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EPW: A program for calculating the electron-phonon coupling using maximally localized Wannier functions

Jesse Noffsinger, Feliciano Giustino, Brad D. Malone, Cheol-Hwan Park, Steven G. Louie, Marvin L. Cohen

arXiv:1005.4418v1cond-mat.othercond-mat.mtrl-sci

TL;DR

Accurate first-principles electron-phonon calculations require costly Brillouin-zone sampling of many scattering processes. EPW combines density-functional perturbation theory with maximally localized Wannier functions and generalized Fourier interpolation to generate dense-grid matrix elements, enabling accurate and efficient calculations that agree well with experiment.

  • Problem

    Accurate first-principles electron-phonon coupling requires many Brillouin-zone matrix elements, making brute-force calculations prohibitively time-consuming.

  • Method

    EPW combines density-functional perturbation theory with maximally localized Wannier functions and generalized Fourier interpolation to calculate electron-phonon coupling on fine Brillouin-zone grids.

  • Results

    EPW enables accurate and efficient calculations of total and mode-resolved electron-phonon coupling, with a calculated λ = 1.41 showing very good agreement with experiment.

  • Takeaways & Limitations

    EPW makes dense-grid electron-phonon calculations practical for studying electron-phonon interactions in large and complex systems.

Abstract

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EPW (Electron-Phonon coupling using Wannier functions) is a program written in FORTRAN90 for calculating the electron-phonon coupling in periodic systems using density-functional perturbation theory and maximally-localized Wannier functions. EPW can calculate electron-phonon interaction self-energies, electron-phonon spectral functions, and total as well as mode-resolved electron-phonon coupling strengths. The calculation of the electron-phonon coupling requires a very accurate sampling of electron-phonon scattering processes throughout the Brillouin zone, hence reliable calculations can be prohibitively time-consuming. EPW combines the Kohn-Sham electronic eigenstates and the vibrational eigenmodes provided by the Quantum-ESPRESSO package [1] with the maximally localized Wannier functions provided by the wannier90 package [2] in order to generate electron-phonon matrix elements on arbitrarily dense Brillouin zone grids using a generalized Fourier interpolation. This feature of EPW leads to fast and accurate calculations of the electron-phonon coupling, and enables the study of the electron-phonon coupling in large and complex systems.

PROGRAM SUMMARY · I. INTRODUCTION · II. FUNCTIONALITIES AND TECHNICAL RELEASE

EPW is a GPL-licensed Fortran 90 post-processing program that uses Quantum-ESPRESSO and wannier90 to calculate first-principles electron–phonon coupling. It addresses the prohibitive cost of dense Brillouin-zone sampling by enabling calculations from Wannier-function representations and provides self-energy, spectral-function, and coupling analyses.

  • PROGRAM SUMMARY: EPW is a Fortran 90 program for first-principles electron–phonon coupling, optimized for 1 to 64 processors and distributed under the GNU Public License.The program uses BLAS, LAPACK, MPI, FFTW, and wannier90-related routines.
  • I. INTRODUCTION: Electron–phonon coupling calculations are challenging because accurate Brillouin-zone integrals require matrix elements between electronic states connected by phonon wavevectors [11].Existing packages provide electronic states and phonon properties, but the coupling calculation requires their matrix elements throughout the Brillouin zone.
  • I. INTRODUCTION: Thousands to millions of phonon wavevectors can make brute-force convergence of electron–phonon Brillouin-zone integrals prohibitively expensive.Lattice-dynamical calculations for each phonon wavevector are at least as costly as self-consistent total-energy minimizations.
  • II. FUNCTIONALITIES AND TECHNICAL RELEASE: EPW calculates electron–phonon coupling using maximally localized Wannier functions and operates as a post-processing tool with Quantum-ESPRESSO [1] and wannier90 [2].Electronic states use DFT [12, 13] with planewaves and norm-conserving or Vanderbilt ultrasoft pseudopotentials.
  • II. FUNCTIONALITIES AND TECHNICAL RELEASE: EPW computes phonon and electron self-energies within the Migdal approximation and the transport electron–phonon spectral function α2FT.These outputs support analysis of electron–phonon interaction effects and transport-related spectral properties.
  • II. FUNCTIONALITIES AND TECHNICAL RELEASE: EPW requires Quantum-ESPRESSO phonon dynamical matrices, self-consistent-potential derivatives, electronic wavefunctions and eigenvalues, pseudopotentials, and a runtime-parameter input file.Only phonon wavevectors in the irreducible Brillouin-zone wedge are required for the dynamical matrices and potential derivatives.
  • II. FUNCTIONALITIES AND TECHNICAL RELEASE: Initial electron and phonon grids must be commensurate so EPW can map ψ_mk+q(r) onto ψ_mk′+G(r).Electronic wavefunctions, dynamical matrices, and potential derivatives are generated on uniform grids using Quantum-ESPRESSO.

