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EPW: Electron-phonon coupling, transport and superconducting properties using maximally localized Wannier functions

Samuel Poncé, Elena R. Margine, Carla Verdi, Feliciano Giustino

arXiv:1604.03525v2cond-mat.mtrl-sci

TL;DR

EPW v4 addresses the calculation of electron–phonon and related electronic and phononic properties, using self-energy formulations and electron-velocity-based transport calculations. It adds anisotropic superconductivity, polar-material interpolation, transport capabilities, and support for spin–orbit coupling, while demonstrating applications including MgB2 superconductivity and Pb transport.

  • Problem

    EPW computes electron and phonon self-energies, linewidths, scattering rates, coupling strengths, transport quantities, and superconducting properties within Migdal–Eliashberg theory.

  • Method

    The code formulates electron and phonon self-energies from phonon-induced potential derivatives and electronic wavefunctions, and evaluates transport through electronic velocities and the Eliashberg transport coupling function.

  • Results

    EPW v4 demonstrates calculations of superconducting gap behavior in MgB2 and transport and electron–phonon coupling convergence in Pb, alongside validation against silicon band and phonon data.

  • Takeaways & Limitations

    The three principal new capabilities are anisotropic superconducting properties, Wannier interpolation for polar materials, and electronic-transport calculations using electronic velocities.

  • Takeaways & Limitations

    Lifetimes are not directly implemented and must be computed as postprocessing, while one velocity expression is valid only for norm-conserving pseudopotentials.

Abstract

from arXiv · show

The EPW (Electron-Phonon coupling using Wannier functions) software is a Fortran90 code that uses density-functional perturbation theory and maximally localized Wannier functions for computing electron-phonon couplings and related properties in solids accurately and efficiently. The EPW v4 program can be used to compute electron and phonon self-energies, linewidths, electron-phonon scattering rates, electron-phonon coupling strengths, transport spectral functions, electronic velocities, resistivity, anisotropic superconducting gaps and spectral functions within the Migdal-Eliashberg theory. The code now supports spin-orbit coupling, time-reversal symmetry in non-centrosymmetric crystals, polar materials, and $\mathbf{k}$ and $\mathbf{q}$-point parallelization. Considerable effort was dedicated to optimization and parallelization, achieving almost a ten times speedup with respect to previous releases. A computer test farm was implemented to ensure stability and portability of the code on the most popular compilers and architectures. Since April 2016, version 4 of the EPW code is fully integrated in and distributed with the Quantum ESPRESSO package, and can be downloaded through QE-forge at http://qe-forge.org/gf/project/q-e.

NEW VERSION PROGRAM SUMMARY

EPW v4 is a Fortran90 code for computing electron–phonon and related superconducting and transport properties using density-functional perturbation theory and maximally localized Wannier functions. The release adds new functionality and code optimization while targeting portable parallel execution.

  • EPW computes electron and phonon self-energies, linewidths, scattering rates, coupling strengths, transport spectral functions, velocities, resistivity, and Migdal-Eliashberg superconducting properties.
  • EPW v4 supersedes the previous version and was submitted to Computer Physics Communications in July 2016.
  • The code relies on density-functional perturbation theory and maximally localized Wannier functions.
  • The new version introduces additional features and optimizes the code.
  • EPW typically runs for several hours on several tens of processors.

1. Introduction

Electron–phonon calculations require finely sampled electron and phonon wavevectors, creating a computational challenge for converged Brillouin-zone integrals. EPW addresses this challenge by combining maximally localized Wannier functions with generalized Fourier interpolation.

  • Electron–phonon coupling contributes to electronic, phononic, optical, superconducting, transport, and excitation-energy phenomena.
  • DFT provides ground-state electronic properties, while DFPT usually supplies phonon frequencies and electron–phonon matrix elements.
  • Converged Brillouin-zone integrals require very fine sampling of electron and phonon wavevectors.
  • Several approaches have been investigated to make fine-grid electron–phonon calculations computationally feasible.
  • EPW computes fine-grid electron–phonon matrix elements using maximally localized Wannier functions and generalized Fourier interpolation.

2. Functionalities and technical release

EPW is a freely available Fortran90 code integrated with Quantum ESPRESSO that uses DFPT and Wannier functions to evaluate electron–phonon properties on fine grids. Version 4 adds support for SOC, time-reversal symmetry, polar materials, superconductivity, transport, and improved portability.

