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Modal analysis of enhanced absorption in silicon nanowire arrays

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Abstract

We analyze the absorption of solar radiation by silicon nanowire arrays, which are being considered for photovoltaic applications. These structures have been shown to have enhanced absorption compared with thin films, however the mechanism responsible for this is not understood. Using a new, semi-analytic model, we show that the enhanced absorption can be attributed to a few modes of the array, which couple well to incident light, overlap well with the nanowires, and exhibit strong Fabry-Pérot resonances. For some wavelengths the absorption is further enhanced by slow light effects. We study the evolution of these modes with wavelength to explain the various features of the absorption spectra, focusing first on a dilute array at normal incidence, before generalizing to a dense array and off-normal angles of incidence. The understanding developed will allow for optimization of simple SiNW arrays, as well as the development of more advanced designs.

©2011 Optical Society of America

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Figures (9)

Fig. 1
Fig. 1 (a) Schematic of SiNW array showing direction of irradiation and nanowire length, h. We indicate the fields by f for plane waves and c for Bloch modes, where ± indicates direction of propagation. (b) Unit cell reduction of SiNW array with radius a and lattice constants d 1,2 marked. For the square array d 1 = d 2 = d.
Fig. 2
Fig. 2 Simulation results for the dilute SiNW array, over the solar spectrum; 310 nm–1127 nm, calculated at 0.5 nm wavelength intervals. Distinct wavelength regions are labeled with roman numerals. (a) Absorption spectrum, with insert showing the absorption coefficient of bulk silicon (α). (b) Fabry-Pérot resonance calculation where resonances occur at the minima of Eq. (4).
Fig. 3
Fig. 3 (a)–(c) Field distributions, of the dominant transverse electric field component, and (d)–(f) energy distributions (Re(ε)||E||2), at the absorption peak wavelength of 617 nm. Field plots are normalized between −1 (blue) and 1 (red), and energy plots are normalized between 0 (blue) and 1 (red). Bloch modes shown are; (a,d) the fundamental mode #1, (b,e) mode #3 and (c,f) the key mode #4.
Fig. 4
Fig. 4 (a) Absorption spectrum and (b) Fabry-Pérot resonances, in region III, for the dilute array. The wavelengths of features are marked and show excellent agreement for almost all points.
Fig. 5
Fig. 5 (a) Energy concentration fraction within nanowires for fundamental mode (blue) and key mode (red), overlaid on absorption spectrum. (b) Modal dispersion curves of Re(k z d), where d = 600 nm, as a function of wavelength. General modes are shown in blue, the key mode is red, and the black line is the light line.
Fig. 6
Fig. 6 Results for the dense array with wires of 125 nm radius. (a) Absorption spectrum and energy concentration fraction for the fundamental mode (blue) and key modes B 1, B 2 (red, green). (b) Modal dispersion curves for general modes (blue) and B 1, B 2 (red, green) with the light line in black.
Fig. 7
Fig. 7 Cut-off wavelengths of key mode B 1 as a function of lattice constant for arrays of differing nanowire radius. Shown are results from both FEM computations and using a dipole approximation (see Sect. 4.3).
Fig. 8
Fig. 8 Absorption spectra at 20° off-normal. The red and blue lines are s and p polarized incident fields respectively and the dashed black curve is for normal incidence.
Fig. 9
Fig. 9 Ultimate efficiency with off-normal angle of incidence for (a) the dilute and (b) the dense arrays. The insert shows the vectors (10) and (11).

Equations (10)

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u m , k , k z = φ m , k , k z ( r ) e i k r e i k z z .
T = T 21 P ( I R 21 P R 21 P ) 1 T 12 ,
η = λ l λ g I ( λ ) A ( λ ) λ λ g d λ λ l λ u I ( λ ) d λ .
det ( I R 21 P R 21 P ) .
𝒞 nanowire Re ( ε ) | | E | | 2 dA unitcell Re ( ε ) | | E | | 2 dA ,
dk z d λ = ω 2 2 π c dk z d ω = ω 2 2 π c 1 v g .
E z ( r , θ ) = { C E J 1 ( k 1 r ) e i θ for r < a [ A E J 1 ( k 2 r ) + B E H 1 ( k 2 r ) ] e i θ for r a
A 1 E , H = S 0 ( k 2 , 0 ) B 1 E , H ,
S 0 ( k 2 , k ) = p 0 H 0 ( k 2 | R p | ) e i k R p .
ε 1 k 1 J 1 ( k 1 a ) J 1 ( k 1 a ) ε 2 k 2 H 1 ( k 2 a ) + S 0 J 1 ( k 2 a ) H 1 ( k 2 a ) + S 0 J 1 ( k 2 a ) = 0 ,
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