1. Introduction

Photovoltaics (PV) is the conversion of solar radiation directly into electricity and could constitute a large fraction of the world energy production in the future [1–3]. The search for new and improved concepts has been one of the most important tasks in the PV research and technology development. Recently, there has been a rapidly increasing interest in the use of vertical arrays of direct band gap III-V semiconductor nanowires (NWs) for photovoltaics [4–15]. In the NW geometry, as compared to planar thin-film structures, there are, due to strain relaxation in the radial direction, not as strict requirements [16] for crystal lattice matching of the materials in different layers when creating NW heterostructures [6, 17]. Therefore, a much larger pallet of epitaxial combinations of III-V materials becomes available. This allows, for example, for an efficient matching [3] of the band gaps of the NW materials to the solar spectrum [6] and for the fabrication [18] of the active NW regions on top of a substrate of a cheap and abundant material. Furthermore, doping of the NWs can be readily performed to design the electrically active regions of the cells [19]. Previously, InP [4, 6] and GaAs [12] NWs have been integrated into functioning prototype solar cells. But optimization is needed for achieving a high efficiency PV performance.

The absorption of incident light is one of the key factors that determines the efficiency of a solar cell [20]. However, the interaction of the incident light with a NW array could not be described properly by simple effective medium theories [21], and a full electrodynamic description is needed [20]. 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.

Previously, several guidelines for choosing the nanowire diameter and the array period have been provided [5, 8, 11, 13, 14, 20, 22–24]. For InP NWs, when L = 2000 nm was fixed, D = 180 nm and p = 400 nm yielded efficient absorption in the NWs [5]. These values were later refined to D = 180 nm and p = 360 nm [11]. Thus, the array should not be too sparse, but not too dense either [11]. In contrast, for Si NWs in the nominal work by Hu and Chen where the period was limited to p < 100 nm [20], an increasing density of NWs was shown to increase monotonously the absorption performance. However, later, when the limit on p was relaxed, D = 540 nm and p = 600nm were identified as an appropriate choice for fixed L = 2330 nm [22], D/p ≈0.8 was given as a general guideline for L = 5000 nm [23], and D = 500 nm and p = 600 nm were indicated as a suitable choice for varying L [24]. Also for GaAs NWs, D = 180 nm was identified as an optimal choice [8], whereas in another work [13], the array period was limited to 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], no general trends in the optimum choice for 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.

Here, we present a theoretical study of the absorption of light in an InP NW array with the NW diameter, the NW length, and the period of the array being varied in a controlled, systematic manner. The study in this large parameter space is performed with an electrodynamic scattering matrix formalism [25]. We have shown previously that results from such modeling are in good agreement with experimentally measured external quantum efficiency of NW solar cells [26] as well as with measured reflectance spectra of both uncoated [21] and Al₂O₃–coated [15] NW arrays, indicating a high validity of the modeling. We find a set of intuitive design rules for absorption-efficient NW array PV structures. First, we will show that an array consisting of InP NWs of less than 100 nm in diameter does not absorb light of long wavelengths effectively and is thus not a recommended design for PV devices. We then show that the absorption of solar energy in the InP NW array exhibits local maxima at NW diameters of approximately 170 and 410 nm, irrespective of the NW length, in line with our group's very recent experimental demonstration of a high efficiency NW array solar cell fabricated from InP NWs of 180 nm in diameter [26].

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

 

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 k_x = k_y = 0, toward the NW array from air on the top side.

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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, 6, 12, 26]. To quantify the broadband absorption of sun light in the NW array, we use the ultimate efficiency η [5, 14, 20, 27]. 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, 14, 20, 27]

 

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 (λ_bg = 925 nm) and thermalization losses. (b) Zoom-in of the intensity usable from the AM1.5 spectrum for an InP solar cell.

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We use an electrodynamic description to take into account the wave behavior of light. The modeling is done with a scattering matrix method [25], 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 n_InP of the InP [30] and for air n = 1 is used.

We show in Fig. 3(a) 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 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.

 

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 η_max = 0.463, the maximum possible ultimate efficiency for InP, is shown (dashed line).

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We start by considering short NWs of length 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 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.

 

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 η_max = 0.463 (dashed line), the maximum possible ultimate efficiency of InP, is shown.

