OSA's Digital Library

Energy Express

Energy Express

  • Editor: Christian Seassal
  • Vol. 21, Iss. S3 — May. 6, 2013
  • pp: A558–A575
« Show journal navigation

Efficient light management in vertical nanowire arrays for photovoltaics

Nicklas Anttu and H. Q. Xu  »View Author Affiliations


Optics Express, Vol. 21, Issue S3, pp. A558-A575 (2013)
http://dx.doi.org/10.1364/OE.21.00A558


View Full Text Article

Acrobat PDF (1784 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

The absorption of incident light is one of the key factors that determines the efficiency of a solar cell [20

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]

]. However, the interaction of the incident light with a NW array could not be described properly by simple effective medium theories [21

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]

], and a full electrodynamic description is needed [20

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]

]. Three important geometrical parameters, the nanowire diameter 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.

The existence of these two optimized diameter values originates from the dispersion relations of the HE11 and HE12 waveguide modes of individual NWs. At the diameter of 170 nm, the HE11 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 HE12 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

We consider a square array of InP NWs of diameter D and length L (see Fig. 1
Fig. 1 Schematic of the modeled InP NW array. The NWs stand on top of an (optically) infinitely thick InP substrate. There is air between and on top of the NWs. The NW diameter is D and the NW length is L. A unit cell in the x-y plane contains one NW and has the period p in both the x and the y direction. A plane wave of light is incident normally, with kx = ky = 0, toward the NW array from air on the top side.
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 kx = ky = 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]

]. Therefore, we believe that the conclusions we reach here for the case of a square array could be applicable also for other array symmetries.

In PV applications, we are interested in the absorptance spectrum 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]

, 6

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

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

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]

]. To quantify the broadband absorption of sun light in the NW array, we use the ultimate efficiency 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

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

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]

, 27

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]

]. The absorptance 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

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

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

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]

, 27

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=0λλλbgIAM1.5(λ)A(λ)dλ0IAM1.5(λ)dλ
(1)
where λbg = 925 nm is the wavelength of photons with an energy equal to 1.34 eV, the band gap energy Eg 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]

]. Here, IAM1.5(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/.

]. In this definition [27

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]

] of 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
Fig. 2 (a) The 1000 W/m2 AM1.5 direct and circumsolar intensity spectrum (higher red values) [29]. Here, also the intensity usable from the AM1.5 spectrum for an InP solar cell is shown (lower green values), which is obtained by taking into account the band gap of InP (λbg = 925 nm) and thermalization losses. (b) Zoom-in of the intensity usable from the AM1.5 spectrum for an InP solar cell.
together with the maximum intensity usable in an InP solar cell after taking the thermalization losses into account. The intensity spectrum IAM1.5 shows a peak at 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, ηmax = 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)].

We use an electrodynamic description to take into account the wave behavior of light. The modeling is done with a scattering matrix method [25

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]

], which solves the Maxwell equations for incident light of a given wavelength 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 nInP of the InP [30

30. O. J. Glembocki and H. Piller, “Indium phosphide (InP),” in Handbook of optical constants of solids, E. D. Palik, ed. (Academic, 1985).

] and for air n = 1 is used.

We show in Fig. 3(a)
Fig. 3 (a) Absorptance spectrum A(λ) of an InP NW array with period p = 680 nm and NWs of length L = 2000 nm on top of an InP substrate. We consider the cases of NWs of diameter D = 100 nm (i), 177 nm (ii), 221 nm (iii), and 441 nm (iv). The incident light is a plane wave incident at normal angle to the array from the top air side. (b) Ultimate efficiency η as a function of D for an InP NW array with p = 680 nm and L = 2000 nm. The circles (i) - (iv) mark the NW arrays whose absorptance spectra A(λ) are shown in (a). Here, also ηmax = 0.463, the maximum possible ultimate efficiency for InP, is shown (dashed line).
the absorptance 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 λbg = 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 < λbg. 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.

