method, Fig. \[figure4\](a) indicates the drift-induced heat current density as a function of drift velocity with both layers at the same temperature. A parabolic dependence between drift-induced heat current and drift velocity is found numerically for small $v_1$. For large drift exceeding $|v_1| > 7.0\times 10^5\,$m/s, we see non-monotonic behavior. From the NEGF point of view, the chemical potential difference between left and right bath can produce an electric current and it is similar to the drift electron one in FE calculation. In Fig. \[figure4\](b), we found that the heat current density also quadratically depends on the chemical potential difference $\mu_{1R}-\mu_{1L}$. In our symmetric setup, we do not expect that the heat transfer will be different if we reverse the direction of the drift velocity or voltage bias, so it must depend on them quadratically. It is tempting to assume that the heat transfer is given by Joule heating caused by Coulomb drag. Unfortunately, this is not quite true (see Appendix \[app-c\]).

Further numerical evidence is shown in Fig. \[figure4\](c) and (d): the distance dependence of drift-induced heat current. Fig. \[figure4\](c) depicts the drift-induced heat current density as a function of distance under FE calculation with fixed drift velocity. We 