Radiative heat transfer as a Landauer-Büttiker problem

We study the radiative heat transfer between two semi-infinite half-spaces, bounded by conductive surfaces in contact with vacuum. This setup is interpreted as a four-terminal mesoscopic transport problem. The slabs and interfaces are viewed as bosonic reservoirs, coupled perfectly to a scattering center consisting of the two planes and vacuum. Using Rytov's fluctuational electrodynamics and assuming Kirchhoff's circuital law, we calculate the heat flow in each bath. This allows for explicit evaluation of a conductance matrix, from which one readily verifies Büttiker symmetry. Thus, radiative heat transfer in layered media with conductive interfaces becomes a Landauer-Büttiker transport problem.

Figure 1

Layered media 1, 3 with conductive interfaces 0, $D$ as a four-terminal Landauer-Büttiker transmission problem. Planar symmetry reduces the interfaces to two dots, serving as a scattering center enclosed by the dashed rectangle.

Figure 2

Two semi-infinite slabs with temperatures $T_{1,3}$ and dielectric functions ${\epsilon }_{1,3}$ separated by a vacuum gap (${\epsilon }_2=1$) at a distance $d$ apart. The interfaces in contact with vacuum are at temperatures $T_{0,D}$ with conductivities ${\sigma }_{0,D}$. The planar cross-sections should be understood to be infinitely large.

Figure 3

Scattering-emission problem: given incident electric field ${\mathit{E}}_{1+}$, fluctuating surface current ${\mathit{K}}^{\mathrm{f}}$, demanding current response ${\sigma }_0{\mathit{E}}_{\perp }$, find the outgoing fields ${\mathit{E}}_{1-}$ and ${\mathit{E}}_{2+}$.

Figure 4

Radiative heat transfer between a gold slab covered by graphene kept at temperature $T_1=T_0=T_L$, and another suspended (${\epsilon }_3=1$) graphene sheet at $T_D=T_R$, separated by a vacuum gap (${\epsilon }_2=1$) at a distance $d$ apart. For mathematical convenience, the planar cross-sections are infinitely large.

Figure 5

Radiative heat exchange with interface $D$ as function of gap separation $d$. Dashed line: from bulk 1. Dotted line: from interface 0. Solid line: to vacuum bulk 3 (${\mathcal{S}}_{3\to D}<0$, graph shown is its magnitude ${\mathcal{S}}_{D\to 3}$). Parameters used are $T_L=373$ K, $T_R=273$ K, $n=5.9\times {10}^{22}{\mathrm{cm}}^{-3}$, $\tau =2.1\times {10}^{-14}$ s, $E_F=0.3$ eV, $\hslash \mathrm{\Gamma }=3.7$ meV.

Figure 6

Scattered electric fields (wiggly arrows) due to incident field (straight arrow) from bulk 1 (shaded bath).

Figure 7

Scattered electric fields (wiggly arrows) due to fluctuation-induced fields (straight arrow) from interface 0 (shaded bath).