## Efficient light management in vertical nanowire arrays for photovoltaics |

Optics Express, Vol. 21, Issue S3, pp. A558-A575 (2013)

http://dx.doi.org/10.1364/OE.21.00A558

Acrobat PDF (1784 KB)

### Abstract

Vertical arrays of direct band gap III-V semiconductor nanowires (NWs) hold the prospect of cheap and efficient next-generation photovoltaics, and guidelines for successful light-management are needed. Here, we use InP NWs as a model system and find, through electrodynamic modeling, general design principles for efficient absorption of sun light in nanowire arrays by systematically varying the nanowire diameter, the nanowire length, and the array period. Most importantly, we discover the existence of specific band-gap dependent diameters, 170 nm and 410 nm for InP, for which the absorption of sun light in the array is optimal, irrespective of the nanowire length. At these diameters, the individual InP NWs of the array absorb light strongly for photon energies just above the band gap energy due to a diameter-tunable nanophotonic resonance, which shows up also for other semiconductor materials of the NWs. Furthermore, we find that for maximized absorption of sun light, the optimal period of the array increases with nanowire length, since this decreases the insertion reflection losses.

© 2013 OSA

## 1. Introduction

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4. H. Goto, K. Nosaki, K. Tomioka, S. Hara, K. Hiruma, J. Motohisa, and T. Fukui, “Growth of core–shell InP nanowires for photovoltaic application by selective-area metal organic vapor phase epitaxy,” Appl. Phys. Express **2**, 035004 (2009). [CrossRef]

15. N. Anttu, K. Namazi, P. Wu, P. Yang, H. Xu, H. Q. Xu, and U. Håkanson, “Drastically increased absorption in vertical semiconductor nanowire arrays: A non-absorbing dielectric shell makes the difference,” Nano Res. **5**(12), 863–874 (2012). [CrossRef]

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3. M. A. Green, “Third generation photovoltaics: Ultra-high conversion efficiency at low cost,” Prog. Photovolt. Res. Appl. **9**(2), 123–135 (2001). [CrossRef]

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18. C. P. T. Svensson, T. Mårtensson, J. Trägårdh, C. Larsson, M. Rask, D. Hessman, L. Samuelson, and J. Ohlsson, “Monolithic GaAs/InGaP nanowire light emitting diodes on silicon,” Nanotechnology **19**(30), 305201 (2008). [CrossRef] [PubMed]

19. J. Wallentin and M. T. Borgstrom, “Doping of semiconductor nanowires,” J. Mater. Res. **26**(17), 2142–2156 (2011). [CrossRef]

4. H. Goto, K. Nosaki, K. Tomioka, S. Hara, K. Hiruma, J. Motohisa, and T. Fukui, “Growth of core–shell InP nanowires for photovoltaic application by selective-area metal organic vapor phase epitaxy,” Appl. Phys. Express **2**, 035004 (2009). [CrossRef]

6. M. T. Borgström, J. Wallentin, M. Heurlin, S. Fält, P. Wickert, J. Leene, M. H. Magnusson, K. Deppert, and L. Samuelson, “Nanowires with promise for photovoltaics,” IEEE J. Sel. Top. Quantum Electron. **17**(4), 1050–1061 (2011). [CrossRef]

12. J. A. Czaban, D. A. Thompson, and R. R. LaPierre, “GaAs core--shell nanowires for photovoltaic applications,” Nano Lett. **9**(1), 148–154 (2009). [CrossRef] [PubMed]

20. L. Hu and G. Chen, “Analysis of optical absorption in silicon nanowire arrays for photovoltaic applications,” Nano Lett. **7**(11), 3249–3252 (2007). [CrossRef] [PubMed]

21. P. M. Wu, N. Anttu, H. Q. Xu, L. Samuelson, and M.-E. Pistol, “Colorful InAs nanowire arrays: From strong to weak absorption with geometrical tuning,” Nano Lett. **12**(4), 1990–1995 (2012). [CrossRef] [PubMed]

20. L. Hu and G. Chen, “Analysis of optical absorption in silicon nanowire arrays for photovoltaic applications,” Nano Lett. **7**(11), 3249–3252 (2007). [CrossRef] [PubMed]

*D*, the nanowire length

*L*, and the period

*p*of the array, need to be taken into account when studying the absorption of solar energy.

5. N. Anttu and H. Q. Xu, “Coupling of light into nanowire arrays and subsequent absorption,” J. Nanosci. Nanotechnol. **10**(11), 7183–7187 (2010). [CrossRef] [PubMed]

8. L. Wen, Z. Zhao, X. Li, Y. Shen, H. Guo, and Y. Wang, “Theoretical analysis and modeling of light trapping in high efficicency GaAs nanowire array solar cells,” Appl. Phys. Lett. **99**(14), 143116 (2011). [CrossRef]

11. J. Kupec, R. L. Stoop, and B. Witzigmann, “Light absorption and emission in nanowire array solar cells,” Opt. Express **18**(26), 27589–27605 (2010). [CrossRef] [PubMed]

13. Z. Gu, P. Prete, N. Lovergine, and B. Nabet, “On optical properties of GaAs and GaAs/AlGaAs core-shell periodic nanowire arrays,” J. Appl. Phys. **109**(6), 064314–064316 (2011). [CrossRef]

14. N. Huang, C. Lin, and M. L. Povinelli, “Broadband absorption of semiconductor nanowire arrays for photovoltaic applications,” J. Opt. **14**(2), 024004 (2012). [CrossRef]

20. L. Hu and G. Chen, “Analysis of optical absorption in silicon nanowire arrays for photovoltaic applications,” Nano Lett. **7**(11), 3249–3252 (2007). [CrossRef] [PubMed]

22. C. Lin and M. L. Povinelli, “Optical absorption enhancement in silicon nanowire arrays with a large lattice constant for photovoltaic applications,” Opt. Express **17**(22), 19371–19381 (2009). [CrossRef] [PubMed]

24. J. Li, H. Yu, and Y. Li, “Solar energy harnessing in hexagonally arranged Si nanowire arrays and effects of array symmetry on optical characteristics,” Nanotechnology **23**(19), 194010 (2012). [CrossRef] [PubMed]

*L*= 2000 nm was fixed,

*D*= 180 nm and

*p*= 400 nm yielded efficient absorption in the NWs [5

5. N. Anttu and H. Q. Xu, “Coupling of light into nanowire arrays and subsequent absorption,” J. Nanosci. Nanotechnol. **10**(11), 7183–7187 (2010). [CrossRef] [PubMed]

*D*= 180 nm and

*p*= 360 nm [11

11. J. Kupec, R. L. Stoop, and B. Witzigmann, “Light absorption and emission in nanowire array solar cells,” Opt. Express **18**(26), 27589–27605 (2010). [CrossRef] [PubMed]

