## Abstract

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.

## Introduction

Lattice dynamics is the study of the collective atomic vibrations in a crystal. The concept of phonons was introduced by Tamm^{1} in 1930 through an observation of the particle-like energetics of atomic vibrations in a crystal, similar to the wave–particle duality in quantum mechanics. Lattice dynamics has since become an important branch of condensed matter physics and is critical for understanding the thermal properties of crystalline solids at finite temperatures.^{2,3} For example, the phonon densities of states are required for evaluating thermodynamic properties of a crystal,^{4,5,6} such as thermal expansion coefficients, heat capacity, entropy and lattice thermal conductivity.^{7,8,9} There exist excellent reference books on lattice dynamics, e.g., by Wallace^{10} and by Born and Huang.^{2}

With the advances in density functional theory calculations,^{11,12,13,14,15} all the input data needed by lattice dynamics can now be obtained by the first-principles approach solely based on the crystal structure and atomic numbers. Currently, there are essentially two implementations in wide use for the first-principles calculations of lattice dynamics: the linear-response approach^{16} and the direct approach.^{17,18} A general review of first-principles approach to phonon theory was given by Baroni *et al.*^{16} focused on the linear-response approach.^{16,19,20}

The linear-response approach directly evaluates the dynamical matrix at a predetermined reference coarse grid in the wave-vector space through the density functional perturbation theory.^{16,19,20} Then the backward Fourier transform of the calculated dynamical matrix at the coarse wave-vector grid is employed to extract the interatomic force constants on the corresponding real-space grid. In contrast, the direct approach first calculates the force constants using a predetermined reference supercell of the primitive cell. In the literature, the direct approach^{4,16,17,21,22} is also referred as the supercell method, the small-displacement method or the frozen-phonon approach. The features of the linear-response approach and the direct approach are compared in Table 1. A collection of phonon/first-principles codes, including YPHON,^{23} ShengBTE,^{8} PhonTS,^{7} Phonopy,^{18} ALAMODE,^{24} PHON,^{21} ATAT,^{4,22} PHONON,^{17,25,26,27} PWSCF/QUANTUM ESPRESSO,^{14} ABINIT,^{15} CASTEP,^{13} CRYSTAL^{28} and VASP (the Vienna *Ab initio* Simulation Package)^{11,12} that can be employed to calculate phonon and related properties are briefed in Table 2.

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.

For certain optical atomic vibration modes, cations and anions vibrate in opposite directions creating dipole–dipole interactions and hence homogeneous electric fields, which have to be treated with caution in phonon calculations. As an illustration, Figure 1 shows the effects of vibration-induced polarization within supercells. The supercell described in Figure 1a is commensurate with the wavelength of the lattice vibration of interest, so the averaged electric polarization is zero, and thus no macroscopic electric field is generated. It should be pointed out that there are still internal dipole–dipole interactions within an individual supercell, but they are already accounted for in first-principles calculations of the interatomic force constants.^{32,33} In Figure 1b, the supercell is incommensurate with the wavelength of a lattice vibration, and the corresponding phonon produces a nonzero-averaged electric polarization and thus a nonzero artificial macroscopic electric field. Figure 1c shows how the artificial electric polarization varies with the supercell size (or the supercell geometry for the three-dimentional case). In existing literature,^{2,34} polar effects were mainly discussed in the long-wavelength limit. It has been shown that the homogeneous field presents significant difficulties in calculating the phonon frequencies of polar materials for the direct approach under the Born-von Kármán boundary conditions.^{35}

In statistical thermodynamics, a phonon represents a quantized vibrational mode characterized by a frequency at a given reciprocal lattice wave-vector point.^{2,10} An accurate thermodynamic calculation^{4,5,6} requires the phonon frequency distribution, i.e., the phonon density of states, calculated over a fine mesh in the wave-vector space. Although in principle the phonon frequencies at any wave-vector points can be calculated by the first-principles approach, it is still computationally too expensive to account for all mesh points in the wave-vector space. To reduce the computational cost, a practical approach is to first calculate either the force constants for a predefined supercell in the real space or the dynamical matrix for a predefined grid in the reciprocal space,^{16} and followed by Fourier interpolations to evaluate phonon properties at arbitrary wave-vector points. For polar solids, such a strategy has only been implemented in the linear-response approach until recently when a mixed-space method becomes available.^{21,36,37} The mixed-space approach makes it possible to accurately calculate the phonon properties for polar solids within the framework of direct approach for phonon calculations.^{9,28} The mixed-space approach has also been extended to lattice thermal conductivity calculations^{7,8,9,38} where the third-order force constants are needed.

By this review, we will show that (i) the Born-von Kármán boundary conditions^{39} still apply when the phonon frequencies are calculated at the exact wave-vector points;^{40} (ii) The effects of vibration-induced polarization on phonons at an arbitrary wave-vector point can be understood in the real space; and (iii) The longitudinal optical–transverse optical (LO–TO) phonon splitting^{34,40} can be predicted accurately at the exact wave-vector points by the direct approach without explicitly handling the effects of macroscopic electric fields.

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'.

