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  • Editor: Christian Seassal
  • Vol. 21, Iss. S6 — Nov. 4, 2013
  • pp: A964–A969
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Absorption enhancing proximity effects in aperiodic nanowire arrays

Björn C. P. Sturmberg, Kokou B. Dossou, Lindsay C. Botten, Ara A. Asatryan, Christopher G. Poulton, Ross C. McPhedran, and C. Martijn de Sterke  »View Author Affiliations


Optics Express, Vol. 21, Issue S6, pp. A964-A969 (2013)
http://dx.doi.org/10.1364/OE.21.00A964


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Abstract

Aperiodic Nanowire (NW) arrays have higher absorption than equivalent periodic arrays, making them of interest for photovoltaic applications. An inevitable property of aperiodic arrays is the clustering of some NWs into closer proximity than in the equivalent periodic array. We focus on the modes of such clusters and show that the reduced symmetry associated with cluster formation allows external coupling into modes which are dark in periodic arrays, thus increasing absorption. To exploit such modes fully, arrays must include tightly clustered NWs that are unlikely to arise from fabrication variations but must be created intentionally.

© 2013 OSA

1. Introduction

Well-designed nanostructures are known to strongly enhance the absorption of photovoltaic devices [1

1. K. R. Catchpole, “Nanostructures in photovoltaics,” Phil. Trans. R. Soc. A 364, 3493–503 (2006). [CrossRef] [PubMed]

5

5. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010). [CrossRef] [PubMed]

], thereby allowing for the reduction of the active layer thickness to only a few micrometers. This significantly reduces costs, as less material is used and the material quality requirements are eased by shorter charge extraction distances. Nanowire (NW) arrays are a prominent class of nanostructures that build on these advantages by incorporating radial p-n junctions, which further reduce the charge carrier diffusion length requirements. Periodic NW arrays have been studied theoretically [6

6. C. Lin and M. L. Povinelli, “Optical absorption enhancement in silicon nanowire arrays with a large lattice constant for photovoltaic applications,” Opt. Express 17, 19371–19381 (2009). [CrossRef] [PubMed]

, 7

7. B. C. P. Sturmberg, K. B. Dossou, L. C. Botten, A. A. Asatryan, C. G. Poulton, C. M. de Sterke, and R. C. McPhedran, “Modal analysis of enhanced absorption in silicon nanowire arrays,” Opt. Express 19, A1067–A1081 (2011). [CrossRef] [PubMed]

] and numerous cost-effective fabrication methods are under development [8

8. K. Seo, Y. J. Yu, P. Duane, W. Zhu, H. Park, M. Wober, and K. B. Crozier, “Si Microwire Solar Cells: Improved Efficiency with a Conformal SiO2Layer,” Nano 7, 5539–5545 (2013).

13

13. O. Gunawan, K. Wang, B. Fallahazad, Y. Zhang, E. Tutuc, and S. Guha, “High performance wire-array silicon solar cells,” Prog. Photovolt. Res. Appl. 19, 307–312 (2010). [CrossRef]

].

It is known that the absorption of NW arrays is increased when they are arranged in aperiodic lattices, which may occur through random fabrication variations or may be designed on purpose [14

14. H. Bao and X. Ruan, “Optical absorption enhancement in disordered vertical silicon nanowire arrays for photo-voltaic applications,” Opt. Lett. 35, 3378–3380 (2010). [CrossRef] [PubMed]

16

16. C. Lin and M. L. Povinelli, “Optimal design of aperiodic, vertical silicon nanowire structures for photovoltaics,” Opt. Express 19, A1148–A1154 (2011). [CrossRef] [PubMed]

]. Studying such aperiodic arrays is difficult due to the large parameter space and because of the broadband nature of the photovoltaic conversion process. The origin of the absorption enhancement of aperiodic arrays has therefore remained unclear.

Here we avoid the difficulties of large parameter spaces by focussing on a common underlying property of aperiodic arrays: the localised clustering of NWs with reduced gaps between adjacent NWs. This approach provides physical insights that apply to all aperiodic arrays; disordered or designed. The importance of clustered NWs was highlighted by Lin and Povinelli [16

16. C. Lin and M. L. Povinelli, “Optimal design of aperiodic, vertical silicon nanowire structures for photovoltaics,” Opt. Express 19, A1148–A1154 (2011). [CrossRef] [PubMed]

], who showed strong field concentrations within NW clusters at peak absorption wavelengths.