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When the NW length is increased to L = 2000 nm (see Fig. 5), 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, 31]). 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.

 

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 η_max = 0.463 (dashed line), the maximum possible ultimate efficiency of InP, is shown.

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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. 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.

 

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, η_max = 0.463, is shown (dashed-dotted black line).

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

 

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 > λ_bg = 925 nm.

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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 R_top, 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 $k_1=2\pi /\lambda \approx 7.4\cdot {10}^7$ m⁻¹ and mode (2) is a (evanescent) diffracted order with $k_2=\sqrt{{(2\pi /\lambda )}^2-{(2\pi /p)}^2}\approx i5.5\cdot {10}^6$ m⁻¹.

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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 λ ≈λ_bg 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 Al_0.42Ga_0.58As [36] that has a band gap energy of 1.95 eV, which is considerably higher than the InP value of E_g = 1.34 eV, and GaSb [37] 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 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_InPD_1,InPE_g,InP/(n_AlGaAsE_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 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 Al_0.42Ga_0.58As NW array, we find also a maximum of η for D_3,AlGaAs ≈500 nm in the full modeling, which is caused by the HE₁₃ 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.

Finally, we comment on why only the HE_1n waveguide modes contribute to the maxima of η and not any of the other TE_n, TM_n, HE_mn or EH_mn modes that exist for a NW of circular cross-section [32]. The TE_n and TM_n 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 HE_mn and EH_mn 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 HE_1n or the EH_1n modes. The (electric) field patterns of the HE_1n modes are similar to the plane wave pattern outside the NW core, whereas the field patterns of the EH_1n modes differ considerably from the plane wave pattern [32]. Thus, the strongest possible excitation is expected for the HE_1n 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 [38–40]. 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] and the conical and base tapered NWs investigated by Diedenhofen et al. [7].

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 > 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 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 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 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 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

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]. Importantly, the propagation constant k_a = Rek_a + iImk_a 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, Rek_a is the phase constant and Imk_a the attenuation constant of the eigenmode. The electric field of the ath eigenmode decays along z as exp(-Imk_az), and the intensity of the mode decays as exp(−2Imk_az). 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 - 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 Imk_a 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 Imk_HE11 and Imk_HE12 of the HE₁₁ and HE₁₂ 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 Imk_HE1n > 0. We note that for a bound waveguide mode, Imk_HE1n = 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 Rek_HE1n ≥ 2π/λ for which the waveguide mode is bound [42]. 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 λ ≈λ_bg for D₁ and D₂ originate, respectively, from the peaks of Imk_HE11(D) and Imk_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 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 ${\widehat{e}}_z$) through the cross-section of one unit cell of area UC of the periodic array is given by

(2)$$P(z)=\frac{1}{4}{\displaystyle {\int }_{UC}(E^{*}\times H}+E\times H^{*})·{\widehat{e}}_z\text{dS}$$

and 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 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 $A=P\left(z=0\right)-P\left(z=L\right)$]. Furthermore, by expanding the electromagnetic field inside the NW array in terms of the eigenmodes of the NW array, it is possible [43] 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, $E_{xy}(x)={[E_x(x),E_y(x)]}^T,$ can be expanded in terms of the optical eigenmodes as [25]

(3)$$E_{xy}(x)={\displaystyle {\sum }_a{E}_a(x,y)(C_a^{+}{e}^{i{k}_{a}z}+C_a^{-}{e}^{-i{k}_{a}z})}$$

where $E_a(x,y)={[E_{a,x}(x),E_{a,y}(x)]}^T$ contains the x and y components of the electric field of the ath eigenmode. The complex-valued propagation constant (that is, the wave vector) k_a of the ath eigenmode is obtained under the rule that Rek_a > 0 if Imk_a = 0 and Imk_a > 0 if Imk_a ≠ 0. Thus, $C_a^{+}$ is the expansion coefficient of the ath eigenmode that propagates forward (or decays exponentially) along the positive z direction, whereas $C_a^{-}$ 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]

(4)$$H_{xy}(x)={\displaystyle {\sum }_a{H}_a(x,y)(C_a^{+}{e}^{i{k}_{a}z}-C_a^{-}{e}^{-i{k}_{a}z})}.$$

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]