## Basic lattice dynamics of polar solids

As phonons represent waves of collective atomic vibrations in a periodic pattern, we can choose to model a system by a repeated parallelepiped, i.e., a supercell. To formulate atomic vibrations within such a supercell, it is convenient to start from equations of motion for the atoms. For a polar solid, the forces can be divided into two additive contributions—analytic and nonanalytic.^{20} The terms analytic and nonanalytic can be traced back to the mathematical definition^{41} that a function is called analytic if and only if its Taylor series expansion about a reference point converges to the function at each point in some neighbourhood of the reference point, otherwise it is nonanalytic. Under the framework of first-principles calculations, the analytic contribution accounts for all the forces under the restricted periodic boundary conditions under which the averaged electric field is assumed to be zero. The nonanalytic contribution accounts the additional forces owing to a nonzero-averaged electric field.

The classical Newton’s second law of motion for describing the atomic vibrations of a polar solid is given by
$$\begin{array}{}\text{(1)}& {m}_{j}\frac{{\partial}^{2}{u}_{\alpha}(t,j;P)}{\partial {t}^{2}}=-\frac{\partial E(\mathbf{U})}{\partial {u}_{\alpha}(t,j;P)}+e{\mathbf{Z}}_{\alpha}(j)\cdot \mathbf{E}.\end{array}$$
where *m*_{j} represents the atomic mass of the *j*th atom in the primitive unit cell, *t*, the time, the *α* (*α*=x, y, z) component of the atomic displacement from its equilibrium position of the *j*th atom in the *P*th reference primitive unit cell within a supercell. The first term in the right-hand side of Equation (1) accounts for the analytic force due to the short-range interatomic interaction, where **U** represent the whole set of atomic displacements and *E*(**U**) the total energy. The second term in the right-hand side of Equation (1), with −*e* the electron charge, accounts for the nonanalytic force due to the long-range coulombic interactions shown as the dot product between the Born effective charge (**Z**, a second-rank tensor, i.e., 3×3 matrix) and the averaged electric field (**E**) induced by the atomic vibration. The Born effective charges can be calculated as the change in electric polarization divided by the amount an ion is displaced.^{20,42}

## The analytic contribution in the Harmonic approximation

Within the harmonic approximation, one can truncate the first term *E*(**U**) in the right-hand side of Equation (1) at the second order in Taylor series expansion
$$\begin{array}{}\text{(2)}& E(\mathbf{U})=\frac{1}{2}\sum _{P,Q}^{N}\sum _{j,k}^{{N}_{P}}\sum _{\alpha ,\beta}^{3}{\Phi}_{\alpha \beta}^{jk}(P,Q){u}_{\alpha}(t,j;P){u}_{\beta}(t,k;Q),\end{array}$$
where *N* represents the number of primitive unit cells contained in the supercell, *N*_{P} the number of atoms in the primitive cell *P*, and *α* and *β* the Cartesian components *x*, *y* and *z*. ${\Phi}_{\alpha \beta}^{jk}(P,Q)$ is the real-space interatomic force constant matrix representing the interactions between the *j*th atom within the primitive unit cell *P* and *k*th atom within the primitive cell *Q* under zero macroscopic electric field.

## The nonanalytic contribution due to the vibration-induced electric field

The nonanalytic contribution accounts for the LO–TO splitting, i.e., the removal of degeneracy between the LO and TO phonons at the Brillouin zone center.^{35} In particular, the LO–TO splitting generally depends upon the direction along which the wave-vector approaches the Brillouin zone center (mainly for non-cubic solids),^{43} making the splitting discontinuous.

