Abstract
The longwavelength behavior of vibrational modes plays a central role in carrier transport, phononassisted optical properties, superconductivity, and thermomechanical and thermoelectric properties of materials. Here, we present general invariance and equilibrium conditions of the lattice potential; these allow to recover the quadratic dispersions of flexural phonons in lowdimensional materials, in agreement with the phenomenological model for longwavelength bending modes. We also prove that for any lowdimensional material the bending modes can have a purely outofplane polarization in the vacuum direction and a quadratic dispersion in the longwavelength limit. In addition, we propose an effective approach to treat invariance conditions in crystals with nonvanishing Born effective charges where the longrange dipoledipole interactions induce a contribution to the lattice potential and stress tensor. Our approach is successfully applied to the phonon dispersions of 158 twodimensional materials, highlighting its critical relevance in the study of phononmediated properties of lowdimensional materials.
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Introduction
The vibrations of atoms inside crystalline solids are fundamental to many aspects of condensed matter physics and materials science. The lattice vibrations play a crucial role in a wealth of phenomena, including phase transitions^{1}, phononmediated superconductivity^{2,3}, and electronic^{4} and thermal transport^{5,6}. The latticedynamical theory of solids was established at the beginning of the twentieth century with seminal works from Einstein, Born, von Kármán and Debye^{7,8,9,10}. With advances in densityfunctional theory (DFT) and densityfunctional perturbation theory (DFPT)^{11,12}, the parameterfree calculations of phonon dispersions from first principles have become a routine task implemented in many software packages^{13,14,15,16,17,18,19}. However, there still exist some controversies in the calculated phonon dispersions of lowdimensional (LD) materials, where the longwavelength behavior of flexural phonons are not fully understood. Firstprinciples calculations performed by different groups showed either a linear or quadratic dispersion relation for the flexural acoustic (ZA) modes in twodimensional (2D) materials and two bending modes in onedimensional (1D) materials (see the detailed discussion in Ref. ^{20}). It has been recently argued^{19,20,21} that the lack of rotational invariance in the harmonic lattice Hamiltonian with nonvanishing external stress from DFT calculations is responsible for the linear dispersion of the ZA mode; it becomes quadratic after including rotational invariance and vanishing stress condition. Like translational invariance, there are various reasons for the broken rotational invariance in the interatomic force constants (IFCs) obtained from DFT and DFPT; these include incomplete basis sets, insufficient Brillouin zone sampling or numerical stability of algorithms. In contrast, IFCs obtained from semiempirical force fields or models do not typically suffer from such issues, since they can satisfy translational and rotational invariances by construction^{22,23,24,25}.
Translational and rotational invariances drive the conservation of total momentum and angular momentum in crystals^{26,27} (Noether’s theorem^{28}); these are known as BornHuang invariance conditions^{10,29}. In the 1960s, several works from Keating^{30,31}, Gazis and Wallis^{32} questioned the scope of such conditions and derived additional conditions for a harmonic lattice to be rotationally invariant. From the requirement of unchanged potential energy under any arbitrary rotation operation, Gaizs and Wallis^{32} derived additional constraints for the rotational invariance of a harmonic lattice. In a similar way, through the potential energy expressed in terms of the rotation parameter up to the second order, Keating^{30,31} also formulated a new set of equations for harmonic IFCs of the simple cubic lattice. However, Sarkar and Sengupta^{33} demonstrated that these are both equivalent to the BornHuang invariance conditions, which are complete. Finally, Pick, Cohen, and Martin^{34} derived the connection between the microscopic theory and the phenomenological approach, and the microscopic acoustic sum rule for bulk crystals.
In this work, we investigate important aspects of the BornHuang invariance conditions as well as the socalled equilibrium conditions (i.e. Huang conditions) in determining the phonon dispersions of materials and their consequence for LD materials. We introduce the polar BornHuang invariances and polar Huang conditions for harmonic IFCs in infraredactive solids where the existing longrange forces from the dipoledipole interactions need to be treated separately. Indeed, Huang had already demonstrated this in the case of bulk ionic crystals^{35,36}, in which the longrange Coulomb interactions need a special treatment. However, this condition seems to have been omitted in recent investigations on the rotational invariance of 2D materials^{19,20,21}. The detailed implementation of these conditions for correcting the IFCs either from realspace small displacements or reciprocalspace DFPT calculations are then presented. Last, based on the secular equation for the longwavelength vibrations, we demonstrate that the bending modes in any LD material should have a purely outofplane polarization in the vacuum direction and a quadratic dispersion in the longwavelength limit. We apply the present approach to silicon, triclinic calcium triphosphide (CaP_{3}), graphene, 2D molybdenum disulfide (MoS_{2}), 1D singlewall (8,0) carbon nanotube (CNT) and (4,4) boron nitride nanotube (BNNT), and find that the fulfillment of the rotational invariance and vanishing stress condition is crucial in LD materials. We also verify that these conditions have negligible effects in 3D materials, as expected. Finally, these invariance conditions are applied to correct the phonon dispersions of 245 candidate 2D materials from the highthroughput database of Ref. ^{37}. In that study, the authors used a simple sum rule, resulting in 187 phonon dispersions presenting soft modes or an incorrect linear dispersion of the ZA mode. With the rotational invariance and equilibrium conditions correctly applied, the phonon dispersions of 158 materials become real and display a quadratic ZA branch in the longwavelength limit. The remaining 87 materials are either dynamically unstable (54) or require tighter numerical convergence (33).