We start by considering short NWs of length L = 500 nm. The ultimate efficiency η is shown in Fig. 4
Fig. 4 Ultimate efficiency η of the InP NW array as a function of the array period p and the NW diameter D for the fixed NW length L = 500 nm. There is one local maximum of η1 = 0.344 at D1 = 191 nm and p1 = 251 nm and a second local maximum of η2 = 0.341 at D2 = 438 nm and p2 = 530 nm. The inset shows a line-cut of η as a function of D for p = 530 nm (solid line). In this inset also ηmax = 0.463 (dashed line), the maximum possible ultimate efficiency of InP, is shown.
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 η1 = 0.344 located at D1 = 191 nm and p1 = 251 nm, and another one of η2 = 0.341 located at D2 = 438 nm and p2 = 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 η2 = 0.341 is found for the NW diameter D2 = 438 nm. A peak can be seen also at D ≈190 nm, but this peak does not correspond to η1 since η increases with decreasing p to reach the local maximum of η1 = 0.344 at p1 = 251 nm and D1 = 191 nm.

When the NW length is increased to L = 2000 nm (see Fig. 5
Fig. 5 Ultimate efficiency η of the InP NW array as a function of the array period p and the NW diameter D for the fixed NW length L = 2000 nm. There is one local maximum of η1 = 0.431 at D1 = 184 nm and p1 = 340 nm, and a second local maximum of η2 = 0.410 at D2 = 441 nm and p2 = 680 nm. The inset shows a line-cut of η as a function of D for p = 340 nm (solid line). In this inset also ηmax = 0.463 (dashed line), the maximum possible ultimate efficiency of InP, is shown.
), η shows again two local maxima: the first one of η1 = 0.431 is located at D1 = 184 nm and p1 = 340 nm, and the second one of η2 = 0.410 is located at D2 = 441 nm and p2 = 680 nm [see Fig. 3(a), curve (iv) for the A(λ) that gives rise to η2 in Fig. 5]. Thus, compared to the positions of the maxima for L = 500 nm, the diameters D1 and D2 have shifted only by 7 nm and 3 nm, respectively. In contrast, the periods p1 and p2 shift much more. The shift of p1 is 89 nm and the shift of p2 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 η1 has increased by 0.087 and the value of η2 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

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

]). The inset in Fig. 5 shows η as a function of D for p = 340 nm and L = 2000 nm. Here, the local maximum of η1 = 0.431 is found for the NW diameter D1 = 184 nm.

To study the two local maxima of η in detail, we have calculated the values of η1 and η2, and their positions (D1, p1) and (D2, p2) for 500 nm < L < 8000 nm. The results are shown in Fig. 6
Fig. 6 (a) NW diameter D1 (solid line) and array period p1 (dashed line), that give the local maximum η1 of the ultimate efficiency η of the InP NW array, for varying NW length L. (b) NW diameter D2 (solid line) and array period p2 (dashed line), that give the local maximum η2 of the ultimate efficiency η of the NW array, for varying NW length L. (c) Maximum ultimate efficiencies η1 (solid red line) and η2 (dashed blue line) plotted against the NW length L. Here, also the maximum possible value of η for InP, ηmax = 0.463, is shown (dashed-dotted black line).
. Figures 6(a) and 6(b) show that both p1 and p2 increase strongly with increasing L. The value of p1 increases from 251 nm to 470 nm [Fig. 6(a)], and the value of p2 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 p1 and p2 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 p1 and p2 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 p1 and p2 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.

In strong contrast to the dependencies of p1 and p2 on L, we find that the diameters D1 and D2 [Figs. 6(a) and 6(b)], which give the local maxima η1 and η2, shift only slightly for this large increase in the length L of the NWs. The value of D1 decreases from 191 nm to 171 nm and the value of D2 decreases from 438 nm to 407 nm when the value of L is increased from 500 nm to 8000 nm. Thus, the diameters D1 ≈170 nm and D2 ≈410 nm optimize the ultimate efficiency, irrespective of the NW length L. We find also in Fig. 6(c) that η1 > η2 for all values of L.