  • EPW computes electron–phonon-related properties on very fine electron and phonon wavevector grids using DFPT and MLWFs.
  • The software is integrated into Quantum ESPRESSO and uses wannier90 as an internal library.
  • Version 4 implements spin-orbit coupling and time-reversal symmetry for noncentrosymmetric crystals.
  • Wannier interpolation of the polar electron–phonon vertex captures the long-wavelength singularity and enables calculations for semiconductors and insulators.
  • New superconducting and transport capabilities include anisotropic Eliashberg quantities, superconducting gaps, Ziman’s resistivity formula, and electronic velocities.
  • A test farm was established to ensure portability across many architectures and compilers.

3. Electron and phonon self-energies and electronphonon coupling strength

EPW formulates electron–phonon matrix elements and self-energies from DFPT quantities over Brillouin-zone integrations. These quantities provide linewidths, spectral functions, coupling strengths, and Fermi-surface nesting information, with approximations specified for different material classes.

  • DFPT provides first-order electron–phonon matrix elements describing scattering between Kohn–Sham states.
  • The matrix-element formulation uses phonon wavevector, branch, frequency, electronic wavefunctions, and eigenvalues.
  • Electron and phonon self-energies are evaluated at temperature T over the Brillouin zone using electronic occupations and phonon populations.
  • The code can calculate a Fermi-surface nesting function, which is non-zero for wavevectors connecting two Fermi-surface points.
  • Electron and phonon linewidths are obtained from the imaginary parts of the corresponding self-energies.
  • EPW computes spectral functions, mode-resolved coupling strengths, and isotropic Eliashberg spectral functions.

4. Electron-phonon coupling in polar materials

EPW treats the long-range divergence of electron-phonon couplings in polar materials by separating short- and long-range contributions before Wannier interpolation. The long-range term is then restored at arbitrary wavevectors.

  • Motivation: Polar-material electron-phonon matrix elements diverge as 1/|q| when |q| approaches zero because of longitudinal optical modes.The divergence motivates separating the coupling into short- and long-range parts.
  • Long-range correction: The long-range correction depends on Born effective charges, vibrational eigendisplacements, and the electronic macroscopic dielectric tensor.These quantities enter the polar coupling expression used to represent the long-range interaction.
  • Interpolation strategy: Wannier interpolation can directly treat only the short-range component because maximally localized Wannier functions are intrinsically localized.The full coupling is available from linear-response calculations, but its long-range part is not directly handled by standard Wannier interpolation.
  • Interpolation strategy: EPW computes the full coupling on a coarse grid, subtracts the long-range term, and interpolates the resulting short-range component.This procedure isolates the part compatible with standard Wannier interpolation.
  • Interpolation strategy: EPW adds the long-range component back after interpolating the short-range coupling, producing values at arbitrary k- and q-points.The required overlap matrices and Wannier rotation matrices can also be evaluated or interpolated at arbitrary points.

5. Phonon-mediated superconductors

EPW implements semiempirical and first-principles approaches for phonon-mediated superconductivity, including anisotropic Migdal-Eliashberg calculations. Its equations use electron-phonon and screened Coulomb interactions under stated approximations to obtain gaps and superconducting observables.

  • Approaches: EPW implements both McMillan-type semiempirical methods and first-principles Migdal-Eliashberg methods for phonon-mediated superconductivity.These are two of the three major approaches identified for ab-initio superconducting-property calculations.
  • Migdal-Eliashberg theory: The anomalous Green’s function represents the Cooper-pair amplitude and is nonzero below the critical temperature.The superconducting state is formulated using generalized Nambu-Gor’kov Green’s functions.
  • Self-consistent solution: The coupled nonlinear equations determine the renormalization function, energy shift, and order parameter, with the electron number equation fixing the Fermi energy.The superconducting gap is obtained as the ratio of the order parameter to the renormalization function.
  • Migdal-Eliashberg theory: The Migdal-Eliashberg pairing self-energy combines electron-phonon and screened Coulomb interactions.The phonon contribution uses a dressed phonon propagator, while the Coulomb term is dynamically screened.
  • Anisotropy and observables: The framework supports Fermi-surface-averaged and fully anisotropic Eliashberg spectral functions to account for multiband effects and reduced dimensionality.The Fermi-surface averaging is motivated by phonon energies being usually much smaller than the Fermi energy.
  • Anisotropy and observables: EPW can compute superconducting observables including the gap, specific heat, and tunneling density of states.The specific heat is obtained from the free-energy difference, while the tunneling density of states follows from the superconducting solution.
  • Self-consistent solution: The critical temperature is the highest temperature at which the coupled equations have nontrivial superconducting-gap solutions.The equations are solved self-consistently after applying standard approximations, including a Coulomb pseudopotential treatment.
  • Real-axis properties: EPW solves the Eliashberg equations on the imaginary axis and obtains real-axis gaps by analytic continuation using Padé approximants or an exact iterative procedure.Real-axis solutions are needed for properties such as tunneling current and heat capacity, while direct real-axis evaluation is computationally demanding.

