Beyond the Drude model: Surface and nonlocal effects in near-field radiative heat transfer and the Casimir puzzle

We study the charge and current response functions $P$ and $\mathrm{\ensuremath{\Pi}}$ in a semi-infinite metal block using the electron surface Green's functions. The surface electrons behave similarly to a two-dimensional Fermi gas, but are strongly damped due to coupling to the bulk. This substantially reduces the region of validity of the Drude model for $P$, which requires the frequency $\ensuremath{\omega}\ensuremath{\gg}max({v}_{F}q,1/\ensuremath{\tau})$, where ${v}_{F}$ is the Fermi velocity, $q$ is the wave vector, and $\ensuremath{\tau}$ is an effective relaxation time. As a consequence, for typical metal in near-field heat transfer, the thermal conductance due to the Coulomb interaction scales as $1/{d}^{4}$ with the distance of the vacuum gap instead of the well-known $1/{d}^{2}$ result of the Drude model. The current response $\mathrm{\ensuremath{\Pi}}$ is shown to be highly anisotropic. The Drude model describes well the transverse directions parallel to the surface, but is very different in the normal direction up to about 100 lattice sites away from the surface. These ideas and the residual diamagnetic effect of a nonzero $\mathrm{\ensuremath{\Pi}}$ on the surface at zero frequency still cannot resolve the Casimir puzzle.