The Effect of Non-Local Electrical Conductivity on Near-Field Radiative Heat Transfer between Graphene Sheets

Graphene's near-field radiative heat transfer is determined from its electrical conductivity, commonly modeled using the local Kubo and Drude formulas. In this letter, we analyze the non-locality of graphene's electrical conductivity using the Lindhard model combined with the Mermin relaxation time approximation. We also study how the variation of electrical conductivity with wavevector affects near-field radiative conductance between two graphene sheets separated by a vacuum gap. It is shown that the variation of electrical conductivity with wavevector, $k_{\rho}$, is appreciable for $k_{\rho}$s greater than $100k_0$, where $k_0$ is the magnitude of the wavevector in the free space. The Kubo electrical conductivity provides an accurate estimation of the spectral radiative conductance between two graphene sheets except for around the surface-plasmon-polariton frequency of graphene and at separation gaps smaller than 20 nm where there is a non-negligible contribution from modes with $k_{\rho}>100k_0$ to the radiative conductance. The Drude formula proves to be inaccurate for modeling the electrical conductivity and radiative conductance of graphene except for at temperatures much below the Fermi temperature and frequencies much smaller than $2{\mu}_c/{\hbar}$, where ${\mu}_c$ and ${\hbar}$ are the chemical potential and reduced Planck's constant, respectively. It is also shown that the electronic scattering processes should be considered in the Lindhard model properly, such that the local electron number is conserved. A substitution of ${\omega}$ by ${\omega}+i{\gamma}$ (${\omega}$, $i$, and ${\gamma}$ being the angular frequency, imaginary unit, and scattering rate, respectively) in the collisionless Lindhard model does not satisfy the conservation of the local electron number and results in significant errors in computing graphene's electrical conductivity and radiative conductance.