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Efficient light management in vertical nanowire arrays for photovoltaics

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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 Optical Society of America

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Figures (10)

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.
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.
Fig. 3
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).
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.
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.
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).
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.
Fig. 8
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 k 1 =2π/λ7.4 10 7 m−1 and mode (2) is a (evanescent) diffracted order with k 2 = (2π/λ) 2 (2π/p) 2 i5.5 10 6 m−1.
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 [ 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.
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.

Equations (17)

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η= 0 λ λ λ bg I AM1.5 (λ)A(λ)dλ 0 I AM1.5 (λ)dλ
P(z)= 1 4 UC ( E * ×H +E× H * )· e ^ z dS
E xy (x)= a E a (x,y)( C a + e i k a z + C a e i k a z )
H xy (x)= a H a (x,y)( C a + e i k a z C a e i k a z ) .
P(z)= ab [ P ab ++ (z) + P ab + (z)+ P ab + (z)+ P ab (z)]
P ab ++ (z)= N ab ++ C a +* C b + e i( k b k a * )z ,
P ab + (z)= N ab + C a +* C b e i( k b k a * )z ,
P ab + (z)= N ab + C a * C b + e i( k b + k a * )z ,
P ab (z)= N ab C a * C b e i( k b + k a * )z ,
N ab ++ (z)= 1 4 UC ( E a * × H b + E b × H a * )· e ^ z dS,
N ab + (z)= 1 4 UC ( E a * × H b + E b × H a * )· e ^ z dS,
N ab + (z)= 1 4 UC ( E a * × H b E b × H a * )· e ^ z dS,
N ab (z)= 1 4 UC ( E a * × H b E b × H a * )· e ^ z dS.
P a + (z) P aa ++ (z)
P a (z) P aa (z)
P(z) a [ P a + (z) + P a (z)]
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),
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