Abstract

We calculate the dc current induced second harmonic generation in doped graphene using the semiconductor Bloch equations under relaxation time approximations. We find that the maximum value of the effective second order susceptibility appears when the fundamental photon energy matches the chemical potential. For a surface current density 1.1 × 103 A/m and a relaxation time at optical frequencies of 13 fs, the effective second order susceptibility χeff(2);xxx can be as large as 10−7m/V for h̄ω = 0.2 eV or 10−8 m/V for h̄ω = 0.53 eV.

© 2014 Optical Society of America

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2014 (4)

C. Matheisen, M. Waldow, B. Chmielak, S. Sawallich, T. Wahlbrink, J. Bolten, M. Nagel, and H. Kurz, “Electro-optic light modulation and THz generation in locally plasma-activated silicon nanophotonic devices,” Opt. Express 22, 5252–5259 (2014).
[Crossref] [PubMed]

W. Liu, L. Wang, and C. Fang, “In-situ weak-beam and polarization control of multidimensional laser sidebands for ultrafast optical switching,” Appl. Phys. Lett. 104, 111114 (2014).
[Crossref]

M. Glazov and S. Ganichev, “High frequency electric field induced nonlinear effects in graphene,” Phys. Rep. 535, 101–138 (2014).
[Crossref]

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order optical nonlinearity of graphene,” New J. Phys. 16, 053014 (2014).
[Crossref]

2013 (5)

M.-L. Ren, X.-L. Zhong, B.-Q. Chen, and Z.-Y. Li, “An all-optical diode based on plasmonic attenuation and nonlinear frequency conversion,” Chinese Phys. Lett. 30, 097301 (2013).
[Crossref]

N. Kumar, J. Kumar, C. Gerstenkorn, R. Wang, H.-Y. Chiu, A. L. Smirl, and H. Zhao, “Third harmonic generation in graphene and few-layer graphite films,” Phys. Rev. B 87, 121406 (2013).
[Crossref]

S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood, “Optical third-harmonic generation in graphene,” Phys. Rev. X 3, 021014 (2013).

N. Kumar, Q. Cui, F. Ceballos, and H. Zhao, “Observation of strong second harmonic generation in monolayer MoS2,” Phys. Rev. B 87, 161403(R) (2013).
[Crossref]

P. Zhang and M. W. Wu, “Hot-carrier transport and spin relaxation on the surface of topological insulator,” Phys. Rev. B 87, 085319 (2013).
[Crossref]

2012 (5)

C.-H. Liu, N. M. Dissanayake, S. Lee, K. Lee, and Z. Zhong, “Evidence for extraction of photoexcited hot carriers from graphene,” ACS Nano 6, 7172–7176 (2012).
[Crossref] [PubMed]

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nat. Photonics 6, 554–559 (2012).
[Crossref]

H. Zhang, S. Virally, Q. Bao, L. K. Ping, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37, 1856–1858 (2012).
[Crossref] [PubMed]

S. Wu, L. Mao, A. M. Jones, W. Yao, C. Zhang, and X. Xu, “Quantum-enhanced tunable second-order optical nonlinearity in bilayer graphene,” Nano Lett. 12, 2032–2036 (2012).
[Crossref] [PubMed]

A. Y. Bykov, T. V. Murzina, M. G. Rybin, and E. D. Obraztsova, “Second harmonic generation in multilayer graphene induced by direct electric current,” Phys. Rev. B 85, 121413 (2012).
[Crossref]

2011 (7)

M. Glazov, “Second harmonic generation in graphene,” JETP Lett. 93, 366–371 (2011).
[Crossref]

S. A. Mikhailov, “Theory of the giant plasmon-enhanced second-harmonic generation in graphene and semiconductor two-dimensional electron systems,” Phys. Rev. B 84, 045432 (2011).
[Crossref]

M. Gandomkar and V. Ahmadi, “Thermo-optical switching enhanced with second harmonic generation in microring resonators,” Opt. Lett. 36, 3825–3827 (2011).
[Crossref] [PubMed]

H. Yang, X. Feng, Q. Wang, H. Huang, W. Chen, A. T. S. Wee, and W. Ji, “Giant two-photon absorption in bilayer graphene,” Nano Lett. 11, 2622–2627 (2011).
[Crossref] [PubMed]

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).
[Crossref] [PubMed]