We now prove that the second term *e***Z**_{α}(*j*)·**E** at the right-hand side of Equation (1) can be explicitly formulated at any wave-vector points, going beyond the long-wavelength limit by Cochran and Cowley.^{34} On the basis of the Born effective charge and atomic displacement, the averaged polarization^{44} can be evaluated by
$$\begin{array}{}\text{(3)}& \mathbf{P}=\frac{e}{N{V}_{\mathrm{P}}}\sum _{k,Q}^{{N}_{P},N}\mathbf{Z}\left(k\right)\cdot \mathbf{u}(t,k;Q),\end{array}$$
where *V*_{P} is the volume of the primitive cell. The vibration-induced polarization is parallel to the direction of the wave-vector $\stackrel{\mathbf{\u02c6}}{\mathbf{q}}$ and can only have effects on the atomic vibrations along the direction of the wave-vector $\stackrel{\mathbf{\u02c6}}{\mathbf{q}}$. 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 $\stackrel{\mathbf{\u02c6}}{\mathbf{q}}$. As a result, the induced electric field by the lattice vibrations can be formulated as^{45}
$$\begin{array}{}\text{(4)}& \mathbf{E}=-\frac{4\pi \mathbf{P}}{\stackrel{\mathbf{\u02c6}}{\mathbf{q}}\cdot {\mathit{\epsilon}}_{\infty}\cdot \stackrel{\mathbf{\u02c6}}{\mathbf{q}}}=-\frac{4\pi e}{N{V}_{P}}\sum _{k,Q}^{{N}_{P},N}\frac{\mathbf{Z}(k)\cdot \mathbf{u}(t,k;Q)}{\stackrel{\mathbf{\u02c6}}{\mathbf{q}}\cdot {\mathit{\epsilon}}_{\infty}\cdot \stackrel{\mathbf{\u02c6}}{\mathbf{q}}},\end{array}$$
where $\stackrel{\mathbf{\u02c6}}{\mathbf{q}}\cdot {\mathit{\epsilon}}_{\infty}\cdot \stackrel{\mathbf{\u02c6}}{\mathbf{q}}$ represents a projection of the macroscopic dielectric constant tensor into $\stackrel{\mathbf{\u02c6}}{\mathbf{q}}$. As **E** is along $\stackrel{\mathbf{\u02c6}}{\mathbf{q}}$, we can also project **Z**_{α}(*j*) into $\stackrel{\mathbf{\u02c6}}{\mathbf{q}}$ in evaluating *e***Z**_{α}(*j*)·**E** and obtain
$$\begin{array}{}\text{(5)}& e{\mathbf{Z}}_{\alpha}(j)\cdot \mathbf{E}=-\frac{4\pi e}{N{V}_{P}}\sum _{k,Q}^{{N}_{P},N}\frac{{\left[\mathbf{Z}(j)\cdot \stackrel{\mathbf{\u02c6}}{\mathbf{q}}\right]}_{\alpha}\left[\mathbf{Z}(k)\cdot \stackrel{\mathbf{\u02c6}}{\mathbf{q}}\right]\cdot \mathbf{u}(t,k;Q)}{\stackrel{\mathbf{\u02c6}}{\mathbf{q}}\cdot {\mathit{\epsilon}}_{\infty}\cdot \stackrel{\mathbf{\u02c6}}{\mathbf{q}}}.\end{array}$$

For a normal mode^{2,10} in which the atoms oscillate at the same frequency, *ω*, in a periodic pattern, the displacements **u**(*t*, *j*; *P*) can be expressed as
$$\begin{array}{}\text{(6)}& {u}_{\alpha}(t,k;P)={u}_{\alpha}(j;\mathbf{q})\mathrm{exp}\{i\mathbf{q}\cdot [\mathbf{R}(P)+\mathbf{r}(j)-i\omega (\mathbf{q})t\left]\right\},\end{array}$$
where **q** is wave-vector, **R**(*P*) the position of the *P*th primitive cell in the supercell, and **r**(*j*) the position of the *j*th atom in the primitive cell.

Then, inserting equation (6) into equation (5), we obtain $$\begin{array}{}\text{(7)}& e{\mathbf{Z}}_{\alpha}(j)\cdot \mathbf{E}=-\frac{4\pi e}{N{V}_{P}}\sum _{k,Q}^{{N}_{P},N}\sum _{\beta}\frac{{\left[\mathbf{Z}(j)\cdot \stackrel{\mathbf{\u02c6}}{\mathbf{q}}\right]}_{\alpha}{\left[\mathbf{Z}(k)\cdot \stackrel{\mathbf{\u02c6}}{\mathbf{q}}\right]}_{\beta}\cdot {u}_{\beta}(k;\mathbf{q})\mathrm{exp}\{i\mathbf{q}\cdot [\mathbf{R}(Q)+\mathbf{r}(k)]-i\omega (\mathbf{q})t\}}{\stackrel{\mathbf{\u02c6}}{\mathbf{q}}\cdot {\mathit{\epsilon}}_{\infty}\cdot \stackrel{\mathbf{\u02c6}}{\mathbf{q}}}.\end{array}$$

## The dynamical matrix

Substituting Equation (2) into Equation (1) together with utilizing Equation (7), the equation of motion for the atoms can be expressed in the reciprocal space, **q**,
$$\begin{array}{}\text{(8)}& -{\omega}^{2}{w}_{\alpha}(j;\mathbf{q})=-\sum _{k}^{{N}_{P}}\sum _{\beta}^{3}{D}_{\alpha \beta}^{jk}(\mathbf{q}){w}_{\beta}(k;\mathbf{q}),\end{array}$$
where we have replaced *u*_{α}(*j*; **q**) by *w*_{α}(*j*; **q**) which is defined as
$$\begin{array}{}\text{(9)}& {w}_{\alpha}(j;\mathbf{q})=\sqrt{{m}_{j}}{u}_{\alpha}(j;\mathbf{q}).\end{array}$$
${D}_{\alpha \beta}^{jk}(\mathbf{q})$ in Equation (8) is called the dynamical matrix,^{10} which takes the form
$$\begin{array}{}\text{(10)}& {D}_{\alpha \beta}^{jk}(\mathbf{q})=\mathrm{exp}\{i\mathbf{q}\cdot [\mathbf{r}(k)-\mathbf{r}(j)\left]\right\}\left[D{A}_{\alpha \beta}^{jk}(\mathbf{q})+D{N}_{\alpha \beta}^{jk}(\mathbf{q})\right].\end{array}$$
$D{A}_{\alpha \beta}^{jk}(\mathbf{q})$ is called the reduced dynamical matrix,^{10} accounting for the analytic contribution under zero-averaged electric field, whereas $D{N}_{\alpha \beta}^{jk}(\mathbf{q})$ results entirely from the effects of the vibration-induced macroscopic field. They have the following forms
$$\begin{array}{}\text{(11)}& D{A}_{\alpha \beta}^{jk}(\mathbf{q})=\sum _{Q}^{N}\frac{{\Phi}_{\alpha \beta}^{jk}(P,Q)}{\sqrt{{m}_{j}{m}_{k}}}\mathrm{exp}\{i\mathbf{q}\cdot [\mathbf{R}(Q)-\mathbf{R}(P)\},\end{array}$$
$$\begin{array}{}\text{(12)}& D{N}_{\alpha \beta}^{jk}(\mathbf{q})=\frac{4\pi e{\left[\mathbf{Z}(j)\cdot \stackrel{\mathbf{\u02c6}}{\mathbf{q}}\right]}_{\alpha}{\left[\mathbf{Z}(k)\cdot \stackrel{\mathbf{\u02c6}}{\mathbf{q}}\right]}_{\beta}}{\sqrt{{m}_{j}{m}_{k}}{V}_{P}\stackrel{\mathbf{\u02c6}}{\mathbf{q}}\cdot {\mathit{\epsilon}}_{\infty}\cdot \stackrel{\mathbf{\u02c6}}{\mathbf{q}}}\phantom{\rule{1mm}{0ex}}\frac{1}{N}\sum _{Q}^{N}\mathrm{exp}\{i\mathbf{q}\cdot [\mathbf{R}(Q)-\mathbf{R}(P)\left]\right\}.\end{array}$$

