Electron–phonon coupling

Electron–phonon coupling (EPC) is the matrix element of the lattice-induced change in the Kohn–Sham potential between Bloch states. It controls phonon-mediated superconductivity (the Eliashberg function and $\lambda$), the temperature and zero-point renormalization of band gaps (Allen–Heine–Cardona), and phonon-limited carrier mobility. The matrix elements come naturally from DFPT — the lattice perturbation is the same one that gives the dynamical matrix — but converging the Brillouin-zone sums that the physics requires is only tractable through Wannier interpolation (the EPW approach). For polar LO phonons the coupling carries a long-range Fröhlich divergence as $\mathbf{q}\to 0$, governed by the same Born-charge / LO–TO physics as the dynamical matrix, and must be treated analytically.

Prerequisites

dynamical matrix & ASR · DFPT · Wannier functions

1. The EPC matrix element

To first order in the displacement of phonon mode $\nu$ at wavevector $\mathbf{q}$, the self-consistent Kohn–Sham potential changes by ${\partial }_{\nu \mathbf{q}}V$, the derivative of $V_{\mathrm{KS}}$ with respect to the collective phonon coordinate (the eigenvector-weighted, mass-weighted sublattice displacement). The EPC matrix element scatters an electron from $|n\mathbf{k}⟩$ to $|m,\mathbf{k}+\mathbf{q}⟩$:

$$\boxed{g_{mn\nu }(\mathbf{k},\mathbf{q})=(\frac{\hslash }{2M_0{\omega }_{\mathbf{q}\nu }}{)}^{1/2}⟨u_{m,\mathbf{k}+\mathbf{q}}|{\partial }_{\nu \mathbf{q}}V|u_{n\mathbf{k}}⟩}$$

with the prefactor the zero-point amplitude of mode $\nu$ ($M_0$ a reference mass folded into the mode normalization). The screened, self-consistent derivative ${\partial }_{\nu \mathbf{q}}V$ — not the bare ionic potential — is precisely the first-order potential the Sternheimer equation of DFPT returns, so DFPT supplies $g$ for free at any commensurate $\mathbf{q}$ in a primitive cell. The corresponding interaction Hamiltonian is

$${\hat{H}}_{\text{ep}}=\frac{1}{\sqrt{N_p}}\sum _{\mathbf{k},\mathbf{q},\nu }\sum _{mn}{g}_{mn\nu }(\mathbf{k},\mathbf{q}){\hat{c}}_{m,\mathbf{k}+\mathbf{q}}^{†}{\hat{c}}_{n\mathbf{k}}({\hat{a}}_{\mathbf{q}\nu }+{\hat{a}}_{-\mathbf{q}\nu }^{†}),$$

the standard vertex of the many-body theory below. Note $g$ is gauge-dependent in the band index (it inherits the Bloch phase freedom); only $|g{|}^2$ summed over degenerate states is physical.

2. The Eliashberg function and $\lambda$

For phonon-mediated superconductivity the relevant object is the isotropic Eliashberg spectral function, the EPC strength resolved by phonon energy and weighted by the electronic density of states at the Fermi level $N({\epsilon }_F)$:

$${\alpha }^{2}F(\omega )=\frac{1}{N({\epsilon }_F)}\sum _{\mathbf{q}\nu }\sum _{mn}\int \frac{d\mathbf{k}}{{\Omega }_{\mathrm{BZ}}}|g_{mn\nu }(\mathbf{k},\mathbf{q}){|}^2\delta ({\epsilon }_{n\mathbf{k}}-{\epsilon }_F)\delta ({\epsilon }_{m\mathbf{k}+\mathbf{q}}-{\epsilon }_F)\delta (\omega -{\omega }_{\mathbf{q}\nu }).$$

Its first inverse moment is the dimensionless mass-enhancement / coupling constant

$$\lambda =2{\int }_0^{\infty }\frac{{\alpha }^{2}F(\omega )}{\omega }d\omega =\sum _{\mathbf{q}\nu }{\lambda }_{\mathbf{q}\nu },$$

the quantity that renormalizes the electron mass ($m^{\ast }/m=1+\lambda$) and sets $T_c$. With $\lambda$ and a logarithmic average phonon frequency ${\omega }_{\log }$, the Allen–Dynes / McMillan formula estimates $T_c$ from $\lambda$, ${\omega }_{\log }$, and the Morel–Anderson Coulomb pseudopotential ${\mu }^{\ast }$; the fully anisotropic, gap-resolved version requires solving the Eliashberg equations on the Fermi surface. The two Dirac deltas pinning both states to ${\epsilon }_F$ are exactly why the $\mathbf{k}$/$\mathbf{q}$ sums need ultradense grids — the convergence problem §5 addresses.

3. Band-gap renormalization — Allen–Heine–Cardona

Phonons also dress the electron self-energy and shift band energies; the leading (Fan–Migdal + Debye–Waller) theory is Allen–Heine–Cardona (AHC). The temperature-dependent self-energy of state $n\mathbf{k}$ has two terms — name both, because keeping only one is a common error:

• the Fan–Migdal term, second-order in $g$ (one phonon emitted and reabsorbed), an $\omega$- and $T$-dependent dynamical self-energy, $${\Sigma }_{n\mathbf{k}}^{\mathrm{FM}}(\omega ,T)=\sum _{m\nu }\int \frac{d\mathbf{q}}{{\Omega }_{\mathrm{BZ}}}|g_{mn\nu }(\mathbf{k},\mathbf{q}){|}^2[\frac{n_{\mathbf{q}\nu }+1-f_{m\mathbf{k}+\mathbf{q}}}{\omega -{\epsilon }_{m\mathbf{k}+\mathbf{q}}-{\omega }_{\mathbf{q}\nu }+i\eta }+\frac{n_{\mathbf{q}\nu }+f_{m\mathbf{k}+\mathbf{q}}}{\omega -{\epsilon }_{m\mathbf{k}+\mathbf{q}}+{\omega }_{\mathbf{q}\nu }+i\eta }],$$ with $n_{\mathbf{q}\nu }$, $f$ the Bose and Fermi occupations;
• the Debye–Waller term, first-order in the second derivative of $V$ (a static, instantaneous shift), which restores translational (rigid-ion) invariance — the EPC acoustic sum rule, cancelling the spurious $\mathbf{q}\to 0$ contribution of the acoustic modes in the Fan term alone — and the correct $T\to 0$ zero-point limit, and is often as large as the Fan term with the opposite sign.