III. COMPUTATIONAL METHODOLOGY · A. Physical quantities (selfen

EPW computes phonon and electron self-energies, mode-resolved and total electron-phonon coupling, Eliashberg spectral functions, and transport spectral functions from Brillouin-zone quantities. The methodology defines the relevant matrix elements, occupations, weights, velocities, and approximations used in these calculations.

  • A. Physical quantities (selfen: EPW calculates the imaginary phonon self-energy within the Migdal approximation [36, 37].The calculation is formulated using electron-phonon matrix elements and phonon-related quantities.
  • A. Physical quantities (selfen: The phonon self-energy uses electronic wavefunctions, eigenvalues, self-consistent-potential derivatives, Fermi occupations, and spin-normalized k-point weights.The potential derivative is associated with phonon wavevector q, branch ν, and frequency ω_qν.
  • A. Physical quantities (selfen: A common approximation to the phonon self-energy neglects the phonon frequencies ω_qν.This approximation simplifies the expression for the calculated quantity.
  • A. Physical quantities (selfen: EPW defines the electron-phonon coupling strength λ_qν for each phonon mode and wavevector using electronic states and a Dirac delta function.In the double-delta approximation, λ_qν is related to the imaginary part of the phonon self-energy Π′′.
  • A. Physical quantities (selfen: The total electron-phonon coupling λ is the Brillouin-zone average of the mode-resolved strengths λ_qν.The average uses phonon-wavevector weights w_q normalized to 1 in the Brillouin zone.
  • A. Physical quantities (selfen: EPW obtains the Eliashberg spectral function α2F from λ_qν and phonon frequencies, and the transport function α2F_T by replacing λ_qν with λ_T,qν.The transport formulation additionally uses electron velocities v_nk = ∇_k ε_nk.
  • A. Physical quantities (selfen: EPW calculates the real and imaginary parts of the electron self-energy Σ_nk.The expressions include Bose occupation factors n(ω_qν).

B. Calculation of the matrix elements on the coarse Brillouin zone grid (elphon

EPW first computes electron–phonon matrix elements on coarse electron and phonon wavevector grids using standard DFT and DFPT methods. Crystal symmetry reduces these calculations to the irreducible Brillouin-zone wedge, while computed data can be stored for reuse.

  • Calculation on the coarse Brillouin zone grid: EPW’s initial step computes first-principles electron–phonon matrix elements on coarse electron and phonon wavevector grids using standard DFT and DFPT methods.These matrix elements are the inputs for subsequent calculations on dense Brillouin-zone grids.
  • Calculation on the coarse Brillouin zone grid: Phonon dynamical matrices and self-consistent potential variations are calculated only in the irreducible Brillouin-zone wedge, then generated elsewhere using crystal symmetry.This reduces work because computing these quantities is generally the most time-consuming part of the electron–phonon calculation.
  • Calculation on the coarse Brillouin zone grid: EPW can write coarse-grid electron–phonon matrix elements and dynamical matrices to prefix.epb files, allowing later executions to reuse them without recalculation.The stored data are read from disk through an input-file option.

C. Interface with wannier90 (pw2wan90epw)

EPW embeds wannier90 directly into its workflow, reducing execution to a single executable while retaining access to wannier90’s MLWF-generation and analysis features [2]. The resulting gauge matrix transforms Bloch functions into maximally localized Wannier functions for EPW calculations.

