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 OSA

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    [CrossRef] [PubMed]
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    [CrossRef]

2011 (4)

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, 1884–1886 (2011).
[CrossRef] [PubMed]

E. D. Kosten, E. L. Warren, and H. A. Atwater, “Ray optical light trapping in silicon microwires: exceeding the 2n2 intensity limit,” Opt. Express 19, 3316–3331 (2011).
[CrossRef] [PubMed]

K. R. Catchpole, S. Mokkapati, and F. J. Beck, “Comparing nanowire, multi-junction and single junction solar cells in the presence of light trapping,” J. Appl. Phys. 109, 084519 (2011).
[CrossRef]

K. Seo, M. Wober, P. Steinvurzel, E. Schonbrun, Y. Dan, T. Ellenbogen, and K. B. Crozier, “Multicolored Vertical Silicon Nanowires.” Nano Lett. 11, 1851–1856 (2011).
[CrossRef] [PubMed]

2010 (6)

O. Gunawan, K. Wang, B. Fallahazad, Y. Zhang, E. Tutuc, and S. Guha, “High performance wire-array silicon solar cells,” Prog. Photovolt. Res. Appl. 10, 1002 (2010).

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices.” Nat. Mater. 9, 205–213 (2010).
[CrossRef] [PubMed]

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of light trapping in grating structures,” Opt. Express 18, 366–380 (2010).
[CrossRef]

E. C. Garnett and P. Yang, “Light trapping in silicon nanowire solar cells,” Nano Lett. 10, 1082–1087 (2010).
[CrossRef] [PubMed]

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, 239–244 (2010).
[CrossRef] [PubMed]

S. E. Han and G. Chen, “Optical absorption enhancement in silicon nanohole arrays for solar photovoltaics,” Nano Lett. 10, 1012–1015 (2010).
[CrossRef] [PubMed]

2009 (5)

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

Z. Fan, H. Razavi, J. Do, A. Moriwaki, O. Ergen, Y.-L. Chueh, P. W. Leu, J. C. Ho, T. Takahashi, L. A. Reichertz, S. Neale, K. Yu, M. Wu, J. W. Ager, and A. Javey, “Three-dimensional nanopillar-array photovoltaics on low-cost and flexible substrates,” Nat. Mater. 8, 648–53 (2009).
[CrossRef] [PubMed]

R. A. Pala, J. S. White, E. Barnard, J. Liu, and M. L. Brongersma, “Design of plasmonic thin-film Solar cells with broadband absorption enhancements,” Adv. Mater. 21, 3504–3509 (2009).
[CrossRef]

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

J. Li, H. Yu, M. Wong, X. Li, and G. Zhang, “Design guidelines of periodic Si nanowire arrays for solar cell application,” Appl. Phys. Lett. 95, 243113 (2009).
[CrossRef]

2008 (4)

E. C. Garnett and P. Yang, “Silicon nanowire radial p-n junction solar cells.” J. Am. Chem. Soc. 130, 9224–9225 (2008).
[CrossRef] [PubMed]

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[CrossRef]

K. R. Catchpole and A. Polman, “Plasmonic solar cells,” Opt. Express 16, 21793–217800 (2008).
[CrossRef] [PubMed]

L. Tsakalakos, “Nanostructures for photovoltaics,” Mater. Sci. Eng. R 62, 175–189 (2008).
[CrossRef]

2007 (4)

N. S. Lewis, “Toward cost-effective solar energy use,” Science 315, 798–801 (2007).
[CrossRef] [PubMed]

P. Bermel, C. Luo, L. Zeng, L. C. Kimerling, and J. D. Joannopoulos, “Improving thin-film crystalline silicon solar cell efficiencies with photonic crystals,” Opt. Express 15, 16986 (2007).
[CrossRef] [PubMed]

B. M. Kayes, M. A. Filler, M. C. Putnam, M. D. Kelzenberg, N. S. Lewis, and H. A. Atwater, “Growth of vertically aligned Si wire arrays over large areas (> 1 cm2) with Au and Cu catalysts,” Appl. Phys. Lett. 91, 103110 (2007).
[CrossRef]

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

2006 (1)

L. Zeng, Y. Yi, C. Hong, J. Liu, N. Feng, X. Duan, and L. C. Kimerling, “Efficiency enhancement in Si solar cells by textured photonic crystal back reflector,” Appl. Phys. Lett. 89, 111111 (2006).
[CrossRef]

2005 (3)

B. M. Kayes, H. A. Atwater, and N. S. Lewis, “Comparison of the device physics principles of planar and radial p-n junction nanorod solar cells,” J. Appl. Phys. 97, 114302 (2005).
[CrossRef]

K. Peng, Y. Xu, Y. Wu, and Y. Yan, “Aligned single-crystalline Si nanowire arrays for photovoltaic applications,” Small 1, 1062–1067 (2005).
[CrossRef]

K. B. Dossou and M. Fontaine, “A high order isoparametric finite element method for the computation of waveguide modes,” Comput. Method Appl. M. 194, 837–858 (2005).
[CrossRef]

2004 (1)