2. Absorption Enhancement

We begin our study with clusters of 4 NWs of radius a, as shown in the inset of Fig. 1. The NWs are initially periodically spaced in a square lattice with period d/2, in a unit cell of width d, before being gradually shifted towards the cell’s centre. As soon as the NWs are moved, the periodicity of the structure becomes d. We measure the clustering level l, as a fraction of the maximum distance the NWs can be moved before touching at t = (d/4 − a). The gap between the NW surfaces is therefore g = (1−l)×(d/2−2a) (see Fig. 1 inset). Throughout the clustering process the volume fraction of the array is constant, f = πa2/(d/2)2. Our simulations are carried out using a combination of the Finite Element Method (FEM) and the scattering matrix method as described earlier [17

17. K. B. Dossou, L. C. Botten, A. A. Asatryan, B. C. P. Sturmberg, M. A. Byrne, C. G. Poulton, R. C. McPhedran, and C. M. de Sterke, “Modal formulation for diffraction by absorbing photonic crystal slabs,” J. Opt. Soc. Am. A 29, 817–831 (2012). [CrossRef]

]. This calculation in the Bloch mode basis is numerically efficient and provides convenient access to physical quantities that are inaccessible in other methods, thereby allowing for greater insights. Throughout this work the absorption spectra are calculated for normally incidence radiation from the short wavelength edge of the solar spectrum at 310 nm to the band-edge of silicon at 1127 nm.

Fig. 1 Absorption spectra of NW arrays with a = 31 nm, f = 30%, with increasing clustering l = 0.0, 0.2, 0.5, 0.8, 0.95. Inset shows cluster geometry where a is the NW radius, d is the unit cell dimension, g is the gap size between NW surfaces, and t is the distance a NW can be moved before touching its neighbour.

In Fig. 1 we show the absorption spectra for NW arrays with radius a = 31 nm in a unit cell of d = 200 nm (f = 30% silicon) as they are clustered with increasing l from 0.0 to 0.95. Note that the absorption increases with increasing clustering level of the arrays, particularly across the λ = 500 − 600 nm range. The parameters of Fig. 1 are chosen to best illustrate the effect of clustering with clear spectra, rather than to achieve high photovoltaic efficiencies. Nonetheless, the increase in total absorption is significant. At λ = 550 nm the absorptance increases from 0.16 in the unclustered array to 0.46 at l = 0.95. The numerous sharp absorption peaks in this range indicate that resonant modes are excited in the clusters that are not present in the equivalent unclustered array (blue curve).

We quantify the efficiency enhancement produced through the ultimate efficiency, η[18

18. W. Shockley and H. J. Queisser, “Detailed balance Limit of Efficiency of p-n Junction Solar Cells,” J. Appl. Phys. 32, 510–519 (1961). [CrossRef]

], which weights the absorption spectrum by the solar spectrum and by the maximum fraction of energy which can be harvested. In Fig. 2 we show the ultimate efficiency enhancement δη = ηclusteredηunclustered for two intersecting slices through the clustered array parameter-space of a, f, l. Figure 2(a) shows δη for arrays with fixed radius arranged with a range of f, while in Fig. 3(b) the radii are varied and f = 50% throughout. This volume fraction produces near-optimal efficiencies when large radii are used, as in Fig. 2(b), and is consistent with multiple studies [6

6. C. Lin and M. L. Povinelli, “Optical absorption enhancement in silicon nanowire arrays with a large lattice constant for photovoltaic applications,” Opt. Express 17, 19371–19381 (2009). [CrossRef] [PubMed]

, 16

16. C. Lin and M. L. Povinelli, “Optimal design of aperiodic, vertical silicon nanowire structures for photovoltaics,” Opt. Express 19, A1148–A1154 (2011). [CrossRef] [PubMed]

, 19

19. B. C. P. Sturmberg, K. B. Dossou, L. C. Botten, A. A. Asatryan, C. G. Poulton, R. C. McPhedran, and C. Martijn de Sterke, “Nanowire array photovoltaics: Radial disorder versus design for optimal efficiency,” Appl. Phys. Lett. 101, 173902 (2012). [CrossRef]

]. When a = 239 and l = 0.95 in Fig. 2(b) the efficiency is η = 22.54%.

