## 3 Photon and Phonon Tunneling through a Vacuum Gap
### Photon Tunneling through a Vacuum Gap  
The heat transfer is greatly enhanced by the tunneling of evanescent electromagnetic waves from one body to another at \(d\ll\lambda_{T}\)[36; 37; 38; 39; 40; 41]. This is due to the fact that the number of states of evanescent electromagnetic modes localized near the surface is much higher than that of propagating waves. For example, consider electromagnetic waves near the boundary between a flat body and a vacuum. For these waves, the wave vector component \(k_{||}\) parallel to the boundary must be the same on both sides of the boundary, as required by the phase-matching boundary condition. In a vacuum, the component of the wave vector perpendicular to the surface is equal to \(k_{z}=\sqrt{k_{0}^{2}-k_{||}^{2}}\), where \(k_{0}=\omega/c\). For evanescent modes, the component \(k_{||}\) can be much larger than the wave number of propagating modes, which is limited by \(k_{0}\). Indeed, for evanescent modes, \(k_{z}\) can be an imaginary number. Then, the energy density associated with evanescent modes can be much larger than that with propagating waves. However, radiation tunneling is significant only at \(k_{||}\lesssim d^{-1}\). Indeed, evanescent waves in the vacuum region decay exponentially as \(\exp(-2|k_{z}|)\) in the \(z\)-direction perpendicular to the surface, since \(k_{z}\) is an imaginary number at \(k_{||}>k_{0}\). The number of evanescent modes available to conduct heat is proportional to the volume of the phase space they occupy. This number is approximately proportional to \(d^{-2}\) since \(k_{||}\in\left(k_{0},d^{-1}\right)\), where \(k_{0}\ll d^{-1}\) at \(d^{-1}\gg\lambda_{T}^{-1}\) and \(\lambda_{T}^{-1}=k_{0}(k_{B}T/\hbar\omega)\). Thus, the effect of tunneling of evanescent waves increases approximately according to the law \(1/d^{2}\) at \(d\to 0\)[37; 38; 39; 40; 41]. Again, it is assumed that \(d\) is much larger than the interatomic distance \(a\). Additionally, in the case of metals, this increase reaches saturation when nonlocal effects become significant, that is, at \(d\) around \(d_{F}\), as determined by Equation (6) or \(d\approx l_{e}\), the electron mean free path [36; 42]:  
\[d_{F}=\frac{v_{F}\hbar}{k_{B}T}, \tag{6}\]  
where \(v_{F}\) is the Fermi velocity [43]. For example, \(d_{F}=25\) nm at \(v_{F}=10^{6}\) m/s and \(l_{e}\) is on the order of tens of nm for pure metals at room temperature, when the electron-phonon scattering predominates, and the contributions to scattering due to crystalline defects are negligible [44]. Since the density of electromagnetic energy increases significantly in the near-field of the surface, the interfacial thermal conductance \(G_{nf}\) associated with near-field heat transfer is much greater than the \(G_{bb}\) associated with black-body radiation. The maximum heat flux due to near-field heat transfer between two planar surfaces is approximately equal to \(\left(k_{B}^{2}/6\hbar d^{2}\right)\left(T_{1}^{2}-T_{2}^{2}\right)\)[37]. Thus, the maximum value of the coefficient \(G_{nf}\) can be represented by Equation (7). \[G_{nf}^{max}(d)=\frac{k_{B}^{2}T}{\hbar d^{2}}\text{B}_{max}, \tag{7}\]  
where \(\text{B}_{max}=1/3\). For example, \(G_{nf}^{max}(d)\) is about \(10^{6}\) W/m\({}^{2}\)K at \(d=10\) nm and \(T=300\) K. However, not every evanescent mode available for thermal conduction effectively participates in the heat transfer. Actually, \(G_{nf}(d)\) is approximately one or two orders of magnitude less than \(G_{nf}^{max}(d)\) for insulators [37; 38; 39; 45; 46; 47] or semiconductors [48; 49; 50; 51; 52] and about two or three orders of magnitude less than \(G_{nf}^{max}(d)\) for metals [37; 42; 45; 53; 54; 55]. However, the near-field heat transfer at the metal-dielectric interface is greater than at the metal-metal interface, and this heat transfer can be further enhanced by an oxide film on the metal surface. This enhancement is significant if the thickness of the oxide films exceeds the size of the interfacial gap [45].