## Designing optimized nano textures for thin-film silicon solar cells |

Optics Express, Vol. 21, Issue S4, pp. A656-A668 (2013)

http://dx.doi.org/10.1364/OE.21.00A656

Acrobat PDF (1765 KB)

### Abstract

Thin-film silicon solar cells (TFSSC), which can be manufactured from abundant materials solely, contain nano-textured interfaces that scatter the incident light. We present an approximate very fast algorithm that allows optimizing the surface morphology of two-dimensional nano-textured interfaces. Optimized nano-textures scatter the light incident on the solar cell stronger leading to a higher short-circuit current density and thus efficiency. Our algorithm combines a recently developed scattering model based on the scalar scattering theory, the Perlin-noise algorithm to generate the nano textures and the simulated annealing algorithm as optimization tool. The results presented in this letter allow to push the efficiency of TFSSC towards their theoretical limit.

© 2013 OSA

## 1. Introduction

3. H. W. Deckman, C. R. Wronski, H. Witzke, and E. Yablonovitch, “Optically enhanced amorphous silicon solar cells,” Appl. Phys. Lett. **42**, 968–970 (1983) [CrossRef] .

4. C. Battaglia, J. Escarré, K. Söderström, M. Charrière, M. Despeisse, F. Haug, and C. Ballif, “Nanomoulding of transparent zinc oxide electrodes for efficient light trapping in solar cells,” Nat. Photonics **5**, 535–538 (2011) [CrossRef] .

## 2. Theory

### 2.1. The scalar scattering model

### 2.2. Generating nano-textures with the Perlin noise algorithm

23. K. Perlin, “An image synthesizer,” SIGGRAPH Comput. Graph. **19**, 287–296 (1985) [CrossRef] .

24. K. Perlin, “Better acting in computer games: the use of procedural methods,” Comput. Graph. **26**, 3–11 (2002) [CrossRef] .

*ℓ*of 1 on a square with side length 10, we have to assign a random number

*z*(

*x*,

*y*) between 0 and 1 to every point of the square with integer coordinates,

*e.g.*(3,5). The values of the points in between these integer coordinates are then given as an interpolation of the neighboring integer coordinates. We used a cosine interpolation in order to assure that the first derivatives of the surface are continuous. Also other interpolations, like linear interpolations could be used, however they would not alter the final result of this work. In the cosine interpolation, the points in the square enclosed by the square (0,0), (0,1), (1,0), (1,1) are given by where The signs before the cosines in

*c*(

*x*) and

*f*(

*x*) are determined by the requirement that the first derivatives on the corners of the square are zero.

*ℓ*is one sixteenth of the side length of the square. (The lateral feature size

*ℓ*is not to be confused with the correlation length.) In this and all subsequent textures the average plane was subtracted. The points in between were interpolated with cosine functions as in Eqs. (1) and (2). When we superpose different

*generations*of Perlin noise with lateral feature size 1, ½, ¼... with respect to the side length, we can generate a fractal surface texture that looks like “cloudy sky”, as illustrated in Fig. 1(b). We just have to assure that the rms roughness

*σ*of each generation scales with its feature size,

_{r}*i.e.*where the subscript denotes the inverse of the feature size as a fraction of the side length. The surfaces in Fig. 1 have no length or height scales because at this point, the scales are completely arbitrary. They only need to be scaled later at a later stage, when they are used for simulating scattering of light.

### 2.3. The simulated annealing algorithm

25. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science **220**, 671–680 (1983) [CrossRef] [PubMed] .

26. V. Černý, “Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm,” J. Optimiz. Theory App. **45**, 41–51 (1985) [CrossRef] .

