Near-Field Radiative Heat Transfer Between Anisotropic Materials
As an anisotropic material, the optical response of hBN is related to the orientation of its optical axis.
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Near-Field Radiative Heat Transfer Between Anisotropic Materials
As an anisotropic material, the optical response of hBN is related to the orientation of its optical axis. Here, we studied the near-field radiative heat transfer between a planar emitter and a planar receiver separated by a vacuum gap as shown in Fig. 5.1. The structures of the emitter and the receiver, which consist of bare hBN or graphenecovered hBN slabs, are essentially the same. For convenience, the optical axis of hBN is considered in the x-z plane of the coordinate system xyz and it is tiled off the z-axis by angle of α 1 and α 2 for the emitter and the receiver, respectively. In addition, the width of vacuum gap and the thickness of hBN are denoted by d and h, respectively. Graphene is modeled as a layer of thickness = 0.3 nm with an effective dielectric function εeff = 1 −
jσs ε0 ω
(5.1)
where ε0 is the absolute permittivity of vacuum and σs is the sheet conductivity that includes the contributions from both the intraband and intranband transitions. In the mid- and far-infrared region, σs is dominated by the intraband transitions and can be approximately written as σs =
τ e2 μ 2 π 1 + jωτ
(5.2)
where e is the electron charge, is the reduced Planck constant, τ is the relaxation time and μ is the chemical potential. Based on the fluctuation–dissipation theorem and the reciprocity of the dyadic green function, the NFRHF between anisotropic media can be expressed as
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 X. Wu, Thermal Radiative Properties of Uniaxial Anisotropic Materials and Their Manipulations, Springer Theses, https://doi.org/10.1007/978-981-15-7823-6_5
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5 Near-Field Radiative Heat Transfer Between Anisotropic Materials
Fig. 5.1 Schematic of near-field radiative heat transfer between two graphene-hBN heterostrucutures. The optical axis (OA) of hBN is in the x-z plane and tilted off the z-axis by an angle
1 Q= 8π 3
∞
2π ∞ ξ (ω, β, φ)βdβdφ
[ (ω, T1 ) − (ω, T2 )] 0
0
(5.3)
0
where (ω, T ) = ω eω/ k B T − 1 is the average energy of a Planck oscillator, φ is the azimuth angle, and ξ (ω, β, φ) is called the energy transmission coefficient or the phonon tunneling probability, which can be expressed as ξ (ω, β, φ) =
Tr I − R2∗ R2 − T∗2T2 D I − R1∗ R1 − T∗1 T1 D∗ , β < k0 (5.4) β > k0 Tr R2∗ − R2 D R1 − R1∗ D∗ e−2|kz |d ,
where β is the wavevector component parallel to the x-y plane. k z = k02 − β 2 is the wavevector component along the z-axis in vacuum. Note that the asterisk denotes conjugate transpose, and Tr(•) takes the trace of a matrix. I is a 2 × 2 unit matrix and
R1,2 =
(1,2) (1,2) (1,2) rss tss(1,2) tsp rsp , T1,2 = (1,2) (1,2) r (1,2) r (1,2) t ps t pp ps pp
(5.5)
and the matrices that include the reflection and transmission coefficients for incident s- and p-polarized plane waves from vacuum to the emitter or the receiver, respectively. The first and second letters of the subscript in
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