Fig. 2 Ultimate efficiency increase δη versus the gap between NW surfaces. (a) Arrays of different volume fractions with fixed NW radius a = 31 nm. (b) Arrays with different NW radii but fixed volume fraction, f = 50%.
Fig. 3 (a)–(d) Electric field vector of Bloch modes where colour and length indicate the field strength at the arrows’ origin. (a) Fundamental mode of the unclustered array; (b) CKM of unclustered array; (c) CKM with l = 0.5; (d) CKM with l = 0.8. (e)–(h) Bloch mode energy Re(ε)|E|2 where red and blue indicates high and low energy density, respectively. (e) Fundamental mode of the unclustered array; (f) CKM with l = 0.5; (g) CKM with l = 0.8; (h) KM of an array with twice the radius, i.e., a = 62 nm. For all figures λ = 550 nm, d = 200 nm and in (a)–(g) a = 31 nm.

In Figs. 2(a),(b) the efficiency monotonically increases with decreasing gap size, moving right to left, but substantial enhancements only occur once the NWs are tightly clustered, typically to a gap distance of less than 20% of the NW radius. Calculations with different volume fractions and radii, not shown here, show that these observations are true in general; accordingly all NW arrays benefit from increased efficiency with increased clustering, while substantial enhancements are only achieved for small gaps. The enhancement in the ultimate efficiency η can be almost 5% for the parameters in Fig. 2, a relative improvement of 85%.

3. Excitation of cluster modes

To explain the existence of the second bright mode we show Bloch mode electric field vectors in Fig. 3(a)–(d). Here both colour and length indicate the field strength at the arrows’ origin. For the unclustered array we show the fundamental bright mode in Fig. 3(a) and a dark mode in Fig. 3(b). The bright mode has all strong field components directed from left to right, while the dark mode has field components of equal strength oriented in opposite directions. The overlap integral with a plane wave therefore vanishes for the dark mode of Fig. 3(b). In contrast it is non-zero for the bright mode of Fig. 3(a) for horizontally polarised incoming light. Vertically polarized plane waves couple to a vertically oriented mode degenerate with that in Fig. 3(a).

Once the NWs are clustered, the symmetry of the dark mode of Fig. 3(b) is lowered as the fields between the cylinders strengthen, and those on the outside of the cluster weaken. As shown in Figs. 3(c) and (d), this results in an electric field that is predominantly oriented from left to right, allowing for coupling to an incoming field. The asymmetry strengthens with increasing clustering, leading to stronger coupling. This is shown in Fig. 4 where the coupling coefficient of the incident plane wave to the fundamental mode at λ = 550 nm decreases with decreasing gap (blue) as the coupling into the second bright mode increases (green).

Fig. 4 Coupling coefficient (left axis) of the incident plane wave to the fundamental mode (blue upward triangles) and the second bright mode (green downward triangles) versus gap between NW surfaces. The ultimate efficiency η of the clusters is also shown (red circles and right axis).

To be highly absorbing a Bloch mode must not only couple to the incident field but also concentrate its energy within the absorbing material [7

7. B. C. P. Sturmberg, K. B. Dossou, L. C. Botten, A. A. Asatryan, C. G. Poulton, C. M. de Sterke, and R. C. McPhedran, “Modal analysis of enhanced absorption in silicon nanowire arrays,” Opt. Express 19, A1067–A1081 (2011). [CrossRef] [PubMed]

]. In Figs. 3(e)–(h) we show the energy concentration Re(ε)|E|2 of modes at λ = 550 nm. At this wavelength, the bright mode of the unclustered array in Fig. 3(e) has little energy within the silicon, resulting in the low absorption of this structure (see Fig. 1). In contrast, the bright mode of the clustered arrays has significant energy in the silicon, as shown in Fig. 3(f) and (g) for l = 0.5 and 0.8, respectively. When combined with strong coupling this produces the enhanced absorption of Fig. 1. Consistent with this, the red curve in Fig. 4 shows that the clusters’ ultimate efficiency improves as the coupling into the second bright mode increases. Though the coupling is shown for a single wavelength, this is a broadband effect which occurs for all wavelengths below the bright mode’s cut-off at λ = 600 nm. The absorption of this mode, which is associated with all absorption peaks in Fig. 1, is enhanced by its low group velocity and strong Fabry-Pérot resonances between the top and bottom interfaces. Both of these effects are characteristics of the Key Modes (KMs) of periodic NW arrays [7

