Flexoelectricity from density-functional perturbation theory
Massimiliano Stengel
备注：
26 pages, 2 figures. The newly added Section VI deals with the nonanalytic properties of the electronic response functions, and generalizes the lattice-dynamical theory of Pick, Cohen and Martin (Ref. 32) to higher orders in q
We derive the complete flexoelectric tensor, including electronic and lattice-mediated effects, of an arbitrary insulator in terms of the microscopic linear response of the crystal to atomic displacements. The basic ingredient, which can be readily calculated from first principles in the framework of density-functional perturbation theory, is the quantum-mechanical probability current response to a long-wavelength acoustic phonon. Its second-order Taylor expansion in the wavevector q around the Gamma (q=0) point in the Brillouin zone naturally yields the flexoelectric tensor. At order one in q we recover Martin's theory of piezoelectricity [R. M. Martin, Phys. Rev. B 5, 1607 (1972)], thus providing an alternative derivation thereof. To put our derivations on firm theoretical grounds, we perform a thorough analysis of the nonanalytic behavior of the dynamical matrix and other response functions in a vicinity of Gamma. Based on this analysis, we find that there is an ambiguity in the specification of the "zero macroscopic field" condition in the flexoelectric case; such arbitrariness can be related to an analytic band-structure term, in close analogy to the theory of deformation potentials. As a byproduct, we derive a rigorous generalization of the Cochran-Cowley formula [W. Cochran and R. A. Cowley, J. Phys. Chem. Solids 23, 447 (1962)] to higher orders in q. This can be of great utility in building reliable atomistic models of electromechanical phenomena, as well as for improving the accuracy of the calculation of phonon dispersion curves. Finally, we discuss the physical interpretation of the various contributions to the flexoelectric response, either in the static or dynamic regime.
Within the framework of density functional perturbation theory (DFPT), we implement and test a novel "metric wave" response-function approach. It consists in the reformulation of an acoustic phonon perturbation in the curvilinear frame that is comoving with the atoms. This means that all the perturbation effects are encoded in the first-order variation of the real-space metric, while the atomic positions remain fixed. This approach can be regarded as the generalization of the uniform strain perturbation of Hamann et al. [D. R. Hamann, X. Wu, K. M. Rabe, and D. Vanderbilt, Phys. Rev. B 71, 0351…
The origin of "giant" flexoelectricity, orders of magnitude larger than theoretically predicted, yet frequently observed, is under intense scrutiny. There is mounting evidence correlating giant flexoelectric-like effects with parasitic piezoelectricity, but it is not clear how piezoelectricity (polarization generated by strain) manages to imitate flexoelectricity (polarization generated by strain gradient) in typical beam-bending experiments, since in a bent beam the net strain is zero. In addition, and contrary to flexoelectricity, piezoelectricity changes sign under space inversion, and this…
We develop a general and unified first-principles theory of piezoelectric and flexoelectric tensor, formulated in such a way that the tensor elements can be computed directly in the context of density-functional calculations, including electronic and lattice contributions. We introduce a practical supercell-based methods for calculating the flexoelectric coefficients from first principles, and demonstrate them by computing the coefficients for a variety of cubic insulating materials, including C, Si, MgO, NaCl, CsCl, BaZrO3, BaTiO3, PbTiO3 and SrTiO3.
