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

Abstract The EPW ( E lectron- P honon coupling using W annier 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 k and 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 . Program summary Program title: EPW Catalogue identifier: AEHA_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEHA_v2_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: GNU General Public Licence 3 No. of lines in distributed program, including test data, etc.: 1635099 No. of bytes in distributed program, including test data, etc.: 22533187 Distribution format: tar.gz Programming language: Fortran 90, MPI. Computer: Non-specific. Operating system: Unix/Linux. RAM: Typically 2GB/core Classification: 7.3, 7.8, 7.9. External routines: LAPACK, BLAS, MPI, FFTW, Quantum- ESPRESSO package [1] Does the new version supersede the previous version?: Yes Nature of problem: Calculation of 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. Solution method: The code relies on density-functional perturbation theory and maximally localized Wannier functions. Reasons for new version: New features (listed in the paper) and optimization of the code. Summary of revisions: Recent developments and new functionalities are described in Section 2 of the paper. Running time: Up to several hours on several tens of processors. References: [1] P. Giannozzi, et al., J. Phys. Condens. Matter 21 (2009), 395502, http://www.quantum-espresso.org/ .

[1]  P. B. Allen Fermi-surface harmonics: A general method for nonspherical problems. Application to Boltzmann and Eliashberg equations , 1976 .

[2]  Holcomb Finite-temperature real-energy-axis solutions of the isotropic Eliashberg integral equations. , 1996, Physical Review B (Condensed Matter).

[3]  M. Calandra,et al.  Wannier interpolation of the electron-phonon matrix elements in polar semiconductors: Polar-optical coupling in GaAs , 2015, 1508.06172.

[4]  S. Louie,et al.  Electron-phonon interaction via electronic and lattice Wannier functions: superconductivity in boron-doped diamond reexamined. , 2007, Physical review letters.

[5]  Marsiglio,et al.  Iterative analytic continuation of the electron self-energy to the real axis. , 1988, Physical review. B, Condensed matter.

[6]  Read,et al.  Calculation of optical matrix elements with nonlocal pseudopotentials. , 1991, Physical review. B, Condensed matter.

[7]  J. Ziman,et al.  In: Electrons and Phonons , 1961 .

[8]  S. Louie,et al.  Phonon-assisted optical absorption in silicon from first principles. , 2012, Physical review letters.

[9]  D. Fruchart,et al.  Evidence for two superconducting energy gaps in MgB(2) by point-contact spectroscopy. , 2001, Physical review letters.

[10]  H. J. Vidberg,et al.  Solving the Eliashberg equations by means ofN-point Padé approximants , 1977 .

[11]  J. Carbotte,et al.  Properties of boson-exchange superconductors , 1990 .

[12]  H. Fröhlich Electrons in lattice fields , 1954 .

[13]  S. Louie,et al.  Electron-phonon interaction using Wannier functions , 2007 .

[14]  S. Poncé,et al.  Many-Body Effects on the Zero-Point Renormalization of the Band Structure , 2014 .

[15]  Leonard Kleinman,et al.  Efficacious Form for Model Pseudopotentials , 1982 .

[16]  Manuel Cardona,et al.  Theory of the temperature dependence of the direct gap of germanium , 1981 .

[17]  C. R. Leavens,et al.  Extension of the N-point padé approximants solution of the eliashberg equations to T∼Tc , 1985 .

[18]  Fujio Izumi,et al.  VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data , 2011 .

[19]  Kurt Stokbro,et al.  First-principles method for electron-phonon coupling and electron mobility: Applications to two-dimensional materials , 2015, 1511.02045.

[20]  J. Nagamatsu,et al.  Superconductivity at 39 K in Magnesium Diboride. , 2001 .

[21]  Francesco Mauri,et al.  Nonlocal pseudopotentials and magnetic fields. , 2003, Physical review letters.

[22]  K P Bohnen,et al.  Phonon dispersion and electron-phonon coupling in MgB2 and AlB2. , 2001, Physical review letters.