The choice of the reference *P* is arbitrary due to the translational invariance by which ${\Phi}_{\alpha \beta}^{jk}(P,Q)$ depends on *P* and *Q* only through the difference **R**(*Q*)−**R**(*P*). Note that the prefactor term in the right-hand side of Equation (12)
$$\begin{array}{}\text{(13)}& {\Delta}_{\alpha \beta}^{jk}(\mathbf{q})=\frac{4\pi e{\left[\mathbf{Z}(j)\cdot \stackrel{\mathbf{\u02c6}}{\mathbf{q}}\right]}_{\alpha}{\left[\mathbf{Z}(k)\cdot \stackrel{\mathbf{\u02c6}}{\mathbf{q}}\right]}_{\beta}}{\sqrt{{m}_{j}{m}_{k}}{V}_{P}\stackrel{\mathbf{\u02c6}}{\mathbf{q}}\cdot {\mathit{\epsilon}}_{\infty}\cdot \stackrel{\mathbf{\u02c6}}{\mathbf{q}}}\end{array}$$
is generally not continuous at **q**=0 (except for cubic crystals), which is the main reason why contribution given by Equation (12) is called ‘*nonanalytic*’.

With Equation (8), determining phonon frequencies is reduced to finding the eigenvalues for the secular equation $$\begin{array}{}\text{(14)}& \mathrm{det}\left|{D}_{\alpha \beta}^{jk}(\mathbf{q})-{\omega}^{2}(\mathbf{q})\right|=0.\end{array}$$

There are generally 3*N*_{P} eigenvalues whose roots *ω*_{j}(**q**) (*j*=1, 2, …, 3*N*_{P}), are the normal phonon frequencies.

## Helmholtz energy and quasiharmonic approximation

Once the phonon frequencies are obtained, all the thermodynamic quantities can be calculated using statistical physics without further approximations. Neglecting the electron–phonon coupling and the thermal electronic contributions, it is a well-demonstrated procedure^{5} to decompose the Helmholtz energy *F* of a system at temperature *T* into two additive contributions as follows
$$\begin{array}{}\text{(15)}& F({V}_{P},T)={E}_{c}({V}_{})+{F}_{vib}({V}_{},T),\end{array}$$
where *E*_{c} is the static total energy per primitive unit cell at 0 K, and *F*_{vib} is the vibrational contribution to the Helmholtz energy given by^{46}
$$\begin{array}{}\text{(16)}& {F}_{vib}(V,T)={k}_{\mathrm{B}}T{\int}_{0}^{\infty}\mathrm{ln}\left[2\mathrm{sinh}\frac{\hslash \omega}{2{k}_{B}T}\right]g(\omega ,V)d\omega ,\end{array}$$
where *k*_{B} is the Boltzmann constant and $g(\omega ,V)$ is the phonon density of states.

The term ‘quasiharmonic approximation’ arises from the approach that for a given volume, *F*_{vib}(*V*,*T*) is calculated under the harmonic approximation, and the anharmonic effects are included solely through the volume dependence of the phonon frequency. Once the Helmholtz energy is calculated as a function of volume and temperature, other thermodynamic quantities can be calculated as usual, such as entropy *S*=−(∂*F*/∂*T*)_{V}, enthalpy *H*=*F*+*TS* and so on.

## The mixed-space approach

The procedure presented in the above section represents a combined solution to the phonon problem for polar materials that (i) the long-ranged coulombic interactions are accounted for through Equation (12) in the reciprocal space through **Z** and **ε**_{∞} calculated at **q**=0; and (ii) short-ranged interatomic interactions are accounted for through Equation (11) in terms of ${\Phi}_{\alpha \beta}^{jk}(P,Q)$ by a supercell in the real space. We therefore refer our solution as mixed-space approach.

