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Non-perturbative theory of the electron-phonon coupling and its first-principles implementation

Raffaello Bianco, Ion Errea

Abstract

The harmonic approximation of ionic fluctuations and the linear coupling between phonons and electrons provide the standard framework to compute, from first principles, the contribution of nuclear dynamics and its interaction with electrons to materials properties. These approaches become questionable when quantum and anharmonic effects are significant, such as in hydrogenous systems, high-$T_c$ superconductors, and systems close to displacive phase transitions. Here we propose a novel non-perturbative approach to compute the electron-phonon interaction from first principles, including non-linear effects and accounting for the quantum nature of nuclei. The method is based on the $GW^{ph}$ approximation for the electron self-energy, given by the nuclei-mediated electron-electron interaction $W^{ph}$ and the electron Green's function $G$. Electrons are treated at a mean-field level, while nuclear dynamics is described by a Gaussian distribution function that captures anharmonic effects, for example within the self-consistent harmonic approximation. The key quantities of the Gaussian $GW^{ph}$ self-energy are renormalized average vertices, computed in supercells using a stochastic approach based on self-consistent electronic potentials for distorted configurations. To validate the method, $GW^{ph}$ calculations are performed on aluminum, where the results reproduce standard linear electron-phonon theory, and on palladium hydride, where strong non-linear contributions emerge, with corrections comparable in magnitude to the linear-order result.

Non-perturbative theory of the electron-phonon coupling and its first-principles implementation

Abstract

The harmonic approximation of ionic fluctuations and the linear coupling between phonons and electrons provide the standard framework to compute, from first principles, the contribution of nuclear dynamics and its interaction with electrons to materials properties. These approaches become questionable when quantum and anharmonic effects are significant, such as in hydrogenous systems, high- superconductors, and systems close to displacive phase transitions. Here we propose a novel non-perturbative approach to compute the electron-phonon interaction from first principles, including non-linear effects and accounting for the quantum nature of nuclei. The method is based on the approximation for the electron self-energy, given by the nuclei-mediated electron-electron interaction and the electron Green's function . Electrons are treated at a mean-field level, while nuclear dynamics is described by a Gaussian distribution function that captures anharmonic effects, for example within the self-consistent harmonic approximation. The key quantities of the Gaussian self-energy are renormalized average vertices, computed in supercells using a stochastic approach based on self-consistent electronic potentials for distorted configurations. To validate the method, calculations are performed on aluminum, where the results reproduce standard linear electron-phonon theory, and on palladium hydride, where strong non-linear contributions emerge, with corrections comparable in magnitude to the linear-order result.
Paper Structure (13 sections, 112 equations, 9 figures, 3 tables)

This paper contains 13 sections, 112 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The nuclei-mediated electron-electron interaction
  3. The linear order
  4. Nuclei with Gaussian space distribution
  5. Nuclei with Gaussian space distribution: a diagrammatic analysis
  6. The Gaussian $GW^{{\text{ph}}}$ approximation
  7. The first-principles implementation
  8. A case of strongly non-linear electron-phonon coupling: palladium hydride
  9. A case of linear electron-phonon coupling: aluminum
  10. Conclusions
  11. From $W^{{\text{ph}}}_{n'\boldsymbol{k}'\,n\boldsymbol{k}}(iw_l)$ to $\lambda_{n\boldsymbol{k}}$
  12. Derivation Eq. \ref{["eq:sum_n'k'"]}
  13. Restriced sum to avoid numerical instabilities

Figures (9)

  • Figure 1: Diagrammatic representation of the nuclei-mediated effective electron-electron interaction between Bloch states. Due to this interaction, at time $t'$ an electron scatters from $\Ket{n\boldsymbol{k}'}$ to $\Ket{l\boldsymbol{k}}$, which causes the scattering of an electron from $\Ket{m\boldsymbol{k}}$ to $\Ket{i\boldsymbol{k}'}$ at a later time $t$.
  • Figure 2: Diagrammatic representation of the nuclei-mediated effective electron-electron interaction between Bloch states at lowest order in the atomic displacement from the equilibrium configuration.
  • Figure 3: Diagrammatic expansion of the nuclei-mediated electron-electron effective interaction $W^{{\text{ph}}}$ as described in Eqs. \ref{['eq:Velph_Gauss_freq_series']}, \ref{['eq:Velph_Gauss_freq_nth']}, and \ref{['eq:convol_phon']}.
  • Figure 4: Average single electron-phonon vertex $\bigl\langle\mkern-1mu g \mkern-1mu \bigr\rangle{}$, given by the superposition of simple $g$, triple $\overset{{{(3)}}}{g}{}$, quintuple $\overset{{{(5)}}}{g}{}$ … phonon vertices with only one external phonon leg, and the other phonon legs paired in loops.
  • Figure 5: Average double electron-phonon vertex $\bigl\langle\mkern-1mu \overset{{{(2)}}}{g} \mkern-1mu \bigr\rangle{}$, given by the superposition of double $\overset{{{(2)}}}{g}{}$, quadrupole $\overset{{{(4)}}}{g}{}$, sixtupole $\overset{{{(6)}}}{g}{}$ … phonon vertices with only one external phonon leg, and the other phonon legs paired in loops.
  • ...and 4 more figures