Inelastic carrier lifetime in a coupled graphene/electron-phonon system: Role of plasmon-phonon coupling

We calculate the inelastic scattering rates and the hot-electron inelastic mean free paths for both monolayer and bilayer graphene on a polar substrate. We study the quasiparticle self-energy by taking into account both electron-electron and electron–surface-optical-phonon interactions. In this calculation the leading-order dynamic screening approximation $({\mathrm{G}}_0\mathrm{W}$ approximation) is used to obtain the quasiparticle self-energy by treating electrons and phonons on an equal footing. We find that the strong coupling between the surface-optical phonon and the plasmon leads to an additional decay channel for the quasiparticle through the emission of the coupled mode, and gives rise to an abrupt increase in the scattering rate, which is absent in the uncoupled system. In monolayer graphene a single jump in the scattering rate occurs, arising from the emission of the low-energy branch of the coupled plasmon-phonon modes. In bilayer graphene the emission of both low- and high-energy branches of the coupled modes contributes to the scattering rate and gives rise to two abrupt changes in the scattering rate. The jumps in the scattering rate can potentially be used in hot-electron devices such as switching devices and oscillators.

Figure 1

(a) Electron-electron Coulomb interaction. (b) Phonon-mediated electron-electron interaction. (c) Series of diagrams corresponding to the RPA for the effective interaction in the presence of both electron-electron and electron-phonon interactions. The wiggly (dashed) line represents the electron-electron Coulomb (SO-phonon-mediated) interaction and ${\Pi }_0$ the bare polarizability.

Figure 2

(a) Calculated scattering rate as a function of the on-shell energy ${\xi }_{\mathit{k}}$ for different carrier densities $n=(1,2,5,10)\times {10}^{11}$ ${\mathrm{cm}}^{-2}$ for (a) uncoupled and (b) coupled monolayer graphene $\left(J=1\right)$, and the calculated energy-loss function for (c) uncoupled and (d) coupled monolayer graphene at $n={10}^{12}$ ${\mathrm{cm}}^{-2}$. The dotted horizontal line in (d) represents the SO-phonon frequency, and the dashed line represents the boundary of the IEEL continua for an electron injected with the energy 140 meV (c) and 106 meV (d). ${\mathrm{SPE}}^{\mathrm{intra}}$ $\left({\mathrm{SPE}}^{\mathrm{inter}}\right)$ represents the single-particle excitation region for the intraband (interband) electron-hole excitations. Note that for the coupled system the plasmon dispersion is partly covered by the IEEL continua and thus a decay process via plasmon emission is available.

Figure 3

Calculated scattering rate as a function of ${\xi }_{\mathit{k}}$ for different carrier densities $n=(1,2,5,10)\times {10}^{11}$ ${\mathrm{cm}}^{-2}$ for (a) uncoupled and (b) coupled bilayer graphene $\left(J=2\right)$, and calculated energy-loss function for (c) uncoupled and (d) coupled bilayer graphene at $n={10}^{12}$ ${\mathrm{cm}}^{-2}$. In (d), the IEEL continua are drawn for two different energies of an injected electron: ${\xi }_{\mathit{k}}=66$ meV (blue dashed line) and ${\xi }_{\mathit{k}}=106.7$ meV (green dashed line). At these energies, the scattering rate increases sharply because of the decay via the emission of the plasmonlike mode and phononlike mode, respectively, as shown in (b).

Figure 4

Calculated mean free path as a function of ${\xi }_{\mathit{k}}$ for different carrier densities $n=(1,2,5,10,20)\times {10}^{11}$ ${\mathrm{cm}}^{-2}$ for (a) monolayer graphene and (b) bilayer graphene. The solid (dotted) lines represent the mean free paths in the presence (absence) of the electron–SO-phonon coupling.

Figure 5

Calculated scattering rate as a function of ${\xi }_{\mathit{k}}$ for different carrier densities $n={10}^{11},{10}^{12},{10}^{13}$ ${\mathrm{cm}}^{-2}$ for monolayer graphene (left column) and for bilayer graphene (right column). The solid (dotted) lines represent the scattering rates with dynamically (statically) screened electron–SO-phonon interaction. Here $\hslash {\omega }_{\mathrm{SO}}=20$ meV is used for this calculation.