where $k_x$ is the wave number along $x$ direction. $k_b$ is the Boltzmann constant. Under the relaxation time approximation, the drift-induced distribution is similar to the distribution with chemical potential of photon in semiconductors. The heat transfer rate per unit area between the layers of graphene is then given under FE as [@rytov1959theory; @RN7]: $$\begin{aligned}
 \label{RHT_general}
    H=\int_0^{\infty} \frac{d\omega}{2\pi}  \int \frac{d^2 {\bf k}_\perp}{(2\pi)^2} 
    \Theta_{12}(\omega, k_x) \mathrm{T}_{12} (\omega, {\bf k}_\perp),\end{aligned}$$ where $\omega$ is the frequency of electromagnetic wave. ${\bf k}_\perp = (k_x, k_y)$ is the wavevector in the graphene plane. $\Theta_{12}(\omega,k_x) \equiv \hbar \omega\bigl[ N_1(\omega, k_x) - N_2(\omega) \bigr] $. $N_2(\omega)$ is the usual Bose function at temperature $T_2$. $\mathrm{T}_{12}(\omega,{\bf k}_\perp)$ is the transmission coefficient.

For an infinitely large suspended double-layer graphene with nanoscale separation, the $p$-polarized wave is the dominant channel for RHT [@PhysRevB.85.155422]. Based on FE, the transmission for evanescent $p$-polarized modes between a pair of two-dimensional materials in a parallel plate geometry can be written as (in the non-retardation limit, the speed of light $c \to \infty$) [@polder1971theory; @pendry1999radiative; @RN7]: $$\begin{aligned}
 \label{Transmisson_p}
    \mathrm{T}_{12} (\omega,{\bf k}_\perp)=\frac{4\,\mathrm{Im}(r_1)\, \mathrm{Im}(r_2)}{|1-r_1r_2e^{-2\gamma d}|^2} e^{-2\gamma d},\end{aligned}$$ where $r_1$ and $r_2$ are the reflection coefficients at the bottom and top interface of the vacuum gap. $d$ is 