Abstract

We investigate the optical absorption in a thin Cu(In,Ga)Se2 solar cell with a Lambertian white paint beneath a transparent back contact. Although this configuration has been proposed more than 30 years ago, it turns out that rigorous simulation of Maxwell’s equations demand powerful numerical calculations. This type of approach is time consuming and does not provide a physical insight in the absorption mechanisms. Here, we use the radiative transfer equation to deal with multiple scattering of the diffuse part of the light. The collimated part is treated accounting for wave effects. Our model is in good agreement with optical measurements.

© 2013 OSA

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2012 (2)

Z. Jehl Li Kao, N. Naghavi, F. Erfurth, J. F. Guillemoles, I. Gérard, A. Etcheberry, J. L. Pelouard, S. Collin, G. Voorwinden, and D. Lincot, “Towards ultrathin copper indium gallium diselenide solar cells: proof of concept study by chemical etching and gold back contact engineering,” Prog. Photovolt: Res. Appl.20, 582–587 (2012).
[CrossRef]

N. Dahan, Z. Jehl, T. Hildebrandt, J.-J. Greffet, J.-F. Guillemoles, D. Lincot, and N. Naghavi, “Optical approaches to improve the photocurrent generation in Cu(In,Ga)Se2 solar cells with absorber thicknesses down to 0.5 μm,” J. Appl. Phys.112, 094902 (2012).
[CrossRef]

2011 (3)

W. Szabelak and W. Nasalski, “Enhancement of cross-polarized beam components at a metamaterial surface,” Appl. Phys. B103, 369–375 (2011).
[CrossRef]

M. Bouttemy, P. Tran-Van, I. Gérard, T. Hildebrandt, A. Causier, J. L. Pelouard, G. Dagher, Z. Jehl, N. Naghavi, G. Voorwinden, B. Dimmler, M. Powalla, J. F. Guillemoles, D. Lincot, and A. Etcheberrya, “Thinning of CIGS solar cells: part I: chemical processing in acidic bromine solutions,” Thin Solid Films519, 7207–7211 (2011).
[CrossRef]

P. Jackson, D. Hariskos, E. Lotter, S. Paetel, R. Wuerz, R. Menner, W. Wischmann, and M. Powalla, “New world record efficiency for Cu(In,Ga)Se2 thin film solar cells beyond 20%,” Prog. Photovolt: Res. Appl.19, 894–897 (2011).
[CrossRef]

2010 (1)

B. Lipovšek, J. Krč, O. Isabella, M. Zeman, and M. Topič, “Modeling and optimization of white paint back reectors for thin-lm silicon solar cells,” J. Appl. Phys.108, 103115 (2010).
[CrossRef]

2009 (1)

J. Rousset, E. Saucedo, and D. Lincot, “Extrinsic doping of electrodeposited zinc oxide films by chlorine for transparent conductive oxide applications,” Chem. Mater.21, 534–540 (2009).
[CrossRef]

2008 (1)

R. Santbergen and R. J. C. van Zolingen, “The absorption factor of crystalline silicon PV cells: a numerical and experimental study,” Sol. Energ. Mat. Sol. Cells92, 432–444 (2008).
[CrossRef]

2007 (1)

A. Čampa, J. Krč, J. Malmström, M. Edoff, F. Smole, and M. Topič, “The potential of textured front ZnO and flat TCO/metal back contact to improve optical absorption in thin Cu(In,Ga)Se2 solar cells,” Thin Solid Films515, 5968–5972 (2007).
[CrossRef]

2005 (2)

P. J. Rostan, J. Mattheis, G. Bilger, U. Rau, and J. H. Werner, “Formation of transparent and ohmic ZnO:Al/MoSe2 contacts for bifacial Cu(In,Ga)Se2 solar cells and tandem structures,” Thin Solid Films480–481, 67–70 (2005).
[CrossRef]

P. C. Y. Chang, J. G. Walker, and K. I. Hopcraft, “Ray tracing in absorbing media,” J. Quant. Spectrosc. Radiat. Transfer96, 327–341 (2005).
[CrossRef]

2004 (3)

J. Caron, C. Andraud, and J. Lafait, “Radiative transfer calculations in multilayer systems with smooth or rough interfaces,” J. Mod. Opt.51, 575–595 (2004).
[CrossRef]