11. J. Kupec, R. L. Stoop, and B. Witzigmann, “Light absorption and emission in nanowire array solar cells,” Opt. Express **18**(26), 27589–27605 (2010). [CrossRef] [PubMed]

*p*< 100 nm [20

**7**(11), 3249–3252 (2007). [CrossRef] [PubMed]

*p*was relaxed,

*D*= 540 nm and

*p*= 600nm were identified as an appropriate choice for fixed

*L*= 2330 nm [22

22. C. Lin and M. L. Povinelli, “Optical absorption enhancement in silicon nanowire arrays with a large lattice constant for photovoltaic applications,” Opt. Express **17**(22), 19371–19381 (2009). [CrossRef] [PubMed]

*D*/

*p*≈0.8 was given as a general guideline for

*L*= 5000 nm [23

23. J. Li, H. Yu, S. M. Wong, X. Li, G. Zhang, P. G.-Q. Lo, and D.-L. Kwong, “Design guidelines of periodic Si nanowire arrays for solar cell application,” Appl. Phys. Lett. **95**(24), 243113 (2009). [CrossRef]

*D*= 500 nm and

*p*= 600 nm were indicated as a suitable choice for varying

*L*[24

24. J. Li, H. Yu, and Y. Li, “Solar energy harnessing in hexagonally arranged Si nanowire arrays and effects of array symmetry on optical characteristics,” Nanotechnology **23**(19), 194010 (2012). [CrossRef] [PubMed]

*D*= 180 nm was identified as an optimal choice [8

8. L. Wen, Z. Zhao, X. Li, Y. Shen, H. Guo, and Y. Wang, “Theoretical analysis and modeling of light trapping in high efficicency GaAs nanowire array solar cells,” Appl. Phys. Lett. **99**(14), 143116 (2011). [CrossRef]

13. Z. Gu, P. Prete, N. Lovergine, and B. Nabet, “On optical properties of GaAs and GaAs/AlGaAs core-shell periodic nanowire arrays,” J. Appl. Phys. **109**(6), 064314–064316 (2011). [CrossRef]

*p*< 150 nm, in which case the value of

*D*= 180 nm was not reached. However, regarding a recommendation for

*p*when

*D*,

*p*, and

*L*were varied simultaneously [14

14. N. Huang, C. Lin, and M. L. Povinelli, “Broadband absorption of semiconductor nanowire arrays for photovoltaic applications,” J. Opt. **14**(2), 024004 (2012). [CrossRef]

*p*as a function of

*L*were identified. Clearly, there exists a need for a detailed study to elucidate the optimum choice for

*D*and

*p*and their dependence on the nanowire length.

25. N. Anttu and H. Q. Xu, “Scattering matrix method for optical excitation of surface plasmons in metal films with periodic arrays of subwavelength holes,” Phys. Rev. B **83**(16), 165431 (2011). [CrossRef]

26. J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Åberg, 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**(6123), 1057–1060 (2013). [CrossRef] [PubMed]

21. P. M. Wu, N. Anttu, H. Q. Xu, L. Samuelson, and M.-E. Pistol, “Colorful InAs nanowire arrays: From strong to weak absorption with geometrical tuning,” Nano Lett. **12**(4), 1990–1995 (2012). [CrossRef] [PubMed]

_{2}O

_{3}–coated [15

15. N. Anttu, K. Namazi, P. Wu, P. Yang, H. Xu, H. Q. Xu, and U. Håkanson, “Drastically increased absorption in vertical semiconductor nanowire arrays: A non-absorbing dielectric shell makes the difference,” Nano Res. **5**(12), 863–874 (2012). [CrossRef]

26. J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Åberg, 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**(6123), 1057–1060 (2013). [CrossRef] [PubMed]

_{11}and HE

_{12}waveguide modes of individual NWs. At the diameter of 170 nm, the HE

_{11}mode is excited strongly by the incident light and absorbed strongly in the NWs at wavelengths close to the band gap wavelength of InP, while at the diameter of 410 nm, the HE

_{12}mode is excited and absorbed strongly at wavelengths close to the band gap wavelength of InP. We show that this mechanism of strong absorption in the NW array for wavelengths close to the band gap wavelength by a suitable choice of the NW diameter is applicable also for other direct band gap semiconductors. It constitutes a general approach for maximizing the absorption of sun light in nanowire arrays, and we have therefore identified a simple, general way to efficiently manage light in nanowire arrays for PV applications.

## 2. Results and discussion

*D*and length

*L*(see Fig. 1 for a schematic). The period of the NW array is

*p*in both the

*x*and the

*y*direction. The NWs stand on top of an InP substrate that is considered to be (optically) infinitely thick, and there is air between and on top of the NWs. A plane wave of light of free-space wavelength

*λ*is incident toward the NW array from the air side. We study a large range of NW array parameters with 0 <

*p*< 1000 nm, 0 <

*D*< 1000 nm, and 500 nm <

*L*< 8000 nm. We will only consider the case in which light is incident normally to the NW array, i.e., with

*k*

_{x}=

*k*

_{y}= 0, which is relevant for direct sunlight illumination where light is incident from a limited solid angle centered around normal incidence. We note that the absorption properties of NW arrays do not appear to be strongly dependent on the type of array symmetry [24

24. J. Li, H. Yu, and Y. Li, “Solar energy harnessing in hexagonally arranged Si nanowire arrays and effects of array symmetry on optical characteristics,” Nanotechnology **23**(19), 194010 (2012). [CrossRef] [PubMed]

*A*(

*λ*) of the NW array, assuming that the active region of the solar cell consists of the NWs only, as is often the case in experiments [4

4. H. Goto, K. Nosaki, K. Tomioka, S. Hara, K. Hiruma, J. Motohisa, and T. Fukui, “Growth of core–shell InP nanowires for photovoltaic application by selective-area metal organic vapor phase epitaxy,” Appl. Phys. Express **2**, 035004 (2009). [CrossRef]

**17**(4), 1050–1061 (2011). [CrossRef]

12. J. A. Czaban, D. A. Thompson, and R. R. LaPierre, “GaAs core--shell nanowires for photovoltaic applications,” Nano Lett. **9**(1), 148–154 (2009). [CrossRef] [PubMed]

26. J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Åberg, 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**(6123), 1057–1060 (2013). [CrossRef] [PubMed]

*η*[5

5. N. Anttu and H. Q. Xu, “Coupling of light into nanowire arrays and subsequent absorption,” J. Nanosci. Nanotechnol. **10**(11), 7183–7187 (2010). [CrossRef] [PubMed]

14. N. Huang, C. Lin, and M. L. Povinelli, “Broadband absorption of semiconductor nanowire arrays for photovoltaic applications,” J. Opt. **14**(2), 024004 (2012). [CrossRef]