[23]  Ab initiotheory of superconductivity in a magnetic field. II. Numerical solution , 2015, 1503.01014.

[24]  M. Calandra,et al.  Universal increase in the superconducting critical temperature of two-dimensional semiconductors at low doping by the electron-electron interaction. , 2014, Physical review letters.

[25]  G. Nilsson,et al.  Study of the Homology between Silicon and Germanium by Thermal-Neutron Spectrometry , 1972 .

[26]  Bo Qiu,et al.  Significant reduction of lattice thermal conductivity by the electron-phonon interaction in silicon with high carrier concentrations: a first-principles study. , 2015, Physical review letters.

[27]  G. Profeta,et al.  Adiabatic and nonadiabatic phonon dispersion in a Wannier function approach , 2010, 1007.2098.

[28]  R. Dynes,et al.  Transition temperature of strong-coupled superconductors reanalyzed , 1975 .

[29]  K. Yamashita,et al.  Ab Initio Study of Temperature and Pressure Dependence of Energy and Phonon-Induced Dephasing of Electronic Excitations in CdSe and PbSe Quantum Dots† , 2008 .

[30]  E K U Gross,et al.  Superconducting properties of MgB2 from first principles. , 2005, Physical review letters.

[31]  Claudia Ambrosch-Draxl,et al.  Wannier interpolation scheme for phonon-induced potentials: Application to bulk MgB 2 , W, and the ( 1 × 1 ) H-covered W(110) surface , 2008 .

[32]  J. Schrieffer,et al.  STRONG-COUPLING SUPERCONDUCTIVITY. I , 1966 .

[33]  R. Fisher,et al.  Specific heat of Mg11B2: evidence for a second energy gap. , 2001, Physical review letters.

[34]  A. Maradudin,et al.  Symmetry Properties of the Normal Vibrations of a Crystal , 1968 .

[35]  K. Novoselov,et al.  Breakdown of the adiabatic Born-Oppenheimer approximation in graphene. , 2007, Nature materials.

[36]  E. Kioupakis,et al.  Auger recombination and free-carrier absorption in nitrides from first principles , 2010 .

[37]  J. Carbotte,et al.  Aspects of optical properties in conventional and oxide superconductors , 1997 .

[38]  W. Kohn,et al.  Self-Consistent Equations Including Exchange and Correlation Effects , 1965 .

[39]  Electron-phonon coupling and phonon self-energy in MgB 2 : Interpretation of MgB 2 Raman spectra , 2004, cond-mat/0406072.

[40]  Fred J. Hickernell,et al.  Algorithm 823: Implementing scrambled digital sequences , 2003, TOMS.

[41]  J Kortus,et al.  Beyond Eliashberg superconductivity in MgB2: anharmonicity, two-phonon scattering, and multiple gaps. , 2001, Physical review letters.

[42]  K. Cheng Theory of Superconductivity , 1948, Nature.

[43]  M. Cardona,et al.  Phonon dispersion curves in wurtzite-structure GaN determined by inelastic x-ray scattering. , 2001, Physical review letters.

[44]  G. C. Asomba,et al.  The effect of interband interactions of phonon and charge fluctuation on the superconducting parameters of MgB2 , 2010, 1706.06668.

[45]  E. R. Margine,et al.  Two-gap superconductivity in heavily n-doped graphene: Ab initio Migdal-Eliashberg theory , 2014, 1407.7005.

[46]  John W. Wilkins,et al.  Effective Tunneling Density of States in Superconductors , 1963 .

[47]  Philip B. Allen,et al.  Theory of Superconducting Tc , 1983 .

[48]  E. Penev,et al.  Temperature dependence of specific heat and penetration depth of anisotropic-gap Bardeen-Cooper-Schrieffer superconductors for a factorizable pairing potential , 2005 .

[49]  J. Bardeen,et al.  FREE-ENERGY DIFFERENCE BETWEEN NORMAL AND SUPERCONDUCTING STATES , 1964 .