F. Ghmari, T. Ghbara, M. Laroche, R. Carminati, and J.-J. Greffet, “Influence of microroughness on emissivity,” J. Appl. Phys.96, 2656–2664 (2004).
[CrossRef]

J. Springer, A. Poruba, and M. Vanecek, “Improved three-dimensional optical model for thin-film silicon solar cells,” J. Appl. Phys.96, 5329–5337 (2004).
[CrossRef]

2003 (2)

J. Krč, F. Smole, and M. Topič, “Analysis of light scattering in amorphous Si:H solar cells by a one-dimensional semi-coherent optical model,” Prog. Photovolt: Res. Appl.11, 15–26 (2003).
[CrossRef]

K. Orgassa, H. W. Schock, and J. H. Werner, “Alternative back contact materials for thin film Cu(In,Ga)Se2 solar cells,” Thin Solid Films431–432, 387–391 (2003).
[CrossRef]

2002 (2)

J. Krč, M. Zeman, F. Smole, and M. Topič, “Optical modeling of a-Si:H solar cells deposited on textured glass/SnO2 substrates,” J. Appl. Phys.92, 749–755 (2002).
[CrossRef]

M. A. Green, “Lambertian light trapping in textured solar cells and light-emitting diodes: analytical solutions,” Prog. Photovolt: Res. Appl.10, 235–241 (2002).
[CrossRef]

2001 (1)

2000 (1)

M. Zeman, R. A. C. M. M. van Swaaij, J. W. Metselaar, and R. E. I. Schropp, “Optical modeling of a-Si:H solar cells with rough interfaces: effect of back contact and interface roughness,” J. Appl. Phys.88, 6436–6443 (2000).
[CrossRef]

1999 (1)

R. Carminati and J.-J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett.82, 1660–1663 (1999).
[CrossRef]

1996 (2)

K. Tang, R. A. Dimenna, and R. O. Buckius, “Regions of validity of the geometric optics approximation for angular scattering from very rough surfaces,” Int. J. Heat Mass Transfer40, 49–59 (1996).
[CrossRef]

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion24, 327–370 (1996).
[CrossRef]

1994 (4)

1991 (1)

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transfer46, 413–423 (1991).
[CrossRef]

1990 (2)

J. Morris, R. R. Arya, J. G. O’Dowd, and S. Wiedeman, “Absorption enhancement in hydrogenated amorphous silicon-based solar cells,” J. Appl. Phys.67, 1079–1087 (1990).
[CrossRef]

S. Kumar, A. Majumdar, and C. L. Tien, “The differential-discrete-ordinate method for solutions of the equation of radiative transfer,” J. Heat Transfer112, 424–429 (1990).
[CrossRef]

1989 (1)

1983 (1)

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]

1982 (1)

Acklin, B.

Andraud, C.

J. Caron, C. Andraud, and J. Lafait, “Radiative transfer calculations in multilayer systems with smooth or rough interfaces,” J. Mod. Opt.51, 575–595 (2004).
[CrossRef]

Arya, R. R.

J. Morris, R. R. Arya, J. G. O’Dowd, and S. Wiedeman, “Absorption enhancement in hydrogenated amorphous silicon-based solar cells,” J. Appl. Phys.67, 1079–1087 (1990).
[CrossRef]

Bilger, G.

P. J. Rostan, J. Mattheis, G. Bilger, U. Rau, and J. H. Werner, “Formation of transparent and ohmic ZnO:Al/MoSe2 contacts for bifacial Cu(In,Ga)Se2 solar cells and tandem structures,” Thin Solid Films480–481, 67–70 (2005).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, New York, 1983).

Bouttemy, M.

M. Bouttemy, P. Tran-Van, I. Gérard, T. Hildebrandt, A. Causier, J. L. Pelouard, G. Dagher, Z. Jehl, N. Naghavi, G. Voorwinden, B. Dimmler, M. Powalla, J. F. Guillemoles, D. Lincot, and A. Etcheberrya, “Thinning of CIGS solar cells: part I: chemical processing in acidic bromine solutions,” Thin Solid Films519, 7207–7211 (2011).
[CrossRef]

Buckius, R. O.

K. Tang, R. A. Dimenna, and R. O. Buckius, “Regions of validity of the geometric optics approximation for angular scattering from very rough surfaces,” Int. J. Heat Mass Transfer40, 49–59 (1996).
[CrossRef]

Campa, A.