7. B. C. P. Sturmberg, K. B. Dossou, L. C. Botten, A. A. Asatryan, C. G. Poulton, C. M. de Sterke, and R. C. McPhedran, “Modal analysis of enhanced absorption in silicon nanowire arrays,” Opt. Express 19, A1067–A1081 (2011). [CrossRef] [PubMed]

]. We therefore refer to the bright mode of the cluster as a Clustered Key Mode (CKM). In Fig. 3(h) we show how the energy concentration of the CKM is similar to the KM of a periodic array with doubled radius NWs (a = 62 nm).

4. Discussion and conclusion

Our results in Fig. 2, and others not shown here, show that, compared to the unclustered periodic case with period d/2, clustering always increases absorption. The degree to which it does so depends not only on the cluster parameters but also on the array’s global properties such as the value of d. It is therefore difficult to draw specific quantitative conclusions. However, we have observed that the clustering only significantly enhances the absorption when the NW are brought close, typically such that the gap between their surfaces is approximately 20% of the NW radius. This is consistent with Fig. 4(b) of Lin and Povinelli [16

16. C. Lin and M. L. Povinelli, “Optimal design of aperiodic, vertical silicon nanowire structures for photovoltaics,” Opt. Express 19, A1148–A1154 (2011). [CrossRef] [PubMed]

]. As a consequence of this, the random variations in NW positioning that arise naturally during fabrication of periodic arrays are unlikely to be sufficient to exploit the efficiency enhancement potential of aperiodic arrays. Rather, the potential of aperiodic arrays may only be harnessed by purpose fabrication of aperiodic arrays with closely clustered NWs.

Acknowledgments

We acknowledge helpful discussions with Dr Kylie Catchpole and Dr Thomas White. This work was supported by the Australian Renewable Energy Agency, and by the Australian Research Council (ARC) Center of Excellence for Ultrahigh bandwidth Devices for Optical Systems (Project CE110001018) and by an ARC Discovery Grant. Computation resources were provided by the National Computational Infrastructure, Australia.

References and links

1.

K. R. Catchpole, “Nanostructures in photovoltaics,” Phil. Trans. R. Soc. A 364, 3493–503 (2006). [CrossRef] [PubMed]

2.

L. Tsakalakos, “Nanostructures for photovoltaics,” Mater. Sci. Eng. R 62, 175–189 (2008). [CrossRef]

3.

O. L. Muskens, J. G. Rivas, R. E. Algra, E. P. A. M. Bakkers, and A. Lagendijk, “Design of Light Scattering in Nanowire Materials for Photovoltaic Applications,” Nano Lett. 8, 2–6 (2008). [CrossRef]

4.

S.-F. Leung, M. Yu, Q. Lin, K. Kwon, K.-L. Ching, L. Gu, K. Yu, and Z. Fan, “Efficient photon capturing with ordered three-dimensional nanowell arrays,” Nano Lett. 12, 3682–3689 (2012). [CrossRef] [PubMed]

5.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010). [CrossRef] [PubMed]

6.

C. Lin and M. L. Povinelli, “Optical absorption enhancement in silicon nanowire arrays with a large lattice constant for photovoltaic applications,” Opt. Express 17, 19371–19381 (2009). [CrossRef] [PubMed]

7.

B. C. P. Sturmberg, K. B. Dossou, L. C. Botten, A. A. Asatryan, C. G. Poulton, C. M. de Sterke, and R. C. McPhedran, “Modal analysis of enhanced absorption in silicon nanowire arrays,” Opt. Express 19, A1067–A1081 (2011). [CrossRef] [PubMed]

8.

K. Seo, Y. J. Yu, P. Duane, W. Zhu, H. Park, M. Wober, and K. B. Crozier, “Si Microwire Solar Cells: Improved Efficiency with a Conformal SiO2Layer,” Nano 7, 5539–5545 (2013).

9.