**7**(11), 3249–3252 (2007). [CrossRef] [PubMed]

27. W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. **32**(3), 510–519 (1961). [CrossRef]

*A*(

*λ*) is defined as the fraction of incident intensity of a given (free-space) wavelength

*λ*absorbed in the NWs, while the ultimate efficiency

*η*is defined as [5

**10**(11), 7183–7187 (2010). [CrossRef] [PubMed]

**14**(2), 024004 (2012). [CrossRef]

**7**(11), 3249–3252 (2007). [CrossRef] [PubMed]

27. W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. **32**(3), 510–519 (1961). [CrossRef]

*λ*

_{bg}= 925 nm is the wavelength of photons with an energy equal to 1.34 eV, the band gap energy

*E*

_{g}of InP [28

28. I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for III-V compound semiconductors and their alloys,” J. Appl. Phys. **89**(11), 5815–5875 (2001). [CrossRef]

*I*(

_{AM1.5}*λ*) is the AM1.5 direct and circumsolar intensity spectrum [29

29. Air Mass 1.5 Spectra, American Society for Testing and Materials, http://rredc.nrel.gov/solar/spectra/am1.5/.

27. W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. **32**(3), 510–519 (1961). [CrossRef]

*η*, we take into account the band gap of InP and the thermalization losses in the energies of the electrons and the holes that are created by photons with an energy greater than the band gap energy. These electrons and holes lose their excess energies by thermalization and relax, respectively, to the bottom of the conduction band and the top of the valence band. The AM1.5 spectrum is shown in Fig. 2 together with the maximum intensity usable in an InP solar cell after taking the thermalization losses into account. The intensity spectrum

*I*shows a peak at

_{AM1.5}*λ*≈500 nm, drops rapidly for shorter wavelengths, and reaches a zero value at

*λ*< 300 nm. In the spectrum of the maximum intensity usable, the peak is dampened [see Fig. 2(b)] due to the thermalization losses. We are thus aiming to maximize the absorptance of the NWs for 300 nm <

*λ*< 925 nm. The maximum possible ultimate efficiency for InP,

*η*

_{max}= 0.463, is obtained when all this usable intensity is absorbed [i.e., assuming

*A*(

*λ*) = 1 for 300 nm <

*λ*< 925 nm in Eq. (1)].

25. N. Anttu and H. Q. Xu, “Scattering matrix method for optical excitation of surface plasmons in metal films with periodic arrays of subwavelength holes,” Phys. Rev. B **83**(16), 165431 (2011). [CrossRef]

*λ*. We calculate the transmittance

*T*(

*λ*) of light into the InP substrate and the reflectance

*R*(

*λ*) of light back into the air on top of the NWs. The absorptance of the NWs is obtained from the energy balance equation:

*A*(

*λ*) = 1 -

*R*(

*λ*) -

*T*(

*λ*). We use tabulated values of the wavelength dependent (complex-valued) refractive index

*n*

_{InP}of the InP [30] and for air

*n*= 1 is used.

*A*(

*λ*) for four NW arrays with the same fixed values of

*p*= 680 nm and

*L*= 2000 nm, but with four different NW diameters of

*D*= 100, 177, 221, and 441 nm. Here, we see very different characteristics of

*A*(

*λ*) at different values of

*D*. The ultimate efficiency

*η*of the system as a function of

*D*is shown in Fig. 3(b). Here,

*η*shows a two-peak structure. First, we find that the NW array with

*D*= 100 nm shows a low value of

*η*. The main reason for this is the low absorptance at long wavelengths of

*λ*> 700 nm. The NW array with

*D*= 177 nm gives a peak value of

*η*. This results from the broad peak of

*A*(

*λ*) at wavelengths close to the band gap wavelength

*λ*

_{bg}= 925 nm. For

*D*= 221 nm,

*η*is at a dip. The main reason for this decrease in

*η*when compared to the case for

*D*= 177 nm is that the peak of

*A*(

*λ*) close to the band gap seen for

*D*= 177 nm has disappeared in the case of

*D*= 221 nm. The NW array with

*D*= 441 nm shows a second peak in

*η*. This clearly results from the high values of

*A*(

*λ*) for all wavelengths of

*λ*<

*λ*

_{bg}. However, the above results are found for the specific case of the NW array with period

*p*= 680 nm and NW length

*L*= 2000 nm. To be able to make more general conclusions, we turn to study the dependencies of

*η*on all the three geometrical parameters

*D*,

*p*, and

*L*.

*L*= 500 nm. The ultimate efficiency

*η*is shown in Fig. 4 as a function of NW diameter

*D*and array period

*p*. The most important discovery from Fig. 4 is the presence of the two local maxima of

*η*: one maximum of

*η*

_{1}= 0.344 located at

*D*

_{1}= 191 nm and

*p*

_{1}= 251 nm, and another one of

*η*

_{2}= 0.341 located at

*D*

_{2}= 438 nm and

*p*

_{2}= 530 nm. It is also seen that for

*D*< 100 nm, the value of

*η*remains low for all values of

*p*until

*p*is so small that the NWs start to touch each other at

*p*=

*D*. The inset in Fig. 4 shows

*η*as a function of

*D*for

*p*= 530 nm and

*L*= 500 nm. Here, the local maximum of

*η*

_{2}= 0.341 is found for the NW diameter

*D*

_{2}= 438 nm. A peak can be seen also at

*D*≈190 nm, but this peak does not correspond to

*η*

_{1}since

*η*increases with decreasing

*p*to reach the local maximum of

*η*

_{1}= 0.344 at

*p*

_{1}= 251 nm and

*D*

_{1}= 191 nm.

*L*= 2000 nm (see Fig. 5),

*η*shows again two local maxima: the first one of

*η*

_{1}= 0.431 is located at

*D*

_{1}= 184 nm and

*p*

_{1}= 340 nm, and the second one of

*η*

_{2}= 0.410 is located at

*D*

_{2}= 441 nm and

*p*

_{2}= 680 nm [see Fig. 3(a), curve (iv) for the

*A*(

*λ*) that gives rise to

*η*

_{2}in Fig. 5]. Thus, compared to the positions of the maxima for

*L*= 500 nm, the diameters

*D*

_{1}and

*D*

_{2}have shifted only by 7 nm and 3 nm, respectively. In contrast, the periods

*p*

_{1}and

*p*

_{2}shift much more. The shift of

*p*

_{1}is 89 nm and the shift of

*p*

_{2}is 150 nm when the value of

*L*is increased from 500 nm to 2000 nm. When the values of

*η*for

*L*= 2000 nm in Fig. 5 are compared to the values of

*η*for

*L*= 500 nm in Fig. 4, we find that the values of

*η*increase for all values of

*D*and

*p*when the length

*L*of the NWs is increased. Specifically, the value of

*η*

_{1}has increased by 0.087 and the value of

*η*

_{2}has increased by 0.069. Similar to the case of

*L*= 500 nm above, we find that the value of

*η*for

*L*= 2000 nm is low for

*D*< 100 nm. The reason for this is that for

*D*< 100 nm the absorptance

*A*is low for

*λ*>700 nm [see Fig. 3(a), curve (i), for the specific case of

*p*= 680 nm and

*D*= 100 nm, and Appendix A for varying

*D*] (we note that this weak absorption can be understood from the electrostatic screening that occurs for normally incident light in small diameter NWs [15