  • Interface with wannier90 (pw2wan90epw): wannier90 generates MLWFs by minimizing the spread of the Berry-phase position operator through a unitary transform [16].Its standalone workflow uses first-principles data, a starting-guess and crystal-structure file, and a runtime input file [2].
  • Interface with wannier90 (pw2wan90epw): EPW embeds wannier90 through its run library routine, so users run one executable that communicates directly with wannier90.This avoids manually coordinating multiple programs and intermediate files between wannier90 and Quantum-ESPRESSO.
  • Interface with wannier90 (pw2wan90epw): The interface accepts additional wannier90 input parameters, preserving access to features such as plotting bandstructures or MLWFs [2].
  • Interface with wannier90 (pw2wan90epw): The gauge matrix U^k_mn produced by wannier90 maps Bloch eigenstates to MLWFs and is written to disk for reuse in subsequent EPW executions.Its dimensions are the number of Bloch bands by MLWFs by electronic wavevectors on the coarse Brillouin-zone grid.

D. Transformation from the Bloch representation on the coarse Brillouin zone grid to the MLWF

EPW transforms coarse-grid Bloch electron, phonon, and electron–phonon quantities into localized Wannier representations for interpolation and subsequent calculations. These transformations cover the electronic Hamiltonian, dynamical matrix, and electron–phonon matrix elements, with derivations in Ref. [15].

  • Transformation overview: EPW transforms the electronic Hamiltonian, phonon dynamical matrix, and electron–phonon matrix elements from coarse-grid Bloch representations into Wannier representations [15].The transformations are applied after calculating matrix elements on coarse electron and phonon Brillouin-zone grids.
  • Electronic Hamiltonian: The electronic Hamiltonian is transformed into the MLWF representation, whose matrix elements reflect the overlap of MLWFs centered in different unit cells.The weights used in the transformation are normalized to 1.
  • Phonon dynamical matrix: The phonon dynamical matrix is transformed into a localized real-space representation using orthonormal dynamical-matrix eigenvectors and mass-scaled force constants.The phonon eigenvectors enter through the dynamical-matrix transformation.
  • Electron–phonon matrix elements: Electron–phonon matrix elements are transformed into the MLWF representation using phonon eigenvectors scaled by atomic masses [15].EPW also checks the spatial decay of the resulting real-space quantities as functions of electron and phonon lattice-cell indices.
  • Implementation: Wannier-representation data can be written to files and reloaded so subsequent calculations restart without repeating earlier computational steps.A run-time option combines all Wannier data into one file.

E. Tranformation from the MLWF representation to the Bloch representation on the fine Brilluin zone grid · F. Summary of the tasks executed by EPW

EPW transforms Wannier-represented Hamiltonians, dynamical matrices, and electron–phonon matrix elements into Bloch quantities on arbitrarily fine electron and phonon grids. The preceding computational workflow is then summarized schematically.

  • E. Tranformation from the MLWF representation to the Bloch representation on the fine Brilluin zone grid: The interpolation accuracy depends on the spatial localization of the MLWFs and phonon perturbations.MLWFs are typically localized within a few Å [39, 40], whereas phonon-perturbation localization depends on dielectric screening and must be verified for each system before interpolation [15].
  • E. Tranformation from the MLWF representation to the Bloch representation on the fine Brilluin zone grid: The Hamiltonian, dynamical matrix, and electron–phonon matrix elements are truncated outside real-space supercells containing Nk and Nq unit cells for electrons and phonons, respectively.Nk and Nq equal the numbers of points in the corresponding coarse Brillouin-zone meshes.
  • E. Tranformation from the MLWF representation to the Bloch representation on the fine Brilluin zone grid: After truncation, EPW Fourier-transforms these quantities back into the Bloch representation at arbitrary electron and phonon wavevectors by inverting Eqs. (11), (12), and (13) [15].This produces the fine-grid electronic eigenstates, phonon frequencies, and electron–phonon matrix elements needed for subsequent calculations.
  • E. Tranformation from the MLWF representation to the Bloch representation on the fine Brilluin zone grid: EPW enables electron–phonon matrix elements to be calculated on extremely fine Brillouin-zone grids specified by the user.The approach is analogous to Fourier interpolation of time series and has also been used for Fermi-surface and anomalous-Hall-effect studies [18, 20, 41].
  • E. Tranformation from the MLWF representation to the Bloch representation on the fine Brilluin zone grid: The fine-grid Hamiltonian, dynamical matrix, and electron–phonon matrix elements are used to compute the physical quantities described in Sec. III A.This is the final computational stage following the transformation to fine electron and phonon wavevector grids.
  • F. Summary of the tasks executed by EPW: The computational steps executed by EPW are summarized schematically in the following section.The summary consolidates the workflow described in the preceding sections.