D. B. Turner-Evans, C. T. Chen, H. Emmer, W. E. McMahon, and H. A. Atwater, “Optoelectronic analysis of multijunction wire array solar cells,” J. Appl. Phys. 114, 014501 (2013). [CrossRef]

10.

J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Aberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgström, “InP Nanowire Array Solar Cells Achieving 13.8% Efficiency by Exceeding the Ray Optics Limit,” Science 339, 1057–1060 (2013). [CrossRef] [PubMed]

11.

E. C. Garnett, M. L. Brongersma, Y. Cui, and M. D. McGehee, “Nanowire Solar Cells,” Annu. Rev. Mater. Res. 41, 269–295 (2011). [CrossRef]

12.

M.-P. Lu, M.-Y. Lu, and L.-J. Chen, “p-Type ZnO nanowires: From synthesis to nanoenergy,” Nano Energy 1, 247–258 (2012). [CrossRef]

13.

O. Gunawan, K. Wang, B. Fallahazad, Y. Zhang, E. Tutuc, and S. Guha, “High performance wire-array silicon solar cells,” Prog. Photovolt. Res. Appl. 19, 307–312 (2010). [CrossRef]

14.

H. Bao and X. Ruan, “Optical absorption enhancement in disordered vertical silicon nanowire arrays for photo-voltaic applications,” Opt. Lett. 35, 3378–3380 (2010). [CrossRef] [PubMed]

15.

Q. G. Du, C. H. Kam, H. V. Demir, H. Y. Yu, and X. W. Sun, “Broadband absorption enhancement in randomly positioned silicon nanowire arrays for solar cell applications,” Opt. Lett. 36, 1884–1886 (2011). [CrossRef] [PubMed]

16.

C. Lin and M. L. Povinelli, “Optimal design of aperiodic, vertical silicon nanowire structures for photovoltaics,” Opt. Express 19, A1148–A1154 (2011). [CrossRef] [PubMed]

17.

K. B. Dossou, L. C. Botten, A. A. Asatryan, B. C. P. Sturmberg, M. A. Byrne, C. G. Poulton, R. C. McPhedran, and C. M. de Sterke, “Modal formulation for diffraction by absorbing photonic crystal slabs,” J. Opt. Soc. Am. A 29, 817–831 (2012). [CrossRef]

18.

W. Shockley and H. J. Queisser, “Detailed balance Limit of Efficiency of p-n Junction Solar Cells,” J. Appl. Phys. 32, 510–519 (1961). [CrossRef]

19.

B. C. P. Sturmberg, K. B. Dossou, L. C. Botten, A. A. Asatryan, C. G. Poulton, R. C. McPhedran, and C. Martijn de Sterke, “Nanowire array photovoltaics: Radial disorder versus design for optimal efficiency,” Appl. Phys. Lett. 101, 173902 (2012). [CrossRef]

OCIS Codes
(040.5350) Detectors : Photovoltaic
(050.0050) Diffraction and gratings : Diffraction and gratings
(350.6050) Other areas of optics : Solar energy
(350.4238) Other areas of optics : Nanophotonics and photonic crystals
(310.6628) Thin films : Subwavelength structures, nanostructures

ToC Category:
Light Trapping for Photovoltaics

History
Original Manuscript: July 31, 2013
Revised Manuscript: September 13, 2013
Manuscript Accepted: September 15, 2013
Published: September 26, 2013

Citation
Björn C. P. Sturmberg, Kokou B. Dossou, Lindsay C. Botten, Ara A. Asatryan, Christopher G. Poulton, Ross C. McPhedran, and C. Martijn de Sterke, "Absorption enhancing proximity effects in aperiodic nanowire arrays," Opt. Express 21, A964-A969 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-S6-A964


