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  • Editor: Alan E. Willner
  • Vol. 37, Iss. 23 — Dec. 1, 2012
  • pp: 4868–4870
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Engineering Gaussian disorder at rough interfaces for light trapping in thin-film solar cells

Piotr Kowalczewski, Marco Liscidini, and Lucio Claudio Andreani  »View Author Affiliations


Optics Letters, Vol. 37, Issue 23, pp. 4868-4870 (2012)
http://dx.doi.org/10.1364/OL.37.004868


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Abstract

A theoretical study of randomly rough interfaces to obtain light trapping in thin-film silicon solar cells is presented. Roughness is modeled as a surface with Gaussian disorder, described using the root mean square of height and the lateral correlation length as statistical parameters. The model is shown to describe commonly used rough substrates. Rigorous calculations, with short-circuit current density as a figure of merit, lead to an optimization of disorder parameters and to a significant absorption enhancement. The understanding and optimization of disorder is believed to be of general interest for various realizations of thin-film solar cells.

© 2012 Optical Society of America

Reducing the thickness of a solar cell active layer improves quality of the film and decreases material consumption [1

1. R. Brendel, Thin-Film Crystalline Silicon Solar Cells(Wiley-VCH, 2003).

], but the device performance deteriorates due to poor absorption. For this reason, increasing optical thickness of the device by light trapping [2

2. E. Yablonovitch, J. Opt. Soc. Am. 72, 899 (1982). [CrossRef]

] is a fundamental issue in thin-film solar-cell design. A particularly interesting approach to obtain light trapping is the use of properly optimized rough interfaces [3

3. S. Fahr, C. Rockstuhl, and F. Lederer, Appl. Phys. Lett. 92, 171114 (2008). [CrossRef]

8

8. F. Lederer, S. Fahr, C. Rockstuhl, and T. Kirchartz, in MRS Proceedings (2012), Vol. 1391.

].

In this Letter, we present a theoretical study of a randomly rough Gaussian surface, which is used as a light-trapping interface within a silicon solar-cell structure. Our purpose is twofold: to show that the model of Gaussian disorder gives a good description of actual rough substrates used in real solar cells, and to determine the optimal roughness parameters yielding the highest absorption enhancement in the active layer.

We consider the case of a single rough interface [see Fig. 1(a)] described in real space by the root mean square (RMS) of height σ and the correlation length lc [9

9. I. Simonsen, Eur. Phys. J. Special Topics 181, 1 (2010). [CrossRef]

], which is defined as the distance at which the correlation function W(|x|)=exp(x2/lc2) decreases by 1/e [10

10. A. A. Maradudin and T. Michel, J. Stat. Phys. 58, 485(1990). [CrossRef]

]. Notice that the average spacing between consecutive maxima/minima of the rough surface is given by 1.28×lc. The algorithm to generate random surface profiles with given statistical parameters σ and lc was derived in [11

11. V. Freilikher, E. Kanzieper, and A. Maradudin, Phys. Rep. 288, 127 (1997). [CrossRef]

]:
ξ(x)=σ2Λm=1g(2πmΛ)×[α2m1sin(2πmxΛ)+α2mcos(2πmxΛ)],
(1)
where α is a random number with the normal distribution, Λ is the period of the surface profile, and g(k)=πlcexp(k2lc2/4) is the power spectrum. This Gaussian surface model is known to give a good representation for a wide class of random surfaces [9

9. I. Simonsen, Eur. Phys. J. Special Topics 181, 1 (2010). [CrossRef]

].

Fig. 1. Sketch of (a) rough interface and (b) solar-cell structure under investigation.

Calculations were done by solving Maxwell equations for unpolarized light using rigorous coupled-wave analysis [12

12. D. M. Whittaker and I. S. Culshaw, Phys. Rev. B 60, 2610 (1999). [CrossRef]

,13

13. M. Liscidini, D. Gerace, L. C. Andreani, and J. E. Sipe, Phys. Rev. B 77, 035324 (2008). [CrossRef]

]. The rough surface profile is discretized in the vertical direction, and we consider a computational cell of length equal to Λ in Eq. (1). We use lengths between 10 and 20 μm, which allow us to neglect the effects of periodicity. Convergence with numbers of plane waves and discretization layers has been carefully checked.

