Radiative heat transfer and nonequilibrium Casimir-Lifshitz force in many-body systems with planar geometry

Ivan Latella, Philippe Ben-Abdallah, Svend-Age Biehs, Mauro Antezza, and Riccardo Messina

Phys. Rev. B 95, 205404 – Published 3 May 2017

Abstract

A general theory of photon-mediated energy and momentum transfer in $N$ -body planar systems out of thermal equilibrium is introduced. It is based on the combination of the scattering theory and the fluctuational-electrodynamics approach in many-body systems. By making a Landauer-like formulation of the heat transfer problem, explicit formulas for the energy transmission coefficients between two distinct slabs as well as the self-coupling coefficients are derived and expressed in terms of the reflection and transmission coefficients of the single bodies. We also show how to calculate local equilibrium temperatures in such systems. An analogous formulation is introduced to quantify momentum transfer coefficients describing Casimir-Lifshitz forces out of thermal equilibrium. Forces at thermal equilibrium are readily obtained as a particular case. As an illustration of this general theoretical framework, we show on three-body systems how the presence of a fourth slab can impact equilibrium temperatures in heat-transfer problems and equilibrium positions resulting from the forces acting on the system.

Received 24 January 2017
Revised 10 April 2017

DOI:https://doi.org/10.1103/PhysRevB.95.205404

Physics Subject Headings (PhySH)

Casimir effect Electromagnetic interactions Transport phenomena

Nanostructures

Particles & FieldsCondensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Ivan Latella^1,*, Philippe Ben-Abdallah^1,2,†, Svend-Age Biehs^3,‡, Mauro Antezza^4,5,§, and Riccardo Messina^4,¶

¹Laboratoire Charles Fabry, UMR 8501, Institut d'Optique, CNRS, Université Paris-Saclay, 2 Avenue Augustin Fresnel, 91127 Palaiseau Cedex, France
²Université de Sherbrooke, Department of Mechanical Engineering, Sherbrooke, PQ J1K 2R1, Canada
³Institut für Physik, Carl von Ossietzky Universität, D-26111 Oldenburg, Germany
⁴Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de Montpellier, F- 34095 Montpellier, France
⁵Institut Universitaire de France, 1 rue Descartes, F-75231 Paris Cedex 05, France

^*ivan.latella@institutoptique.fr
^†pba@institutoptique.fr
^‡s.age.biehs@uni-oldenburg.de
^§mauro.antezza@umontpellier.fr
^¶riccardo.messina@umontpellier.fr

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Vol. 95, Iss. 20 — 15 May 2017

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Images

Figure 1
Representation of the $N$ -body system. The source fields and the total electric field in each region are also indicated.
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Figure 2
Equilibrium temperatures in four-body systems. (a) Equilibrium temperature of body 2 as a function of the separation distance $d_{3}$ with fixed $T_{1} = 400 K$ and $T_{3} = T_{4} = 300 K$ . Bodies 1 and 3 are hBN slabs of width $δ_{1} = δ_{3} = 200 nm$ , whereas bodies 2 and 4 are made of SiC and have widths $δ_{2} = 200 nm$ and $δ_{4} = 5 μ m$ , respectively. The rest of the separation distances are fixed to $d_{1} = d_{2} = 200 nm$ . (b) Equilibrium temperature of body 3 as a function of the separation distance $d_{1}$ with fixed $T_{1} = T_{2} = 400 K$ and $T_{4} = 300 K$ . Bodies 2 and 4 are hBN slabs of width $δ_{2} = δ_{4} = 200 nm$ , whereas bodies 1 and 3 are made of SiC and have widths $δ_{1} = 5 μ m$ and $δ_{3} = 200 nm$ , respectively. The rest of the separation distances are fixed to $d_{2} = d_{3} = 200 nm$ . In (a) and (b), we also show the equilibrium temperature of the intermediate body in a three-body system: $T_{2, 3 B}^{eq}$ is obtained by removing body 4 in (a) and $T_{3, 3 B}^{eq}$ is obtained by removing body 1 in (b). In both cases the environmental temperatures are fixed to $T_{0} = T_{5} = 300 K$ .
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Figure 3
Equilibrium temperatures in four-body systems. Materials, geometry, and background temperature are the same as in Fig. 2, but now the temperatures of two bodies are allowed to reach equilibrium conditions. (a) Equilibrium temperatures of bodies 2 and 4 as a function of the separation distance $d_{3}$ with fixed $T_{1} = 400 K$ and $T_{3} = 300 K$ . (b) Equilibrium temperatures of bodies 1 and 3 as a function of $d_{1}$ with fixed $T_{2} = 400 K$ and $T_{4} = 300 K$ . For comparison, the equilibrium temperature of the intermediate body in the three-body configuration is also included in the plots.
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Figure 4
Transmission coefficients $T^{1, 3}$ in the $(k, ω)$ plane for the four-body configuration corresponding to Fig. 2 and Fig. 3. White dashed lines indicate the light line $ω = c k$ . The upper panel corresponds to TM polarization and the lower one to TE polarization. Bodies 1 and 3 are SiC slabs of width $δ_{1} = 5 μ m$ and $δ_{3} = 200 nm$ , respectively, whereas bodies 2 and 4 are made of hBN and have widths $δ_{2} = δ_{4} = 200 nm$ . In (a) and (c) we set $d_{1} = 100 nm$ , while in (b) and (d) we take $d_{1} = 100 μ m$ . In all cases $d_{2} = d_{3} = 200 nm$ . The horizontal lines indicate the longitudinal and transverse optical frequencies $ω_{L} = 1.83 \times 10^{14} rad$ /s and $ω_{T} = 1.49 \times 10^{14} rad$ /s for SiC, respectively.
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Figure 5
Net pressure $P_{eq}^{2}$ acting on body 2 in a four-body configuration at thermal equilibrium ( $T = 300 K$ ). This pressure is shown as a function of the position of body 2, $z_{2}$ , for several separation distances $d_{3}$ with fixed $d_{1} + d_{2} = 1 μ m$ . Bodies 1 and 3 are SiC slabs of width $δ_{1} = δ_{3} = 100 nm$ , and bodies 2 and 4 are gold slabs of width $δ_{2} = 100 nm$ and $δ_{4} = 10 μ m$ , respectively. The inset shows the variation of the (unstable) equilibrium position of body 2 when $d_{3}$ is varied. This variation $Δ z_{2}$ is measured with respect to the equilibrium position, at $z_{2} = 0$ , in the three-body configuration in which body 4 is removed.
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