Results and discussion
BornHuang invariance conditions
The vibrational properties of solids can be determined by the nuclear Hamiltonian constructed based on the secondorder IFCs. Neglecting higherorder anharmonic contributions, a Taylor expansion of the BornOppenheimer potential energy E in the atomic displacements up to the second order gives:
where E_{0} is the total potential energy of the chosen equilibrium reference structure, and Φ_{ϰα} = ∂E/∂u_{ϰα} and \({\Phi }_{\varkappa \alpha ,\varkappa ^{\prime} \beta }={\partial }^{2}E/\partial {u}_{\varkappa \alpha }\partial {u}_{\varkappa ^{\prime} \beta }\) are the corresponding first and secondorder IFCs, respectively, with atomic indices ϰ and \(\varkappa^{\prime}\) in the Bornvon Kármán supercell and the Cartesian directions α and β; the atom index ϰ ≡ {l, κ} denotes the atomic index κ in the unit cell l, and u_{ϰα} is the corresponding atomic displacement from the equilibrium position. Apart from the space group symmetry of the given crystal type and permutation symmetry, these IFCs are also required to satisfy both the global translational and rotational invariances due to the conservation of total crystal and angular momenta.
The BornHuang invariance conditions provide the acoustic sum rules for both the translational and rotational invariances of a harmonic lattice. These constraint equations on the first and secondorder IFCs read^{27}:
where τ_{ϰα} is the equilibrium position of atom ϰ in the Cartesian direction α and δ_{αβ} represents the Kronecker delta. In these expressions, Equations (2) and (3) are the acoustic sum rules for the translational and rotational invariances of the firstorder IFCs, while Equations (4) and (5) are their counterparts for the secondorder IFCs. When atoms are at their equilibrium positions, this set of constraint equations can be derived by adding a global translation or an infinitesimal rotation to all atoms in the system, which results in an unchanged lattice potential and zero forces acting on atoms. These acoustic sum rules correspond to the conservation of total momentum and angular momentum in the crystal. In general, the rotational sum rules link the (n + 1)th order to the nth order of the Taylorexpanded potential^{27}. For instance, Eq. (5) links the secondorder IFCs to the firstorder.
Equilibrium conditions
For an infinite crystal, the lattice sites are perfectly periodic throughout the whole bulk region, and this requirement has two important consequences. One is the total force on each atom must be zero, which means that the firstorder IFCs in Eq. (1) vanish at the equilibrium. This condition gives rise to the BornHuang invariances imposed on the secondorder IFCs as being rotationally invariant:
The other consequence is the vanishing of the stress tensor that governs the equilibrium volume of the unit cell, eliminating any surface effect as an infinite lattice. The stress tensor for a finite lattice is given by^{27,33}
where Ω is the volume of a finite lattice, and the firstorder IFCs Φ_{ϰα} account for the surface forces and vanish in the bulk region^{27}. In order to derive the stress tensor for an infinite lattice, we multiply the rotational invariance, Eq. (5), by the atomic position τ_{ϰδ} and sum over ϰ. Then, using Eq. (7) we arrive at
where we have introduced
Since T_{αδ,βγ} might not be symmetric near the surface, we define
With the help of the translational invariance, Eq. (4), we can further express Eq. (11) as
where \({\tau }_{\varkappa \varkappa ^{\prime} \gamma }\equiv {\tau }_{\varkappa ^{\prime} \gamma }{\tau }_{\varkappa \gamma }\) is the distance between two atoms along the Cartesian direction γ. By construction, the symmetrized tensor has the following symmetry relations
To derive the stress tensor for an infinite crystal, we further subtract both sides of Eq. (10) by the quantity Ωσ_{γδ}δ_{αβ} to obtain
where the symmetry of the stress tensor σ_{γδ} = σ_{δγ} is used. From Equations (8) and (9), it can be noted that the righthand side of Eq. (14) is invariant by interchanging α with δ and also β with γ. Therefore, the lefthand side of Eq. (14) has the following symmetry
which is the expression of the stress tensor for infinite crystals^{27}. At equilibrium, the stress tensor vanishes, which gives the Huang conditions \({T}_{\alpha \beta ,\gamma \delta }^{{{{\rm{sym}}}}}={T}_{\gamma \delta ,\alpha \beta }^{{{{\rm{sym}}}}}\):
In general, there are 36 Huang conditions due to the fourthrank tensor \({T}_{\alpha \beta ,\gamma \delta }^{{{{\rm{sym}}}}}\), which can be reduced to 15 by exploiting the symmetry relations in Eq. (13). For the anisotropic components of stress tensor, they become
for α ≠ β. We emphasize here that the Huang conditions are not the constraints for a potential to be rotationally invariant but the ones for a vanishing stress tensor. The set of Equations (4), (6) and (16) constitute the complete description of invariance conditions for the lattice dynamics of crystalline solids at the equilibrium. Leaving the translational invariance aside, we focus on the rotational invariance, Eq. (6), and the equilibrium conditions, Eq. (16), which we collectively refers to as invariance conditions.
Invariance conditions to interatomic force constants
In practice, the realspace IFCs can be obtained by fitting the DFT forces using the small displacement method^{13} or a Fourier transformation within DFPT^{11,12}. The IFCs are then transformed into reciprocal space on dense momentum grids, giving rise to the dynamical matrices. This can be achieved by a Fourier interpolation of the IFCs.