We validate this simple model by describing common rough substrates used for thin-film solar cells. To avoid making a material-dependent comparison, we choose the benchmark situation considered in [5

5. C. Rockstuhl, S. Fahr, K. Bittkau, T. Beckers, R. Carius, F.-J. Haug, T. Söderström, C. Ballif, and F. Lederer, Opt. Express 18, A335 (2010). [CrossRef]

], namely a rough interface between a transparent conductive oxide (TCO) with refractive index nTCO=1.915 and a dielectric medium with specified n. In Fig. 2 we show (a) the haze and (b) the angle-resolved scattering (ARS) function calculated with the parameters of two typical substrates: Neuchâtel (σ=81nm, lc=140nm) and Asahi-U (σ=35nm, lc=160nm). These results have been obtained by averaging over 500 surface realizations having the same statistical parameters. Both the haze and the ARS calculated with the Gaussian model are in good agreement with those reported in Fig. 2 of [5

5. C. Rockstuhl, S. Fahr, K. Bittkau, T. Beckers, R. Carius, F.-J. Haug, T. Söderström, C. Ballif, and F. Lederer, Opt. Express 18, A335 (2010). [CrossRef]

], which are obtained by importing the measured topographic profile of real substrates into the finite-difference time-domain calculation (differences in the central dip of the ARS are believed to be due to our use of finer spacing and better resolution for the polar angle θ). The agreement is surprisingly good, considering that the present Gaussian model is one-dimensional (1D), while the real substrates are two-dimensional (2D).

Fig. 2. (a) Haze for the interface between two rough TCO substrates (nTCO=1.915) and a medium with varying refractive index n. Curves are guides to the eye. (b) ARS function for the two interfaces. Substrate parameters: see [5] and text. The wavelength is λ=633nm.

Let us now consider the solar cell structure presented in Fig. 1(b). It consists of a 70 nm thick homogeneous TCO layer (nTCO=1.65, nonabsorbing) on a 1 μm thick crystalline silicon slab [14

14. E. D. Palik, Handbook of Optical Constants of Solids(Academic, 1985).

], and a silver back reflector [14

14. E. D. Palik, Handbook of Optical Constants of Solids(Academic, 1985).

]. The top layer is used as an antireflection (AR) coating, while the rough interface between the silicon and TCO layers is responsible for light trapping. The choice of c-Si is motivated by recent developments on epitaxy-free fabrication of thin c-Si layers and solar cells [15

15. V. Depauw, Y. Qiu, K. Van Nieuwenhuysen, I. Gordon, and J. Poortmans, Prog. Photovoltaics 19, 844 (2011). [CrossRef]

].

Fig. 3. Absorption spectra for structures with lc=100nm and different values of RMS height. The black curve for σ=0 corresponds to the flat structure.

In Fig. 4 we show Jsc as a function of σ for the solar cell structure of Fig. 1(b), assuming lc=100nm. For σ larger than a few tens of nanometers, Jsc increases and saturates around 21.5mA/cm2 for σ larger than about 150 nm. The behavior as a function of σ is very similar to that calculated for a-Si solar cells [7

7. S. Fahr, T. Kirchartz, C. Rockstuhl, and F. Lederer, Opt. Express 19, A865 (2011). [CrossRef]

,8

8. F. Lederer, S. Fahr, C. Rockstuhl, and T. Kirchartz, in MRS Proceedings (2012), Vol. 1391.

]. Again, the 1D Gaussian model captures the same physical behavior that has been previously shown for rough 2D substrates. To perform a more detailed analysis, the inset in Fig. 4 shows the small-σ region calculated for dTCO=70nm [as in Fig. 1(b)] and dTCO=1μm. In the latter case Jsc increases quadratically at small σ, which is reminiscent of the scattering losses in photonic crystal waveguides in the perturbative regime [22

22. D. Gerace and L. C. Andreani, Opt. Lett. 29, 1897 (2004). [CrossRef]

,23

23. S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, Phys. Rev. Lett. 94, 033903 (2005). [CrossRef]

]. When dTCO=70nm, the short-circuit current decreases from the σ=0 value. We interpret this behavior as follows: dTCO=70nm is the optimized width for AR action, but by introducing the roughness the AR property gets partially lost, leading to a decrease of Jsc. When dTCO=1μm, the TCO layer is not optimized as AR coating; thus Jsc starts from a smaller value at σ=0, but it increases monotonically with σ. An important message here is that Jsc is independent of the TCO thickness for σ larger than a few tens of nanometers.