Flexoelectricity has garnered much attention owing to its ability to bring electromechanical functionality to non-piezoelectric materials and its nanoscale significance. In order to move towards a more complete understanding of this phenomenon and improve the efficacy of flexoelectric-based devices, it is necessary to quantify microstructural contributions to flexoelectricity. Here we characterize the flexoelectric response of bulk centrosymmetric LaAlO$_3$ crystals with different twin boundary microstructures. We show that twin boundary flexoelectric contributions are comparable to intrinsic …
Density-functional perturbation theory (DFPT) is nowadays the method of choice for the accurate computation of linear and non-linear response properties of materials from first principles. A notable advantage of DFPT over alternative approaches is the possibility of treating incommensurate lattice distortions with an arbitrary wavevector, ${\bf q}$, at essentially the same computational cost as the lattice-periodic case. Here we show that ${\bf q}$ can be formally treated as a perturbation parameter, and used in conjunction with established results of perturbation theory (e.g. the "2n+1" theor…
We present the derivation and code implementation of a first-principles methodology to calculate the lattice-mediated contributions to the bulk flexoelectric tensor. The approach is based on our recent analytical long-wavelength extension of density-functional perturbation theory [Royo and Stengel, Phys. Rev. X 9, 021050 (2019)], and avoids the cumbersome numerical derivatives with respect to the wave vector that were adopted in previous implementations. To substantiate our results, we revisit and numerically validate the sum rules that relate flexoelectricity and uniform elasticity by general…
In this Chapter we provide an overview of the current first-principles perspective on flexoelectric effects in crystalline solids. We base our theoretical formalism on the long-wave expansion of the electrical response of a crystal to an acoustic phonon perturbation. In particular, we recover the known expression for the piezoelectric tensor from the response at first order in wavevector ${\bf q}$, and then obtain the flexoelectric tensor by extending the formalism to second order in $\bf q$. We put special emphasis on the issue of surface effects, which we first analyze heuristically, and the…
Using a super-operator formulation of linearized time-dependent density-functional theory, the dynamical polarizability of a system of interacting electrons is given a matrix continued-fraction representation whose coefficients can be obtained from the non-symmetric block-Lanczos method. The resulting algorithm allows for the calculation of the {\em full spectrum} of a system with a computational workload which is only a few times larger than that needed for {\em static} polarizabilities within time-independent density-functional perturbation theory. The method is demonstrated with the calcula…
This article reviews the current status of lattice-dynamical calculations in crystals, using density-functional perturbation theory, with emphasis on the plane-wave pseudo-potential method. Several specialized topics are treated, including the implementation for metals, the calculation of the response to macroscopic electric fields and their relevance to long wave-length vibrations in polar materials, the response to strain deformations, and higher-order responses. The success of this methodology is demonstrated with a number of applications existing in the literature.
A real-space formalism for density-functional perturbation theory (DFPT) is derived and applied for the computation of harmonic vibrational properties in molecules and solids. The practical implementation using numeric atom-centered orbitals as basis functions is demonstrated exemplarily for the all-electron Fritz Haber Institute ab initio molecular simulations (FHI-aims) package. The convergence of the calculations with respect to numerical parameters is carefully investigated and a systematic comparison with finite-difference approaches is performed both for finite (molecules) and extended (…
The non-linear response of infinite periodic solids to homogenous electric fields and collective atomic displacements is discussed in the framework of density functional perturbation theory. The approach is based on the 2n + 1 theorem applied to an electric-field-dependent energy functional. We report the expressions for the calculation of the non-linear optical susceptibilities, Raman scattering efficiencies and electrooptic coefficients. Different formulations of third-order energy derivatives are examined and their convergence with respect to the k-point sampling is discussed. We apply our …
Long-range exchange and correlation effects, responsible for the failure of currently used approximate density functionals in describing van der Waals forces, are taken into account explicitly after a separation of the electron-electron interaction in the Hamiltonian into short- and long-range components. We propose a "range-separated hybrid" functional based on a local density approximation for the short-range exchange-correlation energy, combined with a long-range exact exchange energy. Long-range correlation effects are added by a second-order perturbational treatment. The resulting scheme …
The electron-phonon interaction plays a crucial role in many fields of physics and chemistry. Nevertheless, its actual calculation by means of modern many-body perturbation theory is weakened by the use of model Hamiltonians that are based on parameters difficult to extract from the experiments. Such shortcoming can be bypassed by using density-functional theory to evaluate the electron-phonon scattering amplitudes, phonon frequencies and electronic bare energies. In this work, we discuss how a consistent many-body diagrammatic expansion can be constructed on top of density-functional theory. …
The self-consistent evaluation of Hubbard parameters using linear-response theory is crucial for quantitatively predictive calculations based on Hubbard-corrected density-functional theory. Here, we extend a recently-introduced approach based on density-functional perturbation theory (DFPT) for the calculation of the on-site Hubbard $U$ to also compute the inter-site Hubbard $V$. DFPT allows to reduce significantly computational costs, improve numerical accuracy, and fully automate the calculation of the Hubbard parameters by recasting the linear response of a localized perturbation into an ar…
We present a benchmark of the density functional linear response calculation of NMR shieldings within the Gauge-Including Projector-Augmented-Wave method against all-electron Augmented-Plane-Wave$+$local-orbital and uncontracted Gaussian basis set results for NMR shieldings in molecular and solid state systems. In general, excellent agreement between the aforementioned methods is obtained. Scalar relativistic effects are shown to be quite large for nuclei in molecules in the deshielded limit. The small component makes up a substantial part of the relativistic corrections.
