Electron-phonon interaction on the surface of a three-dimensional topological insulator

We analyze the interaction at the surface of a three-dimensional (3D) topological insulator among 2D electron states belonging to the Dirac cone close to the point of the Brillouin zone and the Rayleigh surface phonon mode. The model deals with an elastic continuum in the long-wavelength limit, in Random Phase Approximation (RPA). Screening of the electronic polarization is quite effective at small wave vectors. On the other hand, the absence of backscattering for the Dirac electrons at the Fermi surface is partly responsible for the reduced influence of the electron-phonon interaction in renormalizing the phonon dispersion at finite wave vectors. We infer that softening of the Rayleigh mode appears as unlikely, at least for the case of a clean and defect-free surface to which our approximate treatment applies. The dielectric response to virtual excitation of the Rayleigh phonon could drive the electron-electron interaction attractive at low frequencies, but the average weak coupling pairing interaction is found to be too small to induce a surface superconducting instability.

Figure 1

Quality factor $Q$ as a function of momentum $q$ for different values of the Fermi wave vector: $k_F=1.5{\text{nm}}^{-1}$ (black solid line), $k_F=2{\text{nm}}^{-1}$ (red dashed line), and $k_F=2.5{\text{nm}}^{-1}$ (green dash-dotted line).

Figure 2

(a) Plot of the phonon dispersion ${\omega }_q$ given by Eq. (17), for $\alpha =35\text{eV}$. The solid line and the dashed line are for $k_F=1{\text{nm}}^{-1}$ and $k_F=2.5{\text{nm}}^{-1}$, respectively. The dash-dotted curve is obtained for $k_F=1{\text{nm}}^{-1}$ by setting ${\Lambda }_l\left(q\right)=1$. (Inset) The phonon frequency at $k_F=1.5{\text{nm}}^{-1}$ (in units $v_F{k}_F$) versus $q$ for different values of the strength of the deformation potential $\alpha =105\text{eV}$ (black solid line), $210\text{eV}$ (red dashed line), and $227.5\text{eV}$ (green dash-dotted line). (b) Plot of the first derivative of ${\omega }_q$ versus $q$ for $\alpha =35\text{eV}$ and various $k_F$ values. The discontinuity at $q=2k_F$ signals the Kohn anomaly. The inset shows the corresponding marked discontinuities in the second derivative.

Figure 3

(a) Color plot of the dimensionless quantity $\rho \left(E_F\right)V(q,\omega )$ for $k_F=1{\text{nm}}^{-1}$ in the $q/\omega$ plane ($\alpha =35\text{eV}$). The horizontal black line at $\omega /v_F{k}_F=0.002$ marks a cross section of the plot that is drawn separately in the panel (b).

Figure 4

Plot of the dimensionless product $\rho \left(E_F\right)V(2k_F\sin \theta /2,0)$ of Eq. (23) versus the scattering angle $\theta$ for various values of $k_F$ ($\alpha =35\text{eV}$). The inset shows the dependence of the pairing coupling strength $\lambda$ versus $k_F$.