27. N. Metropolis and S. Ulam, “The Monte Carlo Method,” J. Amer. Stat. Assoc. **44**, 335–341 (1949) [CrossRef] .

**c**, via a

*cost function C*that is minimized during the optimization. An optimized set of system parameters is found by “cooling” the system just as a molten metal crystallizes to configurations of lowest potential energy when it is cooled. By cooling the system fast, small crystals will emerge. Their energy will be slightly higher than that of the large crystals emerging when the system is cooled down slowly. Thus there is a trade-off between the cooling rate and the quality of the optimization. In detail the optimization is done as follows:

**c**

_{0}needs to be chosen, usually with random numbers, and a starting temperature

*T*

_{0}needs to be set. At the beginning of every optimization step one system parameter is chosen randomly that is slightly varied (or “tweaked”) depending on a random number. Next, the cost

*C*is calculated with the changed parameter set

_{i}**c**

*. If*

_{i}*C*<

_{i}*C*, where the subscript a denotes the last accepted set of parameters,

_{a}**c**

*is accepted as new parameter set,*

_{i}**c**

*=*

_{a}**c**

*. Else,*

_{i}*C*is accepted according to the probability This probability is analogous to the Maxwell-Boltzmann distribution exp(−

_{i}*E/kT*). At the end of each step, the system is cooled with a constant factor

*d*,

*T*

_{i}_{+1}=

*dT*. Clearly, the smaller

_{i}*d*the faster the system will cool. Due to this cooling,

*p*decreases as the simulation progresses,

*i.e.*it becomes more unlikely that the parameter set

**c**

*is accepted if*

_{i}*C*>

_{i}*C*. If

_{a}*T*≡ 0 throughout the simulation,

*p*will always be 0. Such a simulation is called

*greedy*.

### 2.4. Optimizing nano-textures

**c**= (

*c*

_{1},

*c*

_{2},

*c*

_{4}...) instead of the fractal set (1, ½, ¼...). To obtain results that are independent of a single generated texture, we performed the optimizations for ten Perlin textures simultaneously. At the beginning of every optimization run we made ten Perlin textures of every generation and we randomly chose an initial set of coefficients

**c**

_{0}. Thus, we had ten different textures generated with the same set of coefficients, where the counter

*j*∈ {1, 2, 3...10}. The total cost then is given as the average cost of the ten textures. We used different cost functions, which are discussed in detail in the next section.

*random search*method. In that method the set of coefficients

**c**is kept constant throughout a simulation run. However, for every optimization step a new set of textures

## 3. Optimization results

21. K. Jäger, M. Fischer, R. A. C. M. M. van Swaaij, and M. Zeman, “A scattering model for nano-textured interfaces and its application in opto-electrical simulations of thin-film silicon solar cells,” J. Appl. Phys. **111**, 083108 (2012) [CrossRef] .

*σ*≈ 40 nm, as illustrated on the right of Fig. 2(a). During this and all subsequent optimizations, the rms roughness of the texture was kept constant at 40 nm.

_{r}*i.e.*the light is scattered into much smaller angles. Therefore also the (simulated) EQE is smaller than that of the reference sample. The EQE was simulated with the

`ASA`software [21

21. K. Jäger, M. Fischer, R. A. C. M. M. van Swaaij, and M. Zeman, “A scattering model for nano-textured interfaces and its application in opto-electrical simulations of thin-film silicon solar cells,” J. Appl. Phys. **111**, 083108 (2012) [CrossRef] .

`ASA`simulations we used the same electrical parameters, optical constants and layer thicknesses. We assumed one scattering layer between the TCO and the silicon, as illustrated on the right of Fig. 2(c). The haze and AID of this layer were calculated with a scattering model publisher by us earlier [21

21. K. Jäger, M. Fischer, R. A. C. M. M. van Swaaij, and M. Zeman, “A scattering model for nano-textured interfaces and its application in opto-electrical simulations of thin-film silicon solar cells,” J. Appl. Phys. **111**, 083108 (2012) [CrossRef] .

**c**. We performed optimizations for TCO-air and TCO-silicon interfaces.

*C*

_{tot}=

*C*(

*λ*

_{1}) +

*C*(

*λ*

_{2}) + ...

*σ*of the texture, at least for values between 40 and 200 nm.

_{r}*α*(

*λ*) is the the absorption coefficient of a-Si:H and

_{i}*d*is the thickness of the layer, in our case

*d*= 300 nm.