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 $$\begin{array}{}\text{(17)}& f(\mathbf{q})=\frac{1}{N}\sum _{Q}^{N}\mathrm{exp}\{i\mathbf{q}\cdot [\mathbf{R}(Q)-\mathbf{R}(P)\left]\right\}.\end{array}$$

The values for *f*(**q**) depends on whether **q** is one of the exact wave-vector points^{10,16,47} in the direct approach.^{21,37} The exact wave-vector points satisfy the following condition
$$\begin{array}{}\text{(18)}& {\mathbf{q}}_{ex}\cdot {\mathbf{S}}_{i}=2\pi \times \mathrm{Integer}\end{array}$$
where **S**_{i} with *i*=1, 2, and 3 represent the three lattice vectors of the supercell in the direct approach.

if

**q**is an exact wave-vector point, i.e., at**q**=**q**_{ex}(see Figure 1a). This is the case that no corrections are needed in calculating the dynamical matrix, i.e., the interatomic force constants calculated by the direct approach can be used directly in calculating the dynamical matrix. The internal dipole–dipole interactions within the supercell are already accounted at all exact wave-vector points except for the point at**q**=0.*f*(**q**)≠0 if**q**is not an exact wave-vector point (see Figure 1b). In this case, phonon properties are determined based on the interatomic force constants obtained with a supercell that is incommensurate with the wavelength corresponding to the wave-vector**q**. As a result, an artificial electric field is introduced by the incomplete supercell and must be accounted for as a correction to those interatomic force constants calculated under zero macroscopic electric field.*f*(0)=1. This is the limiting case as**q**→0 of case (ii) in which all the local polarizations within the supercell would be equal to each other. The long-wavelength limit**q**→0 of Equation (12) recovers the results by Cochran and Cowley.^{34}

## Computational procedure

An actual first-principles phonon calculation of polar solids can be summarized as follows:^{13,37}

Calculate the interatomic force constants ${\Phi}_{\alpha \beta}^{jk}(P,Q)$ in the right-hand side of Equation (11) in the real space,

^{4,48}based on a designated supercell; or calculate the dynamical matrix ${D}_{\alpha \beta}^{jk}(\mathbf{q})$ in Equation (8) in the wave-vector space based on a designated wave-vector grid.^{16}Calculate the dielectric constant and Born effective charge tensors used in Equation (13) based on the primitive cell by employing either the linear-response approach

^{42}or the Berry phase expressions of electric polarization.^{44}Combine the dielectric properties from step (ii) with ${\Phi}_{\alpha \beta}^{jk}(P,Q)$ in step (i) to interpolate the phonon frequencies at any wave-vector points.

^{13,37}

It should be reiterated that in step (i), an implicit condition is that the averaged electric field or the macroscopic electric field is zero. The enforcement of the zero macroscopic electric-field condition is due to the periodic condition adopted in most computer codes for the total electronic energy calculations. For step (iii), a tedious procedure exists in several major computer software packages employing the linear-response approach^{20,49} by separating the dipole–dipole interaction from the short-range interactions. In comparison, except for the codes based on the mixed-space approach,^{36,37} other computational implementations^{25,26,27} of step (iii) in the direct approach are inaccurate.

## Phonon software packages have implemented the mixed-space approach

The mixed-space approach has been adopted in a number of software packages, including YPHON,^{23} ShengBTE,^{8} CRYSTAL,^{28} PhonTS,^{7} Phonopy^{18} and ALAMODE.^{24} In these codes, a generalized force constant ${\Psi}_{\alpha \beta}^{jk}(P,Q)$ was introduced as
$$\begin{array}{}\text{(19)}& {\Psi}_{\alpha \beta}^{jk}(P,Q)={\Phi}_{\alpha \beta}^{jk}(P,Q)+{\varphi}_{\alpha \beta}^{jk}(\stackrel{\mathbf{\u02c6}}{\mathbf{q}}),\end{array}$$
where
$$\begin{array}{}\text{(20)}& {\varphi}_{\alpha \beta}^{jk}(\stackrel{\mathbf{\u02c6}}{\mathbf{q}})=\frac{1}{N}\phantom{\rule{1mm}{0ex}}\frac{4\pi e{[\mathbf{Z}(j)\cdot \stackrel{\mathbf{\u02c6}}{\mathbf{q}}]}_{\alpha}{[\mathbf{Z}(k)\cdot \stackrel{\mathbf{\u02c6}}{\mathbf{q}}]}_{\beta}}{{V}_{P}\stackrel{\mathbf{\u02c6}}{\mathbf{q}}\cdot {\stackrel{\mathbf{\u02c6}}{\mathit{\epsilon}}}_{\infty}\cdot \stackrel{\mathbf{\u02c6}}{\mathbf{q}}}.\end{array}$$

Therefore, the evaluation of the dynamical matrix in Equation (14) becomes $$\begin{array}{}\text{(21)}& {D}_{\alpha \beta}^{jk}(\mathbf{q})=\mathrm{exp}\{i\mathbf{q}\cdot [\mathbf{r}(k)-\mathbf{r}(j)\left]\right\}\sum _{Q}^{N}\frac{{\Psi}_{\alpha \beta}^{jk}(P,Q)}{\sqrt{{m}_{j}{m}_{k}}}\phantom{\rule{1mm}{0ex}}\mathrm{exp}\{i\mathbf{q}\cdot [\mathbf{R}(Q)-\mathbf{R}(P)\}.\end{array}$$

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 field^{51} 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.