The fact that the diameter values D1 and D2 depend only very weakly on L indicates strongly that D1 and D2 are connected to the optical properties of the individual NWs that constitute the NW array. Indeed, we have found that D1 and D2 originate from the properties of the HE11 and HE12 waveguide modes [32

32. J. Bures, Guided optics: Optical fibers and all-fiber components (Wiley-VCH, 2009).

, 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]

] of the individual NWs (see Appendix B for a qualitative and Appendix C for a quantitative analysis of the excitation and absorption of these two modes). By choosing the diameter D1 (D2), the HE11 (HE12) 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
Fig. 7 Absorptance A as a function of wavelength λ and NW diameter D for an InP NW array with period p = 680 nm and NW length L = 2000 nm (see Fig. 1 in the main text for a schematic). A rapid drop to a zero value of the absorptance occurs for λ > λbg = 925 nm.
in Appendix A and Fig. 8(a)
Fig. 8 (a) Absorptance A (solid line) as a function of the NW diameter D for InP NWs of length L = 2000 nm placed in a square array of period p = 680 nm (see Fig. 1 in the main text for a schematic). Here also Rtop, the in-coupling reflection loss of the top air/NW interface, is shown (dashed-dotted line). The light is of 850 nm in wavelength and incident at normal angle to the NW array from the air top side. (b) The attenuation constant Imk of the two eigenmodes (1) and (2) of the NW array that show the lowest values of Imk (solid lines). Here, also the corresponding values of the HE11 [close to the values of eigenmode (1) of the NW array] and the HE12 [close to the values of eigenmode (2) of the NW array] waveguide modes of a single NW are shown (dashed lines). (c) Same as (b) but for Rek, the phase constant of the modes. We note that at D = 0, that is, when the NW array region consists of empty space, mode (1) is the diffracted zeroth order with k1=2π/λ7.4107 m−1 and mode (2) is a (evanescent) diffracted order with k2=(2π/λ)2(2π/p)2i5.5106 m−1.
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 HE1m 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]

] in these vertical NWs reminds of the resonant absorption through the leaky modes in horizontal NWs [34

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]

]. Furthermore, this tuning of the resonant absorption by the NW diameter can be contrasted to the non-resonant tuning of the optical response of microwire arrays [35

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]

] where design principles from geometrical optics (such as inclusion of micron-sized dielectric particles to scatter light within the array to enhance the light path) can be employed.

Since the optimized diameters D1 and D2 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 HE11 (HE12) waveguide mode occurs at λλbg for D = D1 (D = D2)] 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 Al0.42Ga0.58As [36

36. D. E. Aspnes, S. M. Kelso, R. A. Logan, and R. Bhat, “Optical properties of AlxGa1-xAs,” J. Appl. Phys. 60(2), 754–767 (1986). [CrossRef]

] that has a band gap energy of 1.95 eV, which is considerably higher than the InP value of Eg = 1.34 eV, and GaSb [37

37. R. Ferrini, M. Patrini, and S. Franchi, “Optical functions from 0.02 to 6 eV of AlxGa1-xSb/GaSb epitaxial layers,” J. Appl. Phys. 84(8), 4517–4524 (1998). [CrossRef]

] that has a considerably lower band gap energy of 0.7 eV. For the AlGaAs NW array, we have found from full three-dimensional modeling (detailed data not shown) that D1,AlGaAs ≈120 nm. This value is in good agreement with the value of approximately 110 nm obtained from the rescaling of D1,InP using the formula D1,AlGaAsnInPD1,InPEg,InP/(nAlGaAsEg,AlGaAs). Here, we used the refractive index values of nInP(λ = λbg,InP) ≈3.4 and nAlGaAs(λ = λbg,AlGaAs) ≈3.6. Similarly, we find that D2,AlGaAs ≈300 nm (found from full modeling) is in good agreement with the 270 nm obtained from the rescaling of D2,InP. For the GaSb NW array, we find (from full modeling) that D1,GaSb ≈280 nm, which is in excellent agreement with the approximate value of 280 nm obtained from the rescaling of D1,InP when nGaSb(λ = λbg,GaSb) ≈4.0 is used. However, for the GaSb NWs, we did not find in our numerical studies a maximum of η for D2,GaSb ≈670 nm which could be obtained from rescaling the D2 ≈410 nm of InP. This lack of the second maximum of η for GaSb is presumably caused by the corresponding period p2 moving beyond the range of p < 1000 nm used in our numerical analysis.