L. C. Botten, T. P. White, A. A. Asatryan, T. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 1–13 (2004).
[CrossRef]

2000 (1)

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, and K. A. Grubits, “Lattice sums for gratings and arrays,” J. Math. Phys. 41, 7808–16 (2000).
[CrossRef]

1995 (1)

M. A. Green and M. J. Keevers, “Optical properties of intrinsic silicon at 300 K,” Prog. Photovolt. Res. Appl. 3, 189–192 (1995).
[CrossRef]

1984 (1)

G. H. Derrick and R. C. McPhedran, “Coated crossed gratings,” J. Opt. 15, 69–81 (1984).
[CrossRef]

1982 (2)

E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Am. 72, 899–907 (1982).
[CrossRef]

E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Dev. 29, 300–305 (1982).
[CrossRef]

1981 (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

R. C. McPhedran and W. T. Perrins, “Electrostatic and optical resonances of cylinder pairs,” Appl. Phys. 24, 311–318 (1981).
[CrossRef]

1961 (1)

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

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Ager, J. W.

Z. Fan, H. Razavi, J. Do, A. Moriwaki, O. Ergen, Y.-L. Chueh, P. W. Leu, J. C. Ho, T. Takahashi, L. A. Reichertz, S. Neale, K. Yu, M. Wu, J. W. Ager, and A. Javey, “Three-dimensional nanopillar-array photovoltaics on low-cost and flexible substrates,” Nat. Mater. 8, 648–53 (2009).
[CrossRef] [PubMed]

Altermatt, P. P.

P. P. Altermatt, Y. Yang, T. Langer, A. Schenk, and R. Brendel, “Simulation of optical properties of Si wire cells,” Conf. Proc. 34th IEEE PVSC , 972–977 (2009).

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Asatryan, A. A.

L. C. Botten, T. P. White, A. A. Asatryan, T. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 1–13 (2004).
[CrossRef]

Atwater, H. A.

E. D. Kosten, E. L. Warren, and H. A. Atwater, “Ray optical light trapping in silicon microwires: exceeding the 2n2 intensity limit,” Opt. Express 19, 3316–3331 (2011).
[CrossRef] [PubMed]

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, 239–244 (2010).
[CrossRef] [PubMed]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices.” Nat. Mater. 9, 205–213 (2010).
[CrossRef] [PubMed]

B. M. Kayes, M. A. Filler, M. C. Putnam, M. D. Kelzenberg, N. S. Lewis, and H. A. Atwater, “Growth of vertically aligned Si wire arrays over large areas (> 1 cm2) with Au and Cu catalysts,” Appl. Phys. Lett. 91, 103110 (2007).
[CrossRef]

B. M. Kayes, H. A. Atwater, and N. S. Lewis, “Comparison of the device physics principles of planar and radial p-n junction nanorod solar cells,” J. Appl. Phys. 97, 114302 (2005).
[CrossRef]

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[CrossRef]

Barnard, E.

R. A. Pala, J. S. White, E. Barnard, J. Liu, and M. L. Brongersma, “Design of plasmonic thin-film Solar cells with broadband absorption enhancements,” Adv. Mater. 21, 3504–3509 (2009).
[CrossRef]

Beck, F. J.

K. R. Catchpole, S. Mokkapati, and F. J. Beck, “Comparing nanowire, multi-junction and single junction solar cells in the presence of light trapping,” J. Appl. Phys. 109, 084519 (2011).
[CrossRef]

Bermel, P.

Boettcher, S. W.

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, 239–244 (2010).
[CrossRef] [PubMed]

Botten, L. C.

L. C. Botten, T. P. White, A. A. Asatryan, T. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 1–13 (2004).
[CrossRef]

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, and K. A. Grubits, “Lattice sums for gratings and arrays,” J. Math. Phys. 41, 7808–16 (2000).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Brendel, R.

P. P. Altermatt, Y. Yang, T. Langer, A. Schenk, and R. Brendel, “Simulation of optical properties of Si wire cells,” Conf. Proc. 34th IEEE PVSC , 972–977 (2009).

Briggs, R. M.

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, 239–244 (2010).
[CrossRef] [PubMed]

Brongersma, M. L.

R. A. Pala, J. S. White, E. Barnard, J. Liu, and M. L. Brongersma, “Design of plasmonic thin-film Solar cells with broadband absorption enhancements,” Adv. Mater. 21, 3504–3509 (2009).
[CrossRef]

Catchpole, K. R.

K. R. Catchpole, S. Mokkapati, and F. J. Beck, “Comparing nanowire, multi-junction and single junction solar cells in the presence of light trapping,” J. Appl. Phys. 109, 084519 (2011).
[CrossRef]

K. R. Catchpole and A. Polman, “Plasmonic solar cells,” Opt. Express 16, 21793–217800 (2008).
[CrossRef] [PubMed]

Chen, G.

S. E. Han and G. Chen, “Optical absorption enhancement in silicon nanohole arrays for solar photovoltaics,” Nano Lett. 10, 1012–1015 (2010).
[CrossRef] [PubMed]

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

Chueh, Y.-L.