[50]  Andreas Hoffmann,et al.  Zone-boundary phonons in hexagonal and cubic GaN , 1997 .

[51]  Cheng-Chung Chi,et al.  Quasiparticle and phonon lifetimes in superconductors , 1976 .

[52]  A. Marini,et al.  Zero point motion effect on the electronic properties of diamond, trans-polyacetylene and polyethylene , 2012 .

[53]  G. V. Chester,et al.  Solid State Physics , 2000 .

[54]  Paul Bratley,et al.  Algorithm 659: Implementing Sobol's quasirandom sequence generator , 1988, TOMS.

[55]  J. Garland BAND-STRUCTURE EFFECTS IN SUPERCONDUCTIVITY. I. FORMALISM. , 1967 .

[56]  C. Tavernier,et al.  Electron-phonon scattering in Si and Ge: From bulk to nanodevices , 2011, 2011 International Conference on Simulation of Semiconductor Processes and Devices.

[57]  Steven G. Louie,et al.  First-Principles Calculation of the Superconducting Transition in MgB2 within the Anisotropic Eliashberg Formalism , 2002 .

[58]  Manuel Cardona,et al.  Temperature dependence of the direct gap of Si and Ge , 1983 .

[59]  J. Cook,et al.  Thermal conductivity, electrical resistivity, and thermoelectric power of Pb from 260 to 550 K , 1974 .

[60]  A. Marini,et al.  Effect of the quantum zero-point atomic motion on the optical and electronic properties of diamond and trans-polyacetylene. , 2011, Physical review letters.

[61]  Matthieu Verstraete,et al.  Erratum: “Temperature dependence of the electronic structure of semiconductors and insulators” [J. Chem. Phys. 143, 102813 (2015)] , 2017 .

[62]  P. Anderson,et al.  CALCULATION OF THE SUPERCONDUCTING STATE PARAMETERS WITH RETARDED ELECTRON- PHONON INTERACTION , 1962 .

[63]  M. Platt,et al.  Atoms , 2009, Archives of Disease in Childhood.

[64]  G. Caglioti,et al.  Crystal Dynamics of Lead. I. Dispersion Curves at 100°K , 1962 .

[65]  S. Louie,et al.  Velocity renormalization and carrier lifetime in graphene from the electron-phonon interaction. , 2007, Physical review letters.

[66]  I. Tang,et al.  Calculation of Tc and the ratio 2Δ(0)/kBTc of MgB2 within a two-band model of superconductivity , 2004 .

[67]  W. Pickett,et al.  Role of two dimensionality in MgB2 , 2003 .

[68]  X. Xi,et al.  Two-band superconductor magnesium diboride , 2008 .

[69]  G. Bester,et al.  Large nuclear zero-point motion effect in semiconductor nanoclusters , 2013 .

[70]  K. Novoselov,et al.  Born-Oppenheimer Breakdown in Graphene , 2006, cond-mat/0611714.

[71]  Feliciano Giustino,et al.  Fröhlich Electron-Phonon Vertex from First Principles. , 2015, Physical review letters.

[72]  M. Aftabuzzaman,et al.  Superconductivity in Diamond-like BC 3 Phase , 2010 .

[73]  J. C. Ward,et al.  Ground-State Energy of a Many-Fermion System. II , 1960 .

[74]  D. Hinks,et al.  The reduced total isotope effect and its implications on the nature of superconductivity in MgB2 , 2001, cond-mat/0104242.

[75]  M. Lazzeri,et al.  Nonadiabatic Kohn anomaly in a doped graphene monolayer. , 2006, Physical review letters.

[76]  Specific heat in the superconducting and normal state (2–300 K, 0–16 T), and magnetic susceptibility of the 38 K superconductor MgB2: evidence for a multicomponent gap , 2001, cond-mat/0103181.

[77]  S. Boggs,et al.  The intrinsic electrical breakdown strength of insulators from first principles , 2012 .

[78]  Thermodynamics of two-band superconductors : The case of MgB2 , 2005, cond-mat/0502659.