Actually, for the Al0.42Ga0.58As NW array, we find also a maximum of η for D3,AlGaAs ≈500 nm in the full modeling, which is caused by the HE13 waveguide mode. For InP, after rescaling D3,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.

Finally, we comment on why only the HE1n waveguide modes contribute to the maxima of η and not any of the other TEn, TMn, HEmn or EHmn modes that exist for a NW of circular cross-section [32

32. J. Bures, Guided optics: Optical fibers and all-fiber components (Wiley-VCH, 2009).

]. The TEn and TMn modes of a single NW have an angular quantization number of m = 0 and cannot be excited by the normally incident plane wave of m = 1. Similarly, the HEmn and EHmn modes cannot be excited for even 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 HE1n or the EH1n modes. The (electric) field patterns of the HE1n modes are similar to the plane wave pattern outside the NW core, whereas the field patterns of the EH1n modes differ considerably from the plane wave pattern [32

32. J. Bures, Guided optics: Optical fibers and all-fiber components (Wiley-VCH, 2009).

]. Thus, the strongest possible excitation is expected for the HE1n modes, explaining why only they give rise to the maxima of the ultimate efficiency.

We note that an interesting future direction for research on direct band gap NW arrays would be to investigate if random/aperiodic NW arrays can be, for given NW length, more efficient at absorbing light than the periodic arrays studied here. Such an effect has already been indicated for Si NW arrays [38

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

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]

]. Furthermore, the effect on the optimized ultimate efficiency of tapering the NWs would be of interest to investigate. Such future efforts are inspired by the dual-diameter nanopillars investigated by Fan et al. [41

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]

] and the conical and base tapered NWs investigated by Diedenhofen et al. [7

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

In conclusion, we have studied the light-management in InP nanowire arrays for PV applications. In this study, the NW diameter, the NW length, and the period of the array have been varied in a controlled manner. The absorption of light in the array has been investigated and optimal geometrical designs for PV devices have been identified (see Fig. 6). We have found that with increasing NW length, the period of the optimized array increases strongly, leading to a reduction of the insertion reflection losses of the array. Our most important discovery is the existence of specific band-gap-dependent NW diameters that are the most appropriate for PV applications, irrespective of the NW length. These diameters are approximately 170 and 410 nm for InP NWs and originate from the optical properties of the individual NWs. For wavelengths close to the band gap wavelength, the HE11 (HE12) 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

Figure 7 shows the absorptance 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 > λbg = 925 nm. When we consider λ = 850 nm, a wavelength close to the band gap wavelength λbg, 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 D1 ≈ 170 nm and D2 ≈ 410 nm that give, respectively, rise to the ultimate efficiency maxima η1 and η2 of the NW array, irrespective of the NW length L (see Fig. 6 in main text). Thus, there is for λλbg a peak in the absorptance when the diameter D of the NWs is tuned to maximize the ultimate efficiency (that is, when D1 ≈ 170 nm is chosen to give η = η1 or when D2 ≈ 410 nm is chosen to give η = η2).

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

Figure 8(a) shows for 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 D1 ≈ 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 D2 ≈ 410 nm. Between these two peaks there is a dip of A = 0.552 for D = 251 nm.

To investigate if the dip and the two peaks of 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 Rtop of the top air/NW interface. This Rtop 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- Rtop is the limit value of A when L → ∞. We find low values of Rtop for all the values of D considered in Fig. 8(a), and Rtop appears to increase monotonously with D. For D < 100 nm, where A shows very low values, 1- Rtop 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- Rtop > 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

Here, we give a qualitative description of the physical origin of the low values of 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.

