We present an ab-initio density-functional-theory approach for calculating electron-phonon interactions within the projector augmented-wave method. The required electron-phonon matrix elements are defined as the second derivative of the one-electron energies in the PAW method. As the PAW method leads to a generalized eigenvalue problem, the resulting electron-phonon matrix elements lack some symmetries that are usually present for simple eigenvalue problems and all-electron formulations. We discuss the relation between our definition of the electron-phonon matrix element and other formulations. To allow for efficient evaluation of physical properties, we introduce a Wannier-interpolation scheme, again adapted to generalized eigenvalue problems. To explore the method's numerical characteristics, the temperature-dependent band-gap renormalization of diamond is calculated and compared with previous publications. Furthermore, we apply the method to selected binary compounds and show that the obtained zero-point renormalizations agree well with other values found in literature and experiments.
We report a theoretical scheme that enables the calculation of maximally localized Wannier functions in the formalism of projector-augmented-waves (PAW) which also includes the ultrasoft-pseudopotential (USPP) approach. We give a description of the basic underlying formalism and explicitly write all the required matrix elements from the common ingredients of the PAW/USPP theory. We report an implementation of the method in a form suitable to accept the input electronic structure from USPP plane-wave DFT simulations. We apply the method to the calculation of Wannier functions, dipole moments an…
The first order standard perturbation theory combined with ab initio projector augmented wave operator challenges the realization of the standard Sternheimer equation with linear computational efficiency. This efficiency motivates us to describe the electron-phonon interaction in two-dimensional (2D) black phosphorous monolayer using generalized density functional perturbation theory (DFPT) with Boltzmann transport theory (BTE). Subsequently, linear response phonon dynamic behaviour in terms of conductivities, seebeck coefficients and transport properties are focused for its thermoelectric app…
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. Th…
Spin-orbit interaction is an important vehicle for spin relaxation. At finite temperature lattice vibrations modulate the spin-orbit interaction and thus generate a mechanism for spin-phonon coupling, which needs to be incorporated in any quantitative analysis of spin transport. Starting from a density functional theory \textit{ab initio} electronic structure, we calculate spin-phonon matrix elements over the basis of maximally localized Wannier functions. Such coupling terms form an effective Hamiltonian to be used to extract thermodynamic quantities, within a multiscale approach particularly…
We evaluate the zero-point renormalization (ZPR) due to electron-phonon interactions of 28 solids using the projector-augmented-wave (PAW) method. The calculations cover diamond, many zincblende semiconductors, rock-salt and wurtzite oxides, as well as silicate and titania. Particular care is taken to include long-range electrostatic interactions via a generalized Fröhlich model, as discussed in Phys. Rev. Lett. 115, 176401 (2015) and Phys. Rev. B 92, 054307 (2015). The data are compared to recent calculations, npj Computational Materials 6, 167 (2020), and generally very good agreement is fou…
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 pr…
We describe a semiconductor multilayer structure based in acoustic phonon cavities and achievable with MBE technology, designed to display acoustic phonon Bloch oscillations. We show that forward and backscattering Raman spectra give a direct measure of the created phononic Wannier-Stark ladder. We also discuss the use of femtosecond laser impulsions for the generation and direct probe of the induced phonon Bloch oscillations. We propose a gedanken experiment based in an integrated phonon source-structure-detector device, and we present calculations of pump and probe time dependent optical ref…
We generalize the Wannier interpolation of the electron-phonon matrix elements to the case of polar-optical coupling in polar semiconductors. We verify our methodological developments against experiments, by calculating the widths of the electronic bands due to electron-phonon scattering in GaAs, the prototype polar semiconductor. The calculated widths are then used to estimate the broadenings of excitons at critical points in GaAs and the electron-phonon relaxation times of hot electrons. Our findings are in good agreement with available experimental data. Finally, we demonstrate that while t…
Thanks to the nearsightedness principle, the low-energy electronic structure of solids can be represented by localized states such as the Wannier functions. Wannier functions are actively being applied to a wide range of phenomena in condensed matter systems. However, the Wannier-functionbased representation is limited to a small number of bands and thus cannot describe the change of wavefunctions due to various kinds of perturbations, which require sums over an infinite number of bands. Here, we introduce the concept of the Wannier function perturbation, which provides a localized representat…
