[Ziman Principles Of The Theory Of Solids 13
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Although the theory of lattice dynamics was established six decades ago, its accurate implementation for polar solids using the direct (or supercell, small displacement, frozen phonon) approach within the framework of density-function-theory-based first-principles calculations had been a challenge until recently. It arises from the fact that the vibration-induced polarization breaks the lattice periodicity, whereas periodic boundary conditions are required by typical first-principles calculations, leading to an artificial macroscopic electric field. The article reviews a mixed-space approach to treating the interactions between lattice vibration and polarization, its applications to accurately predicting the phonon and associated thermal properties, and its implementations in a number of existing phonon codes.
The present review focuses on the theory of lattice dynamics for polar solids. Here a polar solid implies an insulator or a semiconductor composed of cations with positive charges and anions with negative charges. As a matter of fact, the majorities of modern functional materials are made of polar solids, such as the topological crystalline insulator group-IV tellurides,29 the ferroelectrics and multiferroics,30 and materials for solar cells.31 The accurate descriptions of phonon properties have key roles for the understandings and developments of these materials.
The present paper is organized as follows: 'Basic lattice dynamics of polar solids' describes the basics of lattice dynamics and the fundamentals of the mixed-space approach. 'Helmholtz energy and quasiharmonic approximation' outlines the first-principles thermodynamics based on the phonon theory; More discussions of the mixed-space approach are given in 'The mixed-space approach'. 'Computational procedure' summarizes the common procedures in phonon calculations; 'Phonon software packages have implemented the mixed-space approach' briefs the implementation of the mixed-space approach in several software packages. Extensive applications of the mixed-space approach are summarized in 'Recent calculations using the mixed-space approach'. 'Other phonon software packages' briefs a list of other phonon codes implemented differently from the mixed-space approach for polar solids. 'Software packages for both electronic and phonon calculations' introduces a few widely in use first-principles codes for both electronic and phonon calculations. Finally, the last section is the 'Summary'.
where VP is the volume of the primitive cell. The vibration-induced polarization is parallel to the direction of the wave-vector q ˆ and can only have effects on the atomic vibrations along the direction of the wave-vector q ˆ . It is noticed that a normal mode with a wave-vector q is nothing but a collective vibration of some parallel charged crystal planes with normal along q ˆ . As a result, the induced electric field by the lattice vibrations can be formulated as45
D A α β j k ( q ) is called the reduced dynamical matrix,10 accounting for the analytic contribution under zero-averaged electric field, whereas D N α β j k ( q ) results entirely from the effects of the vibration-induced macroscopic field. They have the following forms
In this section, we discuss how the analytic and the nonanalytic contributions are related to the supercell geometry and the type of wave-vector points for evaluating the normal vibration frequencies of a polar solid. Let us first examine the nonanalytic contribution to the dynamical matrix in Equation (12) from which we can extract the mathematical geometry factor
Calculate the dielectric constant and Born effective charge tensors used in Equation (13) based on the primitive cell by employing either the linear-response approach42 or the Berry phase expressions of electric polarization.44
This greatly simplifies the computational procedure since one only needs to add a constant term to the calculated force constants by the direct approach. In this respect, the mixed-space approach is a generalization of the approach for the specific case of GaAs.50 It should be pointed out that the mixed-space approach is significantly different from previous implementations accounting for the presence of a macroscopic field51 in the linear-response approach,14,20 where rather tedious and expensive mathematical calculations are involved in order to decompose the calculated interatomic force constants into the short-range contributions and the long-range one from the polar effects.
ShengBTE is a software package for computing the lattice thermal conductivity of crystalline materials and nanowires with diffusive boundary conditions by Li et al.8 Both the linear-response approach and the mixed-space approach were implemented in ShengBTE.
Using the mixed-space approach, phonon and associated properties have been studied for a variety of polar (and non-polar) solids. Most of these calculations are based on the direct approach using the output data from first-principles codes such as VASP11,12 as input. Examples are firstly shown for several energy conversion and storage materials.