15. N. Anttu, K. Namazi, P. Wu, P. Yang, H. Xu, H. Q. Xu, and U. Håkanson, “Drastically increased absorption in vertical semiconductor nanowire arrays: A non-absorbing dielectric shell makes the difference,” Nano Res. **5**(12), 863–874 (2012). [CrossRef]

31. N. Anttu, “Geometrical optics, electrostatics, and nanophotonic resonances in absorbing nanowire arrays,” Opt. Lett. **38**(5), 730–732 (2013). [CrossRef] [PubMed]

*η*as a function of

*D*for

*p*= 340 nm and

*L*= 2000 nm. Here, the local maximum of

*η*

_{1}= 0.431 is found for the NW diameter

*D*

_{1}= 184 nm.

*η*in detail, we have calculated the values of

*η*

_{1}and

*η*

_{2}, and their positions (

*D*

_{1},

*p*

_{1}) and (

*D*

_{2},

*p*

_{2}) for 500 nm <

*L*< 8000 nm. The results are shown in Fig. 6. Figures 6(a) and 6(b) show that both

*p*

_{1}and

*p*

_{2}increase strongly with increasing

*L*. The value of

*p*

_{1}increases from 251 nm to 470 nm [Fig. 6(a)], and the value of

*p*

_{2}increases from 530 nm to 870 nm [Fig. 6(b)] when

*L*is increased from 500 nm to 8000 nm. The reason for this large increase of both

*p*

_{1}and

*p*

_{2}can be understood as follows. The insertion reflection losses at the top air/NW interface decrease with increasing

*p*(see Appendix D). However, we must note that an increase of the period

*p*(at constant

*L*and

*D*) is expected to weaken the absorption inside the NW array since the amount of absorbing semiconductor material in the NW array is decreased with increasing

*p*. Thus, the resulting dependencies of

*p*

_{1}and

*p*

_{2}on

*L*stem from the simultaneous minimization of the insertion reflection losses and maximization of the absorption inside the NW array. From the large increase in both

*p*

_{1}and

*p*

_{2}with increasing

*L*, we understand that longer NWs absorb so much stronger (than shorter NWs) that a larger period

*p*can be afforded in order to decrease the reflection losses.

*p*

_{1}and

*p*

_{2}on

*L*, we find that the diameters

*D*

_{1}and

*D*

_{2}[Figs. 6(a) and 6(b)], which give the local maxima

*η*

_{1}and

*η*

_{2}, shift only slightly for this large increase in the length

*L*of the NWs. The value of

*D*

_{1}decreases from 191 nm to 171 nm and the value of

*D*

_{2}decreases from 438 nm to 407 nm when the value of

*L*is increased from 500 nm to 8000 nm. Thus, the diameters

*D*

_{1}≈170 nm and

*D*

_{2}≈410 nm optimize the ultimate efficiency, irrespective of the NW length

*L*. We find also in Fig. 6(c) that

*η*

_{1}>

*η*

_{2}for all values of

*L*.

*D*

_{1}and

*D*

_{2}depend only very weakly on

*L*indicates strongly that

*D*

_{1}and

*D*

_{2}are connected to the optical properties of the individual NWs that constitute the NW array. Indeed, we have found that

*D*

_{1}and

*D*

_{2}originate from the properties of the HE

_{11}and HE

_{12}waveguide modes [32, 33

33. B. Wang and P. W. Leu, “Tunable and selective resonant absorption in vertical nanowires,” Opt. Lett. **37**(18), 3756–3758 (2012). [CrossRef] [PubMed]

*D*

_{1}(

*D*

_{2}), the HE

_{11}(HE

_{12}) waveguide mode of the individual NWs is excited strongly by the incident light and absorbed strongly in the NWs for wavelengths close to the band gap wavelength [see Appendix C]. This leads to a large absorptance of the NW array for wavelengths close to the band gap wavelength [see Fig. 7 in Appendix A and Fig. 8(a) in Appendix B]. We note that the absorption coefficient of the InP material is low for photon energies just above the band gap energy. It is therefore not completely surprising that for optimized absorption of the broadband sun light, a tunable absorption resonance should be placed to this close-to-band-gap region. We note that this resonant absorption through the HE

_{1m}modes [33

33. B. Wang and P. W. Leu, “Tunable and selective resonant absorption in vertical nanowires,” Opt. Lett. **37**(18), 3756–3758 (2012). [CrossRef] [PubMed]

34. L. Cao, J. S. White, J.-S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. **8**(8), 643–647 (2009). [CrossRef] [PubMed]

35. M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nat. Mater. **9**(3), 239–244 (2010). [PubMed]

*D*

_{1}and

*D*

_{2}originate from the properties of the optical modes of individual NWs, the results presented here for InP can be readily extended to other direct band gap semiconductors. This can be done with a simple rescaling of the diameter values in order to take into account the band gap wavelength [since the strong absorption due to the HE

_{11}(HE

_{12}) waveguide mode occurs at

*λ*≈

*λ*

_{bg}for

*D*=

*D*

_{1}(

*D*=

*D*

_{2})] and the refractive index

*n*of the new semiconductor material. Thus, the appropriate diameter values for other direct band gap semiconductors can be approximated from the results presented here for InP, without having to turn to full, numerical three-dimensional modeling. We have verified that this works for NWs of both Al

_{0.42}Ga

_{0.58}As [36

36. D. E. Aspnes, S. M. Kelso, R. A. Logan, and R. Bhat, “Optical properties of Al_{x}Ga_{1-x}As,” J. Appl. Phys. **60**(2), 754–767 (1986). [CrossRef]

*E*

_{g}= 1.34 eV, and GaSb [37

37. R. Ferrini, M. Patrini, and S. Franchi, “Optical functions from 0.02 to 6 eV of Al_{x}Ga_{1-x}Sb/GaSb epitaxial layers,” J. Appl. Phys. **84**(8), 4517–4524 (1998). [CrossRef]

*D*

_{1,AlGaAs}≈120 nm. This value is in good agreement with the value of approximately 110 nm obtained from the rescaling of