A. Čampa, J. Krč, J. Malmström, M. Edoff, F. Smole, and M. Topič, “The potential of textured front ZnO and flat TCO/metal back contact to improve optical absorption in thin Cu(In,Ga)Se2 solar cells,” Thin Solid Films515, 5968–5972 (2007).
[CrossRef]

Carminati, R.

F. Ghmari, T. Ghbara, M. Laroche, R. Carminati, and J.-J. Greffet, “Influence of microroughness on emissivity,” J. Appl. Phys.96, 2656–2664 (2004).
[CrossRef]

R. Carminati and J.-J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett.82, 1660–1663 (1999).
[CrossRef]

Caron, J.

J. Caron, C. Andraud, and J. Lafait, “Radiative transfer calculations in multilayer systems with smooth or rough interfaces,” J. Mod. Opt.51, 575–595 (2004).
[CrossRef]

Causier, A.

M. Bouttemy, P. Tran-Van, I. Gérard, T. Hildebrandt, A. Causier, J. L. Pelouard, G. Dagher, Z. Jehl, N. Naghavi, G. Voorwinden, B. Dimmler, M. Powalla, J. F. Guillemoles, D. Lincot, and A. Etcheberrya, “Thinning of CIGS solar cells: part I: chemical processing in acidic bromine solutions,” Thin Solid Films519, 7207–7211 (2011).
[CrossRef]

Chance, B.

Chang, P. C. Y.

P. C. Y. Chang, J. G. Walker, and K. I. Hopcraft, “Ray tracing in absorbing media,” J. Quant. Spectrosc. Radiat. Transfer96, 327–341 (2005).
[CrossRef]

Collin, S.

Z. Jehl Li Kao, N. Naghavi, F. Erfurth, J. F. Guillemoles, I. Gérard, A. Etcheberry, J. L. Pelouard, S. Collin, G. Voorwinden, and D. Lincot, “Towards ultrathin copper indium gallium diselenide solar cells: proof of concept study by chemical etching and gold back contact engineering,” Prog. Photovolt: Res. Appl.20, 582–587 (2012).
[CrossRef]

Dagher, G.

M. Bouttemy, P. Tran-Van, I. Gérard, T. Hildebrandt, A. Causier, J. L. Pelouard, G. Dagher, Z. Jehl, N. Naghavi, G. Voorwinden, B. Dimmler, M. Powalla, J. F. Guillemoles, D. Lincot, and A. Etcheberrya, “Thinning of CIGS solar cells: part I: chemical processing in acidic bromine solutions,” Thin Solid Films519, 7207–7211 (2011).
[CrossRef]

Dahan, N.

N. Dahan, Z. Jehl, T. Hildebrandt, J.-J. Greffet, J.-F. Guillemoles, D. Lincot, and N. Naghavi, “Optical approaches to improve the photocurrent generation in Cu(In,Ga)Se2 solar cells with absorber thicknesses down to 0.5 μm,” J. Appl. Phys.112, 094902 (2012).
[CrossRef]

Deckman, H. W.

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]

Dimenna, R. A.

K. Tang, R. A. Dimenna, and R. O. Buckius, “Regions of validity of the geometric optics approximation for angular scattering from very rough surfaces,” Int. J. Heat Mass Transfer40, 49–59 (1996).
[CrossRef]

Dimmler, B.

M. Bouttemy, P. Tran-Van, I. Gérard, T. Hildebrandt, A. Causier, J. L. Pelouard, G. Dagher, Z. Jehl, N. Naghavi, G. Voorwinden, B. Dimmler, M. Powalla, J. F. Guillemoles, D. Lincot, and A. Etcheberrya, “Thinning of CIGS solar cells: part I: chemical processing in acidic bromine solutions,” Thin Solid Films519, 7207–7211 (2011).
[CrossRef]

Ding, K.-H.

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, New York, 2000).
[CrossRef]

Dupertuis, M. A.

Edoff, M.

A. Čampa, J. Krč, J. Malmström, M. Edoff, F. Smole, and M. Topič, “The potential of textured front ZnO and flat TCO/metal back contact to improve optical absorption in thin Cu(In,Ga)Se2 solar cells,” Thin Solid Films515, 5968–5972 (2007).
[CrossRef]

Erfurth, F.