Accurate phonon properties have also been predicted for CaF2 and CeO2 with the fluorite structure.69 CaF2 is a typical superionic conductor, its phonons have been studied by the PBE exchange-correlation functional70 using a 192-atom supercell. CeO2 has been used in catalytic converters in automotive applications and as an electrolyte in fuel cells because of its relatively high oxygen ion conductivity. In particular for considering the f-electron system, phonon dispersions of CeO2 have been studied by a HSE06 hybrid functional,71,72 showing better accuracy69 than the previous predictions from e.g., PWSCF73 and ABINIT.74
There are systems such as MnO, NiO, and UO2 (refs 53,66) for which the high-temperature phase is paramagnetic, and the low temperature phase is antiferromagnetic. For these systems, it is too expensive to calculate the phonons of the high-temperature phase accurately. An alternative approximation to calculate the phonons of the high-temperature phase is to use the force constants calculated from the corresponding low-temperature phase. In doing so, one primary problem to solve is the symmetry broken by the magnetic degree of freedom. Using UO2 as an example, the primitive cell of the antiferromagnetic structure contains 6 atoms resulting in 18 phonon dispersions, whereas the primitive cell of the paramagnetic structure contains 3 atoms resulting in 9 phonon dispersions. A solution to the problem is to restore the symmetry by a transformation as
Phonon dispersions of UO2. Open circles: measured data by Pang et al.;52 solid squares: measured data by Dolling et al.110 The lines represent the present calculations. The (red) dot-dashed lines along the [ζζ0] direction emphasize the three phonon dispersions that were not reported by the calculation of Pang et al.52
PHON is an open-source code developed by Alf21 to calculate phonon frequencies following the direct approach by Parliński et al.17 ATAT is a generic name that refers to a collection of open-source alloy theory tools developed by van de Walle et al.4,22 For phonon calculations, it appears that neither codes can yet handle the vibration-induced polarization effects.
The codes widely in use include the open-source packages PWSCF/QUANTUM ESPRESSO,14 and ABINIT,15 and the commercial packages CASTEP,13 CRYSTAL28 and VASP.11,12 For phonon calculations of polar materials, PWSCF, QUANTUM ESPRESSO, ABINIT and CASTEP employ the linear-response approach. If one wants to calculate phonons of a polar solid using CRYSTAL (starting from CRYSTAL1428), then the mixed-space approach is the only choice.
PWSCF is one of the core packages of open-source QUANTUM ESPRESSO.14 We employed PWSCF to perform the linear-response calculations of phonon properties of Ni, Al, NiAl and Ni3Al,5 and MgO.129 Linear-response approach is also employed for to phonon calculations by ABINIT,15 which is an open-source package using pseudopotentials and a planewave basis. The linear-response approach relies on the availability of pseudopotentials in specific formats. For example, the CASTEP13 code requires the use of the Norm-conserving pseudopotentials.130 To account for the polar effects on phonon calculations, these linear-response codes commonly use the following computational procedure14,19,130 for the evaluation of the dynamical matrix at an arbitrary q points:
For an arbitrary q point outside the coarse wave-vector grid, make forward Fourier-transform of the interatomic interaction constants followed by re-adding the long-ranged coulombic contribution to get the dynamical matrix at the arbitrary q points.
This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-FG02-07ER46417 (Wang and Chen) and by National Science Foundation (NSF) through Grant Nos. DMR-1310289 and CHE-1230924 (Wang, Shang, Fang, and Liu). First-principles calculations were carried out partially on the LION clusters at the Pennsylvania State University, partially on the resources of NERSC supported by the Office of Science of the U.S. Department of Energy under contract No. DE-AC02-05CH11231, and partially on the resources of XSEDE supported by NSF with Grant No. ACI-1053575.
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