*D*

_{1,InP}using the formula

*D*

_{1,AlGaAs}≈

*n*

_{InP}

*D*

_{1,InP}

*E*

_{g,InP}/(

*n*

_{AlGaAs}

*E*

_{g,AlGaAs}). Here, we used the refractive index values of

*n*

_{InP}(

*λ*=

*λ*

_{bg,InP}) ≈3.4 and

*n*

_{AlGaAs}(

*λ*=

*λ*

_{bg,AlGaAs}) ≈3.6. Similarly, we find that

*D*

_{2,AlGaAs}≈300 nm (found from full modeling) is in good agreement with the 270 nm obtained from the rescaling of

*D*

_{2,InP}. For the GaSb NW array, we find (from full modeling) that

*D*

_{1,GaSb}≈280 nm, which is in excellent agreement with the approximate value of 280 nm obtained from the rescaling of

*D*

_{1,InP}when

*n*

_{GaSb}(

*λ*=

*λ*

_{bg,GaSb}) ≈4.0 is used. However, for the GaSb NWs, we did not find in our numerical studies a maximum of

*η*for

*D*

_{2,GaSb}≈670 nm which could be obtained from rescaling the

*D*

_{2}≈410 nm of InP. This lack of the second maximum of

*η*for GaSb is presumably caused by the corresponding period

*p*

_{2}moving beyond the range of

*p*< 1000 nm used in our numerical analysis.

_{0.42}Ga

_{0.58}As NW array, we find also a maximum of

*η*for

*D*

_{3,AlGaAs}≈500 nm in the full modeling, which is caused by the HE

_{13}waveguide mode. For InP, after rescaling

*D*

_{3,AlGaAs}, we expect therefore a maximum of

*η*for a diameter of approximately 770 nm. The reason why this diameter did not give a maximum of the ultimate efficiency

*η*for InP NWs in the extensive numerical analysis presented here (for example in Fig. 5) is, again, presumably due to the limit of considered array periods to

*p*< 1000 nm. We note that in Fig. 5, a slight increase of

*η*occurs for

*D*≈770 nm when

*p*approaches 1000 nm, indicating that a peak resides beyond

*p*= 1000 nm.

_{1}

*waveguide modes contribute to the maxima of*

_{n}*η*and not any of the other TE

*, TM*

_{n}*, HE*

_{n}*or EH*

_{mn}*modes that exist for a NW of circular cross-section [32]. The TE*

_{mn}*and TM*

_{n}*modes of a single NW have an angular quantization number of*

_{n}*m*= 0 and cannot be excited by the normally incident plane wave of

*m*= 1. Similarly, the HE

*and EH*

_{mn}*modes cannot be excited for even*

_{mn}*m*, and for odd

*m*> 1 their overlap with the incident plane wave is expected to be smaller than for

*m*= 1. Thus, the largest overlap with the incident plane wave is expected for either the HE

_{1}

*or the EH*

_{n}_{1}

*modes. The (electric) field patterns of the HE*

_{n}_{1}

*modes are similar to the plane wave pattern outside the NW core, whereas the field patterns of the EH*

_{n}_{1}

*modes differ considerably from the plane wave pattern [32]. Thus, the strongest possible excitation is expected for the HE*

_{n}_{1}

*modes, explaining why only they give rise to the maxima of the ultimate efficiency.*

_{n}38. C. Lin and M. L. Povinelli, “Optimal design of aperiodic, vertical silicon nanowire structures for photovoltaics,” Opt. Express **19**(S5Suppl 5), A1148–A1154 (2011). [CrossRef] [PubMed]

40. H. Bao and X. Ruan, “Optical absorption enhancement in disordered vertical silicon nanowire arrays for photovoltaic applications,” Opt. Lett. **35**(20), 3378–3380 (2010). [CrossRef] [PubMed]

41. Z. Fan, R. Kapadia, P. W. Leu, X. Zhang, Y.-L. Chueh, K. Takei, K. Yu, A. Jamshidi, A. A. Rathore, D. J. Ruebusch, M. Wu, and A. Javey, “Ordered Arrays of Dual-Diameter Nanopillars for Maximized Optical Absorption,” Nano Lett. **10**(10), 3823–3827 (2010). [CrossRef] [PubMed]

7. S. L. Diedenhofen, O. T. A. Janssen, G. Grzela, E. P. A. M. Bakkers, and J. Gómez Rivas, “Strong geometrical dependence of the absorption of light in arrays of semiconductor nanowires,” ACS Nano **5**(3), 2316–2323 (2011). [CrossRef] [PubMed]

## 4. Conclusions

_{11}(HE

_{12}) waveguide mode in individual NWs is excited strongly by the incident light and absorbed strongly in the NWs at the diameter of 170 nm (410 nm). This mechanism for strong absorption in the NW array for wavelengths close to the band gap wavelength by a suitable choice of the NW diameter is applicable also for NWs of other semiconductor materials. It constitutes therefore a simple and intuitive approach for maximizing the absorption of broadband sun light in NW arrays. The optimized diameters for other direct band gap semiconductor NW arrays can be estimated simply from the results found here for InP NW arrays by rescaling with the band gap wavelength and refractive index of these new NW materials.

## Appendix A - Peaks of absorptance at wavelengths close to the band gap wavelength

*A*(of the InP NWs) as a function of wavelength

*λ*and NW diameter

*D*for NWs of length

*L*= 2000 nm placed in a square array of period

*p*= 680 nm. We notice the sharp drop of the absorptance

*A*when the photon energy decreases below the band gap energy, that is, when

*λ*>

*λ*

_{bg}= 925 nm. When we consider

*λ*= 850 nm, a wavelength close to the band gap wavelength

*λ*

_{bg}, we find two peaks in

*A*. The first peak is located at

*D*≈ 180 nm and the second at

*D*≈ 440 nm. We have verified (not shown here) that the positions of these peaks are only very weakly dependent on the NW period

*p*and the NW length

*L*, indicating that the peaks originate from the optical properties of the individual NWs that constitute the NW array. These two diameters of 170 and 440 nm coincide closely with the diameters

*D*

_{1}≈ 170 nm and

*D*

_{2}≈ 410 nm that give, respectively, rise to the ultimate efficiency maxima

*η*

_{1}and

*η*

_{2}of the NW array, irrespective of the NW length

*L*(see Fig. 6 in main text). Thus, there is for

*λ*≈

*λ*

_{bg}a peak in the absorptance when the diameter

*D*of the NWs is tuned to maximize the ultimate efficiency (that is, when

*D*

_{1}≈ 170 nm is chosen to give

*η*=

*η*

_{1}or when

*D*

_{2}≈ 410 nm is chosen to give

*η*=

*η*

_{2}).