Fig. 4. Short-circuit current density as a function of RMS height for lc=100nm and dTCO=70nm. Inset: blow-up of small-σ region for 70 nm and 1 μm TCO thickness.

In Fig. 5 we show the short-circuit current density calculated as a function of lc, between 60 and 220 nm, and σ, between 0 and 300 nm. Each point has been averaged over 10 surface realizations. The dependence on lc at fixed σ has a bell-like shape, with a maximum at an optimal value of lc [7

7. S. Fahr, T. Kirchartz, C. Rockstuhl, and F. Lederer, Opt. Express 19, A865 (2011). [CrossRef]

]. This optimal lc is dependent on σ, increasing from less than 80 nm at low σ to about 160 nm for σ=300nm. In Fig. 5 we indicate the position of Neuchâtel and Asahi-U substrates, for the statistical parameters given in [5

5. C. Rockstuhl, S. Fahr, K. Bittkau, T. Beckers, R. Carius, F.-J. Haug, T. Söderström, C. Ballif, and F. Lederer, Opt. Express 18, A335 (2010). [CrossRef]

] (the Jülich substrate, having lc1.4μm, lies outside our calculation range). This shows that (1) there is margin for improving light trapping by optimizing rough substrates, and (2) analyses for specific solar-cell structures should take into account correlations between the disorder parameters σ and lc.

Fig. 5. Short-circuit current density as a function of RMS height and correlation length. The two points denote the parameters of two common rough substrates.

While the results of Fig. 2 suggest the qualitative behavior of the short-circuit current density as a function of σ and lc to be the same for the present 1D model and for the 2D interfaces, the values of Jsc are expected to be higher for 2D rough substrates. The difference can be estimated by calculating the Lambertian light-trapping limit (for a single wavelength, the light path enhancement is πn for 1D, and 4n2 for 2D [2

2. E. Yablonovitch, J. Opt. Soc. Am. 72, 899 (1982). [CrossRef]

,17

17. Z. Yu, A. Raman, and S. Fan, Opt. Express 18, A366 (2010). [CrossRef]

]). The short-circuit current density for a 1 μm thick c-Si layer is calculated to be 22.7mA/cm2 with a 1D, and 28.7mA/cm2 with a 2D Lambertian scatterer. The values reported in Fig. 5 are close to the 1D Lambertian limit. Thus, we expect an optimized 2D roughness to yield a Jsc about 6mA/cm2 higher than for the optimized 1D roughness.

Since the values of Jsc depend on several factors (material absorption, electronic transport, assumptions for solar spectrum), the effect of roughness and comparison with real thin-film solar cells should better be understood in terms of a relative enhancement. Our results for an optimized 1D substrate lead to a relative increase of Jsc by 44%. In [24

24. M. Berginski, J. Hüpkes, M. Schulte, G. Schöpe, H. Stiebig, B. Rech, and M. Wuttig, J. Appl. Phys. 101, 074903 (2007). [CrossRef]

,25

25. H. Sai, H. Fujiwara, M. Kondo, and Y. Kanamori, Appl. Phys. Lett. 93, 143501 (2008). [CrossRef]

], for example, the short-circuit current density of μc-Si:H solar cells of about 1 μm thickness on TCO substrates with various degrees of roughness is determined, and the reported relative increase of Jsc is +33% and +21%, respectively. Thus, in these cases, the calculated Jsc enhancement of a substrate with optimized 1D roughness is larger than for real solar cells with rough 2D substrates.

In conclusion, we performed a theoretical study of a randomly rough Gaussian interface, and we showed that, unexpectedly, this simple 1D model can be used to describe with high accuracy the optical properties of existing 2D rough interfaces. We then applied this model to the study of thin-film c-Si solar cells, and quantified the increase of short-circuit current density by the rough interface as a function of texture height and lateral correlation length. Moreover, we showed that light absorption close to the Lambertian limit can be obtained by the simultaneous optimization of the two statistical parameters. Given the simplicity of this model and its implementation, these results open the path toward the analysis of more complex architectures for light trapping in thin-film solar cells.

