Resonant electron-lattice cooling in graphene

Controlling energy flows in solids through switchable electron-lattice cooling can grant access to a range of interesting and potentially useful energy transport phenomena. Here we discuss a tunable electron-lattice cooling mechanism arising in graphene due to phonon emission mediated by resonant scattering on defects in a crystal lattice, which displays an interesting analogy to the Purcell effect in optics. In that, the electron-phonon cooling rate is enhanced due to hot carrier trapping at resonant defects. Resonant dependence of this process on carrier energy translates into gate-tunable cooling rates, exhibiting strong enhancement of cooling that occurs when the carrier energy is aligned with the electron resonance of the defect.

Figure 1

(a) Schematics showing a hot electron of initial state $|\mathbf{p}\rangle$ being trapped by the impurity forming a resonant state $|{\epsilon }_{\text{LS}}\rangle$, emitting a phonon of energy $\hslash {\omega }_k$ in the process. (b) The same process shown as a Feynman diagram. The vertex represents the matrix element $M$ in Eq. (12). Note that the outgoing state is the resonant state $|{\epsilon }_{\text{LS}}\rangle$.

Figure 2

A plot of the cooling power, in the presence of resonant scatterers vs Fermi energy for several values of the scatterer energy ${\epsilon }_{\text{LS}}$ [Eqs. (26) and (17)]. Local temperature change [Eq. (29)], which is proportional to the cooling power, is shown on the right axis. The curves are sharply peaked on resonance, falling off rapidly away from resonance. The purple dashed line shows the experimental curve from [19], where peaks in cooling power due to resonant scatterers were observed near $\epsilon \approx -22$ meV. The intrinsic contribution [Eq. (24)] vanishes at the Dirac point and remains low, compared to the peaks, throughout the range of Fermi energies plotted. (b) Semilog plot showing the relative contributions of $P_0$, $P_1$, and $P_2$ for ${\epsilon }_{\text{LS}}=-22$ meV. $P_1$ is small throughout the range of $\mu$.