The small displacement method fits the IFCs in the Taylorexpanded interatomic forces:
This can be recast in matrix form as a linear regression problem^{38}
where \({\mathbb{A}}\) is the displacement matrix, Φ is an IFC vector, and \({\mathbb{C}}\) is null space constructed from all symmetry constraints on the IFCs. The constraints contain those from space group symmetry, permutation symmetry and the invariance conditions discussed so far. Through the null space \({\mathbb{C}}\), the IFCs Φ can be projected into the independent IFC parameters ϕ. Then, a leastsquare or compressive sensing technique^{38} can be applied to determine the unknown parameters ϕ, and the IFCs Φ are further obtained via \({{{\mathbf{\Phi }}}}={\mathbb{C}}\cdot {{{\boldsymbol{\phi }}}}\).
In DFPT calculations, the IFCs are corrected to make them fulfill the invariance conditions and then Fourier interpolated. Two approaches exist to correct the IFCs. The first one consists in projecting the IFCs onto a subspace, spanned by the vector basis of the invariance conditions and obtaining the solution by nearest distance minimization^{39}. In order to get the projection Φ^{p} of the IFCs in such a subspace, the orthogonal basis set {g_{i}} of the subspace needs to be built from the constraint matrix of invariance conditions and yields
which is the nearest distance between the original and the corrected IFCs. The corrected IFCs Φ^{c} that satisfy the invariance conditions are obtained via Φ^{c} = Φ − Φ^{p}. We have implemented the invariance conditions through this optimal projection of IFCs in the QUANTUM ESPRESSO distribution^{15,40}.
The second approach adds a correction to the IFCs and then minimizes the correction to remain as close as possible to the original IFCs. We define the deviation from the invariance conditions as
where \({\mathbb{I}}\) is the constraint matrix of the invariance conditions. The null space \({{\mathbb{C}}}^{\prime}\) introduced in Eq. (22) ensures that any correction made will not break other symmetries, such as space group symmetry. The prime on null space indicates that \({{\mathbb{C}}}^{\prime}\) is evaluated excluding the invariance conditions that we want to correct. We apply a small correction Δϕ such that
which means that \({\mathbb{I}}\cdot {{\mathbb{C}}}^{\prime}\cdot \Delta {{{\boldsymbol{\phi }}}}={{{\bf{d}}}}\). We can solve this problem efficiently by utilizing ridge regression^{41} with the help of a penalty term \(\mu \parallel \Delta {{{\boldsymbol{\phi }}}}{\parallel }_{2}^{2}\) where ∥…∥_{2} denotes the ℓ_{2}norm and μ is a hyperparameter with values usually in the range 10^{−2} to 10^{−6}, which ensures that the correction Δϕ on the IFCs is as small as possible:
A similar implementation that applies ridge regression to impose the invariance conditions on the IFCs can be found in the HIPHIVE software^{19}.
In the case of crystals, the periodicboundary conditions need to be taken into account in order to impose the invariance conditions properly, and this can be tackled easily by constructing the WignerSeitz supercell. Thanks to the periodicity of the lattice and translational invariance, all the interactions are considered between the atoms in the chosen central unit cell and the other atoms in the supercell. To construct the WignerSeitz supercell, their relative distances are recalculated as the nearest periodic image. In the case of equal distances among their periodic images, a weight inversely proportional to its cardinality is given. As a result, the sum of the weights is unity because all of the atom pairs in the supercell contribute uniquely to the potential energy. This approach is similar to the one implemented in the EPW code^{42,43} to improve the accuracy of the interpolated dynamical matrix, which optimizes atomic pairs to ensure that the distance between each two of atomic positions are within a cutoff radius.
Fourier interpolation of interatomic force constants
The dynamical matrix \({D}_{\kappa \alpha ,\kappa ^{\prime} \beta }({{{\bf{q}}}})\) is computed via a Fourier interpolation of the realspace IFCs as:
where q is a phonon wavevector defined on an arbitrary grid in reciprocal space. Due to the transitional invariance, the IFCs \({\Phi }_{\varkappa \alpha ,\varkappa ^{\prime} \beta }\) in crystalline solids depend only on the relative position between the two atoms ϰ and \(\varkappa ^{\prime}\). Therefore, it is convenient to consider only the IFCs \({\Phi }_{\kappa \alpha ,\kappa ^{\prime} \beta }({{{\bf{R}}}})\) between the atom labelled by κ within the reference unit cell 0 and the atom labelled by \(\kappa ^{\prime}\) within another unit cell R.
However, in the case of infraredactive materials, the atomic vibrations generate longrange Coulomb interactions that can deteriorate the quality of the Fourier interpolations^{11,12,44}. A common strategy^{12} to overcome this problem is to split the dynamical matrix into a shortrange (\({{{\mathcal{S}}}}\)) and longrange (\({{{\mathcal{L}}}}\)) component as:
such that the shortrange part is a smooth function of q. If we consider only dipoledipole interactions^{12,45}, the longrange part takes the following form:
which is nonanalytic at q = 0 and induces the directiondependent LOTO splitting. To deal with the q = 0 term exactly, the Ewald summation technique^{11,12} is adopted in practice with a sum over reciprocal G vectors. In this expression, e is the elementary charge unit, Ω_{0} is the volume of unit cell, \({{{{\bf{Z}}}}}_{\kappa }^{* }\) is the Born effective charge tensor of the atom κ within the unit cell, and W_{c}(q) is the screened Coulomb potential which depends on the dimensionality of the system as^{44,46,47}
where ϵ is the ionclamped dielectric tensor of 3D system, r_{eff} ≈ ϵb/2 is the inplane matrix of effective screening length, b is the thickness of the 2D material, ϵ_{1D} is the 1D dielectric constant, t is the effective screening radius, I_{n}(x) and K_{n}(x) are the nthorder modified cylindrical Bessel functions, and \({G}_{pq}^{mn}\) is the Meijer Gfunction^{48}. The 1D longrange formula, Eq. (30), has been recently derived and more details can be found in Ref. ^{47}. The longrange part is substracted from the dynamical matrix on the coarse grid, the remaining shortrange part is Fourier transformed into real space to get the IFCs.