YPHON^{23} is an open-source code (c++) for the calculations of phonon dispersions and phonon density of states.

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.

The CRYSTAL package performs *ab initio* calculations of the ground state energy, energy gradient, electronic wave function, and properties of periodic systems. Hartree–Fock or Kohn–Sham Hamiltonians (that adopt an exchange-correlation potential following the postulates of the density functional theory) can be used. Like ShengBTE, PhonTS developed by Chernatynskiy and Phillpot^{7} is a code mainly for thermal conductivity calculations. The mixed-space approach has been implemented in PhonTS exampled by the calculation of phonon lifetime and thermal conductivity of UO_{2}.^{52,53}

Phonopy is a Python code developed by Togo^{18} for phonon, thermodynamic properties and thermal conductivity calculations. The mixed-space approach has been implemented in Phonopy for the phonon properties, replacing that by Parliński *et al*.^{25,26,27} In particular, a formulation for the evaluation of the third-order force constants is also derived by Togo *et al.*,^{9} extending the mixed-space approach.

ALAMODE (Anharmonic LAttice MODEl)^{24} is designed for estimating harmonic and anharmonic properties of lattice vibrations (phonons) in solids.

## Recent calculations using the mixed-space approach

## Phonon and thermodynamic properties

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 VASP^{11,12} as input. Examples are firstly shown for several energy conversion and storage materials.

Polar crystal Li_{2}S with the anti-fluorite structure and band gap of 4.4 eV is a major compound in Li-S batteries.^{54} Phonons of Li_{2}S have been calculated^{55} using the mixed-space approach. In addition to a nearly perfect agreement between experimental and calculated dispersions of Li_{2}S, a large LO–TO splitting was found for the T_{1u} infrared (IR) mode of Li_{2}S (~30% and >120 cm^{−}^{1}).

Chalcogenide Cu_{2}ZnSn(S,Se)_{4} (labelled as CZTSSe) with the kesterite structure (space group $P\overline{4}2c$) and band gap 1.0~1.5 eV is a photovoltaic absorber material that has helped achieving significant recent improvement in photovoltaic device cell efficiency (12.6%).^{56} Phonon frequencies at the Γ point and phonon density of state of CZTS were calculated using the mixed-space approach, a 64-atom supercell, and a combined PBEsol^{57} and HSE06. A gap band in the phonon density of state (from 170~250 cm^{−1}) was predicted for CZTS, and especially a negative thermal expansion was suggested at low temperatures (e.g., <50 K) in terms of phonon density of states.

Li*M*PO_{4} (*M*=Mn, Fe, Co, and Ni) compounds with the olivine structure (space group Pnma) are a class of cathode materials and viable alternatives to the conventional cathode LiCoO_{2}.^{58} By means of the mixed-space approach, a X–C functional of GGA+*U *(refs 59,60; *U* is used to account for the strong on-site Coulomb interaction in transition metals), and a 112-atom supercell, a comparative phonon study has been performed for the antiferromagnetic Li*M*PO_{4}, and in turn, the associated thermodynamics and bonding strength (the strongest one being P–O bonding) have been reported.^{58}

Li_{2}CO_{3} with space group *C*2/c and band gap 5.0 eV has been identified as a main component of the solid electrolyte interphase—a passivating film that forms on Li-ion battery anode surfaces.^{61} Phonons of Li_{2}CO_{3} were predicted using the mixed-space approach (including the LO–TO splitting and using a 96-atom supercell).

TiO_{2} with band gap ~3.0 eV has extensive applications such as solar cells, photocatalysts and storage capacitors. Phonon and associated thermodynamics of six TiO_{2} polymorphs including rutile (space group P4_{2}/mnm), anatase (I4_{1}/amd), TiO_{2}-II phase (Pbcn), baddeleyite (P2_{1}/c), orthorhombic I (Pbca), and cotunnite (Pnma) were obtained using the mixed-space approach and the X–C of the local-density approximation (LDA),^{62} indicating that all TiO_{2} polymorphs are dynamically stable^{62} and the pressure-induced phase transitions and the pressure–temperature phase diagrams of TiO_{2} were predicted.^{63}