This work was supported by the EU through Marie Curie Action FP7-PEOPLE-2010-ITN Project No. 264687 “PROPHET” and Fondazione Cariplo project 2010-0523 “Nanophotonics for thin-film photovoltaics”.

References

1.

R. Brendel, Thin-Film Crystalline Silicon Solar Cells(Wiley-VCH, 2003).

2.

E. Yablonovitch, J. Opt. Soc. Am. 72, 899 (1982). [CrossRef]

3.

S. Fahr, C. Rockstuhl, and F. Lederer, Appl. Phys. Lett. 92, 171114 (2008). [CrossRef]

4.

D. Dominé, F.-J. Haug, C. Battaglia, and C. Ballif, J. Appl. Phys. 107, 044504 (2010). [CrossRef]

5.

C. Rockstuhl, S. Fahr, K. Bittkau, T. Beckers, R. Carius, F.-J. Haug, T. Söderström, C. Ballif, and F. Lederer, Opt. Express 18, A335 (2010). [CrossRef]

6.

X. Sheng, S. G. Johnson, J. Michel, and L. C. Kimerling, Opt. Express 19, A841 (2011). [CrossRef]

7.

S. Fahr, T. Kirchartz, C. Rockstuhl, and F. Lederer, Opt. Express 19, A865 (2011). [CrossRef]

8.

F. Lederer, S. Fahr, C. Rockstuhl, and T. Kirchartz, in MRS Proceedings (2012), Vol. 1391.

9.

I. Simonsen, Eur. Phys. J. Special Topics 181, 1 (2010). [CrossRef]

10.

A. A. Maradudin and T. Michel, J. Stat. Phys. 58, 485(1990). [CrossRef]

11.

V. Freilikher, E. Kanzieper, and A. Maradudin, Phys. Rep. 288, 127 (1997). [CrossRef]

12.

D. M. Whittaker and I. S. Culshaw, Phys. Rev. B 60, 2610 (1999). [CrossRef]

13.

M. Liscidini, D. Gerace, L. C. Andreani, and J. E. Sipe, Phys. Rev. B 77, 035324 (2008). [CrossRef]

14.

E. D. Palik, Handbook of Optical Constants of Solids(Academic, 1985).

15.

V. Depauw, Y. Qiu, K. Van Nieuwenhuysen, I. Gordon, and J. Poortmans, Prog. Photovoltaics 19, 844 (2011). [CrossRef]

16.

A. Chutinan and S. John, Phys. Rev. A 78, 023825 (2008). [CrossRef]

17.

Z. Yu, A. Raman, and S. Fan, Opt. Express 18, A366 (2010). [CrossRef]

18.

C. Battaglia, C.-M. Hsu, K. Söderström, J. Escarré, F.-J. Haug, M. Charrière, M. Boccard, M. Despeisse, D. T. L. Alexander, M. Cantoni, Y. Cui, and C. Ballif, ACS Nano 6, 2790 (2012). [CrossRef]

19.

A. Oskooi, P. A. Favuzzi, Y. Tanaka, H. Shigeta, Y. Kawakami, and S. Noda, Appl. Phys. Lett. 100, 181110 (2012). [CrossRef]

20.

J. Nelson, The Physics of Solar Cells (Imperial College, 2003).

21.

A. Bozzola, M. Liscidini, and L. C. Andreani, Opt. Express 20, A224 (2012). [CrossRef]

22.

D. Gerace and L. C. Andreani, Opt. Lett. 29, 1897 (2004). [CrossRef]

23.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, Phys. Rev. Lett. 94, 033903 (2005). [CrossRef]

24.

M. Berginski, J. Hüpkes, M. Schulte, G. Schöpe, H. Stiebig, B. Rech, and M. Wuttig, J. Appl. Phys. 101, 074903 (2007). [CrossRef]

25.