[79]  D. Chadi,et al.  Special points for Brillouin-zone integrations , 1977 .

[80]  S. Louie,et al.  First-principles study of electron linewidths in graphene. , 2009, Physical review letters.

[81]  Path-integral molecular dynamics simulation of diamond , 2006, cond-mat/0606028.

[82]  D. Newsham,et al.  Energy of formation of lattice vacancies in lead from equilibrium resistivity and quenching studies , 1966 .

[83]  D. Hamann Optimized norm-conserving Vanderbilt pseudopotentials , 2013, 1306.4707.

[84]  S. Louie,et al.  Electron-phonon renormalization of the direct band gap of diamond. , 2010, Physical review letters.

[85]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[86]  P. B. Allen,et al.  Theory of temperature dependence of optical properties of solids , 1977 .

[87]  Blazej Grabowski,et al.  Ab initio study of the thermodynamic properties of nonmagnetic elementary fcc metals: Exchange-correlation-related error bars and chemical trends , 2007 .

[88]  A. Marini,et al.  A many-body perturbation theory approach to the electron-phonon interaction with density-functional theory as a starting point , 2015, 1503.00567.

[89]  N. Marzari,et al.  Maximally localized Wannier functions for entangled energy bands , 2001, cond-mat/0108084.

[90]  Testa,et al.  Green's-function approach to linear response in solids. , 1987, Physical review letters.

[91]  Xavier Gonze,et al.  Dynamical and anharmonic effects on the electron-phonon coupling and the zero-point renormalization of the electronic structure , 2015, 1505.07738.

[92]  S G Louie,et al.  Coupling of nonlocal potentials to electromagnetic fields. , 2001, Physical review letters.

[93]  Z. A. Ibrahim,et al.  Temperature dependence of the optical response: Application to bulk GaAs using first-principles molecular dynamics simulations , 2008 .

[94]  Martins,et al.  Efficient pseudopotentials for plane-wave calculations. II. Operators for fast iterative diagonalization. , 1991, Physical review. B, Condensed matter.

[95]  P. D. Gennes,et al.  Superconductivity of metals and alloys , 1966 .

[96]  P. Hertel TRANSITION TEMPERATURE OF STRONG-COUPLED SUPERCONDUCTORS. , 1971 .

[97]  A. D. Corso Ab initio phonon dispersions of face centered cubic Pb: effects of spin?orbit coupling , 2008 .

[98]  F. Giustino,et al.  Quantum nuclear dynamics in the photophysics of diamondoids , 2013, Nature Communications.

[99]  E. I. Blount Formalisms of Band Theory , 1962 .

[100]  Yannick Gillet,et al.  Precise effective masses from density functional perturbation theory , 2016 .

[101]  D. R. Hamann,et al.  Pseudopotentials that work: From H to Pu , 1982 .

[102]  Bartomeu Monserrat,et al.  Anharmonic vibrational properties in periodic systems: energy, electron-phonon coupling, and stress , 2013, 1303.0745.

[103]  P. Oppeneer,et al.  Ab initio theory of magnetic-field-induced odd-frequency two-band superconductivity in MgB 2 , 2015 .

[104]  P. B. Allen Solids with thermal or static disorder. I. One-electron properties , 1978 .

[105]  V. A. Stepanov,et al.  Direct evidence for two-band superconductivity in MgB2 single crystals from directional point-contact spectroscopy in magnetic fields. , 2002, Physical review letters.

[106]  K. Bohnen,et al.  Effect of spin-orbit coupling on the electron-phonon interaction of the superconductors Pb and Tl , 2010 .

[107]  F. Giustino,et al.  Stochastic Approach to Phonon-Assisted Optical Absorption. , 2015, Physical review letters.

[108]  Paxton,et al.  High-precision sampling for Brillouin-zone integration in metals. , 1989, Physical review. B, Condensed matter.

[109]  L. Ram-Mohan,et al.  Electron-phonon interaction and scattering in Si and Ge: Implications for phonon engineering , 2015 .

