | M.A. Green, K. Emery, Y. Hishikawa, W. Warta, and E.D. Dunlop, “Solar cell efficiency tables (version 43),” Prog. Photovolt.: Res. Appl. |

7. | W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” Journal of applied physics |

8. | A. De Vos, “Detailed balance limit of the efficiency of tandem solar cells,” J. Phys. D |

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. |

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 |

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 |

12. | T. Tiedje, E. Yablonovitch, G. D. Cody, and B. G. Brooks, “Limiting efficiency of silicon solar cells,” IEEE Transactions on Electron Devices |

13. | E. Yablonovitch, “Statistical ray optics,” JOSA |

14. | M. Ghebrebrhan, P. Bermel, Y. Avniel, J. D. Joannopoulos, and S. G. Johnson, “Global optimization of silicon photovoltaic cell front coatings,” Optics express |

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 |

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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 |

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 |

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34. | E. L. Haines and A. B. Whitehead, “Pulse height defect and energy dispersion in semiconductor detectors,” Review of Scientific Instruments |

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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) |

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 |

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 |

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 |

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 |

**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

- R. Margolis, ed. SunShot vision study (U.S. Department of Energy, 2012).
- N.S. Lewis, “Toward Cost-Effective Solar Energy Use,” Science315, 798–801 (2007). [CrossRef] [PubMed]
- 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]
- A. G. Aberle, “Thin-film solar cells,” Thin Solid Films517, 4706–4710 (2009). [CrossRef]
- 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]
- 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]
- 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]
- A. De Vos, “Detailed balance limit of the efficiency of tandem solar cells,” J. Phys. D13, 839–845 (1980).
- 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]
- 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]
- 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]
- 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]
- E. Yablonovitch, “Statistical ray optics,” JOSA72, 899–907 (1982). [CrossRef]
- 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]
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