Z. Fan, H. Razavi, J. Do, A. Moriwaki, O. Ergen, Y.-L. Chueh, P. W. Leu, J. C. Ho, T. Takahashi, L. A. Reichertz, S. Neale, K. Yu, M. Wu, J. W. Ager, and A. Javey, “Three-dimensional nanopillar-array photovoltaics on low-cost and flexible substrates,” Nat. Mater. 8, 648–53 (2009).
[CrossRef] [PubMed]

Cody, G. D.

E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Dev. 29, 300–305 (1982).
[CrossRef]

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Crozier, K. B.

K. Seo, M. Wober, P. Steinvurzel, E. Schonbrun, Y. Dan, T. Ellenbogen, and K. B. Crozier, “Multicolored Vertical Silicon Nanowires.” Nano Lett. 11, 1851–1856 (2011).
[CrossRef] [PubMed]

Dan, Y.

K. Seo, M. Wober, P. Steinvurzel, E. Schonbrun, Y. Dan, T. Ellenbogen, and K. B. Crozier, “Multicolored Vertical Silicon Nanowires.” Nano Lett. 11, 1851–1856 (2011).
[CrossRef] [PubMed]

de Sterke, C. M.

L. C. Botten, T. P. White, A. A. Asatryan, T. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 1–13 (2004).
[CrossRef]

Demir, H. V.

Derrick, G. H.

G. H. Derrick and R. C. McPhedran, “Coated crossed gratings,” J. Opt. 15, 69–81 (1984).
[CrossRef]

Do, J.

Z. Fan, H. Razavi, J. Do, A. Moriwaki, O. Ergen, Y.-L. Chueh, P. W. Leu, J. C. Ho, T. Takahashi, L. A. Reichertz, S. Neale, K. Yu, M. Wu, J. W. Ager, and A. Javey, “Three-dimensional nanopillar-array photovoltaics on low-cost and flexible substrates,” Nat. Mater. 8, 648–53 (2009).
[CrossRef] [PubMed]

Dossou, K. B.

K. B. Dossou and M. Fontaine, “A high order isoparametric finite element method for the computation of waveguide modes,” Comput. Method Appl. M. 194, 837–858 (2005).
[CrossRef]

Du, Q. G.

Duan, X.

L. Zeng, Y. Yi, C. Hong, J. Liu, N. Feng, X. Duan, and L. C. Kimerling, “Efficiency enhancement in Si solar cells by textured photonic crystal back reflector,” Appl. Phys. Lett. 89, 111111 (2006).
[CrossRef]

Ellenbogen, T.

K. Seo, M. Wober, P. Steinvurzel, E. Schonbrun, Y. Dan, T. Ellenbogen, and K. B. Crozier, “Multicolored Vertical Silicon Nanowires.” Nano Lett. 11, 1851–1856 (2011).
[CrossRef] [PubMed]

Ergen, O.

Z. Fan, H. Razavi, J. Do, A. Moriwaki, O. Ergen, Y.-L. Chueh, P. W. Leu, J. C. Ho, T. Takahashi, L. A. Reichertz, S. Neale, K. Yu, M. Wu, J. W. Ager, and A. Javey, “Three-dimensional nanopillar-array photovoltaics on low-cost and flexible substrates,” Nat. Mater. 8, 648–53 (2009).
[CrossRef] [PubMed]

Fallahazad, B.

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L. Zeng, Y. Yi, C. Hong, J. Liu, N. Feng, X. Duan, and L. C. Kimerling, “Efficiency enhancement in Si solar cells by textured photonic crystal back reflector,” Appl. Phys. Lett. 89, 111111 (2006).
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Adv. Mater. (1)

R. A. Pala, J. S. White, E. Barnard, J. Liu, and M. L. Brongersma, “Design of plasmonic thin-film Solar cells with broadband absorption enhancements,” Adv. Mater. 21, 3504–3509 (2009).
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Appl. Phys. (1)

R. C. McPhedran and W. T. Perrins, “Electrostatic and optical resonances of cylinder pairs,” Appl. Phys. 24, 311–318 (1981).
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Appl. Phys. Lett. (3)

L. Zeng, Y. Yi, C. Hong, J. Liu, N. Feng, X. Duan, and L. C. Kimerling, “Efficiency enhancement in Si solar cells by textured photonic crystal back reflector,” Appl. Phys. Lett. 89, 111111 (2006).
[CrossRef]

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J. Li, H. Yu, M. Wong, X. Li, and G. Zhang, “Design guidelines of periodic Si nanowire arrays for solar cell application,” Appl. Phys. Lett. 95, 243113 (2009).
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Comput. Method Appl. M. (1)

K. B. Dossou and M. Fontaine, “A high order isoparametric finite element method for the computation of waveguide modes,” Comput. Method Appl. M. 194, 837–858 (2005).
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IEEE Trans. Electron. Dev. (1)

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J. Am. Chem. Soc. (1)

E. C. Garnett and P. Yang, “Silicon nanowire radial p-n junction solar cells.” J. Am. Chem. Soc. 130, 9224–9225 (2008).
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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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