6. Transport properties

EPW computes electronic transport in metals and doped semiconductors using the Boltzmann transport equation and the Ziman resistivity approximation. The transport formulation uses Eliashberg coupling functions and semiclassical electron velocities, while anisotropic extensions remain future work.

  • Transport formulation: Electronic transport in metals and doped semiconductors is calculated from the Boltzmann transport equation using Ziman’s resistivity formula.Ziman’s formula is described as the simplest and most popular approximation considered here.
  • Transport formulation: The transport resistivity formulation uses an averaged Eliashberg coupling function and introduces an Eliashberg transport coupling function.Mode-resolved coupling strengths are defined analogously to the standard electron-phonon coupling strength.
  • Velocity treatment: The electron velocity is defined as the band-energy derivative v_nk = ∂ε_nk/∂k in the transport expression.A modified expression conserving the norm of the velocity term can also be used.
  • Limitations and extensions: The transport formula uses a semiclassical approximation in which velocities are expectation values of single-electron velocities.A more general treatment would use velocity matrix elements.
  • Limitations and extensions: Anisotropic transport extensions can be implemented using the anisotropic Eliashberg function but are left for future work.The extension depends on availability of the anisotropic electron-phonon coupling information.
  • Velocity treatment: EPW currently supports electron-velocity calculations by finite differences and analytic derivatives in the local approximation.Wannier-based perturbative derivatives were identified as a future integration target.

7. Electronic velocities

EPW evaluates electronic velocities through the periodic velocity operator and offers a local approximation based on momentum and Hamiltonian terms. This approximation is limited for nonlocal pseudopotentials, whose omitted contributions require additional matrix elements or derivative methods.

  • Velocity operator: The periodic velocity operator is represented as the commutator between the Hamiltonian and position operator, with matrix elements including momentum and nonlocal-pseudopotential terms.The Cartesian direction is indexed by α = 1, 2, 3.
  • Local approximation: The local approximation neglects the nonlocal-pseudopotential contribution to the velocity matrix elements.Within this approximation, the velocity expression can be expanded using plane waves.
  • Limitations: The resulting local-approximation velocity expression is valid only for norm-conserving pseudopotentials.This condition defines the immediate applicability boundary of the implemented expression.
  • Limitations: Beyond the local approximation, EPW would need matrix elements involving the nonlocal pseudopotential and position operator, or analytic first-order Hamiltonian derivatives.These alternatives were identified as possible future treatments.

8. Parallelization and Speedup

EPW 4 accelerates electron–phonon calculations through code optimization, Wannier-based workflow improvements, and k- and q-point MPI parallelization. In SiC tests, the EPW portion was about ten times faster sequentially, while overall speedup and scaling improved substantially over EPW 3.

  • Parallelization and Speedup: 294% global speedup separated EPW 4 from the previous code, with sensitive physical quantities differing by less than 6%.The reported quantities include phonon frequencies, self-energies, electron-phonon coupling strengths, and Eliashberg spectral functions.
  • Parallelization and Speedup: A ∼10 times speedup was obtained for the EPW portion in sequential SiC calculations compared with the previous version.The workflow comprised matrix-element construction, Wannierization, and Wannier interpolation, each of which can serve as a restart point.
  • Parallelization and Speedup: EPW 4 used k- and q-point MPI parallelization, while G-vector parallelization remained planned for a future release.The scalability study used denser coarse and fine k-point grids to expose k-point parallelization improvements.
  • Parallelization and Speedup: At 128 processors, interpolation speedup reached 88% of ideal for EPW 4 versus 51% for the previous version.The interpolation maps real-space quantities onto fine k- and q-point grids and received dedicated scaling work.

9. Buildbot-based test farm

EPW is maintained through a Buildbot test farm covering multiple hardware platforms, operating systems, compilers, MPI implementations, and numerical libraries. The suite improves code stability and currently covers 93% of routines and 61% of software blocks.