Inspired by this approach, we formulate the polar BornHuang invariances and polar Huang conditions by separating the total IFCs into shortrange and longrange IFCs,
and
Since the \({\Phi }^{{{{\mathcal{L}}}}}\) are analytically defined, we apply the corrections due to the invariance conditions only on \({\Phi }^{{{{\mathcal{S}}}}}\). Due to the presence of the longrange terms, we have to solve an inhomogeneous set of linear equations which is here achieved by using a ridge regression method^{41}. We then Fourier interpolate the resulting corrected shortrange IFCs at arbitrary momentum points and add back the analytic longrange part. This approach is an alternative to applying the BornHuang invariances, Eq. (6), and Huang conditions, Eq. (16), on the total realspace IFCs. We have verified in Fig. 4 that both methods give the same interpolated 2D phonon band structures. We also reiterate that we have neglected dynamical quadrupoles in the longrange treatments which have been recently shown to be important in some cases^{43,45,46,49} and have been left for future works.
Finally, in the case of lowsymmetry infraredactive solids, we have observed that the total dynamical matrices are not guaranteed to be Hermitian. As observed in Ref. ^{50}, the dipoledipole IFC ansatz (e.g. Eq. (4) of Ref. ^{51} or Eq. (70) of Ref. ^{12}) which is used in the analytic form of the longrange dynamical matrix does not guarantee Hermiticity for a general crystal. Zhou et al.^{50} addressed this issue by adding a realspace shortrange correction to the IFC matrix. Here, for simplicity, we directly symmetrize the longrange part of the onsite dynamical matrix by its conjugate transpose and Fourier transform both the shortrange and longrange parts into real space. We then apply the invariance conditions on the total IFCs and subsequently substract the symmetrized longrange in real space. Last, we Fourier interpolate the shortrange IFCs to arbitrary momentum and add the longrange part symmetrized in the same way.
Bending modes in 2D materials
The lattice dynamics of crystalline solids is governed by the secular equation^{1,10}
where m_{κ} is the mass of the κth atom within the unit cell. The eigenvector e_{ν,κα}(q) is obtained by diagonalizing the massscaled dynamical matrix \({D}_{\kappa \alpha ,\kappa ^{\prime} \beta }({{{\bf{q}}}})/\sqrt{{m}_{\kappa }{m}_{\kappa ^{\prime} }}\) with the corresponding eigenfrequency ω_{ν}(q) for the phonon mode ν at momentum q, and ν ranges from 1 to 3n with n atoms in the unit cell. To analyze the dispersion relation of bending modes in the longwavelength limit (q → 0), we perform a Taylor expansion of Eq. (33) in terms of q. The resulting longwavelength equation for lattice vibrations is^{10,21}:
where \({\omega }_{\nu }^{(1)}({{{\bf{q}}}})\equiv \frac{\partial {\omega }_{\nu }({{{\bf{q}}}})}{\partial {{{\bf{q}}}}}{\left\right.}_{{{{\bf{q}}}} = 0}\cdot {{{\bf{q}}}}\). In this equation, u_{να} is an arbitrary polarization vector which is introduced via the solution for the zeroorder equation of the Taylor expansion, \({e}_{\nu ,\kappa \alpha }^{(0)}({{{\bf{q}}}})=\sqrt{{m}_{\kappa }}{u}_{\nu \alpha }\). The \({T}_{\alpha \beta ,\gamma \delta }^{{{{\rm{sym}}}}}\) defined in Eq. (12) can be rewritten in term of a sum over primitive cells as
and \({T}_{\alpha \gamma ,\beta \delta }^{{{{\rm{int}}}}}\) is the contribution associated with the relaxation of internal coordinates in response to external stress fields^{10}:
with the ν × ν matrix \({\Gamma }_{\kappa \lambda ,\kappa ^{\prime} \mu }\) denoting the inverse of the massscaled IFC matrix \({\sum }_{{{{\bf{R}}}}}{\Phi }_{\kappa \lambda ,\kappa ^{\prime} \mu }({{{\bf{R}}}})/\sqrt{{m}_{\kappa }{m}_{\kappa ^{\prime} }}\). Unfortunately, the IFC matrix is singular because of the acoustic sum rules for translational invariance and the \({\Gamma }_{\kappa \lambda ,\kappa ^{\prime} \mu }\) is thus introduced as
where the pseudoinverse \({\hat{\Gamma }}_{\kappa \lambda ,\kappa ^{\prime} \mu }\) is calculated and satisfies the relation^{10}
Furthermore, the tensor \({T}_{\alpha \gamma ,\beta \delta }^{{{{\rm{int}}}}}\) satisfies the following symmetry relations^{10,36}:
which result from the rotational invariance, Eq. (6), and the fact that \({\Gamma }_{\kappa \lambda ,\kappa ^{\prime} \mu }={\Gamma }_{\kappa ^{\prime} \mu ,\kappa \lambda }\).