The electromagnetic field inside the NW array can be expressed as a superposition of the (optical) eigenmodes of the NW array [25

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]

]. Importantly, the propagation constant ka = Reka + iImka describes the propagation of the ath 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, Reka is the phase constant and Imka the attenuation constant of the eigenmode. The electric field of the ath eigenmode decays along z as exp(-Imkaz), and the intensity of the mode decays as exp(−2Imkaz). The intensity of the light is therefore expected to decay inside the NW array with increasing 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 - Rtop (note that A = 1 - Rtop 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 Imka and therefore set the upper limit on the decay length of the intensity inside the NW array.

We find that the D dependence of A in Fig. 8(a) matches well with the D dependence of Imk1 and Imk2 shown in Fig. 8(b). The low values of A for D < 100 nm coincide with the very low values of Imk1 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 Imk2 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 Imk1, 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 Imk2. At this dip of Imk2, 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 Imk2, 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.

To relate the eigenmodes (1) and (2) of the NW array to more familiar concepts encountered in nano-optics, we show in Fig. 8(b) also the semi-analytically calculated ImkHE11 and ImkHE12 of the HE11 and HE12 waveguide modes [32

32. J. Bures, Guided optics: Optical fibers and all-fiber components (Wiley-VCH, 2009).

] 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 ImkHE1n > 0. We note that for a bound waveguide mode, ImkHE1n = 0 for a non-absorbing dielectric NW). To study the whole dispersion relation k(D), we show in Fig. 8(c) the phase constants, that is, Rek, of the four modes considered in Fig. 8(b). We consider in Fig. 8(b) and Fig. 8(c) only the cases of RekHE1n ≥ 2π/λ for which the waveguide mode is bound [42

42. A. W. Snyder and J. D. Love, Optical waveguide theory (Chapman and Hall, 1983).

]. We note that the fundamental waveguide mode HE11 is bound for all D whereas HE12 is cut off for D < 368 nm. We find that the dispersion relation k1(D) of eigenmode (1) of the NW array follows very closely that of the HE11 mode of a single NW. Similarly, the dispersion relation of mode (2) of the NW array closely follows that of the HE12 mode of a single NW. This shows that the eigenmode (1) [(2)] of the NW array stems from the HE11 [HE12] mode of the individual NWs that constitute the array. Importantly, from this qualitative analysis we understand that the absorptance peaks in Fig. 7 at λλbg for D1 and D2 originate, respectively, from the peaks of ImkHE11(D) and ImkHE12(D) that can be found from the semi-analytic dispersion relations kHE11(D) and kHE12(D) of the waveguide modes of a single NW. Furthermore, since kHE11(D) and kHE12(D) do not depend on the NW array period p or the NW length L, we understand why the diameters D1 and D2, 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

A schematic of the NW array is shown in Fig. 1: The NW axis is in the 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])/p2 where p is the period of the square array.

Eigenmode expansion of the power flow inside the nanowire array

The power flow P(z) along the z direction (with unit vector e^z) through the cross-section of one unit cell of area UC of the periodic array is given by
P(z)=14UC(E*×H+E×H*)·e^zdS
(2)
and we choose 0 < xp and 0 < yp 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 dP(z)/dz. 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 A=P(z=0)P(z=L)]. Furthermore, by expanding the electromagnetic field inside the NW array in terms of the eigenmodes of the NW array, it is possible [43

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]

] to identify which eigenmodes are responsible for the power transport through the NW array. At the same time, it is possible to elucidate how strongly those modes dissipate the power (that is, are absorbed by the NWs) during the transport.

The x and y components of the electric field inside the NW array, Exy(x)=[Ex(x),Ey(x)]T, can be expanded in terms of the optical eigenmodes as [25

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]

]
Exy(x)=aEa(x,y)(Ca+eikaz+Caeikaz)
(3)
where Ea(x,y)=[Ea,x(x),Ea,y(x)]T contains the x and y components of the electric field of the ath eigenmode. The complex-valued propagation constant (that is, the wave vector) ka of the ath eigenmode is obtained under the rule that Reka > 0 if Imka = 0 and Imka > 0 if Imka ≠ 0. Thus, Ca+ is the expansion coefficient of the ath eigenmode that propagates forward (or decays exponentially) along the positive z direction, whereas Ca is the expansion coefficient of the ath 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

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]

]

