Many-Body Correlation Effects in Fröhlich Electron-Phonon Coupling

Abstract

In compound semiconductors and insulators, the polar electron-phonon coupling diverges at long range, known as the Fröhlich interaction. Modern first-principles electron-phonon calculations treat the Fröhlich interaction in a semiclassical electrostatic formalism based on density-functional perturbation theory. Here, using many-body $GW$ perturbation theory, we reveal important electron correlation effects in the Fröhlich-type electron-phonon coupling, which are missed by the prevailing approaches. Going beyond the electrostatic treatment, we derive and implement the $GW$ self-energy contribution to the long-range polar electron-phonon coupling, and demonstrate its critical role and nontrivial behaviors in properties such as electron linewidth and polaron formation. Our work establishes the many-body generalization of the Fröhlich interaction that is essential for accurate electron-phonon calculations at the full $GW$ level combined with Wannier interpolation techniques.

Figures (3)

• Figure 1: (a) Calculated LO-mode $e$-ph matrix elements (symbols) and the Wannier interpolations (lines) at DFPT (blue color) and $GW$PT (red color) levels in SiC. The Wannier interpolations are constructed with direct calculations on coarse wavevectors (open squares), and compared against additional direct calculations on fine wavevectors (solid dots). The gray solid line represents the $GW$PT interpolation without any LR contributions, whereas the gray dashed line represents the $GW$PT interpolation with the DFT-level LR contribution $g^{\mathcal{L}, \text{DFT}}$. The red solid line shows that by correctly incorporating the $GW$-level LR contribution $g^{\mathcal{L}, GW}$ (Eq. \ref{['eq:gw-correction']}), the interpolation excellently agrees with the direct $GW$PT calculations (red dots) both at the SR regime and towards the LR limit. (b) Calculated LO-mode matrix elements (symbols) and the Wannier interpolations (lines) for phonon-induced changes in the exchange-correlation potential $\partial_\textbf{q} V_\text{xc}$ (blue color) and in the $GW$ self-energy $\partial_\textbf{q} \Sigma$ (red color).
• Figure 2: Calculated imaginary part of the phonon-induced electron self-energy ($\text{Im}\Sigma^{e\text{-ph}}_{n\textbf{k}}$) at 20 K. (a)-(c) Hole linewidth near the VBM for SiC, cubic BN, and wurtzite GaN, respectively. (d) Electron linewidth near the CBM for SrTiO$_3$. Vertical gray lines indicate the characteristic LO phonon energies. Four levels of theory are compared: (i) the standard DFT and DFPT baseline; (ii) $GW$ quasiparticle energies combined with conventional DFPT $e$-ph matrix; (iii) $GW$ eigenvalues with $GW$PT corrections only in the SR $e$-ph matrix elements, whereas the LR contribution is accounted for at the DFPT level ($g^{\mathcal{L},\text{DFT}}$); and (iv) the fully $GW$ and $GW$PT treatment, where the LR $e$-ph elements are treated also at the $GW$ level ($g^{\mathcal{L},GW}$).
• Figure 3: Hole polaron formation energy $\Delta E_f$ (symbols) as a function of inverse BvK supercell size and the extrapolation (dotted lines) to the dilute limit (infinite supercell size) for (a) LiF and (b) TiO$_2$. Four levels of theory are presented. Phonon dispersion of (c) LiF and (d) TiO$_2$ overlaid by spectral decomposition of the lattice distortion at the full $GW$ and $GW$PT level (case (iv)).