S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, “Electronic transport in two-dimensional graphene,” Rev. Mod. Phys. 83, 407–470 (2011).
[Crossref]

H. Liu, Y. Liu, and D. Zhu, “Chemical doping of graphene,” J. Mater. Chem. 21, 3335–3345 (2011).
[Crossref]

2010 (5)

J. H. Wülbern, S. Prorok, J. Hampe, A. Petrov, M. Eich, J. Luo, A. K.-Y. Jen, M. Jenett, and A. Jacob, “40 GHz electro-optic modulation in hybrid silicon-organic slotted photonic crystal waveguides,” Opt. Lett. 35, 2753–2755 (2010).
[Crossref] [PubMed]

D. Sun, C. Divin, J. Rioux, J. E. Sipe, C. Berger, W. A. de Heer, P. N. First, and T. B. Norris, “Coherent control of ballistic photocurrents in multilayer epitaxial graphene using quantum interference,” Nano Lett. 10, 1293–1296 (2010).
[Crossref] [PubMed]

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[Crossref]

E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A. Mikhailov, “Coherent nonlinear optical response of graphene,” Phys. Rev. Lett. 105, 097401 (2010).
[Crossref] [PubMed]

J. J. Dean and H. M. van Driel, “Graphene and few-layer graphite probed by second-harmonic generation: Theory and experiment,” Phys. Rev. B 82, 125411 (2010).
[Crossref]

2009 (2)

J. J. Dean and H. M. van Driel, “Second harmonic generation from graphene and graphitic films,” Appl. Phys. Lett. 95, 261910 (2009).
[Crossref]

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109–162 (2009).
[Crossref]

2008 (6)

E. H. Hwang and S. Das Sarma, “Single-particle relaxation time versus transport scattering time in a two-dimensional graphene layer,” Phys. Rev. B 77, 195412 (2008).
[Crossref]

S. A. Mikhailov and K. Ziegler, “Nonlinear electromagnetic response of graphene: frequency multiplication and the self-consistent-field effects,” J. Phys. Condens. Matter 20, 384204 (2008).
[Crossref] [PubMed]

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science 320, 1308 (2008).
[Crossref] [PubMed]

X. Du, I. Skachko, A. Barker, and E. Y. Andrei, “Approaching ballistic transport in suspended graphene,” Nature Nanotechnol. 3, 491–495 (2008).
[Crossref]

I. F. Herbut, V. Juričić, and O. Vafek, “Coulomb interaction, ripples, and the minimal conductivity of graphene,” Phys. Rev. Lett. 100, 046403 (2008).
[Crossref] [PubMed]

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320, 206–209 (2008).
[Crossref] [PubMed]

2007 (6)

K. Nomura and A. H. MacDonald, “Quantum transport of massless Dirac fermions,” Phys. Rev. Lett. 98, 076602 (2007).
[Crossref] [PubMed]

E. H. Hwang and S. Das Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75, 205418 (2007).
[Crossref]

Y.-W. Tan, Y. Zhang, K. Bolotin, Y. Zhao, S. Adam, E. H. Hwang, S. Das Sarma, H. L. Stormer, and P. Kim, “Measurement of scattering rate and minimum conductivity in graphene,” Phys. Rev. Lett. 99, 246803 (2007).
[Crossref]

E. G. Mishchenko, “Effect of electron-electron interactions on the conductivity of clean graphene,” Phys. Rev. Lett. 98, 216801 (2007).
[Crossref] [PubMed]

S. A. Mikhailov, “Non-linear electromagnetic response of graphene,” Europhys. Lett. 79, 27002 (2007).
[Crossref]

J. Moser, A. Barreiro, and A. Bachtold, “Current-induced cleaning of graphene,” Appl. Phys. Lett. 91, 163513 (2007).
[Crossref]

2005 (1)

G. Y. Guo and J. C. Lin, “Second-harmonic generation and linear electro-optical coefficients of bn nanotubes,” Phys. Rev. B 72, 075416 (2005).
[Crossref]

2000 (1)

J. E. Sipe and A. I. Shkrebtii, “Second-order optical response in semiconductors,” Phys. Rev. B 61, 5337–5352 (2000).
[Crossref]

1995 (2)

C. Aversa and J. E. Sipe, “Nonlinear optical susceptibilities of semiconductors: Results with a length-gauge analysis,” Phys. Rev. B 52, 14636–14645 (1995).
[Crossref]

J. B. Khurgin, “Current induced second harmonic generation in semiconductors,” Appl. Phys. Lett. 67, 1113–1115 (1995).
[Crossref]

1994 (1)

1993 (1)

C. Ironside, J. Aitchison, and J. Arnold, “An all-optical switch employing the cascaded second-order nonlinear effect,” IEEE J. Quantum Electron. 29, 2650–2654 (1993).
[Crossref]

1970 (1)

N. Mermin, “Lindhard dielectric function in the relaxation-time approximation,” Phys. Rev. B 1, 2362–2363 (1970).
[Crossref]

Adam, S.