[110]  Ab-initio theory of superconductivity - I: Density functional formalism and approximate functionals , 2004, cond-mat/0408685.

[111]  M. Shur,et al.  Properties of advanced semiconductor materials : GaN, AlN, InN, BN, SiC, SiGe , 2001 .

[112]  A. Migdal 1958 INTERACTION BETWEEN ELECTRONS AND LATTICE VIBRATIONS IN A NORMAL METAL , 2022 .

[113]  D. Vanderbilt,et al.  Spectral and Fermi surface properties from Wannier interpolation , 2007, cond-mat/0702554.

[114]  N. Marzari,et al.  Maximally localized generalized Wannier functions for composite energy bands , 1997, cond-mat/9707145.

[115]  Paul Tangney,et al.  Temperature dependence of the band gap of semiconducting carbon nanotubes. , 2005, Physical review letters.

[116]  W. Pickett Generalization of the theory of the electron-phonon interaction: Thermodynamic formulation of superconducting- and normal-state properties , 1982 .

[117]  Order parameter of MgB(2): a fully gapped superconductor. , 2001, Physical review letters.

[118]  E. Gross,et al.  Ab initio theory of superconductivity in a magnetic field. I. Spin density functional theory for superconductors and Eliashberg equations , 2015, 1503.00985.

[119]  Jia Gao,et al.  Tuning the physical parameters towards optimal polymer-wrapped single-walled carbon nanotubes dispersions , 2012 .

[120]  Yoichmo Namsu,et al.  Quasi-Particles and Gauge Invariance in the Theory of Superconductivity , 2011 .

[121]  N. E. Phillips,et al.  HEAT CAPACITY OF ALUMINUM BETWEEN 0.1 K AND 4.0 K , 1959 .

[122]  Electron - Phonon Superconductivity , 2001, cond-mat/0106143.

[123]  Ab initio theory of superconductivity. II. Application to elemental metals , 2004, cond-mat/0408686.

[124]  M. Scheffler,et al.  Ab initio pseudopotentials for electronic structure calculations of poly-atomic systems using density-functional theory , 1998, cond-mat/9807418.

[125]  R. Graves,et al.  Absolute Seebeck coefficient of platinum from 80 to 340 K and the thermal and electrical conductivities of lead from 80 to 400 K , 1973 .

[126]  X. Gonze,et al.  Verification of first-principles codes: Comparison of total energies, phonon frequencies, electron–phonon coupling and zero-point motion correction to the gap between ABINIT and QE/Yambo , 2013, 1309.0729.

[127]  Wu Li Electrical transport limited by electron-phonon coupling from Boltzmann transport equation: An ab initio study of Si, Al, and MoS 2 , 2015 .

[128]  R. K. Kirby,et al.  Thermophysical Properties of Matter - the TPRC Data Series. Volume 13. Thermal Expansion - Nonmetallic Solids , 1977 .

[129]  G. Grimvall,et al.  The electron-phonon interaction in metals , 1981 .

[130]  D. Johnston,et al.  Elaboration of the α-model derived from the BCS theory of superconductivity , 2013, 1304.2275.

[131]  Yanli Wang,et al.  Quantum ESPRESSO: a modular and open-source software project for quantum simulations of materials , 2009 .

[132]  J Kortus,et al.  Superconductivity of metallic boron in MgB2. , 2001, Physical review letters.

[133]  S. Massidda,et al.  Phononic self-energy effects and superconductivity in CaC6 , 2011, 1108.2800.

[134]  A. Starace Length and Velocity Formulas in Approximate Oscillator-Strength Calculations , 1971 .

[135]  Stefano de Gironcoli,et al.  Phonons and related crystal properties from density-functional perturbation theory , 2000, cond-mat/0012092.

[136]  D. Hinks,et al.  The Complex Nature of Superconductivity in MgB2 as Revealed by the Reduced Total Isotope Effect. , 2001 .

[137]  A. Franceschetti First-principles calculations of the temperature dependence of the band gap of Si nanocrystals , 2007 .