We develop a first-principles scheme to calculate adiabatic and non-adiabatic phonon frequencies in the full Brillouin zone. The method relies on the variational properties of a force-constants functional with respect to the first-order perturbation of the electronic charge density and on the localization of the deformation potential in the Wannier function basis. This allows for calculation of phonon dispersion curves free from convergence issues related to Brillouin zone sampling. In addition our approach justify the use of the static screened potential in the calculation of the phonon linew…
We present a self-consistent, real-space calculation of the Wannier functions of Si and GaAs within density functional theory. We minimize the total energy functional with respect to orbitals which behave as Wannier functions under crystal translations and, at the minimum, are orthogonal. The Wannier functions are used to calculate the total energy, lattice constant, bulk modulus, and the frequency of the zone-center TO phonon of the two semiconductors with the accuracy required nowadays in ab-initio calculations. Furthermore, the centers of the Wannier functions are used to compute the macros…
The electronic ground state of a periodic system is usually described in terms of extended Bloch orbitals, but an alternative representation in terms of localized "Wannier functions" was introduced by Gregory Wannier in 1937. The connection between the Bloch and Wannier representations is realized by families of transformations in a continuous space of unitary matrices, carrying a large degree of arbitrariness. Since 1997, methods have been developed that allow one to iteratively transform the extended Bloch orbitals of a first-principles calculation into a unique set of maximally localized Wa…
By applying Wannier-based extended Kugel-Khomskii model, we carry out first-principles calculations and electronic structure analysis to understand the spin-phonon coupling effect in transition-metal perovskites. We demonstrate the successful application of our approach to SrMnO$_3$ and BiFeO$_3$. We show that both the electron orbitals under crystal field splitting and the electronic configuration should be taken into account in order to understand the large variances of spin-phonon coupling effects among various phonon modes as well as in different materials.
We develop a method for calculating the electron-phonon vertex in polar semiconductors and insulators from first principles. The present formalism generalizes the Fröhlich vertex to the case of anisotropic materials and multiple phonon branches, and can be used either as a post-processing correction to standard electron-phonon calculations, or in conjunction with {\it ab initio} interpolation based on maximally localized Wannier functions. We demonstrate this formalism by investigating the electron-phonon interactions in anatase TiO$_2$, and show that the polar vertex significantly reduces the…
The monolayer of black phosphorous, or phosphorene, has recently emerged as a new 2D semiconductor with intriguing highly anisotropic transport properties. Existing calculations of its intrinsic phonon-limited electronic transport properties so far rely on the deformation potential approximation, which is in general not directly applicable to anisotropic materials since the deformation along one specific direction can scatter electrons traveling in all directions. We perform a first-principles calculation of the electron-phonon interaction in phosphorene based on density functional perturbatio…
The coupling between electrons and phonons in solids plays a central role in describing many phenomena, including superconductivity and thermoelecric transport. Calculations of this coupling are exceedingly demanding as they necessitate integrations over both the electron and phonon momenta, both of which span the Brillouin zone of the crystal, independently. We present here an ab initio method for efficiently calculating electron-phonon mediated transport properties by dramatically accelerating the computation of the double integrals with a dual interpolation technique that combines maximally…
Quantum chemistry is one of the most promising near-term applications of quantum computers. Quantum algorithms such as variational quantum eigen solver (VQE) and variational quantum deflation (VQD) algorithms have been mainly applied for molecular systems and there is a need to implement such methods for periodic solids. Using Wannier tight-binding Hamiltonian (WTBH) approaches, we demonstrate the application of VQE and VQD to accurately predict both electronic and phonon bandstructure properties of several elemental as well as multi-component solid-state materials. We apply VQE-VQD calculatio…
We develop an ab initio formalism for dipolar electron-phonon interactions (EPI) in two-dimensional (2D) materials. Unlike purely longitudinal Fröhlich model, we show that the out-of-plane dipoles also contribute to the long-wavelength non-analytical behavior of EPI. And the 2D dipolar EPI plays an important role not only in the typical polar material MoS$_2$, but also in graphane and fluorinated graphene. By incorporating this formalism into Wannier-Fourier interpolation, we enable accurate EPI calculations for 2D materials and subsequent intrinsic carrier mobility prediction. The results sho…