The sum gives the zero-point ($T=0$) and finite-temperature renormalization of band energies and gaps — typically tenths of an eV, decisive for comparing computed gaps to experiment, and the reason “static-lattice” DFT/GW gaps are systematically off. The rigid-ion / adiabatic approximation behind the standard AHC implementation should be flagged; it can fail for polar materials and near band edges with strong Fröhlich coupling.

4. Phonon-limited mobility

EPC is the intrinsic resistive mechanism in a clean semiconductor. The transport scattering rate follows from $g$ via Fermi’s golden rule,

$$\frac{1}{{\tau }_{n\mathbf{k}}}=\frac{2\pi }{\hslash }\sum _{m\nu }\int \frac{d\mathbf{q}}{{\Omega }_{\mathrm{BZ}}}|g_{mn\nu }(\mathbf{k},\mathbf{q}){|}^2[(n_{\mathbf{q}\nu }+1-f)\delta ({\epsilon }_{n\mathbf{k}}-{\epsilon }_{m\mathbf{k}+\mathbf{q}}-{\omega }_{\mathbf{q}\nu })+(n_{\mathbf{q}\nu }+f)\delta ({\epsilon }_{n\mathbf{k}}-{\epsilon }_{m\mathbf{k}+\mathbf{q}}+{\omega }_{\mathbf{q}\nu })],$$

and the mobility follows from the Boltzmann transport equation (the self-energy relaxation-time approximation, or the full iterative BTE for back-scattering weighting). Converging $\mu (T)$ to within experimental precision is the most grid-demanding EPC observable, again motivating Wannier interpolation.

5. The Fröhlich divergence and Wannier interpolation

Two practical facts dominate any real EPC calculation:

• Fröhlich long-range divergence. For a polar longitudinal-optical phonon, the macroscopic electric field generated by the Born effective charges makes the coupling diverge as $g\sim 1/|\mathbf{q}|$ in the $\mathbf{q}\to 0$ limit, $$g_{mn\nu }^{\mathcal{L}}(\mathbf{k},\mathbf{q}\to 0)\sim i\frac{4\pi e^2}{\Omega }\frac{\mathbf{q}\cdot {\mathbf{Z}}_{\nu }^{\ast }}{\mathbf{q}\cdot {\mathbf{\epsilon }}^{\infty }\cdot \mathbf{q}},$$ built from the same $Z^{\ast }$ and ${\epsilon }^{\infty }$ as the non-analytic LO–TO term of the dynamical matrix. This long-range part is not captured by a short-ranged interpolation and must be subtracted analytically, interpolated in the smooth short-range remainder, then added back — the same split the polar-phonon correction uses. Skipping it corrupts mobilities in polar semiconductors and the $\mathbf{q}\to 0$ contribution to AHC.
• Wannier interpolation (EPW). Every observable above is a dense double integral over $\mathbf{k}$ and $\mathbf{q}$ pinned to the Fermi surface or band edges; explicit DFPT on a fine grid is prohibitive. Because $g$ in the localized Wannier–lattice representation $g_{mn\nu }({\mathbf{R}}_e,{\mathbf{R}}_p)$ is short-ranged in both the electron cell ${\mathbf{R}}_e$ and the phonon cell ${\mathbf{R}}_p$ (after removing the Fröhlich tail), it can be computed on coarse ab-initio grids and interpolated to arbitrarily fine $(\mathbf{k},\mathbf{q})$ at negligible cost — the EPW method, the standard route to converged ${\alpha }^{2}F$, $\lambda$, mobilities, and band-gap renormalization.

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Builds toward: Eliashberg / Migdal–Eliashberg superconductivity · polaron physics & Fröhlich coupling · phonon-assisted optical absorption · DFPT matrix elements

Key references

• EPC theory & Wannier interpolation (review) — F. Giustino, Rev. Mod. Phys. 89, 015003 (2017); F. Giustino, M. L. Cohen & S. G. Louie, Phys. Rev. B 76, 165108 (2007).
• Allen–Heine–Cardona — P. B. Allen & V. Heine, J. Phys. C 9, 2305 (1976); P. B. Allen & M. Cardona, Phys. Rev. B 23, 1495 (1981); review of zero-point renormalization: S. Poncé et al., J. Chem. Phys. 143, 102813 (2015).
• Eliashberg / superconductivity — G. M. Eliashberg, Sov. Phys. JETP 11, 696 (1960); P. B. Allen & R. C. Dynes, Phys. Rev. B 12, 905 (1975).
• Fröhlich / polar EPC interpolation — C. Verdi & F. Giustino, Phys. Rev. Lett. 115, 176401 (2015); J. Sjakste, N. Vast, M. Calandra & F. Mauri, Phys. Rev. B 92, 054307 (2015).
• EPW code — S. Poncé, E. R. Margine, C. Verdi & F. Giustino, Comput. Phys. Commun. 209, 116 (2016).