Besides phonons in energy materials, a long-standing issue has been resolved regarding the occurrence of imaginary phonon frequencies in cubic perovskites. These are in fact spurious as they result from the methodology employed.^{64} For example, in perovskites EuTiO_{3 }(ref. 64) and SrTiO_{3},^{65} a dynamic short-range ordering model using the mixed-space approach as well as the cubic force constants calculated from the low-temperature tetragonal phases was used for phonon calculations. It was seen that the spurious imaginary phonon frequencies in SrTiO_{3} and EuTiO_{3} disappear, resulting in a remarkably good agreement with experiments. A similar idea was used for Mott–Hubbard insulators MnO and NiO,^{66} i.e., the dynamic matrices with the ideal cubic symmetry recovered from the distorted antiferromagnetic structures, the LO–TO splittings estimated using the mixed-space approach, and the strong electron correlations accounted for by the GGA+*U* method,^{59} which produced accurate phonon dispersions for MnO and NiO. For room temperature multiferroic BiFeO_{3},^{67} the challenge in calculating its phonon properties is due to the fact that BiFeO_{3} is a Mott–Hubbard insulator with band gap ~2.5 eV, involving a polar effect and strong correlation among the *d* electrons of Fe. The mixed-space approach together with the GGA+*U* method^{60} accurately predicted the phonon dispersions of BiFeO_{3} and suggested that no gapped magnon modes^{68} exist and contribute to the heat capacity of BiFeO_{3} in the temperature range 5–30 K.

Accurate phonon properties have also been predicted for CaF_{2} and CeO_{2} with the fluorite structure.^{69} CaF_{2} is a typical superionic conductor, its phonons have been studied by the PBE exchange-correlation functional^{70} using a 192-atom supercell. CeO_{2} 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 CeO_{2} have been studied by a HSE06 hybrid functional,^{71,72} showing better accuracy^{69} than the previous predictions from e.g., PWSCF^{73} and ABINIT.^{74}

In addition to the prototype α-Al_{2}O_{3},^{37} phonon and associated thermodynamics were successfully predicted using LDA+*U* and the mixed-space approach for another dense and continuous coating material of Cr_{2}O_{3}.^{75} Furthermore, phonon-related properties have been investigated using the mixed-space approach for many other polar solids such as nanograined half-Heusler semiconductors,^{76} calcium fluoride at high pressure,^{32} the phase diagram of bismuth ferrite,^{33} cubic SiC and hexagonal BN,^{37} ZnO,^{77} UN,^{78,79} Bi_{2}S_{3},^{80} Si, Ge, InAs and GaAs,^{38} GaN,^{81} CrN,^{82} WS_{2},^{83} ZnSe,^{84} ZnS, ZnSe and ZnTe,^{85} SnO_{2},^{86} TaO_{3},^{87} UO_{2},^{52,53,88} CaCO_{3},^{89} Bi_{2}SiO_{5 }(ref. 90) and BaZrO_{3}.^{91} layered antimony telluride,^{92} Phonon transport in SrTiO_{3},^{93} self-consistent phonon calculations for cubic SrTiO_{3} (ref. 94) and Si(Se_{x}S_{1−x})_{2}.^{95}

## Thermal conductivities of polar solids

The main heat carriers in nonmagnetic crystals are phonons and electrons, with phonons dominating in semiconductors and insulators. The phonon contribution to the total thermal conductivity is the lattice thermal conductivity. One important approach to studying phonon transport in solids is the Boltzmann transport equation (BTE).^{96} However, many solutions of BTE rely on the relaxation time approximation along with the Debye approximation, neglecting the true phonon dispersions, and several parameters are introduced to treat different scattering mechanisms. Li *et al.*^{8} and Chernatynskiy *et al.*^{7} implemented parameter-free iterative solutions in their software packages (ShengBTE^{8} and PhonTS^{7}) to solve the BTE based on the inputs from firs principles. The programs compute converged sets of phonon scattering rates and use them to obtain the lattice thermal conductivity and many related quantities. The two main inputs needed by their software packages are sets of second-order (harmonic) and third-order (anharmonic) interatomic force constants (IFCs) for a given crystal structure. Our mixed-space approach has been implemented in both software packages to derive the second-order IFCs, so as to account for the long-range electrostatic interactions in polar compounds. To date, *ab initio* calculations of lattice thermal conductivity have been applied by Li *et al.* for many bulk systems such as Mg_{2}Si, Mg_{2}Sn and Mg_{2}Si* _{x}*Sn

_{1−x},

^{97}two-dimensional systems such as MoS

_{2}(ref. 98) and nanowires made of Si, diamond,

^{99}InAs, AlN and BeO

^{100}under the diffusive boundary conditions. All these applications show excellent agreement with experimental measurements, and an accurate description of polar–polar interactions was found crucial for theoretical predictions to be in line with experiments. As claimed by Li

*et al*.

^{8,100}in their work for InAs of a well-known direct-band gap III–V semiconductor, ‘Therefore it is ideally well suited for validating our approach when isotope scattering and polar bonds are introduced into the picture’. More examples of lattice thermal conductivity calculations have been reported for SnSe,

^{101}MgO, GaAs, SiC, BN, BP, BSb, BAs, BeTe, and BeSe,

^{102}InN,

^{103}phosphorene

^{104,105}and Si, Ge, InAs and GaAs alloys.