We assess the validity of various exchange-correlation functionals for computing the structural, vibrational, dielectric, and thermodynamical properties of materials in the framework of density-functional perturbation theory (DFPT). We consider five generalized-gradient approximation (GGA) functionals (PBE, PBEsol, WC, AM05, and HTBS) as well as the local density approximation (LDA) functional. We investigate a wide variety of materials including a semiconductor (silicon), a metal (copper), and various insulators (SiO$_2$ $α$-quartz and stishovite, ZrSiO$_4$ zircon, and MgO periclase). For the…
We present an implementation of time-dependent density functional perturbation theory for spin fluctuations, based on planewaves and pseudopotentials. We compute the dynamic spin susceptibility self-consistently by solving the time-dependent Sternheimer equation, within the adiabatic local density approximation to the exchange and correlation kernel. We demonstrate our implementation by calculating the spin susceptibility of representative elemental transition metals, namely bcc Fe, fcc Ni and bcc Cr. The calculated magnon dispersion relations of Fe and Ni are in agreement with previous work. …
The methods of density-functional perturbation theory may be used to calculate various physical response properties of insulating crystals including elastic, dielectric, Born charge, and piezoelectric tensors. These and other important tensors may be defined as second derivatives of the total energy with respect to atomic-displacement, electric-field, or strain perturbations, or as mixed derivatives with respect to two of these perturbations. The resulting tensor quantities tend to be coupled in complex ways in polar crystals, giving rise to a variety of variant definitions. For example, it is…
We present a new method to model spin-wave excitations in magnetic solids, based on the Liouville-Lanczos approach to time-dependent density functional perturbation theory. This method avoids computationally expensive sums over empty states and naturally deals with the coupling between spin and charge fluctuations, without ever explicitly computing charge-density susceptibilities. Spin-wave excitations are obtained with one Lanczos chain per magnon wave-number and polarization, avoiding the solution of the linear-response problem for every individual value of frequency, as other state-of-the-a…
Applied magnetic fields can couple to atomic displacements via generalized Lorentz forces, which are commonly expressed as gyromagnetic $g$ factors. We develop an efficient first-principles methodology based on density-functional perturbation theory to calculate this effect in both molecules and solids to linear order in the applied field. Our methodology is based on two linear-response quantities: the macroscopic polarization response to an atomic displacement (i.e., Born effective charge tensor), and the antisymmetric part of its first real-space moment (the symmetric part corresponding to t…
Optical absorption spectra of metal oxides from time-dependent density functional theory and many-body perturbation theory based on optimally-tuned hybrid functionals
The calculation of quasi-particle spectra based on the GW approximation is extended to systems of hundreds of atoms, thus expanding the size range of current approaches by more than one order of magnitude. This is achieved through an optimal strategy, based on the use of Wannier-like orbitals, for evaluating the polarization propagator. Our method is validated by calculating the vertical ionization energies of the benzene molecule and the band structure of crystalline silicon. Its potentials are then demonstrated by addressing the quasi-particle spectrum of a model structure of vitreous silica…
The Liouville-Lanczos approach to linear-response time-dependent density-functional theory is generalized so as to encompass electron energy-loss and inelastic X-ray scattering spectroscopies in periodic solids. The computation of virtual orbitals and the manipulation of large matrices are avoided by adopting a representation of response orbitals borrowed from (time-independent) density-functional perturbation theory and a suitable Lanczos recursion scheme. The latter allows the bulk of the numerical work to be performed at any given transferred momentum only once, for a whole extended frequen…
We introduce HP, an implementation of density-functional perturbation theory, designed to compute Hubbard parameters (on-site $U$ and inter-site $V$) in the framework of DFT+$U$ and DFT+$U$+$V$. The code does not require the use of computationally expensive supercells of the traditional linear-response approach; instead, unit cells are used with monochromatic perturbations that significantly reduce the computational cost of determining Hubbard parameters. HP is an open-source software distributed under the terms of the GPL as a component of Quantum ESPRESSO. As with other components, HP is opt…
The diffusion of large databases collecting different kind of material properties from high-throughput density functional theory calculations has opened new paths in the study of materials science thanks to data mining and machine learning techniques. Phonon calculations have already been employed successfully to predict materials properties and interpret experimental data, e.g. phase stability, ferroelectricity and Raman spectra, so their availability for a large set of materials will further increase the analytical and predictive power at hand. Moving to a larger scale with density functiona…