Z. Jehl Li Kao, N. Naghavi, F. Erfurth, J. F. Guillemoles, I. Gérard, A. Etcheberry, J. L. Pelouard, S. Collin, G. Voorwinden, and D. Lincot, “Towards ultrathin copper indium gallium diselenide solar cells: proof of concept study by chemical etching and gold back contact engineering,” Prog. Photovolt: Res. Appl.20, 582–587 (2012).
[CrossRef]

Etcheberry, A.

Z. Jehl Li Kao, N. Naghavi, F. Erfurth, J. F. Guillemoles, I. Gérard, A. Etcheberry, J. L. Pelouard, S. Collin, G. Voorwinden, and D. Lincot, “Towards ultrathin copper indium gallium diselenide solar cells: proof of concept study by chemical etching and gold back contact engineering,” Prog. Photovolt: Res. Appl.20, 582–587 (2012).
[CrossRef]

Etcheberrya, A.

M. Bouttemy, P. Tran-Van, I. Gérard, T. Hildebrandt, A. Causier, J. L. Pelouard, G. Dagher, Z. Jehl, N. Naghavi, G. Voorwinden, B. Dimmler, M. Powalla, J. F. Guillemoles, D. Lincot, and A. Etcheberrya, “Thinning of CIGS solar cells: part I: chemical processing in acidic bromine solutions,” Thin Solid Films519, 7207–7211 (2011).
[CrossRef]

Evans, K. F.

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transfer46, 413–423 (1991).
[CrossRef]

Feitcknecht, L.

J. Springer, A. Poruba, A. Fejfar, M. Vanecek, L. Feitcknecht, N. Wyrsch, J. Meier, and A. Shah, “Nanotextured thin film silicon solar cells: optical model,” in 16th European Photovoltaic Solar Energy Conference, James and James, ed. (Alden, Glasgow, UK, 2000), pp. 434–437.

Fejfar, A.

J. Springer, A. Poruba, A. Fejfar, M. Vanecek, L. Feitcknecht, N. Wyrsch, J. Meier, and A. Shah, “Nanotextured thin film silicon solar cells: optical model,” in 16th European Photovoltaic Solar Energy Conference, James and James, ed. (Alden, Glasgow, UK, 2000), pp. 434–437.

Fung, A. K.

A. K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, Norwood, Mass., 1994).

Gérard, I.

Z. Jehl Li Kao, N. Naghavi, F. Erfurth, J. F. Guillemoles, I. Gérard, A. Etcheberry, J. L. Pelouard, S. Collin, G. Voorwinden, and D. Lincot, “Towards ultrathin copper indium gallium diselenide solar cells: proof of concept study by chemical etching and gold back contact engineering,” Prog. Photovolt: Res. Appl.20, 582–587 (2012).
[CrossRef]

M. Bouttemy, P. Tran-Van, I. Gérard, T. Hildebrandt, A. Causier, J. L. Pelouard, G. Dagher, Z. Jehl, N. Naghavi, G. Voorwinden, B. Dimmler, M. Powalla, J. F. Guillemoles, D. Lincot, and A. Etcheberrya, “Thinning of CIGS solar cells: part I: chemical processing in acidic bromine solutions,” Thin Solid Films519, 7207–7211 (2011).
[CrossRef]

Ghbara, T.

F. Ghmari, T. Ghbara, M. Laroche, R. Carminati, and J.-J. Greffet, “Influence of microroughness on emissivity,” J. Appl. Phys.96, 2656–2664 (2004).
[CrossRef]

Ghmari, F.

F. Ghmari, T. Ghbara, M. Laroche, R. Carminati, and J.-J. Greffet, “Influence of microroughness on emissivity,” J. Appl. Phys.96, 2656–2664 (2004).
[CrossRef]

Green, M. A.

M. A. Green, “Lambertian light trapping in textured solar cells and light-emitting diodes: analytical solutions,” Prog. Photovolt: Res. Appl.10, 235–241 (2002).
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J. Springer, A. Poruba, A. Fejfar, M. Vanecek, L. Feitcknecht, N. Wyrsch, J. Meier, and A. Shah, “Nanotextured thin film silicon solar cells: optical model,” in 16th European Photovoltaic Solar Energy Conference, James and James, ed. (Alden, Glasgow, UK, 2000), pp. 434–437.

