The role of vector potential coupling in hot electron cooling power in bilayer graphene at low temperature

We have studied, in bilayer graphene (BLG), the hot electron cooling power $F_{VP} (T, n_s)$ due to acoustic phonons via vector potential (VP) coupling. It is calculated as a function of electron concentration $ns$ and temperature $T$ and compared with $F_{DP} (T, n_s)$ that due to deformation potential coupling. For the $n_s$ around $1\times10^{12} cm^{-2}$, $F_{VP} (T, n_s)$ is much smaller than $F_{DP} (T, n_s)$ . With increase of $n_s$, $F_{DP} (T, n_s)$ decreases faster than $F_{VP} (T, n_s)$ does. A cross over is predicted and dominant contribution of $F_{VP} (T, n_s)$ can be observed at large $n_s$. In the Bloch- Gr\"uneisen (BG) regime $F_{VP} (T, n_s)~ n_s^{-1/2}$ and $F_{DP} (T, n_s)~ n_s^{-3/2}$. Both $F_{VP} (T, n_s)$ and $F_{DP} (T, n_s)$ have the same $T$ dependence with $T^4$ power law in BG regime. Behaviour of $F_{DP} (T, n_s)~ n_s^{-3/2}$ and $T^4$ is in agreement with the experimental results at moderate $n_s$. Besides, in BG regime, we have predicted, for both the VP and DP coupling, a relation between $F(T, n_s)$ and the acoustic phonon limited mobility $\mu_p$, opening a new door to determine $\mu_p$ from the measurements of $F(T, n_s)$.