For the ZA bending modes of 2D materials in the longwavelength limit, if they were not purely polarized along the vacuum direction z and coupled with the inplane vibrations, the leading linear term in the ZA dispersion would read:
where the phonon band index ν ≡ ZA is omitted for clarity. By using the Huang conditions and the symmetry of \({T}_{\alpha \gamma ,\beta \delta }^{{{{\rm{int}}}}}\), the identity \({T}_{z\beta ,\gamma \delta }^{{{{\rm{sym}}}}}+{T}_{z\gamma ,\beta \delta }^{{{{\rm{int}}}}}={T}_{\gamma \delta ,z\beta }^{{{{\rm{sym}}}}}+{T}_{\gamma z,\delta \beta }^{{{{\rm{int}}}}}\) holds. As demonstrated in Supplementary Note 1^{52}, we have \({T}_{\gamma \delta ,z\beta }^{{{{\rm{sym}}}}}+{T}_{\gamma z,\delta \beta }^{{{{\rm{int}}}}}=0\) even in case of nonpurely outofplane modes if the rotational invariance of the lattice potential is fullfilled and the system is free of stress, leading to the vanishing linear dispersion term. In the longwavelength limit, the dispersion of ZA modes thus becomes quadratic with a purely outofplane polarization. Having proven this, we can therefore define the bending conditions as:
which can be imposed in case of numerical noise in the calculations to ensure a quadratic dispersion of the flexural modes.
More generally, these conditions for recovering quadratic flexural modes are also applicable to 1D materials, where there are two acoustic bending modes polarizing in two vacuum directions. In 1D materials with periodicity in the z direction, the bending conditions thus become \({T}_{\gamma \delta ,x\beta }^{{{{\rm{sym}}}}}={T}_{\gamma x,\delta \beta }^{{{{\rm{int}}}}}\) and \({T}_{\gamma \delta ,y\beta }^{{{{\rm{sym}}}}}={T}_{\gamma y,\delta \beta }^{{{{\rm{int}}}}}\) for polarizations in the x and y directions, respectively. We note that if LD materials are subjected to external stress fields, the linear term will then dominate the phonon dispersion in the longwavelength limit. In bulk materials, no such bending conditions exist due to the periodicity in all three directions.
Application to bulk materials
The influence of the rotational invariance on the lattice dynamics of bulk materials is almost unexplored. We show the phonon dispersion of silicon in Fig. 1 and find that both the BornHuang invariances (red line) alone and the combination of BornHuang invariances and Huang conditions (blue line) have little effect, compared with the phonon dispersion obtained from the IFCs without the aforementioned conditions imposed (black dashed line). This result is not surprising given that the IFCs of highsymmetry bulk crystals with space inversion automatically fulfill rotational invariance^{53}.
As an example of infraredactive bulk materials, we calculate the phonon dispersion of triclinic CaP_{3} shown in Fig. 2, with the LOTO splitting included. The crystal symmetry of CaP_{3} with the space group of \({{{\rm{P}}}}\bar{1}\) is low: only the identity and the spatial inversion operations. For a 2 × 2 × 2 supercell with 64 atoms, there are 160 unique atomic pairs which means 1440 IFC components. The total number of independent IFC parameters in CaP_{3} is 1215 after considering the space group symmetry, the permutation symmetry and the translational invariance. When we add the BornHuang invariances on the IFCs, the number of independent IFCs is reduced to 1179. The number of independent IFC parameters can further decrease to 1164 if the Huang conditions are included as symmetry constraints. Besides, there is a large deviation from these two invariance conditions reflected by the Frobenius norm of d in Eq. (22); it vanishes with a magnitude of 10^{−6} after the invariance conditions are enforced, which indicates that the BornHuang invariances and Huang conditions are now satisfied. However, the phonon dispersions of CaP_{3} in Fig. 2 are almost unchanged. There is only a small modification of the lowestlying transverse acoustic (TA) branch. Hence, we conclude that the violation of the rotational invariance does not affect the phonon dispersion of bulk materials at equilibrium, and the corrections of the invariance conditions on the IFCs of bulk materials are not important, even for lowsymmetry crystals. This can be rationalized from the fact that bulk materials are periodic along all directions and thus do not have any rigid rotation. Nonetheless, for the elastic tensor c_{ij} (in Voigt notation) determined directly from the secondorder IFCs, the broken symmetry relation (i.e. c_{ij} ≠ c_{ji}) of lowsymmetry solids can be observed due to the violation of the rotational invariance and equilibrium conditions^{53}.