*λ*− Δ

_{i}*λ*/2,

*λ*+ Δ

_{i}*λ*/2) and Δ

*λ*=

*λ*

_{i}_{+1}−

*λ*. The multiplication with

_{i}*λ*is done since we want to maximize the number of absorbed photons.

_{i}*n*

_{Si}shorter than in air, where the refractive index in Si

*n*

_{Si}is about 4. Table 1 summarizes the optimal lateral feature size for TCO-air and TCO-silicon systems.

## 4. Parameter study

*ℓ*and

*σ*on the scattering parameters and the performance of solar cells in more detail. Figures 3(a) and 4(a) show the haze and AID of Perlin textures with different lateral feature sizes

_{r}*ℓ*at TCO-air interfaces. The rms roughness of the Perlin textures was constant,

*σ*= 40 nm. To obtain optimized textures, a short random search was performed for every value of

_{r}*ℓ*. While the haze increases slightly with

*ℓ*, the AID decays much faster. We therefore have to take the trade-off between increasing haze and faster decaying AID into account. At very small lateral feature sizes the haze decays because the light does not see a nano texture anymore but experiences the surface as an effective medium [18

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,” Opt. Express **18**, A335–A341 (2010) [CrossRef] [PubMed] .

*ℓ*when

*σ*is kept constant, we also investigated the influence of a changing

_{r}*σ*when

_{r}*ℓ*is constant in Figs. 3(b) and 4(b). As expected, the haze reacts strongly upon changing

*σ*. However, the AID

_{r}*does not change the shape at all, it merely shifts towards higher intensities.*

_{T}*is controlled by lateral features while the vertical features (and especially*

_{T}*σ*) control the haze. However, larger values of

_{r}*ℓ*are beneficial for the haze as well.

*ℓ*size but a high

*σ*(

_{r}*i.e.*textures with sharp spikes) are very interesting from an optical point of view but have a detrimental effect on the electrical properties of the solar cell [4

4. C. Battaglia, J. Escarré, K. Söderström, M. Charrière, M. Despeisse, F. Haug, and C. Ballif, “Nanomoulding of transparent zinc oxide electrodes for efficient light trapping in solar cells,” Nat. Photonics **5**, 535–538 (2011) [CrossRef] .

29. H. Sakai, T. Yoshida, T. Hama, and Y. Ichikawa, “Effects of surface morphology of transparent electrode on the open-circuit voltage in a-Si:H solar cells,” Jpn. J. Appl. Phys. **29**, 630–635 (1990) [CrossRef] .

30. M. Python, D. Dominé, T. Söderström, F. Meillaud, and C. Ballif, “Microcrystalline silicon solar cells: effect of substrate temperature on cracks and their role in post-oxidation,” Prog. Photovolt: Res. Appl. **18**, 491–499 (2010) [CrossRef] .

*e.g.*by Isabella

*et al.*[31

31. O. Isabella, J. Krč, and M. Zeman, “Modulated surface textures for enhanced light trapping in thin-film silicon solar cells,” Appl. Phys. Lett. **97**, 101106 (2010) [CrossRef] .

32. O. Isabella, F. Moll, J. Krč, and M. Zeman, “Modulated surface textures using zinc-oxide films for solar cells applications,” Phys. Status Solidi A **207**, 642–646 (2010) [CrossRef] .

*σ*and

_{r}*ℓ*are superposed with textures with high

*σ*and

_{r}*ℓ*. With this approach high

*σ*values and small lateral features can be combined without the creation of sharp spikes. Figures 3(c) and 4(c) show the haze and AID

_{r}*of textures that were created by superposing the texture with*

_{T}*ℓ*= 312 nm and

*σ*= 40 nm with a texture with

_{r}*ℓ*= 1250 nm such that the total

*σ*value is 80, 120 and 160 nm, respectively. We observe that the haze of these structures is higher than that of the

_{r}*ℓ*= 312 nm structures shown in Fig. 3(b). However, the AID

*is only higher at narrow angles; at large angles the superposed texture shows hardly any effect. This shows that MST indeed can have a beneficial effect on the haze and the AID*

_{T}*at small angles, however, they do not improve scattering into large angles.*

_{T}*ℓ*and

*σ*= 40 nm. According to Table 1, the optimal lateral feature size is 78 nm. There seems to be hardly any difference between 78 and 156 nm feature size. The simulated EQE of a solar cell with Asahi-U [see Fig. 2(c)] would be nearly identical to that of the 156 nm Perlin texture, which can be understood from the fact that the Asahi-U correlation length of 175 nm is close to this value. Thus, the lateral dimensions of Asahi-U are very optimized.