## Appendix B - Physical origin of absorptance peaks close to the band gap wavelength

*λ*= 850 nm the

*D*dependence of

*A*for the NW array with

*L*= 2000 nm and

*p*= 680 nm (the absorptance for varying

*λ*is shown in Fig. 7). We notice first that

*A*shows low values for

*D*< 100 nm. In contrast, we find a peak of

*A*= 0.995 at

*D*= 177 nm, which is close to

*D*

_{1}≈ 170 nm. Thus, for this diameter of 177 nm, there is for

*λ*= 850 nm almost perfect in-coupling of light into the NW array and almost total absorption of light inside the NW array. There is a second, lower peak of

*A*= 0.932 for

*D*= 437 nm, which is close to

*D*

_{2}≈ 410 nm. Between these two peaks there is a dip of

*A*= 0.552 for

*D*= 251 nm.

*A*can be explained by a diameter dependence of the insertion reflection losses of the NW array, we show in Fig. 8(a) also the reflectance

*R*

_{top}of the top air/NW interface. This

*R*

_{top}is the in-coupling (reflection) loss that limits the absorptance

*A*from reaching the value of 1 when all the light that is coupled into the array is absorbed. Thus, 1-

*R*

_{top}is the limit value of

*A*when

*L*→ ∞. We find low values of

*R*

_{top}for all the values of

*D*considered in Fig. 8(a), and

*R*

_{top}appears to increase monotonously with

*D*. For

*D*< 100 nm, where

*A*shows very low values, 1-

*R*

_{top}shows values very close to 1. Clearly, in-coupling losses do not limit

*A*for this range of

*D*. Furthermore, at the dip of

*A*when

*D*= 251 nm, 1-

*R*

_{top}> 0.9 and thus considerably higher than the dip of

*A*= 0.552. Thus, the low values of

*A*for

*D*< 100 nm, the two peaks of

*A*, and the dip of

*A*originate from a strong

*D*dependence of the absorption properties of the NW array, which we will elucidate below.

### Qualitative description of the origin of the absorptance peaks

*A*for

*D*< 100 nm, the absorptance peaks at

*D*= 177 nm and

*D*= 437 nm, and the dip of

*A*at

*D*= 251 nm found in Fig. 8(a) for

*λ*= 850 nm. We verify the conclusions reached in this subsection by employing in Appendix C a rigorous, but much more tedious, analysis of the propagation and absorption of light inside the NW array.

25. N. Anttu and H. Q. Xu, “Scattering matrix method for optical excitation of surface plasmons in metal films with periodic arrays of subwavelength holes,” Phys. Rev. B **83**(16), 165431 (2011). [CrossRef]

*k*

_{a}= Re

*k*

_{a}+

*i*Im

*k*

_{a}describes the propagation of the

*a*th eigenmode in the direction parallel to the NW axis (

*z*direction in the schematic in Fig. 1), that is, in the direction through the NW array. Here, Re

*k*is the phase constant and Im

_{a}*k*the attenuation constant of the eigenmode. The electric field of the

_{a}*a*th eigenmode decays along

*z*as exp(-Im

*k*

_{a}

*z*), and the intensity of the mode decays as exp(−2Imk

*). The intensity of the light is therefore expected to decay inside the NW array with increasing*

_{a}z*z*along the NW axis at least as quickly as that of the eigenmode of the NW array that shows the lowest value of the attenuation constant. Thus, to understand why the absorptance

*A*in Fig. 8(a) is lower than 1 -

*R*

_{top}(note that

*A*= 1 -

*R*

_{top}when the light that is coupled into the NW array is absorbed before reaching the NW/substrate interface), we turn to analyze the two optical eigenmodes [denoted as mode (1) and mode (2)] of the NW array that show the lowest values of Im

*k*

_{a}and therefore set the upper limit on the decay length of the intensity inside the NW array.

*D*dependence of

*A*in Fig. 8(a) matches well with the

*D*dependence of Im

*k*

_{1}and Im

*k*

_{2}shown in Fig. 8(b). The low values of

*A*for

*D*< 100 nm coincide with the very low values of Im

*k*

_{1}for

*D*< 100 nm. Here, the mode (1) has a very long decay length inside the NW array and the fraction of incident light coupled into this mode can propagate through the NW array without strong absorption losses. The very low values of

*A*indicate that incident light is coupled predominantly into mode (1): The values of Im

*k*

_{2}for

*D*< 100 nm correspond to an intensity decay length on the order of 100 nm, and the fraction of incident light intensity coupled into this mode will decay strongly before reaching

*z*=

*L*= 2000 nm. Noticeable excitation of mode (2) would therefore show up as an increase of

*A*from the values of close to 0 observed in Fig. 8(a) for

*D*< 100 nm. Next, we notice that the peak of

*A*at

*D*= 177 nm coincides with the peak of Im

*k*

_{1}, indicating that this peak of

*A*originates from a strong excitation of mode (1) by the incident light and a strong absorption of mode (1) in the NW array. The dip of

*A*at

*D*= 251 nm coincides with the dip of Im

*k*

_{2}. At this dip of Im

*k*

_{2}, the decay length of the intensity of mode (2) is on the order of 5000 nm and the fraction of incident light intensity coupled into this mode can transfer without strong absorption through the NW array of

*L*= 2000 nm in thickness. Finally, the second peak of

*A*at

*D*≈440 nm is located close to the peak of Im

*k*

_{2}, indicating that this peak of

*A*originates from a strong excitation of mode (2) by the incident light and a strong absorption of mode (2) in the NW array.

*k*

_{HE11}and Im

*k*

_{HE12}of the HE

_{11}and HE

_{12}waveguide modes [32] of an individual InP NW of diameter

*D*(we have used the complex-valued refractive index

*n*of InP in the semi-analytical calculations in order to include the effect of light absorption, that is, Ohmic losses, giving rise to Im

*k*

_{HE1n}> 0. We note that for a bound waveguide mode, Im

*k*

_{HE1}

*= 0 for a non-absorbing dielectric NW). To study the whole dispersion relation*

_{n}*k*(

*D*), we show in Fig. 8(c) the phase constants, that is, Re

*k*, of the four modes considered in Fig. 8(b). We consider in Fig. 8(b) and Fig. 8(c) only the cases of Re

*k*

_{HE1n}≥ 2π/

*λ*for which the waveguide mode is bound [42]. We note that the fundamental waveguide mode HE

_{11}is bound for all

*D*whereas HE

_{12}is cut off for

*D*< 368 nm. We find that the dispersion relation

*k*

_{1}(

*D*) of eigenmode (1) of the NW array follows very closely that of the HE

_{11}mode of a single NW. Similarly, the dispersion relation of mode (2) of the NW array closely follows that of the HE

_{12}mode of a single NW. This shows that the eigenmode (1) [(2)] of the NW array stems from the HE

_{11}[HE

_{12}] mode of the individual NWs that constitute the array. Importantly, from this qualitative analysis we understand that the absorptance peaks in Fig. 7 at

*λ*≈

*λ*

_{bg}for

*D*

_{1}and

*D*

_{2}originate, respectively, from the peaks of Im

*k*

_{HE11}(

*D*) and Im

*k*

_{HE12}(

*D*) that can be found from the semi-analytic dispersion relations

*k*

_{HE11}(

*D*) and

*k*

_{HE12}(

*D*) of the waveguide modes of a single NW. Furthermore, since

*k*

_{HE11}(

*D*) and

*k*

_{HE12}(

*D*) do not depend on the NW array period

*p*or the NW length

*L*, we understand why the diameters

*D*

_{1}and

*D*

_{2}, which optimize

*η*, show a very weak dependence on

*L*[see Figs. 6(a) and 6(b)].