We study the light-absorbing states of the mixed-halide perovskite CH$_{3}$NH$_{3}$PbI$_2$Cl and tri-iodide perovskite CH$_{3}$NH$_{3}$PbI$_3$ with density functional and many-body calculations to explain the desirable photovolatic features of these materials. The short-lived electron-hole bound states produced in this photovoltaic material are of halide to lead electron transfer character, with a Wannier-type exciton. Bethe-Salpeter (GW+BSE) calculations of the absorption cross section reveal strong screening of the electron-hole interaction. The atomic character of the exciton retains ligand…
The description of realistic strongly correlated systems has recently advanced through the combination of density functional theory in the local density approximation (LDA) and dynamical mean field theory (DMFT). This LDA+DMFT method is able to treat both strongly correlated insulators and metals. Several interfaces between LDA and DMFT have been used, such as (N-th order) Linear Muffin Tin Orbitals or Maximally localized Wannier Functions. Such schemes are however either complex in use or additional simplifications are often performed (i.e., the atomic sphere approximation). We present an alt…
Meta-optimization of maximally-localized Wannier functions
Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学研究所）；Department of Physics, The University of Texas at Austin（德克萨斯大学奥斯汀分校物理系）
The interaction between electrons and lattice vibrations determines key physical properties of materials, including their electrical and heat transport, excited electron dynamics, phase transitions, and superconductivity. We present a new ab initio method that employs atomic orbital (AO) wavefunctions to compute the electron-phonon (e-ph) interactions in materials and interpolate the e-ph coupling matrix elements to fine Brillouin zone grids. We detail the numerical implementation of such AO-based e-ph calculations, and benchmark them against direct density functional theory calculations and W…
EPW is an open-source software for $\textit{ab initio}$ calculations of electron-phonon interactions and related materials properties. The code combines density functional perturbation theory and maximally-localized Wannier functions to efficiently compute electron-phonon coupling matrix elements on ultra-fine Brillouin zone grids. This data is employed for predictive calculations of temperature-dependent properties and phonon-assisted quantum processes in bulk solids and low-dimensional materials. Here, we report on significant new developments in the code that occurred during the period 2016…
Many-Body Correlation Effects in Fröhlich Electron-Phonon Coupling
University of Southern California（南加州大学）；Tamkang University（淡江大学）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
A systematic method is presented for constructing effective Hamiltonians for general phonon-related structural transitions. The key feature is the application of group theoretical methods to identify the subspace in which the effective Hamiltonian acts and construct for it localized basis vectors, which are the analogue of electronic Wannier functions. The results of the symmetry analysis for the perovskite, rocksalt, fluorite and A15 structures and the forms of effective Hamiltonians for the ferroelectric transition in $PbTiO_3$ and $BaTiO_3$, the oxygen-octahedron rotation transition in $SrT…
We present a first-principles framework to investigate the electron scattering channels and transport properties for polar material by combining the exact solution of linearized electron-phonon (e-ph) Boltzmann transport equation in its integral-differential form associated with the e-ph coupling matrices obtained from polar Wannier interpolation scheme. No ad hoc parameter is required throughout this calculation, and GaAs, a well-studied polar material, is used as an example to demonstrate this method. In this work, the long-range and short-range contributions as well as the intravalley and i…
The $α$ phase of $Ga_{2}O_{3}$ is an ultra-wideband semiconductor with potential power electronics applications. In this work, we calculate the low field electron mobility in $α-Ga_{2}O_{3}$ from first principles. The 10 atom unit cell contributes to 30 phonon modes and the effect of each mode is taken into account for the transport calculation. The phonon dispersion and the Raman spectrum are calculated under the density functional perturbation theory formalism and compared with experiments. The IR strength is calculated from the dipole moment at the $Γ$ point of the Brillouin zone. The elect…
We investigate the high-field transport in monoclinic \(β\)-Ga\textsubscript{2}O\textsubscript{3}} using a combination of ab initio calculations and full band Monte Carlo (FBMC) simulation. Scattering rate calculation and the final state selection in the FBMC simulation use complete wave-vector (both electron and phonon) and crystal direction dependent electron phonon interaction (EPI) elements. We propose and implement a semi-coarse version of the Wannier-Fourier interpolation method [F. Giustino, M. L. Cohen, and S. G. Louie, Physical Review B, vol. 76, no. 16, 2007] for short-range non-pola…
Motivated by the recent discovery of superconductivity in Ca- and Yb-intercalated graphite (CaC$_{6}$ and YbC$_{6}$) and from the ongoing debate on the nature and role of the interlayer state in this class of compounds, in this work we critically study the electron-phonon properties of a simple model based on primitive graphite. We show that this model captures an essential feature of the electron-phonon properties of the Graphite Intercalation Compounds (GICs), namely, the existence of a strong dormant electron-phonon interaction between interlayer and $π^{\ast}$ electrons, for which we provi…
