a function of frequency and (b) $g(k_x)$ as a function of wave vector, with different drift velocities: no drift (blue dash-dot line), total heat current density $H=-0.84\,$MW/m$^2$, $v_1=5.0 \times 10^5 \ \mathrm{m/s}$ (red dash line), $-0.30\,$MW/m$^2$, and $v_1=9.0 \times 10^5 \ \mathrm{m/s}$ (black solid line), $+0.09\,$MW/m$^2$. The temperatures are $T_1=300\,$K and $T_2=320\,$K. The chemical potential of graphene $\mu$ is set as 0.1eV. Gap distance $d$ is set as 10nm. The damping parameter is $\eta = 9\,$meV.[]{data-label="figure2"}](figure2){width="\columnwidth"}

To understand the drift effects quantitatively, we define two integrated spectral transfer functions as follows: $$\begin{aligned}
 \label{ISF}
    &f(\omega) =  \int \frac{d^2{\bf k}_\perp}{(2\pi)^3} \Theta_{12}(\omega,k_x)  \mathrm{T}_{12} (\omega,{\bf k}_\perp), \\
    &g(k_x)=\int_0^\infty \frac{d\omega}{2\pi} \int \frac{dk_y}{(2\pi)^2} \Theta_{12}(\omega,k_x)  \mathrm{T}_{12} (\omega, {\bf k}_\perp).\end{aligned}$$ Figure \[figure2\](a) shows the spectral transfer function $f(\omega)$ as a function of frequency with different drift velocities. In the undriven case (blue dash-dot), $f(\omega)$ is strictly negative and the heat current flows from top layer to bottom layer due to the temperature difference ($T_1 =300$K and $T_2=320$K). It is the usual RHT between two graphene layers and the RHT can be tuned by doping or top gating (varying $\mu$). However, when the electrons in the bottom layer are drifted with a velocity $v_1=5.0\times 10^5 \mathrm{m/s}$ (red dash line) 