Electron-phonon scattering in topological insulator thin films

We present a theoretical study of electron-phonon scattering effects in thin films made of a strong topological insulator. The phonons are modeled by isotropic elastic continuum theory with stress-free boundary conditions, and the interaction with the helical surface Dirac fermions is mediated by the deformation potential. We determine the temperature-dependent electrical resistivity $\rho \left(T\right)$ and the quasiparticle decay rate $\Gamma \left(T\right)$ observable in photoemission. The low- and high-temperature power laws for both quantities are obtained analytically. Detailed estimates covering the full temperature range are provided for Bi${}_2$Se${}_3$.

Figure 1

Electronic eigenstates for Bi${}_2$Se${}_3$ from Eqs. (1) and (2). Main panel: Densities ${\rho }_{\tau }\left(z\right)$ in Eq. (14) for $L=4$ QL. Inset: Gap $\Delta$ vs thickness $L$. Note the semilogarithmic scale.

Figure 2

Phonon dispersion relation ${\Omega }_{\Lambda }$ vs $q$ for the symmetric ($\lambda =S$) mode (red solid curves). Shown are the ten lowest branches corresponding to the index $n$. Dashed lines separate regions I, II, and III (see text). The dash-dotted line gives the dispersion relation in Eq. (24); note that the $n=1$ mode coincides with the Rayleigh mode for $qL\gg 1$.

Figure 3

Phonon contribution to the resistivity $\rho$ vs temperature $T$ for a TI film of width $L=4$ QL and several values of the chemical potential $\mu$. Dashed straight lines indicate the analytical results for low [Eq. (36)] and high [Eq. (38)] temperatures. Note the double-logarithmic scale.

Figure 4

Width ($L$) dependence of the phonon contribution to the resistivity $\rho$ for $\mu =0.2$ eV and several temperatures. The dashed horizontal line indicates one-quarter of the resistivity ${\rho }_{\infty }\left(T\right)$ in the semi-infinite geometry with otherwise identical parameters (Ref. 11).

Figure 5

Main panel: $T$ dependence of the decay rate $\Gamma$ of a TI film of width $L=4$ QL for $k=k_F$ and $k=0.5k_F$. For $k=0.5k_F$, only the $\mu =0.2$ eV result is displayed. Dashed lines indicate the low- and high-temperature laws ($\Gamma \propto T^2$ and $\Gamma \propto T$), respectively. Inset: $k$ dependence of $\Gamma$ for $\mu =0.2$ eV and two different temperatures: $T=3$ K (solid line) and $T=300$ K (dashed line; the shown result has to be multiplied by 10).

Figure 6

Width ($L$) dependence of the effective electron-phonon coupling constant at the Fermi level ${\lambda }_{k_F}$ for $\mu =0.2$ eV. The dashed horizontal line indicates one-half of the effective coupling constant in the semi-infinite geometry with otherwise identical parameters (Ref. 11).