Quantifying materials' dynamical responses to external electromagnetic fields is central to understanding their physical properties. Here we present an implementation of the density functional perturbation theory for the computation of linear susceptibilities using the projector augmented-wave method. The Sternheimer equations are solved self-consistently through a nested iterative procedure to compute the first-order wavefunctions, from which the linear susceptibilities are obtained. As a demonstration, we compute the spin wave spectral functions of two magnetic metals. The computed magnon sp…
Many technological applications depend on the response of materials to electric fields, but available databases of such responses are limited. Here, we explore the infrared, piezoelectric and dielectric properties of inorganic materials by combining high-throughput density functional perturbation theory and machine learning approaches. We compute Γ-point phonons, infrared intensities, Born-effective charges, piezoelectric, and dielectric tensors for 5015 non-metallic materials in the JARVIS-DFT database. We find 3230 and 1943 materials with at least one far and mid-infrared mode, respectively.…
The ability to perform first-principles calculations of electronic and vibrational properties of two-dimensional heterostructures in a field-effect setup is crucial for the understanding and design of next-generation devices. We present here an implementation of density functional perturbation theories tailored for the case of two-dimensional heterostructures in field-effect configuration. Key ingredients are the inclusion of a truncated Coulomb interaction in the direction perpendicular to the slab and the possibility of simulating charging of the slab via field-effects. With this implementat…
The knowledge of effective masses is a key ingredient to analyze numerous properties of semiconductors, like carrier mobilities, (magneto-)transport properties, or band extrema characteristics yielding carrier densities and density of states. Currently, these masses are usually calculated using finite-difference estimation of density functional theory (DFT) electronic band curvatures. However, finite differences require an additional convergence study and are prone to numerical noise. Moreover, the concept of effective mass breaks down at degenerate band extrema. We assess the former limitatio…
We present a new release of the turboTDDFT code featuring an implementation of hybrid functionals, a recently introduced pseudo-Hermitian variant of the Liouville-Lanczos approach to time-dependent density-functional perturbation theory, and a newly developed Davidson-like algorithm to compute selected interior eigenvalues/vectors of the Liouvillian super-operator. Our implementation is thoroughly validated against benchmark calculations performed on the cyanin (C$_{21}$O$_{11}$H$_{21}$) molecule using the Gaussian09 and turboTDDFT 1.0 codes.
The direct calculation of the elastic and piezoelectric tensors of solids can be accomplished by treating homogeneous strain within the framework of density-functional perturbation theory. By formulating the energy functional in reduced coordinates, we show that the strain perturbation enters only through metric tensors, and can be treated in a manner exactly paralleling the treatment of other perturbations. We present an analysis of the strain perturbation of the plane-wave pseudopotential functional, including the internal strain terms necessary to treat the atomic-relaxation contributions. …
Machine learning thermodynamic perturbation theory (MLPT) is a promising approach to compute finite temperature properties when the goal is to compare several different levels of ab initio theory and/or to apply highly expensive computational methods. Indeed, starting from a production molecular dynamics trajectory, this method can estimate properties at one or more target levels of theory from only a small number of additional fixed-geometry calculations, which are used to train a machine learning model. However, as MLPT is based on thermodynamic perturbation theory (TPT), inaccuracies might …
We present a constrained density functional perturbation theory scheme for the calculation of structural and harmonic vibrational properties of insulators in the presence of an excited and thermalized electron-hole plasma. The method is ideal to tame ultrafast light induced structural transitions in the regime where the photocarriers thermalize faster than the lattice, the electron-hole recombination time is longer than the phonon period and the photocarrier concentration is large enough to be approximated by an electron-hole plasma. The complete derivation presented here includes total energy…
Application of the exact-factorization density-functional perturbation approach to pentacene crystal and monolayer MoS2
Weizmann Institute of Science（魏茨曼科学研究所）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Tsientang Institute of Advanced Study（紫江高等研究院）；The Hebrew University of Jerusalem（耶路撒冷希伯来大学）；Ruđer Bošković Institute（鲁杰·博什科维奇研究所）
We present a theoretical study of the ionization potential in small anionic gold clusters, using density functional theory, with and without exact-exchange, and many body perturbation theory, namely the G0W0 approach. We find that G0W0 is the best approach and correctly describes the first ionization potential with an accuracy of about 0.1 eV.