S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, “Electronic transport in two-dimensional graphene,” Rev. Mod. Phys. 83, 407–470 (2011).
[Crossref]

Y.-W. Tan, Y. Zhang, K. Bolotin, Y. Zhao, S. Adam, E. H. Hwang, S. Das Sarma, H. L. Stormer, and P. Kim, “Measurement of scattering rate and minimum conductivity in graphene,” Phys. Rev. Lett. 99, 246803 (2007).
[Crossref]

Ahmadi, V.

Aitchison, J.

C. Ironside, J. Aitchison, and J. Arnold, “An all-optical switch employing the cascaded second-order nonlinear effect,” IEEE J. Quantum Electron. 29, 2650–2654 (1993).
[Crossref]

Andrei, E. Y.

X. Du, I. Skachko, A. Barker, and E. Y. Andrei, “Approaching ballistic transport in suspended graphene,” Nature Nanotechnol. 3, 491–495 (2008).
[Crossref]

Arnold, J.

C. Ironside, J. Aitchison, and J. Arnold, “An all-optical switch employing the cascaded second-order nonlinear effect,” IEEE J. Quantum Electron. 29, 2650–2654 (1993).
[Crossref]

Assanto, G.

Aversa, C.

C. Aversa and J. E. Sipe, “Nonlinear optical susceptibilities of semiconductors: Results with a length-gauge analysis,” Phys. Rev. B 52, 14636–14645 (1995).
[Crossref]

Bachtold, A.

J. Moser, A. Barreiro, and A. Bachtold, “Current-induced cleaning of graphene,” Appl. Phys. Lett. 91, 163513 (2007).
[Crossref]

Bai, X.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).
[Crossref] [PubMed]

Bao, Q.

Barker, A.

X. Du, I. Skachko, A. Barker, and E. Y. Andrei, “Approaching ballistic transport in suspended graphene,” Nature Nanotechnol. 3, 491–495 (2008).
[Crossref]

Barreiro, A.

J. Moser, A. Barreiro, and A. Bachtold, “Current-induced cleaning of graphene,” Appl. Phys. Lett. 91, 163513 (2007).
[Crossref]

Berger, C.

D. Sun, C. Divin, J. Rioux, J. E. Sipe, C. Berger, W. A. de Heer, P. N. First, and T. B. Norris, “Coherent control of ballistic photocurrents in multilayer epitaxial graphene using quantum interference,” Nano Lett. 10, 1293–1296 (2010).
[Crossref] [PubMed]

Bian, F.

R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).
[Crossref] [PubMed]

Blake, P.

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science 320, 1308 (2008).
[Crossref] [PubMed]

Blount, E. I.

E. I. Blount, Solid State Physics, Advances in Research and Application (Academic, New York, 1962), vol. 13, chap. Formalism of Band Theory, p. 305. And references cited therein.

Bolotin, K.

Y.-W. Tan, Y. Zhang, K. Bolotin, Y. Zhao, S. Adam, E. H. Hwang, S. Das Sarma, H. L. Stormer, and P. Kim, “Measurement of scattering rate and minimum conductivity in graphene,” Phys. Rev. Lett. 99, 246803 (2007).
[Crossref]

Bolten, J.

Bonaccorso, F.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[Crossref]

Booth, T. J.

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science 320, 1308 (2008).
[Crossref] [PubMed]

Bykov, A. Y.

A. Y. Bykov, T. V. Murzina, M. G. Rybin, and E. D. Obraztsova, “Second harmonic generation in multilayer graphene induced by direct electric current,” Phys. Rev. B 85, 121413 (2012).
[Crossref]

Castro Neto, A. H.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109–162 (2009).
[Crossref]

Ceballos, F.

N. Kumar, Q. Cui, F. Ceballos, and H. Zhao, “Observation of strong second harmonic generation in monolayer MoS2,” Phys. Rev. B 87, 161403(R) (2013).
[Crossref]

Chen, B.-Q.

M.-L. Ren, X.-L. Zhong, B.-Q. Chen, and Z.-Y. Li, “An all-optical diode based on plasmonic attenuation and nonlinear frequency conversion,” Chinese Phys. Lett. 30, 097301 (2013).
[Crossref]

Chen, W.

H. Yang, X. Feng, Q. Wang, H. Huang, W. Chen, A. T. S. Wee, and W. Ji, “Giant two-photon absorption in bilayer graphene,” Nano Lett. 11, 2622–2627 (2011).
[Crossref] [PubMed]

Cheng, J. L.

J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order optical nonlinearity of graphene,” New J. Phys. 16, 053014 (2014).
[Crossref]

Chiu, H.-Y.

N. Kumar, J. Kumar, C. Gerstenkorn, R. Wang, H.-Y. Chiu, A. L. Smirl, and H. Zhao, “Third harmonic generation in graphene and few-layer graphite films,” Phys. Rev. B 87, 121406 (2013).
[Crossref]

Chmielak, B.

Crommie, M.

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[Crossref]

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H. Yang, X. Feng, Q. Wang, H. Huang, W. Chen, A. T. S. Wee, and W. Ji, “Giant two-photon absorption in bilayer graphene,” Nano Lett. 11, 2622–2627 (2011).
[Crossref] [PubMed]

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[Crossref]

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R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).
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Wang, Z.

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P. Zhang and M. W. Wu, “Hot-carrier transport and spin relaxation on the surface of topological insulator,” Phys. Rev. B 87, 085319 (2013).
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R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).
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S. Wu, L. Mao, A. M. Jones, W. Yao, C. Zhang, and X. Xu, “Quantum-enhanced tunable second-order optical nonlinearity in bilayer graphene,” Nano Lett. 12, 2032–2036 (2012).
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R. Wu, Y. Zhang, S. Yan, F. Bian, W. Wang, X. Bai, X. Lu, J. Zhao, and E. Wang, “Purely coherent nonlinear optical response in solution dispersions of graphene sheets,” Nano Lett. 11, 5159–5164 (2011).
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H. Yang, X. Feng, Q. Wang, H. Huang, W. Chen, A. T. S. Wee, and W. Ji, “Giant two-photon absorption in bilayer graphene,” Nano Lett. 11, 2622–2627 (2011).
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T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nat. Photonics 6, 554–559 (2012).
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F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320, 206–209 (2008).
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Figures (1)

Fig. 1
Fig. 1

w dependence of Sxxyy(w, w, γ) [(a) real part and (c) imaginary part] and Sxyyx(w, w, γ) [(b) real part and (d) imaginary part] for different γ. Insets in (c) and (d) illustrate the resonant optical transitions of the peak at w = 1 and w = 2 respectively. In the inset, the band structure is illustrated around the Dirac cones, the arrows are for photons with frequency given aside, the short green lines indicate the intermediate states, the orange dashed lines show the chemical potential without dc field, and the filled orange regions indicate the shifted distribution under dc field.

Equations (20)