Hxy(x)=aHa(x,y)(Ca+eikazCaeikaz).
(4)

Next, we use the electric field expansion [Eq. (3)] and the magnetic field expansion [Eq. (4)] in the equation for the power flow [Eq. (2)]. This results in [43

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)=ab[Pab++(z)+Pab+(z)+Pab+(z)+Pab(z)]
(5)
where
Pab++(z)=Nab++Ca+*Cb+ei(kbka*)z,
(6)
Pab+(z)=Nab+Ca+*Cbei(kbka*)z,
(7)
Pab+(z)=Nab+Ca*Cb+ei(kb+ka*)z,
(8)
Pab(z)=NabCa*Cbei(kb+ka*)z,
(9)
and

Nab++(z)=14UC(Ea*×Hb+Eb×Ha*)·e^zdS,
(10)
Nab+(z)=14UC(Ea*×Hb+Eb×Ha*)·e^zdS,
(11)
Nab+(z)=14UC(Ea*×HbEb×Ha*)·e^zdS,
(12)
Nab(z)=14UC(Ea*×HbEb×Ha*)·e^zdS.
(13)

We can identify from Eq. (5)
Pa+(z)Paa++(z)
(14)
as the self power of the ath forward propagating eigenmode. This is the power that is transported inside the NW array when only the ath forward propagating eigenmode is excited (that is, when Ca+0, Cb+=0 for all ba, and Cd=0for all d). Similarly,
Pa(z)Paa(z)
(15)
is the self-power of the ath backward propagating eigenmode.

We note that in a dissipative medium, the power flow is in general not simply the sum of the self-powers of the forward and backward propagating eigenmodes, that is,
P(z)a[Pa+(z)+Pa(z)]
(16)
in general [43

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]

]. Instead, the cross-powers Pab+, Pab+, Pab++, and Pab couple pairs of eigenmodes in the power flow 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 dP(z)/dz is proportional to the absorption].

Notice that in this analysis, first, the NW diameter and the array period affect the electromagnetic field distributions Ea(x,y)and Ha(x,y) and the propagation constants ka of the eigenmodes (thus, this analysis encompasses the photonic bandstructure of the NW array, including possibly emerging air bands and dielectric bands [44

44. J. Joannopoulos, R. Meade, and J. Winn, Photonic crystals: Molding the flow of light (Princeton University, 1995).

]). 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

Here, we corroborate the qualitative claims made in Appendix B regarding the importance of eigenmodes (1) and (2) of the NW array on the absorptance A for λλbg. 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
Fig. 9 Power P(z) [that is, the intensity integrated over the cross-section of one unit cell] of an InP NW array of period p = 680 nm. The NWs are of length L = 2000 nm and we consider varying NW diameters of D = 100 (a), 177 (c), 251 nm (e), and 437 nm (g). The air/NW top interface is located at z = 0 and the NW/substrate bottom interface is located at z = 2000 nm (see Fig. 1 for a Schematic). Here, light of a wavelength of 850 nm is incident at normal angle from the air top side. In (b), (d), (f), and (h) the forward [P1+and P2+] and backward [P1and P2] propagating self-powers of eigenmodes (1) and (2) of the NW array are shown [see Fig. 8 for the correspondence between mode (1) of the NW array and the HE11 waveguide mode of a single NW; and the correspondence between array mode (2) and the HE12 waveguide mode]. Here, also P12ct, the sum of all cross-powers between modes (1) and (2), is shown. All powers are expressed in the unit of Watt and the incident intensity is (1 [W])/p2. Thus, 0 ≤ P(z) ≤ 1.
, 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 P1+(z), P2+(z), P1(z), and P2(z), as well as
P12ct(z)=P11+(z)+P11+(z)+P22+(z)+P22+(z)+P12++(z)+P21++(z)+P12(z)+P21++(z)+P12+(z)+P21+(z)+P12+(z)+P21+(z),
(17)
the sum of all cross-powers between mode (1) and (2).

We start by considering the case of 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 P1+(z) and P1(z) differ noticeably from zero]. We note that this mode is excited also in the backward propagating direction [that is, P1(z)0] showing that the reflection at the NW/substrate interface excites this mode.