Application to 2D materials
As revealed by a few recent studies^{19,20,21}, the violation of the rotational invariance and vanishing stress condition are expected to have a remarkable effect on the latticedynamical properties of 2D materials, which can lead to an unphysical dispersion relation of the outofplane ZA branch. The influence of such two invariance conditions on the phonon dispersions of infraredinactive 2D materials is illustrated in Fig. 3, where we take graphene as a representative material. In general, the lack of these two invariance conditions can result in a ZA mode either displaying imaginary frequencies or a nonquadratic dispersion relation in the longwavelength limit. We confirm these findings by fitting the IFCs obtained from randomly displaced configurations. As can be seen in the inset of Fig. 3, the pristine phonon dispersion (black dashed line) of graphene without the BornHuang invariances and Huang conditions exhibits a linear ZA branch close to the zone center. When we add the BornHuang invariances to make the IFCs satisfy the global rotational invariance, the result (blue line) remains unchanged with respect to the pristine one. This happens because the crystal symmetry of graphene is high and the rotational invariance of its lattice potential is automatically satisfied. In contrast, only the red and yellow lines, where the additional Huang conditions are fulfilled, show the expected parabolic dispersion of the ZA branch around the Γ point. More generally, both the BornHuang invariances and Huang conditions must be satisfied to observe the physically correct ZA phonons with quadratic dispersions in the longwavelength limit. To illustrate this, we consider an MoS_{2} monolayer (see Supplementary Fig. 1^{52}), which lacks the additional inversion symmetry. We see that fulfilling the vanishing stress condition alone is not sufficient for obtaining a parabolic ZA branch, and that the rotational invariance is required as well. We therefore deduce that residual stress remains after structural optimization and that this equilibrium condition enforces strict null stress to recover the quadratic dispersion relation of the ZA phonons. In particular, there is still a nonnegligible stress in the vacuum direction even after a highquality structural optimization with a Coulomb cutoff for 2D materials^{44,54}, owing to the existing weak layer interactions from the periodic images. This observation is consistent with the dispersion model of longwavelength bending modes in Eq. (40) which indicates that any external stress will give rise to a nonvanishing linear dispersion term and hence suppresses the nextleading quadratic dispersion. Overall, both the BornHuang rotational invariance and Huang conditions (vanishing stress condition) are of great importance for 2D materials to have the expected behavior of the ZA branch in the longwavelength limit.
Most 2D insulators and semiconductors are infraredactive with nonvanishing Born effective charges and longrange interactions. In such cases, the longrange Coulomb interactions provide additional contributions to the IFCs^{35,36}. The enforcement of the invariance conditions for infraredactive materials should therefore be performed with caution. We take the MoS_{2} monolayer as an example to showcase this special treatment. In 2D materials, the LOTO splitting vanishes at the zone center, and the LO dispersions become linear in q with a finite slope^{44,54}. As shown in Fig. 4, the phonon dispersions of MoS_{2} monolayer calculated directly from the pristine IFCs from DFPT (black dashed line) exhibit small imaginary frequencies in the ZA branch close to the zone center, as a consequence of the broken invariance conditions. If we add the corrections according to the invariance conditions from Equations (6) and (16) on the shortrange IFCs only (blue line), there still exist some imaginary frequencies around the Γ point, belonging to the ZA phonons. This can be understood by realizing that the longrange IFCs indeed contribute to the stress tensor and the rotational invariance of the lattice potential. When the longrange interactions are added back analytically, the resulting total dynamical matrices in reciprocal space will not fulfill the BornHuang invariances and Huang conditions. This is further confirmed by looking the yellow line in Fig. 4, where the nonanalytic correction corresponding to the longrange interactions are removed, and the result presents a stable ZA branch in the parabolic shape near the Γ point (i.e. the shortrange IFCs satisfying the invariance conditions). To deal with the invariance conditions properly in infraredactive solids, we need to consider Eq. (31) for the polar BornHuang invariances and Eq. (32) for the polar Huang conditions to make the total IFCs meet the invariance conditions (green line). The longrange IFCs contributed from the dipoledipole interactions are considered when imposing the invariance conditions on the shortrange IFCs only, and an analytic longrange correction is further introduced back in reciprocal space to recover the LOTO splitting. As indicated by the red line in Fig. 4, an equivalent approach is to apply the normal invariance conditions on the total IFCs, substract the longrange part obtained by Fourier transformation, and Fourier interpolate the resulting shortrange IFCs with an analytic longrange contribution added back. Both approaches yield the correct ZA branch and the 2D LOTO splitting shown in the inset of Fig. 4. Apart from the effects on the LOTO splitting of optical phonons, we observe that the longrange dipoledipole interactions also modify the ZA modes by lowering their energies when comparing the results in yellow and red lines. In the rest of the manuscript, we always use the last approach (red line) in infraredactive materials.
Application to 1D materials
Another important class of LD materials are 1D materials. The effects of the invariance conditions on their latticedynamical properties have not yet received much attention. We choose the singlewall (8,0) CNT and (4,4) BNNT as representative infraredinactive and active 1D cases, respectively. We illustrate in Fig. 5 the corrections from the invariance conditions on the phonon dispersion of (8,8) CNT, where we can see that the results with only the translational invariance applied yield only three vanishing acoustic frequencies at the Brillouin zone center. In 1D systems with more than one atom in the unit cell, four acoustic branches are expected and correspond to the three global translational degrees of freedom in the three cartesian directions and one rotational degree of freedom along the 1D axis. This is correctly recovered when applying the BornHuang invariances as shown with a red line in Fig. 5. As shown previously^{25,39,55,56}, the frequency of the twisting acoustic mode due to the global rotation along the nanotube axis can become finite at the Γ point (around 1.0 THz in our case) if the BornHuang rotational invariance is not enforced. Also, the dispersion relation of the two acoustic branches corresponding to the bending motions becomes parabolic in the longwavelength limit after adding the corrections from the BornHuang rotational invariance. The Huang conditions for vanishing external stress seem to have no influence on the case of singlewall (8,8) CNT, i.e. the blue line coinciding with the red one. The conditions of vanishing stress have been already satisfied after a tight structural optimization. In general, both the BornHuang invariances and Huang conditions should be fulfilled in order to find the correct dispersions of two bending modes in 1D materials. It is the case for the infraredactive (4,4) BNNT shown in Fig. 6, where the two bending modes with quadratic dispersions are only recovered when both invariance conditions are imposed. Furthermore, there are longrange Coulomb interactions due to the nonvanishing Born effective charges in BNNT. The quadratic phonon dispersions of two bending modes and the emergence of twisting acoustic modes with vanishing frequency at the zone center can only be observed with the polar invariance conditions introduced in Equations (31) and (32). Overall, our proposed invariance scheme can be applied to any material ranging from bulk to 2D and to 1D systems to recover exact latticedynamical properties.