## Appendix C - Rigorous eigenmode-based description of optical dissipation

*z*direction,

*z*= 0 is located at the air/NW top interface, and

*z*=

*L*is at the NW/substrate bottom interface. Thus,

*z*is the direction of power transport through the NW array. In the analysis, we fix the incident intensity from the air top-side to (1 [W])/

*p*

^{2}where

*p*is the period of the square array.

### Eigenmode expansion of the power flow inside the nanowire array

*P*(

*z*) along the

*z*direction (with unit vector

*UC*of the periodic array is given byand we choose 0 <

*x*≤

*p*and 0 <

*y*≤

*p*to define the cross-section of one of the unit cells. The decay of

*P*(

*z*) along

*z*inside the NW array is caused by the absorption of light in the NWs: The absorption per unit length in the

*z*direction is given by d

*P*(

*z*)/d

*z*. Thus, by studying how

*P*(

*z*) varies inside the NW array, we can obtain information of where, along the

*z*direction, the absorption occurs [note that

43. A. A. Barybin, “Modal expansions and orthogonal complements in the theory of complex media waveguide excitation by external sources for isotropic, anisotropic, and bianisotropic media,” Prog. Electromagnetics Res. **19**, 241–300 (1998). [CrossRef]

*x*and

*y*components of the electric field inside the NW array,

**83**(16), 165431 (2011). [CrossRef]

*x*and

*y*components of the electric field of the

*a*th eigenmode. The complex-valued propagation constant (that is, the wave vector)

*k*

_{a}of the

*a*th eigenmode is obtained under the rule that Re

*k*

_{a}> 0 if Im

*k*

_{a}= 0 and Im

*k*

_{a}> 0 if Im

*k*

_{a}≠ 0. Thus,

*a*th eigenmode that propagates forward (or decays exponentially) along the positive z direction, whereas

*a*th eigenmode that propagates backward (or decays exponentially) along the negative

*z*direction. Similarly, the magnetic field inside the NW array can be expanded as [25

**83**(16), 165431 (2011). [CrossRef]

43. A. A. Barybin, “Modal expansions and orthogonal complements in the theory of complex media waveguide excitation by external sources for isotropic, anisotropic, and bianisotropic media,” Prog. Electromagnetics Res. **19**, 241–300 (1998). [CrossRef]

*a*th forward propagating eigenmode. This is the power that is transported inside the NW array when only the

*a*th forward propagating eigenmode is excited (that is, when

*b*≠

*a*, and

*d*). Similarly,is the self-power of the

*a*th backward propagating eigenmode.

43. A. A. Barybin, “Modal expansions and orthogonal complements in the theory of complex media waveguide excitation by external sources for isotropic, anisotropic, and bianisotropic media,” Prog. Electromagnetics Res. **19**, 241–300 (1998). [CrossRef]

*P*(

*z*) [see Eq. (5)]. This is not surprising considering that in the absorption, which is proportional to |

**E**|

^{2}, cross terms between different eigenmodes show up. This leads to cross-terms between different eigenmodes also in

*P*(

*z*) [since d

*P*(

*z*)/d

*z*is proportional to the absorption].

*k*

_{a}of the eigenmodes (thus, this analysis encompasses the photonic bandstructure of the NW array, including possibly emerging air bands and dielectric bands [44]). Next, all three geometrical parameters (NW diameter, array period, and NW length) can in principle affect the excitation of the eigenmodes (that is, the self–powers and cross–powers).

### Rigorous study of dissipation for p = 680 nm, L = 2000 nm, and λ = 850 nm

*A*for

*λ*≈

*λ*

_{bg}. We consider the same cases of

*D*= 100, 177, 251, and 437 nm for

*p*= 680 nm,

*L*= 2000 nm, and

*λ*= 850 nm as studied qualitatively in Appendix B. In Fig. 9, we show for 0 <

*z*< 2000 nm (that is, inside the NW array), the total power flow

*P*(

*z*), the forward and backward propagating self-powers

*D*= 100 nm for which the absorptance

*A*in Fig. 8(a) is low. As expected from this low value of

*A*, from Fig. 9(a) we find that

*P*(

*z*) shows only a very weak decay when going from

*z*= 0 to

*z*=

*L*= 2000 nm. In Fig. 9(b) we see that the reason for this is, as qualitatively deduced above, that only mode (1), which is absorbed weakly here, is excited noticeably [i.e., only

*D*= 177 nm for which the absorptance shows a peak in Fig. 8(a), we find that there is almost perfect incoupling into the NW array [i.e.,

*P*(

*z*= 0) ≈1 in Fig. 9(c)] and almost complete absorption of light occurs inside the NW array before the light reaches the NW/substrate interface located at

*z*=

*L*[i.e.,

*P*(

*z*) decays rapidly inside the NW array to reach a very low value at

*z*=

*L*]. When we consider the excitation of the eigenmodes [Fig. 9(d)] we find that both mode (1) and mode (2) are excited in the forward direction and both show fast decay [i.e., both

*z*= 0 and decay rapidly inside the NW array to reach very low values at

*z*=

*L*]. As compared to the case of the low absorptance at

*D*= 100 nm above, mode (1) is absorbed strongly since the peak of the attenuation constant of mode (1) is located close to

*D*= 177 nm [see Fig. 8(b)]. Thus, for

*D*= 177 nm the mode (1), which corresponds to the HE

_{11}waveguide mode of the single NWs that constitute the array, is excited strongly by the incident light and absorbed strongly inside the NW array. This allows for strong absorption at

*λ*≈

*λ*

_{bg}(as seen in Fig. 7) and enables the maximum of the ultimate efficiency

*η*for

*D*=

*D*

_{1}≈170 nm (see Fig. 6).

*D*= 251 nm we find, as compared to the case of

*D*= 177 nm above, a slow decrease in

*P*(

*z*) and thus weak absorption inside the NW array [Fig. 9(e)]. We find that both mode (1) and mode (2) are excited in the forward direction [i.e., both

*z*= 0, see Fig. 9(f)]. Mode (1) decays much faster than mode (2) since Im

*k*

_{1}> Im

*k*

_{2}as seen in Fig. 8(b). A large fraction of the incident power is coupled into the weakly absorbed mode (2) [i.e.,

*D*= 251 nm in Fig. 8(a).