_{r}*σ*constant. The calculated EQE is highest for the cell with the pure 78 nm texture and lowest for cell with the pure 312 nm texture. The EQE of the cell with a superposed texture lies in between those values, confirming the optimization results that a texture with one optimized lateral feature size is superior to a texture consisting of a superposition of textures with different

_{r}*ℓ*values.

*ℓ*and

*σ*= 80 nm. Here the 78 nm texture is slightly superior to the 156 nm texture. In an experiment, however, the sharper spikes of the 78 nm texture could induce higher electrical losses making the 156 nm texture more optimal in the end.

_{r}*ℓ*= 39 nm nano texture show the highest current density (

*J*=14.9 mA/cm

_{sc}^{2}), which decreases with increasing lateral feature size.

## 5. Discussion

*et al.*found that for one-dimensional rectangular gratings a period (height) of 300 nm (300 nm) is optimal [34

34. A. Čampa, O. Isabella, R. van Erven, P. Peeters, H. Borg, J. Krč, M. Topič, and M. Zeman, “Optimal design of periodic surface texture for thin-film a-Si:H solar cells,” Prog. Photovolt: Res. Appl. **18**, 160–167 (2010) [CrossRef] .

*et al.*reported an optimal period (height) of 400 nm (300 nm) for one-dimensional rectangular gratings and 500 nm (450 nm) for two-dimensional gratings [20

20. O. Isabella, S. Solntsev, D. Caratelli, and M. Zeman, “3-D optical modeling of thin-film silicon solar cells on diffraction gratings,” Prog. Photovolt: Res. Appl. **21**, 94–108 (2013) [CrossRef] .

*et al.*and Isabella

*et al.*To resolve this discrepancy we take another look at Perlin textures,

*e.g.*in Fig. 6(a). We observe that most features are clustered in small groups of similar height that we call grains. Two of these grains are indicated in Fig. 6(a). The scattering is mainly controlled by the grains and not by the small features that build up the grains.

*threshold*algorithm: The points with a height above 50% of the maximal height of the texture belong to grains. In (c), the

*watershed*algorithm was used. In this algorithm, a drop of water is placed at every point of the surface. Then the drops will flow together at the local minima (in this case the tops) and form little lakes – the grains. Both algorithms are described by Klapetek [35, 36

36. P. Klapetek, D. Nečas, and C. Anderson, Gwyddion user guide (2012). http://www.gwyddion.net.

*r*that a disk with the area of the grain would have.

*r̄*for the textures with different

*ℓ*. We see that the differences between

*r̄*obtained by the two methods are very small. Roughly speaking,

*r̄*≈ 2

*ℓ*. As we see from Fig. 6(c) the texture can be interpreted as a

*randomized*two-dimensional grating with period 2

*r̄*, which is approximately 4

*ℓ*. The optimal feature sizes between 78 and 156 nm therefore correspond to periods between 312 and 625 nm, which is in agreement with the findings by Čampa

*et al.*[34

34. A. Čampa, O. Isabella, R. van Erven, P. Peeters, H. Borg, J. Krč, M. Topič, and M. Zeman, “Optimal design of periodic surface texture for thin-film a-Si:H solar cells,” Prog. Photovolt: Res. Appl. **18**, 160–167 (2010) [CrossRef] .

*et al.*[20

20. O. Isabella, S. Solntsev, D. Caratelli, and M. Zeman, “3-D optical modeling of thin-film silicon solar cells on diffraction gratings,” Prog. Photovolt: Res. Appl. **21**, 94–108 (2013) [CrossRef] .

