Ab Initio Electron-Phonon Interactions in Correlated Electron Systems

Electron-phonon ($e\text{−}\mathrm{ph}$) interactions are pervasive in condensed matter, governing phenomena such as transport, superconductivity, charge-density waves, polarons, and metal-insulator transitions. First-principles approaches enable accurate calculations of $e\text{−}\mathrm{ph}$ interactions in a wide range of solids. However, they remain an open challenge in correlated electron systems (CES), where density functional theory often fails to describe the ground state. Therefore reliable $e\text{−}\mathrm{ph}$ calculations remain out of reach for many transition metal oxides, high-temperature superconductors, Mott insulators, planetary materials, and multiferroics. Here we show first-principles calculations of $e\text{−}\mathrm{ph}$ interactions in CES, using the framework of Hubbard-corrected density functional theory ($\mathrm{DFT}+U$) and its linear response extension ($\mathrm{DFPT}+U$), which can describe the electronic structure and lattice dynamics of many CES. We showcase the accuracy of this approach for a prototypical Mott system, CoO, carrying out a detailed investigation of its $e\text{−}\mathrm{ph}$ interactions and electron spectral functions. While standard DFPT gives unphysically divergent and short-ranged $e\text{−}\mathrm{ph}$ interactions, $\mathrm{DFPT}+U$ is shown to remove the divergences and properly account for the long-range Fröhlich interaction, allowing us to model polaron effects in a Mott insulator. Our work establishes a broadly applicable and affordable approach for quantitative studies of $e\text{−}\mathrm{ph}$ interactions in CES, a novel theoretical tool to interpret experiments in this broad class of materials.

Figure 1

(a) Comparison between the $e\text{−}\mathrm{ph}$ matrix elements in CoO computed with $\mathrm{DFPT}+U$ (orange) and standard DFPT (blue). Shown is $|g_{\nu }^{\sigma }(\mathit{q})|\equiv ({\sum }_{mn}|g_{mn\nu }^{\sigma }(\mathit{k}=0,\mathit{q}){|}^2/N_b{)}^{1/2}$ for all phonon modes $\nu$, where the phonon wave vector $\mathit{q}$ is varied along high-symmetry BZ lines and the summation runs over the $N_b=10$ spin-up Co $3d$ Wannier bands [56]. (b) CoO phonon dispersions overlaid with a log-scale color map of $|g_{\nu }^{\sigma }(\mathit{q})|$ computed with $\mathrm{DFPT}+U$ (colored line). The CoO phonon dispersions from standard DFPT (gray line) are given for comparison, with imaginary frequencies shown as negative values.

Figure 2

Comparison between the $e\text{−}\mathrm{ph}$ coupling from the KS potential contribution alone (cyan line) and the total result including the Hubbard correction (orange line). In each case, we show the gauge-invariant $e\text{−}\mathrm{ph}$ coupling strength [41], $D_{\mathrm{tot}}^{\nu \sigma }(\mathit{q})=(2{\omega }_{\nu \mathit{q}}{M}_{\mathrm{uc}}{\sum }_{nm}|g_{nm\nu }^{\sigma }(\mathit{k}=0,\mathit{q}){|}^2/N_b{)}^{1/2}$, computed, respectively, with $e\text{−}\mathrm{ph}$ matrix elements ${\tilde{g}}_{\mathrm{KS}}$ and $g_{\mathrm{tot}}$, summing over all $N_b=13$ highest spin-up valence bands. The BZ labeling refers to an equivalent (distorted) rocksalt structure [36]. Direct $\mathrm{DFPT}+U$ calculations (circles), shown as a benchmark, validate the Wannier interpolation. The arrows indicate the divergence due to the Fröhlich interaction.

Figure 3

(a) Band structure of CoO overlaid with the Hubbard contribution to the imaginary part of the $e\text{−}\mathrm{ph}$ self-energy, $\mathrm{Im}({\mathrm{\Sigma }}_{\mathrm{Hub}})$, for the representative case of the spin-up bands. The right panel shows the projected density of states (PDOS) of the partially and completely filled $3d$ orbitals. (b),(c) Comparison between the spatial decay of the real-space $e\text{−}\mathrm{ph}$ matrix elements for (b) the partially filled and (c) the completely filled $3d$ orbitals. Shown are the contributions from the KS (black squares) and Hubbard (red circles) $e\text{−}\mathrm{ph}$ perturbations [see Eq. (2)] to the maximum value of the Wannier basis matrix elements [8], $\parallel g({\mathbf{r}}_p)\parallel ={\max }_{ij}|g_{ij}({\mathbf{r}}_p)|$, normalized using the KS contribution. The inset in (b) is a schematic of the $e\text{−}\mathrm{ph}$ matrix elements in the Wannier basis, showing the atomic displacement perturbation at distance $|{\mathbf{r}}_p|$.

Figure 4

Electron spectral function in CoO, computed at three temperatures from 100 to 300 K, for the highest valence band at crystal momentum $\mathit{k}=\mathbf{F}$. In each panel, the zero of the electron energy $\omega$ is set to the band energy obtained from $\mathrm{DFT}+U$ calculations.