We extend density functional perturbation theory for lattice dynamics with fully relativistic ultrasoft pseudopotentials to magnetic materials. Our approach is based on the application of the time-reversal operator to the Sternheimer linear system and to its self-consistent solutions. Moreover, we discuss how to include in the formalism the symmetry operations of the magnetic point group which require the time-reversal operator. We validate our implementation by comparison with the frozen phonon method in fcc Ni and in a monatomic ferromagnetic Pt wire.
Degenerate perturbation theory from quantum mechanics is inadequate in density functional theory (DFT) because of nonlinearity in the Kohn-Sham potential. Herein, we develop the fully general perturbation theory for open-shell, degenerate systems in Kohn-Sham DFT, without assuming the presence of symmetry or equal occupation of degenerate orbitals. To demonstrate the resulting methodology, we apply it to the iron atom in the central field approximation, perturbed by an electric quadrupole. This system was chosen because it displays both symmetry required degeneracy, between the five 3\textit{d…
Fractional occupation numbers can produce open-shell degeneracy in density functional theory. We develop the corresponding perturbation theory by requiring that a differentiable map connects the initial and perturbed states. The degenerate state connects to a single perturbed state which extremizes, but does not necessarily minimize or maximize, the energy with respect to occupation numbers. Using a system of three electrons in a harmonic oscillator potential, we relate the counterintuitive sign of first-order occupation numbers to eigenvalues of the electron-electron interaction Hessian.
We introduce turboEELS, an implementation of the Liouville-Lanczos approach to linearized time-dependent density-functional theory, designed to simulate electron energy loss and inelastic X-ray scattering spectra in periodic solids. turboEELS is open-source software distributed under the terms of the GPL as a component of Quantum ESPRESSO. As with other components, turboEELS is optimized to run on a variety of different platforms, from laptops to massively parallel architectures, using native mathematical libraries (LAPACK and FFTW) and a hierarchy of custom parallelization layers built on top…
We propose a method for the evaluation of magnetic exchange couplings based on noncollinear spin-density functional calculations. The method employs the second derivative of the total Kohn-Sham energy of a single reference state, in contrast to approximations based on Kohn-Sham total energy differences. The advantage of our approach is twofold: It provides a physically motivated picture of the transition from a low-spin to a high-spin state, and it utilizes a perturbation scheme for the evaluation of magnetic exchange couplings. The latter simplifies the way these parameters are predicted usin…
For a set of ten crystalline materials (oxides and semiconductors), we compute the electronic band structures using the Tran-Blaha [Phys. Rev. Lett. 102, 226401 (2009)] (TB09) functional. The band widths and gaps are compared with those from the local-density approximation (LDA) functional, many-body perturbation theory (MBPT), and experiments. At the density-functional theory (DFT) level, TB09 leads to band gaps in much better agreement with experiments than LDA. However, we observe that it globally underestimates, often strongly, the valence (and conduction) band widths (more than LDA). MBPT…
For the paradigmatic case of H2-dissociation we compare state-of-the-art many-body perturbation theory (MBPT) in the GW approximation and density-functional theory (DFT) in the exact-exchange plus random-phase approximation for the correlation energy (EX+cRPA). For an unbiased comparison and to prevent spurious starting point effects both approaches are iterated to full self-consistency (i.e. sc-RPA and sc-GW). The exchange-correlation diagrams in both approaches are topologically identical, but in sc-RPA they are evaluated with non-interacting and in sc-GW with interacting Green functions. Th…
Constrained electronic-structure theories enable the construction of effective low-energy models consisting of partially dressed particles. However, the interpretation and physical content of these theories is not straightforward. Here, we carefully explore the properties of downfolding theories for electron-ion problems, in particular constrained density-functional perturbation theory (cDFPT). We show that the dipole selection rules determine whether the partially dressed phonons satisfy Goldstone's theorem, and we prove that electronic screening always lowers the phonon frequencies. We illus…
Low-temperature behavior of density-functional theory for metals based on density-functional perturbation theory and Sommerfeld expansion
European Theoretical Spectroscopy Facility, Institute of Condensed Matter and Nanosciences, Université catholique de Louvain（欧洲理论光谱设施，凝聚态与纳米科学研究所，天主教鲁汶大学）；Center for Integrative Petroleum Research, King Fahd University of Petroleum and Minerals（综合石油研究中心，法赫德国王石油与矿产大学）；Laboratoire de mathématiques d’Orsay, Université Paris-Saclay（奥赛数学实验室，巴黎萨克雷大学）