Next, at 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 P1+(z) and P2+(z) differ noticeably from zero at 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 HE11 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 λλbg (as seen in Fig. 7) and enables the maximum of the ultimate efficiency η for D = D1 ≈170 nm (see Fig. 6).

For 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 P1+(z) and P2+(z) differ from zero at z = 0, see Fig. 9(f)]. Mode (1) decays much faster than mode (2) since Imk1 > Imk2 as seen in Fig. 8(b). A large fraction of the incident power is coupled into the weakly absorbed mode (2) [i.e., P2+(z=0) is close to one], explaining why the absorptance of the NW array shows a dip for D = 251 nm in Fig. 8(a).

Finally, for 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 P1+(z) and P2+(z) differ noticeably from zero at 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 HE12 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 λλbg (as seen in Fig. 7) and enables the maximum of the ultimate efficiency η for D = D2 ≈410 nm (see Fig. 6).

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

We show in Fig. 10
Fig. 10 The ηR-loss, the insertion reflection loss of ultimate efficiency η due to the reflection of incident light at the top NW/air interface, as a function of the diameter D and the period p of the InP NW array (see Fig. 1 for a schematic). These values are calculated by increasing L in the numerical modeling until further increase of L does not alter the results. In this limit case, the light intensity that is coupled into the NW array is absorbed in the NWs. Thus, ηR-loss is the in-coupling loss that limits η from reaching the value of ηmax = 0.463. The inset shows a line-cut of ηR-loss as a function of D for p = 680 nm (solid line). In this inset, we show also the value of 0.151 (dashed line), which is the value of the efficiency loss due to the reflection of light at a planar air/InP interface.
the insertion reflection losses of the NW array.

Acknowledgments

This work was supported by the Swedish Research Council (VR), the Swedish Foundation for Strategic Research (SSF) through the Nanometer Structure Consortium at Lund University (nmC@LU), EU program AMON-RA (No. 214814), Nordic Innovation program NANORDSUN, E.ON AG as part of the E.ON International Research Initiative, and the National Basic Research Program of the Ministry of Science and Technology of China (Nos. 2012CB932703 and 2012CB932700).

References and links

1.

K. W. J. Barnham, M. Mazzer, and B. Clive, “Resolving the energy crisis: nuclear or photovoltaics?” Nat. Mater. 5(3), 161–164 (2006). [CrossRef]

2.

A. Luque, “Will we exceed 50% efficiency in photovoltaics?” J. Appl. Phys. 110(3), 031301–031319 (2011). [CrossRef]

3.

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

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]

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]

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]

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]

9.

P. Kailuweit, M. Peters, J. Leene, K. Mergenthaler, F. Dimroth, and A. W. Bett, “Numerical simulations of absorption properties of InP nanowires for solar cell applications,” Prog. Photovolt. Res. Appl. 20(8), 945–953 (2012). [CrossRef]

10.

J. Kupec and B. Witzigmann, “Dispersion, wave propagation and efficiency analysis of nanowire solar cells,” Opt. Express 17(12), 10399–10410 (2009). [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]

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]

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]

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]

16.

G. Kästner and U. Gösele, “Stress and dislocations at cross-sectional heterojunctions in a cylindrical nanowire,” Philos. Mag. 84, 3803–3824 (2004). [CrossRef]

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. 7(6), 1817–1822 (2007). [CrossRef] [PubMed]

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]

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]

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]

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]

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]

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]

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]

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]

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 Handbook of optical constants of solids, E. D. Palik, ed. (Academic, 1985).

31.

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

32.

J. Bures, Guided optics: Optical fibers and all-fiber components (Wiley-VCH, 2009).

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]

36.

D. E. Aspnes, S. M. Kelso, R. A. Logan, and R. Bhat, “Optical properties of AlxGa1-xAs,” J. Appl. Phys. 60(2), 754–767 (1986). [CrossRef]

37.