H. Sai, H. Fujiwara, M. Kondo, and Y. Kanamori, Appl. Phys. Lett. 93, 143501 (2008). [CrossRef]

OCIS Codes
(040.5350) Detectors : Photovoltaic
(050.1950) Diffraction and gratings : Diffraction gratings

ToC Category:
Diffraction and Gratings

History
Original Manuscript: September 24, 2012
Manuscript Accepted: October 5, 2012
Published: November 22, 2012

Citation
Piotr Kowalczewski, Marco Liscidini, and Lucio Claudio Andreani, "Engineering Gaussian disorder at rough interfaces for light trapping in thin-film solar cells," Opt. Lett. 37, 4868-4870 (2012)
http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-37-23-4868


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References

  1. R. Brendel, Thin-Film Crystalline Silicon Solar Cells(Wiley-VCH, 2003).
  2. E. Yablonovitch, J. Opt. Soc. Am. 72, 899 (1982). [CrossRef]
  3. S. Fahr, C. Rockstuhl, and F. Lederer, Appl. Phys. Lett. 92, 171114 (2008). [CrossRef]
  4. D. Dominé, F.-J. Haug, C. Battaglia, and C. Ballif, J. Appl. Phys. 107, 044504 (2010). [CrossRef]
  5. C. Rockstuhl, S. Fahr, K. Bittkau, T. Beckers, R. Carius, F.-J. Haug, T. Söderström, C. Ballif, and F. Lederer, Opt. Express 18, A335 (2010). [CrossRef]
  6. X. Sheng, S. G. Johnson, J. Michel, and L. C. Kimerling, Opt. Express 19, A841 (2011). [CrossRef]
  7. S. Fahr, T. Kirchartz, C. Rockstuhl, and F. Lederer, Opt. Express 19, A865 (2011). [CrossRef]
  8. F. Lederer, S. Fahr, C. Rockstuhl, and T. Kirchartz, in MRS Proceedings (2012), Vol. 1391.
  9. I. Simonsen, Eur. Phys. J. Special Topics 181, 1 (2010). [CrossRef]
  10. A. A. Maradudin and T. Michel, J. Stat. Phys. 58, 485(1990). [CrossRef]
  11. V. Freilikher, E. Kanzieper, and A. Maradudin, Phys. Rep. 288, 127 (1997). [CrossRef]
  12. D. M. Whittaker and I. S. Culshaw, Phys. Rev. B 60, 2610 (1999). [CrossRef]
  13. M. Liscidini, D. Gerace, L. C. Andreani, and J. E. Sipe, Phys. Rev. B 77, 035324 (2008). [CrossRef]
  14. E. D. Palik, Handbook of Optical Constants of Solids(Academic, 1985).
  15. V. Depauw, Y. Qiu, K. Van Nieuwenhuysen, I. Gordon, and J. Poortmans, Prog. Photovoltaics 19, 844 (2011). [CrossRef]
  16. A. Chutinan and S. John, Phys. Rev. A 78, 023825 (2008). [CrossRef]
  17. Z. Yu, A. Raman, and S. Fan, Opt. Express 18, A366 (2010). [CrossRef]
  18. C. Battaglia, C.-M. Hsu, K. Söderström, J. Escarré, F.-J. Haug, M. Charrière, M. Boccard, M. Despeisse, D. T. L. Alexander, M. Cantoni, Y. Cui, and C. Ballif, ACS Nano 6, 2790 (2012). [CrossRef]
  19. A. Oskooi, P. A. Favuzzi, Y. Tanaka, H. Shigeta, Y. Kawakami, and S. Noda, Appl. Phys. Lett. 100, 181110 (2012). [CrossRef]
  20. J. Nelson, The Physics of Solar Cells (Imperial College, 2003).
  21. A. Bozzola, M. Liscidini, and L. C. Andreani, Opt. Express 20, A224 (2012). [CrossRef]
  22. D. Gerace and L. C. Andreani, Opt. Lett. 29, 1897 (2004). [CrossRef]
  23. S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, Phys. Rev. Lett. 94, 033903 (2005). [CrossRef]
  24. M. Berginski, J. Hüpkes, M. Schulte, G. Schöpe, H. Stiebig, B. Rech, and M. Wuttig, J. Appl. Phys. 101, 074903 (2007). [CrossRef]
  25. H. Sai, H. Fujiwara, M. Kondo, and Y. Kanamori, Appl. Phys. Lett. 93, 143501 (2008). [CrossRef]

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