Electron-phonon coupling in two-dimensional silicene and germanene

Following the work in graphene, we report a first-principles study of electron-phonon coupling (EPC) in low-buckled monolayer silicene and germanene. Despite the similar honeycomb atomic arrangement and linear-band dispersion, the EPC matrix-element squares of the $\Gamma$-$E_g$ and $K$-$A_1$ modes in silicene are only about 50% of those in graphene. However, the smaller Fermi velocity in silicene compensates for this reduction by providing a larger joint electronic density of states near the Dirac point, giving rise to comparable phonon linewidths. We predict that Kohn anomalies associated with these two optical modes are significant in silicene. In addition, the EPC-induced frequency shift and linewidth of the Raman-active $\Gamma$-$E_g$ mode in silicene are calculated as a function of doping. The results are comparable to those in graphene, indicating a similar nonadiabatic dynamical origin. In contrast, the EPC in germanene is found to be much reduced.

Figure 1

(a) Band dispersions of silicene (red solid) and germanene (blue solid). The band structure of graphene (dotted black) is also shown for comparison. Inset: Schematic plot of the honeycomb lattice of germanene (top view). (b)–(d) The band-decomposed charge density distribution of the $\pi$ (${\pi }^{*}$) bands near the Fermi level on the $\{$110$\}$ plane [indicated in the inset of (a)] in graphene, silicene, and germanene, respectively.[26] The Fermi level is shifted to zero in (a).

Figure 2

Phonon dispersions of (a) silicene, with the highest optical modes around (b) $q=\Gamma$ and (c) $q=K$ as a function of smearing parameter $\sigma$. Phonon dispersions of (d) germanene, with the highest optical modes around (e) $q=\Gamma$ and (f) $q=K$ as a function of smearing parameter $\sigma$. The calculated frequencies are marked, with the lines being guides to the eye.

Figure 3

Calculated phonon linewidth $\gamma$ (left axis) and frequency shift $\Delta \omega$ (right axis) in silicene as a function of the Fermi level $E_F$ for the $\Gamma$-$E_g$ mode with $\hslash {\omega }_0$ = 70 meV. The neutrality point has been shifted to zero.

Figure 4

Schematic plot of the effect of the band slope on the phonon linewidth. With ${\beta }_1>$ ${\beta }_2$, system 1 has fewer states in the $k$ space that satisfy the energy conservation and are allowed to couple with the phonon mode than system 2, resulting in a smaller contribution to the self-energy calculation in Eq. (3).