R. Ferrini, M. Patrini, and S. Franchi, “Optical functions from 0.02 to 6 eV of AlxGa1-xSb/GaSb epitaxial layers,” J. Appl. Phys. 84(8), 4517–4524 (1998). [CrossRef]

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]

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. 36(10), 1884–1886 (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]

42.

A. W. Snyder and J. D. Love, Optical waveguide theory (Chapman and Hall, 1983).

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]

44.

J. Joannopoulos, R. Meade, and J. Winn, Photonic crystals: Molding the flow of light (Princeton University, 1995).

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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. K. W. J. Barnham, M. Mazzer, and B. Clive, “Resolving the energy crisis: nuclear or photovoltaics?” Nat. Mater.5(3), 161–164 (2006). [CrossRef]
  2. A. Luque, “Will we exceed 50% efficiency in photovoltaics?” J. Appl. Phys.110(3), 031301–031319 (2011). [CrossRef]
  3. M. A. Green, “Third generation photovoltaics: Ultra-high conversion efficiency at low cost,” Prog. Photovolt. Res. Appl.9(2), 123–135 (2001). [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. Express2, 035004 (2009). [CrossRef]
  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]
  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]
  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 Nano5(3), 2316–2323 (2011). [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]
  9. P. Kailuweit, M. Peters, J. Leene, K. Mergenthaler, F. Dimroth, and A. W. Bett, “Numerical simulations of absorption properties of InP nanowires for solar cell applications,” Prog. Photovolt. Res. Appl.20(8), 945–953 (2012). [CrossRef]
  10. J. Kupec and B. Witzigmann, “Dispersion, wave propagation and efficiency analysis of nanowire solar cells,” Opt. Express17(12), 10399–10410 (2009). [CrossRef] [PubMed]
  11. J. Kupec, R. L. Stoop, and B. Witzigmann, “Light absorption and emission in nanowire array solar cells,” Opt. Express18(26), 27589–27605 (2010). [CrossRef] [PubMed]
  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]
  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]
  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]
  16. G. Kästner and U. Gösele, “Stress and dislocations at cross-sectional heterojunctions in a cylindrical nanowire,” Philos. Mag.84, 3803–3824 (2004). [CrossRef]
  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.7(6), 1817–1822 (2007). [CrossRef] [PubMed]
  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,” Nanotechnology19(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]
  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]
  22. C. Lin and M. L. Povinelli, “Optical absorption enhancement in silicon nanowire arrays with a large lattice constant for photovoltaic applications,” Opt. Express17(22), 19371–19381 (2009). [CrossRef] [PubMed]
  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]
  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,” Nanotechnology23(19), 194010 (2012). [CrossRef] [PubMed]
  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. B83(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,” Science339(6123), 1057–1060 (2013). [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]
  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]
  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 Handbook of optical constants of solids, E. D. Palik, ed. (Academic, 1985).
  31. N. Anttu, “Geometrical optics, electrostatics, and nanophotonic resonances in absorbing nanowire arrays,” Opt. Lett.38(5), 730–732 (2013). [CrossRef] [PubMed]
  32. J. Bures, Guided optics: Optical fibers and all-fiber components (Wiley-VCH, 2009).
  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]
  36. D. E. Aspnes, S. M. Kelso, R. A. Logan, and R. Bhat, “Optical properties of AlxGa1-xAs,” J. Appl. Phys.60(2), 754–767 (1986). [CrossRef]
  37. R. Ferrini, M. Patrini, and S. Franchi, “Optical functions from 0.02 to 6 eV of AlxGa1-xSb/GaSb epitaxial layers,” J. Appl. Phys.84(8), 4517–4524 (1998). [CrossRef]
  38. C. Lin and M. L. Povinelli, “Optimal design of aperiodic, vertical silicon nanowire structures for photovoltaics,” Opt. Express19(S5Suppl 5), A1148–A1154 (2011). [CrossRef] [PubMed]
  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.36(10), 1884–1886 (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]
  42. A. W. Snyder and J. D. Love, Optical waveguide theory (Chapman and Hall, 1983).
  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]
  44. J. Joannopoulos, R. Meade, and J. Winn, Photonic crystals: Molding the flow of light (Princeton University, 1995).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited