Abstract

Near-field thermophotovoltaic (TPV) systems with carefully tailored emitter-PV properties show large promise for a new temperature range (600 – 1200K) solid state energy conversion, where conventional thermoelectric (TE) devices cannot operate due to high temperatures and far-field TPV schemes suffer from low efficiency and power density. We present a detailed theoretical study of several different implementations of thermal emitters using plasmonic materials and graphene. We find that optimal improvements over the black body limit are achieved for low bandgap semiconductors and properly matched plasmonic frequencies. For a pure plasmonic emitter, theoretically predicted generated power density of 14Wcm2 and efficiency of 36% can be achieved at 600K (hot-side), for 0.17eV bandgap (InSb). Developing insightful approximations, we argue that large plasmonic losses can, contrary to intuition, be helpful in enhancing the overall near-field transfer. We discuss and quantify the properties of an optimal near-field photovoltaic (PV) diode. In addition, we study plasmons in graphene and show that doping can be used to tune the plasmonic dispersion relation to match the PV cell bangap. In case of graphene, theoretically predicted generated power density of 6(120)Wcm2 and efficiency of 35(40)% can be achieved at 600(1200)K, for 0.17eV bandgap. With the ability to operate in intermediate temperature range, as well as high efficiency and power density, near-field TPV systems have the potential to complement conventional TE and TPV solid state heat-to-electricity conversion devices.

© 2012 Optical Society of America

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2011 (2)

M. Francoeur, R. Vaillon, and M. P. Mengüç, “Thermal impacts on the performance of nanoscale-gap thermophotovoltaic power generators,” IEEE Trans. Energy Convers. 26, 686–698 (2011).
[Crossref]

A. I. Volokitin and B. N. J. Persson, “Near-field radiative heat transfer between closely spaced graphene and amorphous SiO2,” Phys. Rev. B 83, 241407 (2011).
[Crossref]

2010 (5)

B. N. J. Persson and H. Ueba, “Heat transfer between graphene and amorphous sio 2,” J. Phys. Condens. Matter 22, 462201 (2010).
[Crossref]

P. West, S. Ishii, G. Naik, N. Emani, V. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. 4, 795–808 (2010).
[Crossref]

S. H. Mousavi, A. B. Khanikaev, B. Neuner, Y. Avitzour, D. Korobkin, G. Ferro, and G. Shvets, “Highly confined hybrid spoof surface plasmons in ultrathin metal-dielectric heterostructures,” Phys. Rev. Lett. 105, 176803 (2010).
[Crossref]

K. Joulain, J. Drevillon, and P. Ben-Abdallah, “Noncontact heat transfer between two metamaterials,” Phys. Rev. B 81, 165119 (2010).
[Crossref]

P. Bermel, M. Ghebrebrhan, W. Chan, Y. X. Yeng, M. Araghchini, R. Hamam, C. H. Marton, K. F. Jensen, M. Soljačić, J. D. Joannopoulos, S. G. Johnson, and I. Celanovic, “Design and global optimization of high-efficiency thermophotovoltaic systems,” Opt. Express 18, A314–A334 (2010).
[Crossref] [PubMed]

2009 (5)

E. Rephaeli and S. Fan, “Absorber and emitter for solar thermo-photovoltaic systems to achieve efficiency exceeding the Shockley-Queisser limit,” Opt. Express 17, 15145–15159 (2009).
[Crossref] [PubMed]

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009).
[Crossref]

S. Basu, Z. M. Zhang, and C. J. Fu, “Review of near-field thermal radiation and its application to energy conversion,” Int. J. Energy Res. 33, 1203–1232 (2009).
[Crossref]

S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9, 2909–2913 (2009).
[Crossref] [PubMed]

E. Rousseau, A. Siria, G. Jourdan, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photonics 3, 514–517 (2009).
[Crossref]

2008 (3)

A. Narayanaswamy, S. Shen, and G. Chen, “Near-field radiative heat transfer between a sphere and a substrate,” Phys. Rev. B 78, 115303 (2008).
[Crossref]

K. Park, S. Basu, W. King, and Z. Zhang, “Performance analysis of near-field thermophotovoltaic devices considering absorption distribution,” J. Quant. Spectrosc. Radiat. Transfer 109, 305–316 (2008).
[Crossref]

L. A. Falkovsky, “Optical properties of graphene,” J. Phys. Conf. Ser. 129, 012004 (2008).
[Crossref]

2007 (2)

S. Basu, Y.-B. Chen, and Z. M. Zhang, “Microscale radiation in thermophotovoltaic devices – a review,” Int. J. Energy Res. 31, 689–716 (2007).
[Crossref]

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

2006 (2)

M. Laroche, R. Carminati, and J.-J. Greffet, “Near-field thermophotovoltaic energy conversion,” J. Appl. Phys. 100, 063704 (2006).
[Crossref]

B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8, 318 (2006).
[Crossref]

2005 (2)

R. G. Yang, A. Narayanaswamy, and G. Chen, “Surface-plasmon coupled nonequilibrium thermoelectric refrigerators and power generators,” J. Comput. Theor. Nanosci. 2, 75–87 (2005).

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59–112 (2005).
[Crossref]

2004 (4)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004).
[Crossref] [PubMed]

J. B. Pendry, L. Martn-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004).
[Crossref] [PubMed]

S. H. Brewer and S. Franzen, “Calculation of the electronic and optical properties of indium tin oxide by density functional theory,” Chem. Phys. 300, 285–293 (2004).
[Crossref]

I. Celanovic, F. O’Sullivan, M. Ilak, J. Kassakian, and D. Perreault, “Design and optimization of one-dimensional photonic crystals for thermophotovoltaic applications,” Opt. Lett. 29, 863–865 (2004).
[Crossref] [PubMed]

2003 (1)

A. Narayanaswamy and G. Chen, “Surface modes for near field thermophotovoltaics,” Appl. Phys. Lett. 82, 3544–3546 (2003).
[Crossref]

2002 (3)

M. Whale and E. Cravalho, “Modeling and performance of microscale thermophotovoltaic energy conversion devices,” IEEE Trans. Energy Convers. 17, 130–142 (2002).
[Crossref]

C. H. Park, H. A. Haus, and M. S. Weinberg, “Proximity-enhanced thermal radiation,” J. Phy. D: Appl. Phys. 35, 2857–2863 (2002).
[Crossref]

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophys. Eng. 6, 209–222 (2002).
[Crossref]

2001 (2)

R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O’Quinn, “Thin-film thermoelectric devices with high room-temperature figures of merit,” Nature 413, 597–602 (2001).
[Crossref] [PubMed]

M. Zenker, A. Heinzel, G. Stollwerck, J. Ferber, and J. Luther, “Efficiency and power density potential of combustion-driven thermophotovoltaic systems using GaSb photovoltaic cells,” IEEE Trans. Electron Devices 48, 367–376 (2001).
[Crossref]

2000 (2)

J. Pan, H. Choy, and C. G. Fonstad, “Very large radiative transfer over small distances from a black body for thermophotovoltaic applications,” IEEE Trans. Electron Devices 47, 241–249 (2000).
[Crossref]

J. L. Pan, “Radiative transfer over small distances from a heated metal,” Opt. Lett. 25, 369–371 (2000).
[Crossref]

1999 (1)

J. B. Pendry, “Radiative exchange of heat between nanostructures,” J. Phys.: Condens. Matter 11, 6621–6633 (1999).
[Crossref]

1998 (1)

1994 (1)

J. J. Loomis and H. J. Maris, “Theory of heat transfer by evanescent electromagnetic waves,” Phys. Rev. B 50, 18517–18524 (1994).
[Crossref]

1987 (1)

1986 (1)

I. Hamberg and C. G. Granqvist, “Evaporated Sn-doped In2O3 films: Basic optical properties and applications to energy-efficient windows,” J. Appl. Phys. 60, R123–R160 (1986).
[Crossref]

1971 (1)

D. Polder and M. Van Hove, , “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4, 3303–3314 (1971).

1969 (1)

C. Hargreaves, “Anomalous radiative transfer between closely-spaced bodies,” Phys. Lett. A 30, 491–492 (1969).
[Crossref]

1961 (2)

J. R. Dixon and J. M. Ellis, “Optical properties of n-type indium arsenide in the fundamental absorption edge region,” Phys. Rev. 123, 1560–1566 (1961).
[Crossref]

H. A. Haus, “Thermal noise in dissipative media,” J. Appl. Phys. 32, 493–500 (1961).
[Crossref]

1960 (2)

G. W. Gobeli and H. Y. Fan, “Infrared absorption and valence band in indium antimonide,” Phys. Rev. 119, 613–620 (1960).
[Crossref]

W. Steinmann, “Experimental verification of radiation of plasma oscillations in thin silver films,” Phys. Rev. Lett. 5, 470–472 (1960).
[Crossref]

Araghchini, M.

Avitzour, Y.

S. H. Mousavi, A. B. Khanikaev, B. Neuner, Y. Avitzour, D. Korobkin, G. Ferro, and G. Shvets, “Highly confined hybrid spoof surface plasmons in ultrathin metal-dielectric heterostructures,” Phys. Rev. Lett. 105, 176803 (2010).
[Crossref]

Azarkevich, J.

R. DiMatteo, P. Greiff, D. Seltzer, D. Meulenberg, E. Brown, E. Carlen, K. Kaiser, S. Finberg, H. Nguyen, J. Azarkevich, P. Baldasaro, J. Beausang, L. Danielson, M. Dashiell, D. DePoy, H. Ehsani, W. Topper, and K. Rahner, “Micron-gap thermophotovoltaics (MTPV),” Proc. 6th Conf. Thermophotovoltaic Generation of Electricity (2004).

Baldasaro, P.

R. DiMatteo, P. Greiff, D. Seltzer, D. Meulenberg, E. Brown, E. Carlen, K. Kaiser, S. Finberg, H. Nguyen, J. Azarkevich, P. Baldasaro, J. Beausang, L. Danielson, M. Dashiell, D. DePoy, H. Ehsani, W. Topper, and K. Rahner, “Micron-gap thermophotovoltaics (MTPV),” Proc. 6th Conf. Thermophotovoltaic Generation of Electricity (2004).

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebessen, “Surface plasmon subwavelength optics,” Nature 424, 824–830.
[PubMed]

Basu, S.

S. Basu, Z. M. Zhang, and C. J. Fu, “Review of near-field thermal radiation and its application to energy conversion,” Int. J. Energy Res. 33, 1203–1232 (2009).
[Crossref]

K. Park, S. Basu, W. King, and Z. Zhang, “Performance analysis of near-field thermophotovoltaic devices considering absorption distribution,” J. Quant. Spectrosc. Radiat. Transfer 109, 305–316 (2008).
[Crossref]

S. Basu, Y.-B. Chen, and Z. M. Zhang, “Microscale radiation in thermophotovoltaic devices – a review,” Int. J. Energy Res. 31, 689–716 (2007).
[Crossref]

Beausang, J.

R. DiMatteo, P. Greiff, D. Seltzer, D. Meulenberg, E. Brown, E. Carlen, K. Kaiser, S. Finberg, H. Nguyen, J. Azarkevich, P. Baldasaro, J. Beausang, L. Danielson, M. Dashiell, D. DePoy, H. Ehsani, W. Topper, and K. Rahner, “Micron-gap thermophotovoltaics (MTPV),” Proc. 6th Conf. Thermophotovoltaic Generation of Electricity (2004).

Ben-Abdallah, P.

K. Joulain, J. Drevillon, and P. Ben-Abdallah, “Noncontact heat transfer between two metamaterials,” Phys. Rev. B 81, 165119 (2010).
[Crossref]

Bermel, P.

Boltasseva, A.

P. West, S. Ishii, G. Naik, N. Emani, V. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. 4, 795–808 (2010).
[Crossref]

Brewer, S. H.

S. H. Brewer and S. Franzen, “Calculation of the electronic and optical properties of indium tin oxide by density functional theory,” Chem. Phys. 300, 285–293 (2004).
[Crossref]

Brown, E.

R. DiMatteo, P. Greiff, D. Seltzer, D. Meulenberg, E. Brown, E. Carlen, K. Kaiser, S. Finberg, H. Nguyen, J. Azarkevich, P. Baldasaro, J. Beausang, L. Danielson, M. Dashiell, D. DePoy, H. Ehsani, W. Topper, and K. Rahner, “Micron-gap thermophotovoltaics (MTPV),” Proc. 6th Conf. Thermophotovoltaic Generation of Electricity (2004).

Buljan, H.

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009).
[Crossref]

Carlen, E.

R. DiMatteo, P. Greiff, D. Seltzer, D. Meulenberg, E. Brown, E. Carlen, K. Kaiser, S. Finberg, H. Nguyen, J. Azarkevich, P. Baldasaro, J. Beausang, L. Danielson, M. Dashiell, D. DePoy, H. Ehsani, W. Topper, and K. Rahner, “Micron-gap thermophotovoltaics (MTPV),” Proc. 6th Conf. Thermophotovoltaic Generation of Electricity (2004).

Carminati, R.

M. Laroche, R. Carminati, and J.-J. Greffet, “Near-field thermophotovoltaic energy conversion,” J. Appl. Phys. 100, 063704 (2006).
[Crossref]

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59–112 (2005).
[Crossref]

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophys. Eng. 6, 209–222 (2002).
[Crossref]

Celanovic, I.

Chan, W.

Chen, G.

S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9, 2909–2913 (2009).
[Crossref] [PubMed]

A. Narayanaswamy, S. Shen, and G. Chen, “Near-field radiative heat transfer between a sphere and a substrate,” Phys. Rev. B 78, 115303 (2008).
[Crossref]

R. G. Yang, A. Narayanaswamy, and G. Chen, “Surface-plasmon coupled nonequilibrium thermoelectric refrigerators and power generators,” J. Comput. Theor. Nanosci. 2, 75–87 (2005).

A. Narayanaswamy and G. Chen, “Surface modes for near field thermophotovoltaics,” Appl. Phys. Lett. 82, 3544–3546 (2003).
[Crossref]

Chen, Y.-B.

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Figures (8)

Fig. 1
Fig. 1 Schematic illustration of a near-field TPV system. Plasmonic emitter, characterized by the plasma frequency ωp and damping γ, operates at temperature T1. Next to it, a distance D away, is the photovoltaic cell at temperature T2, characterized by the bandgap energy ωg, photon absorption coefficient α0, and the refractive index n. The parallel and the perpendicular wave vectors are q and kz, respectively.

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Fig. 2
Fig. 2 Contour plot of Π(ω,q) from Eq. (1) for (a) γ = 5 × 10−4eV, (b) γ = 5 × 10−3eV. Dashed line shows the dispersion relation from Eq. (17), and the solid cyan line shows the dispersion relation of a surface plasmon in air. In (c) and (d), solid (dashed) line corresponds to the spectral transfer function Π(ω,q) calculated using Eq. (1) (Eq. (5)), for four different values of q. On a separate scale, β is plotted as a function of ω0 (magenta), where the x-axis for ω0 and ω is shared. Plasma frequency, PV gap frequency and PV absorption coefficient are ωp = 0.6eV, ωg = 0.36eV and α0 = 1.3 × 104cm−1, respectively. Separation is D = 10nm (1/D ≈ 20eV/h̄c).

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Fig. 3
Fig. 3 Contour plot of logarithm of transfer ratio versus ωp, ωg, for the plasmonic emitter-PV system in Fig. 1. Plasmon damping is γ = 5 × 10−3eV, and temperature is T1 = 600K, with other parameters being the same as in Fig. 2.

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Fig. 4
Fig. 4 Plot of χ′ (blue) and χ″ (red) for a semiconductor as a function of frequency, with ωg = 0.36eV and n = 3.51. Three lines correspond to the absorption coefficient α calculated using square-root dependence, Eq. (3), with α0 = 1.3 × 104cm−1, experimental values for InAs from Ref. [19], and using step-like dependence, Eq. (8), with same α0. We see that both PV cell approximations, χ′ ≈ 0.85, and χ″/χ′ ≪ 1, are satisfied in this frequency range.

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Fig. 5
Fig. 5 Radiation transfer ratio, as a function of emitter temperature T1 for a pure plasmonic emitter-PV cell near-field TPV system. The ratio is plotted for several values of ωp and γ, with other parameters being the same as in Fig. 2, namely ωg = 0.36eV, D = 10nm.

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Fig. 6
Fig. 6 Silver-PV near-field TPV system. (a) Radiation transfer ratio as a function of PV gap frequency ωg and separation D, for T1 = 1235K. (b) Contour plot of integrand in Eq. (1) and Eq. (2), as a function of ω, q in regions below and above the vacuum light line, respectively. This plot corresponds to Htot for ωg = 0.36eV, D = 10nm point from plot (a). The y-axis is shared between two plots. Two dashed lines correspond to light lines for n = 1 (vacuum) and n = 3.51 (PV cell).

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Fig. 7
Fig. 7 Graphene-PV near-field TPV system. (a) Contour plot of integrand H(ω,q) in Eq. (1) as a function of ω, q for parameters T1 = 600K, D = 10nm, μ = 0.2eV and τ = 10−13s. PV cell parameters are ωg = 0.17eV, α0 = 0.7 × 104cm−1. Solid (magenta) line is the vacuum surface plasmon dispersion relation, Eq. (14), for the graphene sheet. (b) H(ω) evaluated for different parameters with T1, D same as in (a). For comparison, black ωp-line demonstrates H(ω) for a pure plasmonic emitter analyzed in section 2, with ωp = 0.3eV. (c) Contour plot of the heat transfer ratio vs. the two black bodies in the far field, as a function of T1 and D. In (c) and (d), τ = 10−13s and doping is μ = 0.25eV. (d) Electric power generated PPV as a function of T1 where the voltage across the PV diode terminals is Vo = 0.08V. The x-axis is shared between plots (a), (b) and plots (c), (d), respectively.

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Fig. 8
Fig. 8 Optimization of the flux ratio Hevan/Hbb for graphene-PV near-field TPV system, where ωg = 0.17eV, D = 10nm.

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Tables (2)

Tables Icon

Table 1 Ideal plasmonic emitter-PV cell near-field TPV system: radiated power exchange, Prad, generated electrical power PPV, efficiency, and the flux ratio (relative to the transferred power between two black bodies in the far field), tabulated for a set of parameters, up to two significant digits. We assume D = 10nm, γ = 0.01eV.

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Tables Icon

Table 2 Comparison between a silicon-PV and a graphene-PV near field TPV system: radiated power exchange, Prad, generated electrical power PPV, efficiency, and the flux ratio (relative to the transferred power between two black bodies in the far field), tabulated for a set of parameters, up to two significant digits. We assume D = 10nm, τ = 10−13s.

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Equations (33)

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H evan s , p = 1 π 2 ∫ 0 ∞ d ω [ Θ ( ω , T 1 ) − Θ ( ω , T 2 ) ] ∫ ω / c ∞ d q q Im ( r 01 s , p ) Im ( r 02 s , p ) | 1 − r 01 s , p r 02 s , p e 2 i k z 0 D | 2 e − 2 | k z 0 | D
H prop s , p = 1 π 2 ∫ 0 ∞ d ω [ Θ ( ω , T 1 ) − Θ ( ω , T 2 ) ] ∫ 0 ω / c d q q ( 1 − | r 01 s , p | 2 ) ( 1 − | r 02 s , p | 2 ) 4 | 1 − r 01 s , p r 02 s , p e 2 i k z 0 D | 2
ε 2 ( ω ) = ( n + i α 2 k 0 ) 2 where α ( ω ) = { 0 , ω < ω g α 0 ω − ω g ω g , ω > ω g
H evan p = 1 π 2 ∫ 0 ∞ d ω [ Θ ( ω , T 1 ) − Θ ( ω , T 2 ) ] ∫ ω / c ∞ d q q Π ( ω , q )
Π ( ω , q ) = γ β ( ω , q ) 4 [ ω − ω 0 ( q ) ] 2 + [ γ + β ( ω , q ) ] 2 where β ( ω , q ) ≡ ω p 2 − 2 ω 0 ( q ) 2 2 ω 0 ( q ) χ ″ ( ω ) χ ′ ( ω )
Π ( ω , q ) = γ β ( ω , q ) 4 [ ω − ω 0 ( q ) ] 2 + [ γ + β ( ω , q ) ] 2 ≈ γ β ( ω , q ) 4 [ ω − ω 0 ( q ) ] 2 + β ( ω , q ) 2
A ( q ) = ∫ ω g ∞ d ω Θ ( ω , T 1 ) Π ( ω , q )
ε 2 ( ω ) = ( n + i α 2 k 0 ) 2 where α ( ω ) = { 0 , ω < ω g α 0 , ω > ω g
P rad = 1 π 2 ∫ 0 ∞ d q q [ ∫ 0 ∞ d ω Π ( ω , q ) h ¯ ω e h ¯ ω k T 1 − 1 − ∫ ω g ∞ d ω Π ( ω , q ) h ¯ ω e h ¯ ω − e V o k T 2 − 1 ]
P P V = 1 π 2 ∫ 0 ∞ d q q [ ∫ ω g ∞ d ω Π ( ω , q ) e V o e h ¯ ω k T 1 − 1 − ∫ ω g ∞ d ω Π ( ω , q ) e V o e h ¯ ω − e V o k T 2 − 1 ]
η T P V = P P V P rad
σ intra ( ω , T ) = i ω + i / τ e 2 2 k b T π h ¯ 2 ln   [ 2 cosh μ 2 k b T ]
σ inter ( ω , T ) = e 2 4 h ¯ [ G   ( h ¯ ω 2 ) + i 4 h ¯ ω π ∫ 0 ∞ G ( ε ) − G ( h ¯ ω / 2 ) ( h ¯ ω ) 2 − 4 ε 2 d ε ]
q = ε 0 2 i ω σ ( ω , T )
Π ( ω , q ) = Im   ( ε 1 − 1 ε 1 + 1 )   Im   ( ε 2 − 1 ε 2 + 1 ) | 1 − ( ε 1 − 1 ε 1 + 1 )   ( ε 2 − 1 ε 2 + 1 ) e − 2 q D | 2 e − 2 q D
ε 1 ( ω ) − 1 ε 1 ( ω ) + 1 = { ω p 2 ω p 2 − 2 ω 2 [ 1 + 2 i ω γ ω p 2 − 2 ω 2 ] if  γ ≪ ω p 2 − 2 ω 2 2 ω i ω p 2 2 ω γ [ 1 − i ( ω p 2 − 2 ω 2 ) 2 ω γ ] if  γ ≫ ω p 2 − 2 ω 2 2 ω
ω 0 ( q ) = ω p 2 1 − χ ′ e − 2 q D
ω 0 ( q ) = ω p 2 1 − e − q D
Π ( ω , q ) = ( γ / 2 ) 2 4 ( ω − ω 0 ( q ) ) 2 + γ 2
∫ d ω Π ( ω , q ) = γ 4   [ π + 2 tan − 1   ( 2 ω 0 γ ) ]
E ( r , ω ) = ∫ d 3 r ′ G E ( r , r ′ , ω ) j ( r ′ )
H ( r , ω ) = ∫ d 3 r ′ G H ( r , r ′ , ω ) j ( r ′ )
G E ( r , r ′ , ω ) = − ω μ 0 8 π 2 ∫ d 2 q 1 k z 1 ( s ^ T s s ^ + p ^ 2 + T p p ^ 1 + ) e i q ( r q − r q ′ ) + i k z 2 ( z − D ) − i k z 1 z ′
G H ( r , r ′ , ω ) = − n 2 ω 8 π 2 c ∫ d 2 q 1 k z 1 ( − p ^ 2 + T s s ^ + s ^ T p p ^ 1 + ) e i q ( r q − r q ′ ) + i k z 2 ( z − D ) − i k z 1 z ′
T = t 10 t 02 e i k z 0 D 1 − r 02 r 01 e 2 i k z 0 D
〈 j α * ( r , ω ) j β ( r ′ , ω ′ ) 〉 = Θ ( ω , T ) 2 π [ σ ( ω ) + σ * ( ω ) ] δ ( r − r ′ ) δ ( ω − ω ′ ) δ α β
S z = 2 Re   ( E x H y * − E y H x * )
E x H y * − E y H x * = n 2 * ω 2 μ 0 c ( 8 π 2 ) 2 ∫ d 3 r ′ ∫ d 2 q d 2 q ′ 1 | k z 1 | 2 [ g x α E g y α H * − g y α E g x α H * ] × Θ ( ω , T 1 ) 2 π [ σ + σ * ] δ ( z ′ ) e i ( q − q ′ ) ( r q − r q ′ ) e i k 21 ( z − D ) e − i k z 1 z ′ e − i k z 2 ′ * ( z − D )
g E = s ^ T s s ^ + p ^ 2 + T p p ^ 1 +
g H = − p ^ 2 + T s s ^ + s ^ T p p ^ 1 +
g x α E g y α H * − g y α E g x α H * = | T s | 2 k z 2 * n 2 * k 0 + | T p | 2 k z 2 n 2 k 0 | k z 1 | 2 k 0 2
r p G = 1 − ε G ε G , t p G = 1 ε G , where ε G = 1 + σ k z 0 2 ε 0 ω
2 ω Im ( r p G ) Im ( k z 0 ) = | t p G | 2 Re ( σ ) ε 0

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