Fröhlich Electron-Phonon Vertex from First Principles

We develop a method for calculating the electron-phonon vertex in polar semiconductors and insulators from first principles. The present formalism generalizes the Fröhlich vertex to the case of anisotropic materials and multiple phonon branches, and can be used either as a postprocessing correction to standard electron-phonon calculations, or in conjunction with ab initio interpolation based on maximally localized Wannier functions. We demonstrate this formalism by investigating the electron-phonon interactions in anatase ${\mathrm{TiO}}_2$, and show that the polar vertex significantly reduces the electron lifetimes and enhances the anisotropy of the coupling. The present work enables ab initio calculations of carrier mobilities, lifetimes, mass enhancement, and pairing in polar materials.

Figure 1

(a) Calculated phonon dispersions in anatase ${\mathrm{TiO}}_2$ along high-symmetry lines in the Brillouin zone. The LO phonons discussed in the main text are highlighted by arrows. (b) Calculated electron-phonon matrix elements, with (blue solid lines) and without (red dashed lines) the polar coupling $g^{\mathcal{L}}$ from Eq. (4). The calculations using Wannier-Fourier interpolations (lines) are compared to direct DFPT calculations at each wave vector (filled discs). Here we show the gauge-invariant trace of $|g{|}^2$ over degenerate states. In the calculation of $g_{mn\nu }(\mathbf{k},\mathbf{q})$ we set the initial electronic state $|{\psi }_{n\mathbf{k}}⟩$ to the bottom of the conduction band at $\mathrm{\Gamma }$, the final electronic state $|{\psi }_{m\mathbf{k}+\mathbf{q}}⟩$ to the bottom of the conduction band, and the phonon branch to be the highest (LO) optical mode. (c) The spherical average of the electron-phonon matrix elements, $4\pi q^2|g{|}^2$, with (blue) and without (red) the polar coupling, and using the simple Fröhlich model in Eq. (1) (gray).

Figure 2

(a) Diagram of the Migdal self-energy $\mathrm{\Sigma }$ used to calculate electron lifetimes. The straight and wiggly lines represent the electron and phonon Green’s functions, respectively. The circles are the electron-phonon matrix elements. (d) Close-up of the conduction bands of anatase ${\mathrm{TiO}}_2$ near the band bottom at $\mathrm{\Gamma }$, taken as the zero of the energy. (b), (e) Electron linewidths arising from electron-phonon scattering, $\mathrm{Im}\mathrm{\Sigma }$, along the $\mathrm{\Gamma }Z$ and the $\mathrm{\Gamma }X$ lines, respectively, near the band bottom. The energies of the LO phonons shown in Fig. 1 are indicated by vertical dashed lines. (c),(f) Electron lifetimes from (b) and (e). In (b),(e) and (c),(f) the blue solid lines are computed using the complete vertex, while the red dashed lines are obtained using standard Wannier interpolation [33].