Perturbo is a software package for first-principles calculations of charge transport and ultrafast carrier dynamics in materials. The current version focuses on electron-phonon interactions and can compute phonon-limited transport properties such as the conductivity, carrier mobility and Seebeck coefficient. It can also simulate the ultrafast nonequilibrium electron dynamics in the presence of electron-phonon scattering. Perturbo uses results from density functional theory and density functional perturbation theory calculations as input, and employs Wannier interpolation to reduce the computat…
We present a first-principles study of the temperature- and density-dependent intrinsic electrical resistivity of graphene. We use density-functional theory and density-functional perturbation theory together with very accurate Wannier interpolations to compute all electronic and vibrational properties and electron-phonon coupling matrix elements; the phonon-limited resistivity is then calculated within a Boltzmann-transport approach. An effective tight-binding model, validated against first-principles results, is also used to study the role of electron-electron interactions at the level of ma…
Electron-phonon(e-ph) interaction in Ca2N monolayer, the first electrene material with two-dimensional(2D) electron gas floating in free space, is expected to be very weak and such a character can be used to design weak-scattering transport channels. Therefore, it is highly desirable to quantitatively evaluate the carrier mobility of electrene. In this study, e-ph interaction in Ca2N monolayer is investigated using a precise Wannier interpolation-based first principles technique. The calculated e-ph coupling matrix elements of Ca2N monolayer are indeed small compared to other 2D materials such…
By means of the first-principles density-functional theory calculation and Wannier interpolation, electron-phonon coupling and superconductivity are systematically explored for boron-doped LiBC (i.e. LiB$_{1+x}$C$_{1-x}$), with $x$ between 0.1 and 0.9. Hole doping introduced by boron atoms is treated through virtual-crystal approximation. For the investigated doping concentrations, our calculations show the optimal doping concentration corresponds to 0.8. By solving the anisotropic Eliashberg equations, we find that LiB$_{1.8}$C$_{0.2}$ is a two-gap superconductor, whose superconducting transi…
Motivated by the recent experimental discovery of strongly surface-plane-dependent superconductivity at surfaces of KTaO$_3$ single crystals, we calculate the electron-phonon coupling strength, $λ$, of doped KTaO$_3$ along the reciprocal-space high-symmetry directions. Using the Wannier-function approach implemented in the EPW package, we calculate $λ$ across the experimentally covered doping range and compare its mode-resolved distribution along the [001], [110] and [111] reciprocal-space directions. We find that the electron-phonon coupling is strongest in the optical modes around the $Γ$ po…
Realistic prediction of the superconducting transition temperature (Tc) for PdH is a long-standing challenge, because it depends on robust calculations of the electron and phonon band structures to obtain the electron-phonon scattering matrix element. To date, first-principles calculations employing density functional theory (DFT) have been based on selected exchange-correlation and core-electron approximations. Incorporating anharmonicity produced a more realistic value of Tc that nevertheless still disagreed strongly with experiment unless adjustable parameters were introduced. Here we consi…
Recently, a honeycomb borophene was reported to grow successfully on Al(111) surface. Since the metallic $σ$-bonding bands of honeycomb boron sheet play a crucial role in the 39 K superconductivity of MgB$_2$, it is physically interesting to examine whether similar property exists in this material. We have calculated the electronic structures and the electron-phonon coupling for honeycomb borophene by explicitly considering the substrate effect using first-principles density functional theory in conjunction with the Wannier interpolation technique. We find that the $sp^2$-hybridized $σ$-bondin…
The intrinsic charge transport of stanene is investigated by using density function theory and density function perturbation theory coupled with Boltzmann transport equations from first principles. The accurate Wannier interpolations are applied to calculate the charge carrier scatterings with all branches of phonons with dispersion contribution. The intrinsic carrier mobilities are predicted to be 2~3$\times10^3$ cm$^2$/(V s) at 300 K, and we find that the intervalley scatterings from the out-of-plane and transverse acoustic phonon modes dominate the carrier relaxation. In contrast, the intri…
Coupling between $σ$-bonding electrons and phonons is generally very strong. To metallize $σ$-electrons provides a promising route to hunt for new high-T$_c$ superconductors. Based on this picture and first-principles density functional calculation with Wannier interpolation for electronic structure and lattice dynamics, we predict that trilayer film LiB$_2$C$_2$ is a good candidate to realize this kind of high-T$_c$ superconductivity. By solving the anisotropic Eliashberg equations, we find that free-standing trilayer LiB$_2$C$_2$ is a phonon-mediated superconductor with T$_c$ exceeding the l…