2D materials database
A recent highthroughput screening by Mounet et al.^{37} on experimentally known layered compounds has identified 1825 materials that can be easily or potentially exfoliated. The vibrational properties of 245 of these 2D materials were calculated and explored, with all the data available on the Materials Cloud^{57}. However, a large proportion of the calculated phonon dispersions showed imaginary frequencies, which suggests that the predicted 2D materials would be dynamically unstable. Since most of the unstable phonon branches are ZA modes in the zone center region, we apply here our invariance scheme to see if these can be stabilized.
We select four representative 2D materials from the database, namely FeO_{2}, PrOI, InSe and BiOCl, and report their phonon dispersions in Fig. 7. As can be seen, after the invariance conditions are enforced, all of the four phonon dispersions become stable with a quadratic ZA mode in the longwavelength limit. Besides, FeO_{2} and PrOI are metals, while InSe and BiOCl are inraredactive semiconductors with nonzero Born effective charges. The results in Fig. 7 highlight the effectiveness of the invariance conditions described in this work. Finally, our simulations shed a new light on the origin of imaginary frequencies existing in the 2D material database and shows that in most cases this was due to a violation of some invariance conditions. We therefore expect that most 2D materials reported are stable and will be synthesized in the future.
The updated database is available on the Materials Could archive^{58}, and the phonon dispersions for all of 245 materials can be found in Supplementary Information^{52} of this study. After investigation, there are 54 materials that are not dynamically stable, with pronounced imaginary frequencies not located around the Γ point in the database (see Supplementary Table 1^{52}). After these truly unstable 2D structures are excluded, we still have 33 candidates that exhibit an unstable ZA branch with small imaginary frequencies near the Γ point after the invariance conditions are enforced (see the detailed information of these 2D materials in Supplementary Table 2^{52}). There are mainly two reasons why the correction of the invariance conditions fails to recover a stable and quadratic ZA mode at the zone center. One of them is the numerical accuracy of the structure optimization and DFPT calculation, which might not be sufficient to produce accurate realspace IFCs. The other is more subtle: one might need to expand Eq. (1) also in terms of strainelectric field^{59} and include the treatment of higherorder multipolar interactions^{43,45,46,49}, which we neglect here for simplicity. To reveal the first problem, we have selected a few of cases including HfS_{2}, NbF_{4}, CoO_{2}, TaS_{2} and C where the structure optimization and subsequent DFPT were repeated with more stringent calculation parameters. The corresponding phonon dispersions based on these new IFCs with the invariance conditions applied become stable and display a quadratic ZA mode in the longwavelength limit (see the results with the legend this work in Supplementary Note 4^{52}). In a few cases involving magnetic 2D materials, the selfconsistent solution found might not be the lowestenergy ground state. We did not investigate these cases further. For the second case, we observe that black phosphorus and arsenene have a nonvanishing yz components of the Born effective charge tensor, resulting in small soft modes due to the neglect of electric field in calculating IFCs and higherorder longrange electrostatic fields in the interpolation of IFCs, which could be important in the presence of piezoelectricity or flexoelectricity^{45,59,60,61}.
In conclusion, we have presented a systematic study on the invariance and equilibrium conditions in the lattice dynamics of crystalline solids, as well as its implementation in the framework of modern firstprinciples simulations. The results highlight the importance of enforcing both the BornHuang invariances and Huang conditions in order to recover the quadratic dispersion relation of the ZA phonons in LD materials around the Brillouin zone center. This is in agreement with the theoretical model discussed in Ref. ^{21} and this work for bending waves, where the firstorder dispersion vanishes as the bending conditions regardless of the crystal type of LD solids. As expected, such two invariance conditions are found to have negligible influence on the lattice dynamics of bulk materials, even for lowsymmetry crystals. Besides, we have introduced polar BornHuang invariances and polar Huang conditions, which allow to separately treat the shortrange and longrange IFCs for infraredactive solids. This scheme can work effectively in infraredactive 2D materials where the longrange dipoledipole interactions contribute to the fulfillment of the invariance conditions. To demonstrate the effectiveness of the method developed, we apply this invariance scheme to over two hundred 2D material candidates from a highthroughput database, most of which originally displayed an unstable phonon dispersion with the imaginary ZA modes close to the Γ point. With four types of representative 2D materials from the database illustrated here, we demonstrate that most of their phonon dispersions become stable with a physically quadratic ZA branch in the longwavelength limit, after the invariance conditions are imposed. We believe the proposed rotational invariance and equilibrium conditions will have wide applications in the field of LD materials, benefiting future latticedynamical studies of novel materials, and the calculations of their mechanical, thermodynamic, vibrational, and spectroscopic properties.