## 6. Conclusions

*ℓ*that were generated with the Perlin noise algorithm. An optimization performed with the simulated annealing algorithm revealed that textures with one optimized

*ℓ*have a broader AID than textures made up from superpositions of different

*ℓ*, if the rms roughness

*σ*is kept constant. These results were confirmed by

_{r}`ASA`simulations of the EQE and short circuit current. Further, the haze increases if

*σ*increases. However, a combination of optimized

_{r}*ℓ*and high

*σ*may lead to sharp spikes that deteriorate the solar cell. Modulated surface textures are a way to overcome this problem by combining textures with large

_{r}*ℓ*and

*σ*with textures with optimized small

_{r}*ℓ*. Such textures have a high haze and strong forward scattering. However, scattering into large angles is mainly controlled by the small features and cannot be increased with the MST concept.

## Acknowledgment

## References and links

1. | C. Winneker, ed., |

2. | M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 40),” Prog. Photovolt: Res. Appl. |

3. | H. W. Deckman, C. R. Wronski, H. Witzke, and E. Yablonovitch, “Optically enhanced amorphous silicon solar cells,” Appl. Phys. Lett. |

4. | C. Battaglia, J. Escarré, K. Söderström, M. Charrière, M. Despeisse, F. Haug, and C. Ballif, “Nanomoulding of transparent zinc oxide electrodes for efficient light trapping in solar cells,” Nat. Photonics |

5. | M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. |

6. | M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. |

7. | M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A |

8. | J. Chandezon, G. Raoult, and D. Maystre, “A new theoretical method for diffraction gratings and its numerical application,” J. Optics |

9. | J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. |

10. | S. Fahr, C. Rockstuhl, and F. Lederer, “Engineering the randomness for enhanced absorption in solar cells,” Appl. Phys. Lett. |

11. | R. Dewan, V. Jovanov, C. Haase, H. Stiebig, and D. Knipp, “Simple and fast method to optimize nanotextured interfaces of thin-film silicon solar cells,” Appl. Phys. Express |

12. | M. Peters, M. Rüdiger, H. Hauser, M. Hermle, and B. Bläsi, “Diffractive gratings for crystalline silicon solar cells — optimum parameters and loss mechanisms,” Prog. Photovolt: Res. Appl. |

13. | K. Bittkau and T. Beckers, “Near-field study of light scattering at rough interfaces of a-Si:H/μc-Si:H tandem solar cells,” Phys. Status Solidi A |

14. | E. R. Martins, J. Li, Y. Liu, J. Zhou, and T. F. Krauss, “Engineering gratings for light trapping in photovoltaics: the supercell concept,” Phys. Rev. B |

15. | K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE T. Antenn. Propag. |

16. | A. Taflove and S. C. Hagness, |

17. | J. Jin, |

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,” Opt. Express |

19. | R. Dewan, I. Vasilev, V. Jovanov, and D. Knipp, “Optical enhancement and losses of pyramid textured thin-film silicon solar cells,” J. Appl. Phys. |

20. | O. Isabella, S. Solntsev, D. Caratelli, and M. Zeman, “3-D optical modeling of thin-film silicon solar cells on diffraction gratings,” Prog. Photovolt: Res. Appl. |

21. | K. Jäger, M. Fischer, R. A. C. M. M. van Swaaij, and M. Zeman, “A scattering model for nano-textured interfaces and its application in opto-electrical simulations of thin-film silicon solar cells,” J. Appl. Phys. |

22. | K. Jäger, O. Isabella, R. A. C. M. M. van Swaaij, and M. Zeman, “Angular resolved scattering measurements of nano-textured substrates in a broad wavelength range,” Meas. Sci. Technol. |

23. | K. Perlin, “An image synthesizer,” SIGGRAPH Comput. Graph. |

24. | K. Perlin, “Better acting in computer games: the use of procedural methods,” Comput. Graph. |

25. | S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science |

26. | V. Černý, “Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm,” J. Optimiz. Theory App. |

27. | N. Metropolis and S. Ulam, “The Monte Carlo Method,” J. Amer. Stat. Assoc. |

28. | K. Sato, Y. Gotoh, Y. Wakayama, Y. Hayashi, K. Adachi, and N. Nishimura, “Highly Textured SnO |