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Pelouard, J. L.

Z. Jehl Li Kao, N. Naghavi, F. Erfurth, J. F. Guillemoles, I. Gérard, A. Etcheberry, J. L. Pelouard, S. Collin, G. Voorwinden, and D. Lincot, “Towards ultrathin copper indium gallium diselenide solar cells: proof of concept study by chemical etching and gold back contact engineering,” Prog. Photovolt: Res. Appl.20, 582–587 (2012).
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J. Springer, A. Poruba, A. Fejfar, M. Vanecek, L. Feitcknecht, N. Wyrsch, J. Meier, and A. Shah, “Nanotextured thin film silicon solar cells: optical model,” in 16th European Photovoltaic Solar Energy Conference, James and James, ed. (Alden, Glasgow, UK, 2000), pp. 434–437.

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P. Jackson, D. Hariskos, E. Lotter, S. Paetel, R. Wuerz, R. Menner, W. Wischmann, and M. Powalla, “New world record efficiency for Cu(In,Ga)Se2 thin film solar cells beyond 20%,” Prog. Photovolt: Res. Appl.19, 894–897 (2011).
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[CrossRef]

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P. J. Rostan, J. Mattheis, G. Bilger, U. Rau, and J. H. Werner, “Formation of transparent and ohmic ZnO:Al/MoSe2 contacts for bifacial Cu(In,Ga)Se2 solar cells and tandem structures,” Thin Solid Films480–481, 67–70 (2005).
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J. Rousset, E. Saucedo, and D. Lincot, “Extrinsic doping of electrodeposited zinc oxide films by chlorine for transparent conductive oxide applications,” Chem. Mater.21, 534–540 (2009).
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K. Orgassa, H. W. Schock, and J. H. Werner, “Alternative back contact materials for thin film Cu(In,Ga)Se2 solar cells,” Thin Solid Films431–432, 387–391 (2003).
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M. Zeman, R. A. C. M. M. van Swaaij, J. W. Metselaar, and R. E. I. Schropp, “Optical modeling of a-Si:H solar cells with rough interfaces: effect of back contact and interface roughness,” J. Appl. Phys.88, 6436–6443 (2000).
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J. Springer, A. Poruba, A. Fejfar, M. Vanecek, L. Feitcknecht, N. Wyrsch, J. Meier, and A. Shah, “Nanotextured thin film silicon solar cells: optical model,” in 16th European Photovoltaic Solar Energy Conference, James and James, ed. (Alden, Glasgow, UK, 2000), pp. 434–437.

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I. R. Howell, R. Siegel, and M. P. Mengüç, Thermal Radiation and Heat Transfer (CRC, Boca Raton, Florida, 2010).

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A. Čampa, J. Krč, J. Malmström, M. Edoff, F. Smole, and M. Topič, “The potential of textured front ZnO and flat TCO/metal back contact to improve optical absorption in thin Cu(In,Ga)Se2 solar cells,” Thin Solid Films515, 5968–5972 (2007).
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J. Krč, M. Zeman, F. Smole, and M. Topič, “Optical modeling of a-Si:H solar cells deposited on textured glass/SnO2 substrates,” J. Appl. Phys.92, 749–755 (2002).
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J. Krč, M. Topič, M. Vukadinović, and F. Smole, “Optical modeling of a-Si:H-based solar cells with smooth and rough boundaries,” in 16th European Photovoltaic Solar Energy Conference, James and James, ed. (Alden, Glasgow, UK, 2000), pp. 522–525.

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J. Springer, A. Poruba, and M. Vanecek, “Improved three-dimensional optical model for thin-film silicon solar cells,” J. Appl. Phys.96, 5329–5337 (2004).
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J. Springer, A. Poruba, A. Fejfar, M. Vanecek, L. Feitcknecht, N. Wyrsch, J. Meier, and A. Shah, “Nanotextured thin film silicon solar cells: optical model,” in 16th European Photovoltaic Solar Energy Conference, James and James, ed. (Alden, Glasgow, UK, 2000), pp. 434–437.

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J. Malmström, O. Lundberg, and L. Stolt, “Potential for light trapping in Cu(In,Ga)Se2 solar cells,” in 3rd World Conference on Photovoltaic Energy Conversion, K. Kurokawa, ed. (Arisumi, Osaka, Japan, 2003), pp. 344–347.