Methods
The DFT and DFPT calculations are performed with QUANTUM ESPRESSO^{15,40} and the v1.1 SSSP efficiency pseudopotentials’ library^{62}. To describe the exchange and correlation of electrons, we employ the Perdew, Burke, and Ernzerhof’s parametrization for solids (PBEsol)^{63} of the generalized gradient approximation (GGA). In the case of 2D materials, the Coulomb cutoff for 2D system implemented by Sohier et al.^{54} is adopted to remove the longrange interactions in the vacuum direction, which is important to obtain the correct electronic screening and LOTO splitting in 2D materials^{44}. To compute the IFCs directly in real space from the small displacement method, we use an inhouse code which is interfaced with QUANTUM ESPRESSO. The subsequent Fourier interpolation to reciprocal space is obtained with PHONOPY^{14} to compute the phonon dispersions.
Silicon
The optimized lattice parameter is 5.43 Å, with convergence thresholds for pressure and forces smaller than 10^{−2} kbar and 10^{−4} Ry/Bohr, respectively. The same criteria for other materials are used unless stated. To calculate the realspace IFCs, we use a planewave kinetic energy cutoff of 60 Ry with a Γpoint grid computed on a 4 × 4 × 4 supercell and perform a random displacement with the magnitude of 0.01 Å for all atoms. A leastsquare fit of the Taylorexpanded forces to the DFT forces is carried out to find the best solution for the IFCs.
Triclinic CaP_{3}
The structure is relaxed with a planewave kinetic energy cutoff of 30 Ry and a 10 × 10 × 10 electronic grid. Similar to the case of silicon, we build a 2 × 2 × 2 supercell, and the realspace DFT calculations based on a 2 × 2 × 2 grid are carried out for 30 displaced structures to get the entire IFCs. In addition, a DFPT calculation at q = 0 is performed to get the dielectric tensor and Born effective charges.
Graphene
To simulate graphene as a 2D material, a vacuum of 20 Å is set in the outofplane direction to remove the interaction between periodic images. The structure is relaxed using a planewave kinetic energy cutoff of 80 Ry and a 20 × 20 × 20 electronic grid. The optimized lattice constant of graphene is 2.46 Å. To calculate the phonon dispersions, the same realspace procedure is adopted. We create a 6 × 6 × 1 supercell and the corresponding DFT calculations for 10 displaced supercells, with an electronic sampling of 3 × 3 × 3.
2D MoS_{2}
We choose a vacuum of 25 Å to avoid the spurious interactions in the outofplane direction. With a planewave kinetic energy cutoff of 55 Ry and a 14 × 14 × 1 electronic grid, we obtain the optimized lattice constant as 3.14 Å based on tighter relaxation criteria for pressure (smaller than 10^{−3} kbar) and forces (smaller than 10^{−5} Ry/Bohr). For the latticedynamical calculations, we perform a DFPT calculation on a 6 × 6 × 1 phonon grid, and the IFCs are obtained through the inverse Fourier transform of the dynamical matrices.
1D materials
For the singlewall (8,0) CNT, the periodic direction is chosen along the z direction with a vacuum of 20 Å in both the x and y directions, and the lattice constant of the optimized structure is 4.26 Å based on the relaxation criteria of pressure (smaller than 10^{−2} kbar) and force (smaller than 10^{−5} Ry/Bohr). We create a 1 × 1 × 4 supercell and generated 60 displaced configurations using the same procedure as before to obtain the realspace IFCs. The DFT calculations for these displaced supercells are performed with a planewave kinetic energy cutoff of 80 Ry and a Γpoint electronic grid. For the infraredactive singlewall (4,4) BNNT, the IFCs are obtained by DFPT calculations as in Ref. ^{47}, with the appropriate Coulomb cutoff for 1D systems.
Data availability
The material structures, pseudopotentials, interatomic force constants and related data in order to reproduce this study can be found on the Materials Cloud Archive^{58}.
Code availability
The code developed and used in this work is integrated into the Q2R.X and MATDYN.X executables of the QUANTUM ESPRESSO distribution, which will be released in the next version of QUANTUM ESPRESSO.
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Acknowledgements
We would like to thank Norma Rivano and Thibault Sohier for sharing with us the results of Ref. ^{47} prior to publication, and for the calculations of the interatomic force constants of the singlewall (4,4) boron nitride nanotube. We are also grateful to Alexander Croy for providing us with additional information on Ref. ^{21}. We also thank Junfeng Qiao, Elsa Passaro, Giovanni Pizzi, Yanbo Li, Jian Han, Xavier Gonze, Jiongzhi Zheng, Raffaele Resta, Massimiliano Stengel and Miquel Royo for useful discussions. This work is supported by the Sinergia project of the Swiss National Science Foundation (No. CRSII5_189924). S.P. acknowledges financial support from the Belgian F.R.S.FNRS.
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C.L. implemented the invariance conditions and performed the calculations. S.P. and N.M. supervised the project. All authors analyzed the results and contributed to writing the manuscript.
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Lin, C., Poncé, S. & Marzari, N. General invariance and equilibrium conditions for lattice dynamics in 1D, 2D, and 3D materials. npj Comput Mater 8, 236 (2022). https://doi.org/10.1038/s41524022009206
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DOI: https://doi.org/10.1038/s41524022009206
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