29. | H. Sakai, T. Yoshida, T. Hama, and Y. Ichikawa, “Effects of surface morphology of transparent electrode on the open-circuit voltage in a-Si:H solar cells,” Jpn. J. Appl. Phys. |

30. | M. Python, D. Dominé, T. Söderström, F. Meillaud, and C. Ballif, “Microcrystalline silicon solar cells: effect of substrate temperature on cracks and their role in post-oxidation,” Prog. Photovolt: Res. Appl. |

31. | O. Isabella, J. Krč, and M. Zeman, “Modulated surface textures for enhanced light trapping in thin-film silicon solar cells,” Appl. Phys. Lett. |

32. | O. Isabella, F. Moll, J. Krč, and M. Zeman, “Modulated surface textures using zinc-oxide films for solar cells applications,” Phys. Status Solidi A |

33. | M. Boccard, C. Battaglia, S. Hänni, K. Söderström, J. Escarré, S. Nicolay, F. Meillaud, M. Despeisse, and C. Ballif, “Multiscale transparent electrode architecture for efficient light management and carrier collection in solar cells,” Nano Lett. |

34. | A. Čampa, O. Isabella, R. van Erven, P. Peeters, H. Borg, J. Krč, M. Topič, and M. Zeman, “Optimal design of periodic surface texture for thin-film a-Si:H solar cells,” Prog. Photovolt: Res. Appl. |

35. | P. Klapetek, “Characterization of randomly rough surfaces in nanometric scale using methods of modern metrology,” Ph.D. thesis, Masaryk University, Brno, Czech Republic (2003). |

36. | P. Klapetek, D. Nečas, and C. Anderson, Gwyddion user guide (2012). http://www.gwyddion.net. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(290.5880) Scattering : Scattering, rough surfaces

(350.6050) Other areas of optics : Solar energy

(310.7005) Thin films : Transparent conductive coatings

**ToC Category:**

Photovoltaics

**History**

Original Manuscript: February 25, 2013

Revised Manuscript: April 12, 2013

Manuscript Accepted: April 15, 2013

Published: May 24, 2013

**Citation**

Klaus Jäger, Marinus Fischer, René A.C.M.M. van Swaaij, and Miro Zeman, "Designing optimized nano textures for thin-film silicon solar cells," Opt. Express **21**, A656-A668 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-S4-A656

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### References

- C. Winneker, ed., Global Market Outlook (European Photovoltaic Industry Association, May2013).
- M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 40),” Prog. Photovolt: Res. Appl.20, 606–614 (2012). [CrossRef]
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- C. Battaglia, J. Escarré, K. Söderström, M. Charrière, M. Despeisse, F. Haug, and C. Ballif, “Nanomoulding of transparent zinc oxide electrodes for efficient light trapping in solar cells,” Nat. Photonics5, 535–538 (2011). [CrossRef]
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- M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A12, 1077–1085 (1995). [CrossRef]
- J. Chandezon, G. Raoult, and D. Maystre, “A new theoretical method for diffraction gratings and its numerical application,” J. Optics11, 235–241 (1980). [CrossRef]
- J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am.72, 839–846 (1982). [CrossRef]
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- M. Peters, M. Rüdiger, H. Hauser, M. Hermle, and B. Bläsi, “Diffractive gratings for crystalline silicon solar cells — optimum parameters and loss mechanisms,” Prog. Photovolt: Res. Appl.20, 862–873 (2012). [CrossRef]
- K. Bittkau and T. Beckers, “Near-field study of light scattering at rough interfaces of a-Si:H/μc-Si:H tandem solar cells,” Phys. Status Solidi A207, 661–666 (2010). [CrossRef]
- E. R. Martins, J. Li, Y. Liu, J. Zhou, and T. F. Krauss, “Engineering gratings for light trapping in photovoltaics: the supercell concept,” Phys. Rev. B86, 041404 (2012). [CrossRef]
- K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE T. Antenn. Propag.14, 302–307 (1966). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
- J. Jin, The finite element method in electromagnetics(John Wiley & Sons, 2002).
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