1. Electron eigenstates and eigenvalues are read from disk, · IV. PARALLELIZATION

EPW constructs electron–phonon quantities by combining coarse-grid Bloch calculations, MLWF transformations, and Fourier interpolation onto arbitrarily dense grids. Its parallelization distributes wavevector-dependent work across processor pools, with efficiency demonstrated for hole-doped SiC.

  • 1. Electron eigenstates and eigenvalues are read from disk,: The workflow computes wannier90 overlap matrices A_mn and M_mn, calls wannier90 as a library, and stores the resulting MLWF localization matrix on disk.Most wannier90 inputs, including lattice and reciprocal vectors, are passed through Quantum-ESPRESSO and EPW.
  • 1. Electron eigenstates and eigenvalues are read from disk,: EPW computes electron–phonon matrix elements on coarse electron and phonon wavevector grids before transforming them from the Bloch to the MLWF representation.The Hamiltonian and dynamical matrix undergo the same Bloch-to-MLWF transformation.
  • 1. Electron eigenstates and eigenvalues are read from disk,: EPW Fourier-transforms the MLWF-represented Hamiltonian, dynamical matrix, and electron–phonon matrix elements onto arbitrarily dense electron and phonon wavevector grids.This enables subsequent processing of electronic eigenvalues, phonon frequencies, and matrix elements for quantities such as self-energies and coupling strengths.
  • IV. PARALLELIZATION: EPW distributes electronic wavevectors across processor pools and can split parallel work over electron or phonon wavevectors.Within pools, reciprocal-space G-vectors for wavefunctions are also distributed when multiple processors share a pool.
  • IV. PARALLELIZATION: Overlap-matrix elements are computed independently across coarse-grid k-point pools, so this step’s execution time scales inversely with the number of pools.wannier90 itself runs serially on each processor through a library call, while MLWF determination is negligible compared with coarse-grid matrix-element computation.
  • IV. PARALLELIZATION: Coarse-grid electron–phonon matrix elements are computed sequentially by phonon wavevector, with electron wavevectors distributed across processor pools.Fine-grid calculations parallelize over electronic or phonon wavevectors according to the final quantity, such as phonon or electron self-energy.
  • IV. PARALLELIZATION: Figure 1 evaluates EPW parallelization efficiency for hole-doped SiC across MLWF generation, coarse-grid matrix elements, fine-grid interpolation, phonon self-energy, and the complete computation.The complete computation includes initialization and intermediate I/O.

A. Calculations “on the fly” · V. EXAMPLES · A. Lead

EPW supports memory-saving on-the-fly electron-phonon self-energy calculations and demonstrates accurate bulk-Pb coupling results that agree favorably with experiment. The examples span strong-coupling metals and representative semiconductors.

  • A. Calculations “on the fly”: On-the-fly mode computes self-energies as matrix elements are generated and overwrites quantities instead of storing matrix elements on disk.This mode avoids the storage burden associated with retaining all fine-grid electron-phonon matrix elements.
  • V. EXAMPLES: EPW illustrates its capabilities using Pb, graphene, and boron-doped silicon carbide as prototypical metallic, zero-gap, and wide-gap systems.Pb is a strong-coupling superconductor, graphene has linear quasiparticle dispersions near the Fermi level, and doped silicon carbide becomes a low-temperature superconductor.
  • A. Lead: The Pb calculation used LDA with Quantum-ESPRESSO, wannier90, and EPW, plus a norm-conserving scalar-relativistic pseudopotential containing four valence electrons.The study compares the calculated electron-phonon coupling with experimental data.
  • A. Lead: Four symmetrically localized sp3-like Wannier states with 2.17 Å spread reconstructed Pb’s electronic structure near the Fermi level.The MLWF representation was used to examine the spatial decay of the electron Hamiltonian, phonon dynamical matrix, and electron-phonon matrix elements.
  • A. Lead: λ = 1.41 for Pb, while calculated Tc = 4–6 K compares favorably with the experimental 7.2 K.The calculation used 106 k-points, 8000 q-points, 30 meV smearing, the modified McMillan equation [47], and µ∗ between 0.10 and 0.15.
  • A. Lead: Generalized Fourier interpolation transformed the Pb Hamiltonian, dynamical matrix, and electron-phonon matrix elements from the MLWF to Bloch representation for linewidth and coupling calculations.The Eliashberg spectral function from EPW is compared with the result from Ref..