(5)$$P(z)={\displaystyle {\sum }_{ab}[P_{ab}^{++}(z)}+P_{ab}^{+-}(z)+P_{ab}^{-+}(z)+P_{ab}^{--}(z)]$$

where

(6)$$P_{ab}^{++}(z)=N_{ab}^{++}{C}_a^{+*}{C}_b^{+}{e}^{i(k_b-k_a^{*})z},$$

(7)$$P_{ab}^{+-}(z)=N_{ab}^{+-}{C}_a^{+*}{C}_b^{-}{e}^{i(-k_b-k_a^{*})z},$$

(8)$$P_{ab}^{-+}(z)=N_{ab}^{-+}{C}_a^{-*}{C}_b^{+}{e}^{i(k_b+k_a^{*})z},$$

(9)$$P_{ab}^{--}(z)=N_{ab}^{--}{C}_a^{-*}{C}_b^{-}{e}^{i(-k_b+k_a^{*})z},$$

and

(10)$$N_{ab}^{++}(z)=\frac{1}{4}{\displaystyle {\int }_{UC}(E_a^{*}\times H_b}+E_b\times H_a^{*})·{\widehat{e}}_z\text{dS,}$$

(11)$$N_{ab}^{+-}(z)=\frac{1}{4}{\displaystyle {\int }_{UC}(-E_a^{*}\times H_b}+E_b\times H_a^{*})·{\widehat{e}}_z\text{dS,}$$

(12)$$N_{ab}^{-+}(z)=\frac{1}{4}{\displaystyle {\int }_{UC}(E_a^{*}\times H_b}-E_b\times H_a^{*})·{\widehat{e}}_z\text{dS,}$$

(13)$$N_{ab}^{--}(z)=\frac{1}{4}{\displaystyle {\int }_{UC}(-E_a^{*}\times H_b}-E_b\times H_a^{*})·{\widehat{e}}_z\text{dS}\text{.}$$

We can identify from Eq. (5)

(14)$$P_a^{+}(z)\equiv P_{aa}^{++}(z)$$

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 $C_a^{+}\ne 0$, $C_b^{+}=0$ for all b ≠ a, and $C_d^{-}=0$for all d). Similarly,

(15)$$P_a^{-}(z)\equiv P_{aa}^{--}(z)$$

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,

(16)$$P(z)\ne {\displaystyle {\sum }_a[P_a^{+}(z)}+P_a^{-}(z)]$$

in general [43]. Instead, the cross-powers $P_{ab}^{+-}$, $P_{ab}^{-+}$, $P_{ab}^{++}$, and $P_{ab}^{--}$ 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 $E_a(x,y)$and $H_a(x,y)$ and the propagation constants 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

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 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 $P_1^{+}(z)$, $P_2^{+}(z)$, $P_1^{-}(z)$, and $P_2^{-}(z)$, as well as

(17)$$\begin{array}{c}{P}_{12}^{ct}(z)=P_{11}^{+-}(z)+P_{11}^{-+}(z)+P_{22}^{+-}(z)+P_{22}^{-+}(z)+P_{12}^{++}(z)+P_{21}^{++}(z)+P_{12}^{--}(z)\\ +P_{21}^{++}(z)+P_{12}^{+-}(z)+P_{21}^{+-}(z)+P_{12}^{-+}(z)+P_{21}^{-+}(z),\end{array}$$

the sum of all cross-powers between mode (1) and (2).

 

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 [$P_1^{+}$and $P_2^{+}$] and backward [$P_1^{-}$and $P_2^{-}$] 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 $P_{12}^{ct}$, 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.

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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 $P_1^{+}(z)$ and $P_1^{-}(z)$ differ noticeably from zero]. We note that this mode is excited also in the backward propagating direction [that is, $P_1^{-}(z)\ne 0$] 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 $P_1^{+}(z)$ and $P_2^{+}(z)$ 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 λ ≈λ_bg (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 $P_1^{+}(z)$ and $P_2^{+}(z)$ 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., $P_2^{+}(z=0)$ 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 $P_1^{+}(z)$ and $P_2^{+}(z)$ 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 λ ≈λ_bg (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 the insertion reflection losses of the NW array.

 

Fig. 10 The η_R-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, η_R-loss is the in-coupling loss that limits η from reaching the value of η_max = 0.463. The inset shows a line-cut of η_R-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.

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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).

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