distance $d$, and emitting thermal radiation at temperatures $T_{1(2)}$. Figure 1 schematically illustrates the configuration under our investigation. There is an electric current induced by a static voltage applied across the bottom layer of graphene. Due to the high mobility of graphene, the drift velocity of electrons in graphene can be on the order of the Fermi velocity of graphene, i.e., $10^6\, \mathrm{m/s}$.

![Scheme of a double-layer system under investigation: two graphene layers separated with a vacuum gap $d$. The chemical potential difference ($\mu_{1R}-\mu_{1L}$) induces electron drift in the bottom graphene layer, (a) perspective view, (b) top view.[]{data-label="figure1"}](Figure1.pdf)

Fluctuational electrodynamics description
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We generalize the usual formula for heat transfer, for which local equilibrium in each layer is assumed. Due to the current in the bottom layer, the system is intrinsically not in local thermal equilibrium. This nonequilibrium situation is taken care by a simple hypothesis of an energy shift in the distribution functions. When the bottom layer 1 is driven by a drift velocity $v_1$ in $x$ direction and there is no drift in the top layer, the photon distribution is Doppler shifted on the bottom layer. Thus, the Bose function becomes: $$\begin{aligned}
    N_1(\omega, k_x) = \frac{1}{e^{\frac{\hbar (\omega-k_xv_1)}{k_b T_1}}-1},\end{aligned}$$ 