[138]  B. Monserrat Vibrational averages along thermal lines , 2015, 1512.06377.

[139]  E. R. Margine,et al.  Electron-phonon interaction and pairing mechanism in superconducting Ca-intercalated bilayer graphene , 2016, Scientific Reports.

[140]  N. Marzari,et al.  Electron-phonon interactions and the intrinsic electrical resistivity of graphene. , 2014, Nano letters.

[141]  S. Louie,et al.  ab initio study of hot carriers in the first picosecond after sunlight absorption in silicon. , 2014, Physical review letters.

[142]  Xavier Gonze,et al.  Theoretical approaches to the temperature and zero‐point motion effects on the electronic band structure , 2011 .

[143]  Martins,et al.  Efficient pseudopotentials for plane-wave calculations. , 1991, Physical review. B, Condensed matter.

[144]  The Origin of the Anomalous Superconducting Properties of MgB2 , 2002 .

[145]  S. Louie,et al.  Anisotropic Eliashberg theory of MgB2: Tc, isotope effects, superconducting energy gaps, quasiparticles, and specific heat , 2003 .

[146]  Steven G. Louie,et al.  EPW: A program for calculating the electron-phonon coupling using maximally localized Wannier functions , 2010, Comput. Phys. Commun..

[147]  Comparing electron-phonon coupling strength in diamond, silicon, and silicon carbide: First-principles study , 2014, 1406.0654.

[148]  Andrea Marini,et al.  Ab initio finite-temperature excitons. , 2007, Physical review letters.

[149]  S. Pantelides,et al.  First-principles calculations of electron mobilities in silicon: Phonon and Coulomb scattering , 2009 .

[151]  Quasiparticle lifetimes and the conductivity scattering rate , 1997, cond-mat/9709242.

[152]  Bo Qiu,et al.  First-principles simulation of electron mean-free-path spectra and thermoelectric properties in silicon , 2014, 1409.4862.

[153]  Philip B. Allen,et al.  Theory of the temperature dependence of electronic band structures , 1976 .

[154]  W. L. Mcmillan,et al.  LEAD PHONON SPECTRUM CALCULATED FROM SUPERCONDUCTING DENSITY OF STATES , 1965 .

[155]  A. Zunger,et al.  Self-interaction correction to density-functional approximations for many-electron systems , 1981 .

[156]  Samuel Poncé,et al.  Erratum: Temperature dependence of electronic eigenenergies in the adiabatic harmonic approximation [Phys. Rev. B 90 , 214304 (2014)] , 2017 .

[157]  P. Hohenberg,et al.  Inhomogeneous electron gas , 1964 .

[158]  Francois Gygi,et al.  Optimization algorithm for the generation of ONCV pseudopotentials , 2015, Comput. Phys. Commun..

[159]  B. Alder,et al.  THE GROUND STATE OF THE ELECTRON GAS BY A STOCHASTIC METHOD , 2010 .

[160]  S. Sy Linear response calculations of lattice dynamics using muffin-tin basis sets. , 1992 .

[161]  Two-gap state density in MgB(2): a true bulk property or a proximity effect? , 2001, Physical review letters.

[162]  E. R. Margine,et al.  Anisotropic Migdal-Eliashberg theory using Wannier functions , 2012, 1211.3345.

[163]  J. Noffsinger,et al.  Superconductivity and electron-phonon coupling in lithium at high pressures , 2010 .

[164]  Xavier Gonze,et al.  First-principles responses of solids to atomic displacements and homogeneous electric fields: Implementation of a conjugate-gradient algorithm , 1997 .

[165]  G. M. Éliashberg,et al.  Interactions between electrons and lattice vibrations in a superconductor , 1960 .

[166]  Jackson,et al.  Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. , 1992, Physical review. B, Condensed matter.

[167]  M. Calandra,et al.  Giant nonadiabatic effects in layer metals: raman spectra of intercalated graphite explained. , 2008, Physical review letters.