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B. Lipovšek, J. Krč, O. Isabella, M. Zeman, and M. Topič, “Modeling and optimization of white paint back reectors for thin-lm silicon solar cells,” J. Appl. Phys.108, 103115 (2010).
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J. Krč, M. Topič, M. Vukadinović, and F. Smole, “Optical modeling of a-Si:H-based solar cells with smooth and rough boundaries,” in 16th European Photovoltaic Solar Energy Conference, James and James, ed. (Alden, Glasgow, UK, 2000), pp. 522–525.

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M. Bouttemy, P. Tran-Van, I. Gérard, T. Hildebrandt, A. Causier, J. L. Pelouard, G. Dagher, Z. Jehl, N. Naghavi, G. Voorwinden, B. Dimmler, M. Powalla, J. F. Guillemoles, D. Lincot, and A. Etcheberrya, “Thinning of CIGS solar cells: part I: chemical processing in acidic bromine solutions,” Thin Solid Films519, 7207–7211 (2011).
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[CrossRef]

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[CrossRef]

Z. Jehl Li Kao, N. Naghavi, F. Erfurth, J. F. Guillemoles, I. Gérard, A. Etcheberry, J. L. Pelouard, S. Collin, G. Voorwinden, and D. Lincot, “Towards ultrathin copper indium gallium diselenide solar cells: proof of concept study by chemical etching and gold back contact engineering,” Prog. Photovolt: Res. Appl.20, 582–587 (2012).
[CrossRef]

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[CrossRef]

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K. Orgassa, H. W. Schock, and J. H. Werner, “Alternative back contact materials for thin film Cu(In,Ga)Se2 solar cells,” Thin Solid Films431–432, 387–391 (2003).
[CrossRef]

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[CrossRef]

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[CrossRef]

P. J. Rostan, J. Mattheis, G. Bilger, U. Rau, and J. H. Werner, “Formation of transparent and ohmic ZnO:Al/MoSe2 contacts for bifacial Cu(In,Ga)Se2 solar cells and tandem structures,” Thin Solid Films480–481, 67–70 (2005).
[CrossRef]

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[CrossRef]

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J. Krč, M. Topič, M. Vukadinović, and F. Smole, “Optical modeling of a-Si:H-based solar cells with smooth and rough boundaries,” in 16th European Photovoltaic Solar Energy Conference, James and James, ed. (Alden, Glasgow, UK, 2000), pp. 522–525.

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Figures (8)

Fig. 1
Fig. 1

Schematic description of interface scattering between layers m − 1 and m. The diffuse intensity emerging from the surface can be due to four contributions: (a) transmission from diffuse light, T d +; (b) reflection from diffuse light, R d ; (c) transmission from collimated light, T c d +; and (d) reflection from collimated light, R c d .

Fig. 2
Fig. 2

The mixed Poynting’s vector term (ℳ) as a function of wavelength and incident angle when unpolarized light illuminates from CIGS in case of: (a) CIGS - ZnO:Al and (b) CIGS - CdS interface.

Fig. 3
Fig. 3

Transmission from medium 1 at θi = 60° to medium 2 where n1 = 2 and n2 = 1.5 + in2. (a) n2 = 0: RF = 1 and TF = 0 at θt = 90°. (b) n2 = 0.1: RF = 0.6864 and TF = 0.3136 at θt = 84.4°.

Fig. 4
Fig. 4

(a) Spectral absorption by a CIGS solar cell as depicted in the inset. Curve (A) denotes measurement (solid) and calculation (dash) of solar cell without white paint (w-p) beneath the back contact (A = 1 − RT). Curve (B) indicates measurement (solid) and calculation (dots) when the w-p is attached to the back contact with air gap between them (A = 1−R). Curve (C) is when the w-p is adhered to the back contact using index matching media (IMM). Dense oscillations are obtained due to the thick glass superstrate (gray line). Numerical averaged is performed over a spectral window of 10 nm (dots), as the resolution of the spectrometer (for all the curves). (b) Spectral absorption in the different layers. For clarity, in the CIGS layer we show the absorbed light by the coherent and the diffuse parts. ‘FC’ (‘BC’) stands for the ZnO:Al in the front (back) contact.