B. Graphene · C. Silicon carbide

The section demonstrates EPW calculations for graphene and 4% hole-doped cubic silicon carbide, combining density-functional inputs with Wannier representations to evaluate electron-phonon effects. For SiC, the calculation yields λ = 0.34 and Tc = 1 K, consistent with experiment at Tc = 1.4 K [52].

  • B. Graphene: Graphene is motivated by its Dirac-like electronic states near the Fermi level and the need to characterize electron-phonon contributions to band-structure renormalization.These effects have been investigated using angle-resolved photoemission experiments.
  • B. Graphene: The graphene calculation used LDA, a norm-conserving carbon pseudopotential, a 60 Ry plane-wave cutoff, and 8 Å of vacuum between periodic replicas.The calculation was performed for a freestanding graphene sheet [23].
  • B. Graphene: Graphene’s electronic structure was represented with bonding, pz-like, and out-of-plane s-like maximally localized Wannier functions.The corresponding spatial spreads were 0.565 Å, 0.782 Å, and 1.726 Å, respectively.
  • B. Graphene: The graphene study evaluated the real and imaginary electron self-energy and the electron-phonon coupling parameter as functions of Fermi energy.The real and imaginary self-energies near the Dirac point and λ versus doping are presented in Fig. 5.
  • C. Silicon carbide: The SiC test used a rigid-band model with 4% hole doping, 60 Ry plane-wave cutoff, and an 8×8×8 coarse electron and phonon mesh.The relaxed lattice parameter was 4.34 Å, close to the experimental 4.36 Å.
  • C. Silicon carbide: λ = 0.34 and Tc = 1 K were obtained for 4% hole-doped cubic SiC, agreeing with the experimental Tc = 1.4 K [52].The result used 250,000 k-points, 8,000 q-points, and μ⋆ = 0.1.
  • C. Silicon carbide: For SiC, the MLWF representation captures the spatial decay of the electron Hamiltonian, phonon dynamical matrix, and electron-phonon matrix elements.These decay properties were shown in Fig. 6.

VI. CONCLUSION

EPW is introduced as a first-principles code using density functional perturbation theory and maximally localized Wannier functions to calculate electron–phonon properties accurately and efficiently. It is distributed under the GNU General Public License, with planned extensions to additional superconductivity and optical-response theories.

  • VI. CONCLUSION: EPW calculates mode-resolved and total electron–phonon coupling strengths, phonon linewidths, and electron linewidths with high accuracy and efficiency.The code uses density functional perturbation theory and maximally localized Wannier functions.
  • VI. CONCLUSION: EPW is available from the project website under the GNU General Public License.The passage identifies the distribution website as http://epw.org.uk.
  • VI. CONCLUSION: Planned extensions include anisotropic Eliashberg theory, density functional theory for superconductors, and phonon-assisted optical responses.These extensions are described as future implementation plans for EPW.

VII. APPENDIX · A. Specifying a unique gauge for the electronic wavefunctions (setphases)

The appendix addresses the machine dependence of electron–phonon matrix elements by specifying a unique gauge for electronic wavefunctions in EPW. This enables meaningful use of these matrix elements in applications such as phonon sidebands and phonon-assisted Auger recombination.

  • A. Specifying a unique gauge for the electronic wavefunctions (setphases): Electron–phonon matrix elements are machine dependent because nondegenerate wavefunctions have arbitrary phases and degenerate wavefunctions can mix through unitary transformations.
  • A. Specifying a unique gauge for the electronic wavefunctions (setphases): A unique gauge is needed when using EPW-generated electron–phonon matrix elements outside EPW.
  • A. Specifying a unique gauge for the electronic wavefunctions (setphases): Such external uses include studying phonon sidebands in excitonic spectra.
  • A. Specifying a unique gauge for the electronic wavefunctions (setphases): Another application is phonon-assisted Auger recombination.
  • A. Specifying a unique gauge for the electronic wavefunctions (setphases): EPW identifies subsets of degenerate wavefunctions at each wavevector on the coarse mesh before fixing their gauge.
  • A. Specifying a unique gauge for the electronic wavefunctions (setphases): It artificially lifts degeneracies by diagonalizing each degenerate subset under a small external perturbation, chosen in practice as a nonlinear combination of data from prefix.dvscf files.
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