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ρ k ( t ) t = i h ¯ [ k e E tot ( t ) ξ k , ρ k ( t ) ] e h ¯ E tot ( t ) k ρ k ( t ) ρ k ρ k 0 τ ,
0 = i h ¯ [ k e E dc ξ k , ρ k dc ] e h ¯ E dc k ρ k dc ρ k dc ρ k 0 τ 1 ,
δ ρ s s k D = e h ¯ E dc c τ 1 n s k k c ,
δ ρ s 1 s 2 k N = e h ¯ E dc c ξ s 1 s 2 k c ( n s 1 k n s 2 k ) ω s 1 s 2 k i τ 1 1 ,
ρ k op t = i h ¯ [ k , ρ k op ] + i e h ¯ [ E dc + E ( t ) ] ( [ ξ k , ρ k op ] + i k ρ k op ) + i e h ¯ E ( t ) ( [ ξ k , ρ k dc ] + i k ρ k dc ) ρ k op ( t ) τ 2 .
ρ k op t = i h ¯ [ k , ρ k op ] + i e h ¯ E ( t ) ( [ ξ k , ρ k op ] + i k ρ k op ) + i e h ¯ E ( t ) ( [ ξ k , δ ρ k D ] + i k δ ρ k D ) ρ k op ( t ) τ 2 ,
ρ k op ( t ) = ρ k op ; ( 1 ) ( t ) + ρ k op ; ( 2 ) ( t ) +
ρ k op ; ( 1 ) t = i h ¯ [ k , ρ k op ; ( 1 ) ] + i e h ¯ E ( t ) ( [ ξ k , δ ρ k D ] + i k δ ρ k D ) ρ k op ; ( 1 ) ( t ) τ 2 ,
ρ k op ; ( 2 ) t = i h ¯ [ k , ρ k op ; ( 2 ) ] + i e h ¯ E ( t ) ( [ ξ k , ρ k op ; ( 1 ) ] + i k ρ k op ; ( 1 ) ) ρ k op ; ( 2 ) ( t ) τ 2 .
ρ k op ; ( 1 ) ( t ) = d ω 2 π e h ¯ E ω a e i ω t 𝒫 k ( 1 ) ; a ( ω ) ,
ρ k op ; ( 2 ) ( t ) = d ω 1 d ω 2 ( 2 π ) 2 e 2 h ¯ 2 E ω 1 a E ω 2 b e i ω 0 t 𝒫 k ( 2 ) ; a b ( ω 1 , ω 2 ) ,
𝒫 k ( 1 ) ; b ( ω ) = I ω 2 + ( [ ξ k b , δ ρ k D ] + i δ ρ k D k b ) ,
𝒫 k 2 ; ( a b ) ( ω 1 , ω 2 ) = I ω 0 + ( [ ξ k a , 𝒫 k ( 1 ) ; b ( ω 2 ) + i 𝒫 k ( 1 ) ; b ( ω 2 ) k a ] ) .
J ( 2 ) ; d ( t ) = e k Tr [ v k d ρ k op ; ( 2 ) ( t ) ] = d ω 1 d ω 2 ( 2 π ) 2 e i ω 0 t σ eff ; J ( 2 ) ; d a b ( ω 1 , ω 2 ) E ω 1 a E ω 2 b ,
σ eff ; J ( 2 ) , d a b ( ω 1 , ω 2 ) = 1 2 [ σ ˜ eff ; J ( 2 ) ; d a b ( ω 1 , ω 2 ) + σ ˜ eff ; J ( 2 ) ; d b a ( ω 2 , ω 1 ) ] , σ eff ; J ( 2 ) , d a b ( ω 1 , ω 2 ) = e 3 k Tr [ v k d 𝒫 k ( 2 ) ; a b ( ω 1 , ω 2 ) ] .
δ ρ s s k D = e h ¯ E dc a τ 1 n s k k a , δ ρ s s ¯ k N = e E dc a ξ s s ¯ k a Δ n k ε k i s h ¯ τ 1 1 ,
σ dc x x = σ 0 τ 1 | μ | h ¯ + 2 π σ 0 arctan ( h ¯ 2 | μ | τ 1 ) .
S ( w 1 , w 2 , γ ) = 4 [ ( w 1 + ) 2 4 ] w 2 ( 𝒜 1 w 2 𝒜 3 w 0 + ) + 4 [ ( w 2 + ) 2 4 ] w 1 ( 𝒜 2 w 1 𝒜 3 w 0 + ) + 4 ( w 0 + ) 2 4 [ 1 w 0 + ( 𝒜 0 w 1 + + 𝒜 0 w 2 + + 𝒜 3 w 1 + 𝒜 3 w 2 ) 𝒜 2 w 1 2 𝒜 1 w 2 2 ] + 8 w 0 + [ ( w 0 + ) 2 4 ] 2 ( 𝒜 1 w 2 + 𝒜 2 w 1 + 𝒜 2 w 1 + 𝒜 1 w 2 ) .
S ( w , w , γ ) = 4 [ ( w + i γ ) 2 4 ] w ( 0 w 𝒜 2 w + i γ ) 8 ( 2 w + i γ ) [ ( 2 w + i γ ) 2 4 ] 2 i γ 0 w ( w + i γ ) + 4 ( 2 w + i γ ) 2 4 [ 2 ( 2 w + i γ ) ( w + i γ ) ( i γ w 𝒜 3 0 ) 0 w 2 ]
J op ( 2 ) ( 2 ω ) = 2 σ 1 E ω E ω J dc + σ 2 J dc E ω E ω ,

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