^{106}

## Phonon dispersions for systems with symmetry broken by magnetism

There are systems such as MnO, NiO, and UO_{2} (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 UO_{2} 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
$$\begin{array}{}\text{(22)}& {\Phi}_{st}^{\alpha \beta}(\mathrm{high}\phantom{\rule{0ex}{0ex}}\mathrm{symmetry})=\frac{1}{S}\sum _{r=1}^{N}{\mathbf{O}}_{r}^{-1}{\Phi}_{st}^{\alpha \beta}(\mathrm{low}\phantom{\rule{0ex}{0ex}}\mathrm{symmetry}){\mathbf{O}}_{r},\end{array}$$
where ${\Phi}_{st}^{\alpha \beta}$ (high symmetry) represents the force constant matrix of the high-symmetry structure, ${\Phi}_{st}^{\alpha \beta}$ (low symmetry) the force constant matrix calculated from the low symmetry structure, **O**_{r} the space group operation of the high-symmetry structure, and *S* the number of **O**_{r}’s. Figure 2 shows the phonon dispersions of UO_{2}. It is observed that our calculated phonon dispersions show great improvements over the previous calculations^{52,107,108,109} by comparing them with the experimental data.^{52,110}

## Other phonon software packages

PHON is an open-source code developed by Alfè^{21} 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.

PHONON is a commercial code for phonon and thermal properties developed by Parliński *et al*.^{17,25,26,27} As an example, Figure 3 illustrates the phonon dispersions of α-Al_{2}O_{3} calculated using the PHONON package by Lodziana and Parliński^{111} in comparison with those calculated using YPHON^{37} and measured from the inelastic neutron scattering by Schober *et al*.^{112} It can be seen that the application of PHONON code for phonon dispersion calculations of polar solids appears inaccurate due to an artificial implementation of the Gaussian smear extrapolation^{25,26,27} in accounting for the vibration-induced polarization effects. Practically, PHONON code was applied to many polar solids including, for example, ZrO_{2},^{17} alkaline-earth metals and their hydrides,^{113} LiBeH_{3},^{114} CoO,^{115} Fe_{2}SiO_{4} spinel,^{116} Li_{2}O and Li_{2}CO_{3},^{117} LiF,^{118} CsCl and BaCl,^{119} hafnia and zirconia,^{120} PuO_{2},^{121} CeO_{2}, ThO_{2} and (Ce,Th)O_{2} alloys,^{122,123} LiFePO_{4},^{124} MnO,^{125} ZnO,^{126} BeO^{127} and BiFeO_{3}.^{128}

## Software packages for both electronic and phonon calculations

The codes widely in use include the open-source packages PWSCF/QUANTUM ESPRESSO,^{14} and ABINIT,^{15} and the commercial packages CASTEP,^{13} CRYSTAL^{28} 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 CRYSTAL14^{28}), then the mixed-space approach is the only choice.

VASP only calculates the phonon frequencies at the exact wave-vector points. Although VASP can calculate the dielectric constant and Born effective charge tensors, the current version of VASP (version 5.4.1.05 Feb16) does not report the LO–TO splitting.^{34,40}

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 Ni_{3}Al,^{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 CASTEP^{13} 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 procedure^{14,19,130} for the evaluation of the dynamical matrix at an arbitrary **q** points:

Calculate the dynamical matrices at a predetermined reference coarse grid

Remove the long-ranged coulombic contribution from the dynamical matrices that are calculated at the coarse wave-vector grid

Make backward Fourier-transform of the dynamical matrices to obtain the interatomic interaction constants and

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.

These steps were based on the belief from previous conclusions^{14,19} that for solids having TO–LO splitting (e.g.: polar semiconductors), as the force constants in real space involves long-ranged interatomic interactions due to the nonanalytic term, Fourier interpolation is no longer possible. In contrast to this belief, we note that by using the mixed-space approach, the Fourier interpolation is possible and computationally efficient. One example was given by Zhao *et al*.^{80} in their phonon calculations of Bi_{2}S_{3} nanostructures using QUANTUM ESPRESSO. Instead of using the linear-response approach as implemented in QUANTUM ESPRESSO, Zhao *et al*.^{80} employed the mixed-space approach and they found the mixed-space approach is more efficient without sacrificing the accuracy.

## Summary

This paper reviews a mixed-space formulation to treat the contribution of vibration-induced polarization to phonon dispersions in the direct or supercell approach. It decomposes the interatomic force constants into two contributions: one for all internal interactions within the supercell under zero macroscopic electric field and the other one accounts for the effects of nonzero macroscopic electric field arising from supercells that are incommensurate with the wavelengths of the lattice vibrations. The theory naturally gives rise to the analytic and the nonanalytic contributions to the total force constants at any wave-vector points, without the assumption of the long-wavelength limit made by Cochran and Cowley.^{34} It provides a useful methodology separating the dipole–dipole interaction from those short-range interactions for Fourier interpolation.^{16,20,49} It has been successfully applied in calculating the phonon and thermal properties of a wide range of polar materials and implemented in several broadly used software packages for calculating phonon properties.

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## Acknowledgements

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.

## Author information

## Affiliations

### Department of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania, PA, USA

- Yi Wang
- , Shun-Li Shang
- , Huazhi Fang
- , Zi-Kui Liu
- & Long-Qing Chen

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## Competing interests

The authors declare no conflict of interest.

## Corresponding author

Correspondence to Yi Wang.

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