*D*= 437 nm,

*P*(

*z*) decays rapidly inside the NW array [Fig. 9(g)], as expected from the peak of the absorptance for

*D*= 437 nm in Fig. 8(a). Both mode (1) and mode (2) are excited [Fig. 9(h)] and both decay rapidly [i.e., both

*z*= 0 and decay rapidly inside the NW array to reach very low values at

*z*=

*L*]. As compared to the case of the low absorptance at

*D*= 251 nm above, mode (2) is absorbed strongly since the peak of the attenuation constant of mode (2) is located close to

*D*= 437 nm [see Fig. 8(b)]. Thus, for

*D*= 437 nm the mode (2), which corresponds to the HE

_{12}waveguide mode of the single NWs that constitute the array, is excited strongly by the incident light and absorbed strongly inside the NW array. This allows for strong absorption at

*λ*≈

*λ*

_{bg}(as seen in Fig. 7) and enables the maximum of the ultimate efficiency

*η*for

*D*=

*D*

_{2}≈410 nm (see Fig. 6).

## Appendix D - Reflection losses at the top NW/air interface

## Acknowledgments

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13. | Z. Gu, P. Prete, N. Lovergine, and B. Nabet, “On optical properties of GaAs and GaAs/AlGaAs core-shell periodic nanowire arrays,” J. Appl. Phys. |

14. | N. Huang, C. Lin, and M. L. Povinelli, “Broadband absorption of semiconductor nanowire arrays for photovoltaic applications,” J. Opt. |

15. | N. Anttu, K. Namazi, P. Wu, P. Yang, H. Xu, H. Q. Xu, and U. Håkanson, “Drastically increased absorption in vertical semiconductor nanowire arrays: A non-absorbing dielectric shell makes the difference,” Nano Res. |

16. | G. Kästner and U. Gösele, “Stress and dislocations at cross-sectional heterojunctions in a cylindrical nanowire,” Philos. Mag. |

17. | K. A. Dick, S. Kodambaka, M. C. Reuter, K. Deppert, L. Samuelson, W. Seifert, L. R. Wallenberg, and F. M. Ross, “The morphology of axial and branched nanowire heterostructures,” Nano Lett. |

18. | C. P. T. Svensson, T. Mårtensson, J. Trägårdh, C. Larsson, M. Rask, D. Hessman, L. Samuelson, and J. Ohlsson, “Monolithic GaAs/InGaP nanowire light emitting diodes on silicon,” Nanotechnology |

19. | J. Wallentin and M. T. Borgstrom, “Doping of semiconductor nanowires,” J. Mater. Res. |

20. | L. Hu and G. Chen, “Analysis of optical absorption in silicon nanowire arrays for photovoltaic applications,” Nano Lett. |

21. | P. M. Wu, N. Anttu, H. Q. Xu, L. Samuelson, and M.-E. Pistol, “Colorful InAs nanowire arrays: From strong to weak absorption with geometrical tuning,” Nano Lett. |

22. | C. Lin and M. L. Povinelli, “Optical absorption enhancement in silicon nanowire arrays with a large lattice constant for photovoltaic applications,” Opt. Express |

23. | J. Li, H. Yu, S. M. Wong, X. Li, G. Zhang, P. G.-Q. Lo, and D.-L. Kwong, “Design guidelines of periodic Si nanowire arrays for solar cell application,” Appl. Phys. Lett. |

24. | J. Li, H. Yu, and Y. Li, “Solar energy harnessing in hexagonally arranged Si nanowire arrays and effects of array symmetry on optical characteristics,” Nanotechnology |

25. | N. Anttu and H. Q. Xu, “Scattering matrix method for optical excitation of surface plasmons in metal films with periodic arrays of subwavelength holes,” Phys. Rev. B |

26. | J. Wallentin, N. Anttu, D. Asoli, M. Huffman, I. Åberg, 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 |

27. | W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. |

28. | I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for III-V compound semiconductors and their alloys,” J. Appl. Phys. |

29. | Air Mass 1.5 Spectra, American Society for Testing and Materials, http://rredc.nrel.gov/solar/spectra/am1.5/. |

30. | O. J. Glembocki and H. Piller, “Indium phosphide (InP),” in |

31. | N. Anttu, “Geometrical optics, electrostatics, and nanophotonic resonances in absorbing nanowire arrays,” Opt. Lett. |

32. | J. Bures, |

33. | B. Wang and P. W. Leu, “Tunable and selective resonant absorption in vertical nanowires,” Opt. Lett. |

34. | L. Cao, J. S. White, J.-S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. |

35. | M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater, “Enhanced absorption and carrier collection in Si wire arrays for photovoltaic applications,” Nat. Mater. |

36. | D. E. Aspnes, S. M. Kelso, R. A. Logan, and R. Bhat, “Optical properties of Al |

37. | R. Ferrini, M. Patrini, and S. Franchi, “Optical functions from 0.02 to 6 eV of Al |

38. | C. Lin and M. L. Povinelli, “Optimal design of aperiodic, vertical silicon nanowire structures for photovoltaics,” Opt. Express |

39. | 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. |

40. | H. Bao and X. Ruan, “Optical absorption enhancement in disordered vertical silicon nanowire arrays for photovoltaic applications,” Opt. Lett. |

41. | Z. Fan, R. Kapadia, P. W. Leu, X. Zhang, Y.-L. Chueh, K. Takei, K. Yu, A. Jamshidi, A. A. Rathore, D. J. Ruebusch, M. Wu, and A. Javey, “Ordered Arrays of Dual-Diameter Nanopillars for Maximized Optical Absorption,” Nano Lett. |

42. | A. W. Snyder and J. D. Love, |

43. | A. A. Barybin, “Modal expansions and orthogonal complements in the theory of complex media waveguide excitation by external sources for isotropic, anisotropic, and bianisotropic media,” Prog. Electromagnetics Res. |

44. | J. Joannopoulos, R. Meade, and J. Winn, |

**OCIS Codes**

(160.6000) Materials : Semiconductor materials

(350.6050) Other areas of optics : Solar energy

(310.6628) Thin films : Subwavelength structures, nanostructures

**ToC Category:**

Photovoltaics

**History**

Original Manuscript: January 28, 2013

Revised Manuscript: April 3, 2013

Manuscript Accepted: April 22, 2013

Published: May 1, 2013

**Citation**

Nicklas Anttu and H. Q. Xu, "Efficient light management in vertical nanowire arrays for photovoltaics," Opt. Express **21**, A558-A575 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-S3-A558

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