going into it is quite large, see Fig. 4(b). From the view of energy transport, the energy can be transferred from an electrically driven layer to the closely spaced but electrically isolated layer: the drift electrons are dragged by the Coulomb interaction between two layers, and the energy can be transferred at the cost of the kinetic energy of drift electrons.

![(a) FE calculation of drift-induced heat current density as a function of drift velocity. Vacuum gap distance $d$ is set as 10nm. The chemical potential of graphene $\mu$ is set as 0.1eV. (b) NEGF calculation of drift-induced heat current density as a function of chemical potential difference $\mu_{1R}-\mu_{1L}$ symmetrically biased and $\mu_{2L}=\mu_{2R}=0.1\,\mathrm{eV}$. Vacuum gap distance $d$ is set as 1nm. (c) FE drift-induced heat current density as a function of gap distance with drift velocity $v_1=7.0 \times 10^5\,$m/s. (d) NEGF drift-induced heat current density as a function of gap distance with $\mu_{1R}=0.15$eV, $\mu_{1L}=0.05\,$eV and $\mu_{2L}=\mu_{2R}=0.1\,$eV. There is no temperature difference between the double-layer graphene, $T_1=T_2=300\,$K.[]{data-label="figure4"}](figure4){width="\columnwidth"}

Huge heat transfer appears even for $T_1 = T_2$, which is clearly due to strong Coulomb interactions at short distances. A comparison of FE and NEGF results is presented in Fig. \[figure4\]. Using FE 