Electron-phonon heat transfer in monolayer and bilayer graphene

We calculate the heat transfer between electrons to acoustic and optical phonons in monolayer and bilayer graphene (MLG and BLG) within the quasiequilibrium approximation. For acoustic phonons, we show how the temperature-power laws of the electron-phonon heat current for BLG differ from those previously derived for MLG and note that the high-temperature (neutral-regime) power laws for MLG and BLG are also different, with a weaker dependence on the electronic temperature in the latter. In the general case we evaluate the heat current numerically. We suggest that a measurement of the heat current could be used for an experimental determination of the electron-acoustic-phonon coupling constants, which are not accurately known. However, in a typical experiment heat dissipation by electrons at very low temperatures is dominated by diffusion and we estimate the crossover temperature at which acoustic-phonon coupling takes over in a sample with Joule heating. At even higher temperatures optical phonons begin to dominate. We study some examples of potentially relevant types of optical modes, including, in particular, the intrinsic in-plane modes and additionally the remote surface phonons of a possible dielectric substrate.

Figure 1

(Color online) Geometrical structure of bilayer graphene with AB, or Bernal, stacking and the Slonczewski-Weiss-McClure band parameters ${\gamma }_{0,1,3,4}$. The A1, B1, A2, and B2 atoms of the two sublattices and layers are indicated. The nearest-neighbor distance is $a\approx 0.14\text{nm}$ and the interlayer distance is $d\approx 0.37\text{nm}$.

Figure 2

(Color online) Thermal conductance $G_{e\text{−}ac}\left(\mu ,T\right)$ for (a) MLG and (b) BLG. The unit is $G_0=g_{e}A{D}^2\hslash {\left|\mu \right|}^4{k}_B/\left[8{\pi }^2\rho {\left(\hslash v\right)}^6\right]$ for MLG and $G_0=g_{e}A{D}^2\hslash {\gamma }_1{\left|\mu \right|}^3{k}_B/\left[16{\pi }^2\rho {\left(\hslash v\right)}^6\right]$ for BLG. The parameters are $c/v=0.02$ and ${\gamma }_1/\left|\mu \right|=10$ in (b). The solid lines (magenta) indicate the full numerical solutions and the dashed lines (black) the analytical approximations.

Figure 3

(Color online) Prefactors $q_{e\text{−}op}^{\left(\text{LT}\right)}$ and $q_{e\text{−}op}^{\left(\text{rem}\right)}$ of the electron-optical-phonon heat currents for different doping levels. (a) and (c) are for MLG, and (b) and (d) for BLG. Upper panels are for the intrinsic LT phonons and the lower ones for the remote phonons, with $z=0$.

Figure 4

(Color online) The ratio of (LT) optical-phonon thermal conductance $G_{e\text{−}op}^{\left(\text{LT}\right)}\left(\mu ,T\right)$ and that of acoustic phonons $G_{e\text{−}ac}\left(\mu ,T\right)$ for (a) MLG and (b) BLG. The parameters are $D=30\text{eV}$ and $\left|{\gamma }_0^{\prime }\right|=40\text{eV}/\text{nm}$ (Ref. 27). In both cases the crossover point $G_{e\text{−}op}^{\left(\text{LT}\right)}/G_{e\text{−}ac}=1$ moves to higher temperature with increasing $\left|\mu \right|$.