Fig. 5
Fig. 5

Schematic description of light trapping effect. Assuming n3 > n2 > n1, zone I, II, III corresponds to light cone in n1, n2, and n3, respectively. Therefore, the intensity in zone III equals zero, and on the interface from solar-cell to air, light in zone II is totally reflected.

Fig. 6
Fig. 6

Spectral absorption in the different layers of the CIGS solar cell described in Fig. 4, with an antireflection layer (MgF2 – 100 nm thickness) instead of the glass superstrate, and (a) without IMM layer. (b) Without IMM layer and the back contact itself reflects the light as a white paint. As a result, the diffuse light in the CIGS layer is isotropic.

Fig. 7
Fig. 7

Calculated short circuit current as a function of CIGS thickness for different back contacts.

Fig. 8
Fig. 8

Geometry of the system and schematic notation of the different variables.

Equations (39)

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d F = I ( z , u ^ ) cos θ d Ω .
F ± ( l m ) = ± 2 π I d ± ( l m ; u ^ ) cos θ d Ω .
A d = F ( l n ) F ( l m ) F inc ,
R d ( u ^ , u ^ ) = R d 0 ( u ^ , u ^ ) δ ( u ^ u ^ ) + R d d ( u ^ , u ^ ) T d ( u ^ , u ^ ) = T d 0 ( u ^ , u ^ ) δ ( u ^ u ^ t ) + T d d ( u ^ , u ^ )
I d + ( 0 m , u ^ ) n m 2 = + 2 π T d + ( u ^ , u ^ ) I d + ( L m 1 , u ^ ) n m 1 2 μ d Ω + 2 π R d ( u ^ , u ^ ) I d ( 0 m , u ^ ) n m 2 μ d Ω + T c d + ( u ^ , u ^ ) I c + ( L m 1 , u ^ ) + R c d ( u ^ , u ^ ) I c ( 0 m , u ^ )
R F + T F = 1
R F + T F = 1 + ; = 2 Im ( Q i ) Re ( Q i ) Im ( r ) ,
I t + ( 0 2 , u ^ ) I inc + ( L 1 , u ^ ) = T F ( u ^ , u ^ ) μ 1 d Ω 1 μ 2 d Ω 2
J sc = q A λ ( m CIGS ) I solar ( λ ) λ h c d λ
μ 2 = 1 ( n 1 n 2 ) 2 ( 1 μ 1 2 )
w 2 = ( n 1 n 2 ) 2 ( μ 1 μ 2 ) w 1 ,
N ( j ) = N 0 n j .
N ( j ) = N ( j 1 ) + N ( j )
N 0 , min = 2 Δ n min
I ( μ ) μ d μ j N I ( μ j ) μ j w j .
F = I ( μ , ϕ ) μ d μ d ϕ = 2 π I ( μ ) μ d μ .
I d ( L m 1 , u ^ ) n m 1 2 = 2 π T d ( u ^ , u ^ ) I d ( 0 m , u ^ ) n m 2 μ d Ω + + 2 π R d + ( u ^ , u ^ ) I d + ( L m 1 , u ^ ) n m 1 2 μ d Ω + T c d ( u ^ , u ^ ) I c ( 0 m , u ^ ) + R c d + ( u ^ , u ^ ) I c + ( L m 1 , u ^ ) .
I d ( l m , u ^ ) = I d ( 0 m , u ^ ) exp ( τ m / μ )
I d + ( l m , u ^ ) n m 2 = e τ m / μ + 2 π T d + ( u ^ , u ^ ) I d + ( L m 1 , u ^ ) n m 1 2 μ d Ω + e τ m / μ 2 π R d ( u ^ , u ^ ) I d ( l m , u ^ ) n m 2 e τ m / μ μ d Ω + e τ m / μ [ T c d + ( u ^ , u ^ ) I c + ( L m 1 , u ^ ) + R c d ( u ^ , u ^ ) I c ( 0 m , u ^ ) ]