  • Buildbot-based test farm: The test farm supports continuous maintenance and portability checks across compilers, operating systems, MPI options, and optimization libraries.Buildbot maintains the test infrastructure and exposes current code-coverage information through the EPW website.
  • Buildbot-based test farm: The Buildbot test farm runs EPW across four physical hardware systems with differing processors and memory capacities.The systems range from Intel Xeon machines with 16 Gb of RAM to an AMD Opteron system with 2 Gb and an Intel Core 2 Duo with 4 Gb.
  • Buildbot-based test farm: Ubuntu, CentOS, openSUSE, and Scientific Linux are tested with gcc, Intel, PGI, and NAG compilers.The farm also checks openmpi, mpich, Intel MPI, and mvapich implementations, plus internal, MKL, and FFTW libraries.
  • Buildbot-based test farm: Testing and debugging with the Fortran NAG compiler greatly improved EPW stability.The compiler and architecture combinations are part of the continuously maintained test-farm configuration.
  • Buildbot-based test farm: 10 tests cover 93% of routines and 61% of the 25,600 software blocks.Incomplete coverage is partly attributed to features still in development, with additional tests planned.

10. Examples

The examples demonstrate EPW’s convergence, electron–phonon, transport, polar-material, and superconductivity capabilities across several materials. Results include physically interpretable linewidth and self-energy features, improved agreement with experiment from spin–orbit coupling, and converged MgB2 superconducting properties.

  • 10.1. Spectral functions and linewidths of B-doped diamond: 10^6 random k/q-points accurately describe the Brillouin-zone integrals for B-doped diamond linewidths.This conclusion holds across the tested samplings and Gaussian broadenings.
  • 10.1. Spectral functions and linewidths of B-doped diamond: The electron linewidth in B-doped diamond is parabolic overall with two Fermi-wavevector dips, while phonon linewidths vanish for |q| > 2kF and at the zone center.The electron features follow the density of states and energy-conservation selection rules; large-q phonons cannot generate electron-hole pairs.
  • 10.1. Spectral functions and linewidths of B-doped diamond: The B-doped diamond electron self-energy produces long Fermi-level lifetimes, phonon-energy kinks, and a sign-changing energy renormalization.The renormalization is negative below the phonon energy and positive above it.
  • 10.3. Spectral function and electronic resistivity of Pb with and without spin-orbit coupling: Including spin–orbit coupling significantly improves the calculated Pb resistivity’s agreement with experiment, while SOC also brings phonon frequencies closer to 100 K neutron-scattering data.The resistivity uses Ziman’s formula and the Eliashberg transport function.
  • 10.5. Superconductivity in MgB2: For MgB2, EPW finds dominant and secondary Eliashberg-function peaks near 60 and 86 meV, with total isotropic coupling λ = 0.75.The coupling agrees with prior values ranging from 0.71 to 0.78; the calculated gap structure shifts and flattens toward zero as temperature rises.

11. Conclusions

EPW v4 expands electron-phonon calculations with new superconductivity, polar-material, transport, symmetry, and performance capabilities. It is integrated into Quantum ESPRESSO and supported by testing infrastructure.

  • EPW is fully integrated into Quantum ESPRESSO while retaining a dedicated website and user-support forum.
  • The code supports spin-orbit coupling and time-reversal symmetry, with optimization, parallelization, and a test farm improving stability and portability.
  • EPW computes anisotropic superconducting properties within Migdal-Eliashberg theory, interpolates electron-phonon interactions in polar materials, and evaluates electronic transport through electronic velocities.
  • Five physically relevant examples demonstrate the new features, and four are distributed as EPW tutorials.
  • Electronic mobilities from refined Boltzmann or Kubo theories, exact velocities, Coulomb-potential evaluation, and additional parallelization capabilities remain planned for a future release.

13. Appendix: Time-reversal and separable pseudopotential in EPW

The appendix describes symmetry-based and pseudopotential-specific extensions used to construct electron-phonon matrix elements, then develops dimensionless BCS gap equations and their fitting limits.

  • Time-reversal symmetry: Time-reversal symmetry in non-centrosymmetric crystals reduces the number of irreducible q-points required on the coarse grid.
  • Time-reversal symmetry: The electron-phonon matrix element for momentum transfer Sq is obtained by applying crystal symmetries to the screened potential and ionic pseudopotential.
  • Separable pseudopotentials: For separable Kleinman-Bylander pseudopotentials, plane-wave completeness enables the rotated matrix element to be rewritten in a form implemented using time reversal.
  • BCS gap model: The superconducting gap ratio is expressed using dimensionless variables and solved by fixing the temperature ratio τ and determining the gap ratio δ.
  • BCS gap model: The BCS gap-ratio equation is expected to apply to weak-coupling superconductors with λ ≤ 0.3 and has no solution near τ = 1 for MgB2.
  • BCS gap model: For MgB2, λ = 0.75 and the fitted low-τ exponent is p = 3.3.
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