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References

  1. K. R. Catchpole, “Nanostructures in photovoltaics,” Phil. Trans. R. Soc. A364, 3493–503 (2006). [CrossRef] [PubMed]
  2. L. Tsakalakos, “Nanostructures for photovoltaics,” Mater. Sci. Eng. R62, 175–189 (2008). [CrossRef]
  3. O. L. Muskens, J. G. Rivas, R. E. Algra, E. P. A. M. Bakkers, and A. Lagendijk, “Design of Light Scattering in Nanowire Materials for Photovoltaic Applications,” Nano Lett.8, 2–6 (2008). [CrossRef]
  4. S.-F. Leung, M. Yu, Q. Lin, K. Kwon, K.-L. Ching, L. Gu, K. Yu, and Z. Fan, “Efficient photon capturing with ordered three-dimensional nanowell arrays,” Nano Lett.12, 3682–3689 (2012). [CrossRef] [PubMed]
  5. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater.9, 205–213 (2010). [CrossRef] [PubMed]
  6. C. Lin and M. L. Povinelli, “Optical absorption enhancement in silicon nanowire arrays with a large lattice constant for photovoltaic applications,” Opt. Express17, 19371–19381 (2009). [CrossRef] [PubMed]
  7. B. C. P. Sturmberg, K. B. Dossou, L. C. Botten, A. A. Asatryan, C. G. Poulton, C. M. de Sterke, and R. C. McPhedran, “Modal analysis of enhanced absorption in silicon nanowire arrays,” Opt. Express19, A1067–A1081 (2011). [CrossRef] [PubMed]
  8. K. Seo, Y. J. Yu, P. Duane, W. Zhu, H. Park, M. Wober, and K. B. Crozier, “Si Microwire Solar Cells: Improved Efficiency with a Conformal SiO2Layer,” Nano7, 5539–5545 (2013).
  9. D. B. Turner-Evans, C. T. Chen, H. Emmer, W. E. McMahon, and H. A. Atwater, “Optoelectronic analysis of multijunction wire array solar cells,” J. Appl. Phys.114, 014501 (2013). [CrossRef]
  10. J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Aberg, M. H. Magnusson, G. Siefer, P. Fuss-Kailuweit, F. Dimroth, B. Witzigmann, H. Q. Xu, L. Samuelson, K. Deppert, and M. T. Borgström, “InP Nanowire Array Solar Cells Achieving 13.8% Efficiency by Exceeding the Ray Optics Limit,” Science339, 1057–1060 (2013). [CrossRef] [PubMed]
  11. E. C. Garnett, M. L. Brongersma, Y. Cui, and M. D. McGehee, “Nanowire Solar Cells,” Annu. Rev. Mater. Res.41, 269–295 (2011). [CrossRef]
  12. M.-P. Lu, M.-Y. Lu, and L.-J. Chen, “p-Type ZnO nanowires: From synthesis to nanoenergy,” Nano Energy1, 247–258 (2012). [CrossRef]
  13. O. Gunawan, K. Wang, B. Fallahazad, Y. Zhang, E. Tutuc, and S. Guha, “High performance wire-array silicon solar cells,” Prog. Photovolt. Res. Appl.19, 307–312 (2010). [CrossRef]
  14. H. Bao and X. Ruan, “Optical absorption enhancement in disordered vertical silicon nanowire arrays for photo-voltaic applications,” Opt. Lett.35, 3378–3380 (2010). [CrossRef] [PubMed]
  15. Q. G. Du, C. H. Kam, H. V. Demir, H. Y. Yu, and X. W. Sun, “Broadband absorption enhancement in randomly positioned silicon nanowire arrays for solar cell applications,” Opt. Lett.36, 1884–1886 (2011). [CrossRef] [PubMed]
  16. C. Lin and M. L. Povinelli, “Optimal design of aperiodic, vertical silicon nanowire structures for photovoltaics,” Opt. Express19, A1148–A1154 (2011). [CrossRef] [PubMed]
  17. K. B. Dossou, L. C. Botten, A. A. Asatryan, B. C. P. Sturmberg, M. A. Byrne, C. G. Poulton, R. C. McPhedran, and C. M. de Sterke, “Modal formulation for diffraction by absorbing photonic crystal slabs,” J. Opt. Soc. Am. A29, 817–831 (2012). [CrossRef]
  18. W. Shockley and H. J. Queisser, “Detailed balance Limit of Efficiency of p-n Junction Solar Cells,” J. Appl. Phys.32, 510–519 (1961). [CrossRef]
  19. B. C. P. Sturmberg, K. B. Dossou, L. C. Botten, A. A. Asatryan, C. G. Poulton, R. C. McPhedran, and C. Martijn de Sterke, “Nanowire array photovoltaics: Radial disorder versus design for optimal efficiency,” Appl. Phys. Lett.101, 173902 (2012). [CrossRef]

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