I d ( L m 1 , u ^ ) n m 1 2 = 2 π T d ( u ^ , u ^ ) I d ( l m , u ^ ) n m 2 e τ m / μ μ d Ω + + 2 π R d + ( u ^ , u ^ ) I d + ( L m 1 , u ^ ) n m 1 2 μ d Ω + T c d ( u ^ , u ^ ) I c ( 0 m , u ^ ) + R c d + ( u ^ , u ^ ) I c + ( L m 1 , u ^ ) .
( I d + ( m ) I d ( m 1 ) ) = [ T d + ( m 1 , m ) R d ( m 1 , m ) R d + ( m 1 , m ) T d ( m 1 , m ) ] ( I d + ( m 1 ) I ( m ) ) + ( C + ( m 1 , m ) C ( m 1 , m ) )
A c ( m ) = [ S ( 0 m ) S ( L m ) ] z ^ S inc z ^
F ± = S ± z ^ = | S ± | cos θ ,
F ± = ± 2 π I c ± ( u ^ ) δ ( u ^ u ^ ) μ d Ω = I c ± ( u ^ ) cos θ .
R d 0 + ( u ^ , u ^ ) = I d ( L 1 , u ^ ) / n 1 2 I d + ( L 1 , u ^ ) / n 1 2 = I d ( L 1 , u ^ ) I d + ( L 1 , u ^ ) = R F ( u ^ , u ^ ) T d 0 + ( u ^ , u ^ ) = I d + ( 0 2 , u ^ ) / n 2 2 I d + ( L 1 , u ^ ) / n 1 2 = I d + ( 0 2 , u ^ ) I d + ( L 1 , u ^ ) ( n 1 n 2 ) 2 = T F ( u ^ , u ^ ) .
( I d + ( m ) I d ( m ) ) = [ T d + ( 0 , m ) R d ( 0 , m ) R d + ( 0 , m ) T d ( 0 , m ) ] ( I d + ( 0 ) I d ( m ) ) + ( C + ( 0 , m ) C ( 0 , m ) ) .
T + ( 0 , m ) = T + ( m 1 , m ) Q + ( 0 , m ) T + ( 0 , m 1 ) R ( 0 , m ) = R ( m 1 , m ) + T + ( m 1 , m ) Q + ( 0 , m ) R ( 0 , m 1 ) T ( m 1 , m ) C + ( 0 , m ) = C + ( m 1 , m ) + T + ( m 1 , m ) Q + ( 0 , m ) × [ C + ( 0 , m 1 ) + R ( 0 , m 1 ) C ( m 1 , m ) ] Q + ( 0 , m ) = { 𝟙 R ( 0 , m 1 ) R + ( m 1 , m ) } 1
T ( 0 , m ) = T ( 0 , m 1 ) Q ( 0 , m ) T ( m 1 , m ) R + ( 0 , m ) = R + ( 0 , m 1 ) + T ( 0 , m 1 ) Q ( 0 , m ) R + ( m 1 , m ) T + ( 0 , m 1 ) C ( 0 , m ) = C ( 0 , m 1 ) + T ( 0 , m 1 ) Q ( 0 , m ) × [ C ( m 1 , m ) + R + ( m 1 , m ) C + ( 0 , m 1 ) ] Q ( 0 , m ) = { 𝟙 R + ( m 1 , m ) R ( 0 , m 1 ) } 1 .
I ( z ; u ^ ) = I ( z ; u ^ j ) = ( I ( z ; u ^ 1 ) I ( z ; u ^ 2 ) I ( z ; u ^ N ( m ) ) ) .
S ˜ ( m 1 , m ) = [ T d + ( m 1 , m ) R d ( m 1 , m ) R d + ( m 1 , m ) T d ( m 1 , m ) ]
( I d + ( l m ; u ^ j ) I d ( L 0 ; u ^ j ) ) = S ˜ ( L 0 , l m ) ( I d + ( L 0 ; u ^ j ) I d ( l m ; u ^ j ) ) + ( C + ( L 0 , l m ) C ( L 0 , l m ) ) .
( I d + ( 0 M ; u ^ j ) I d ( l m ; u ^ j ) ) = S ˜ ( l m , 0 M ) ( I d + ( l m ; u ^ j ) 0 ) + ( C + ( l m , 0 M ) C ( l m , 0 M ) ) .
F ± ( l m ) = 2 π j N ( m ) I d + ( l m ; u ^ j ) μ j w j
A ( m ) = A d ( m ) + A c ( m ) .
k j = ( k | | , k z , j )
k | | = Re ( k j ) sin θ j
k z , j = k j 2 k | | 2
r i j = Q i Q j Q i + Q j , t i j = 2 Q i Q i + Q j
R F = | r i j | 2 , T F = Re ( Q j ) Re ( Q i ) | t i j | 2 .

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