Acoustic Sum Rule Research Articles

Edge states of topological acoustic phonons in graphene zigzag nanoribbons

The tight-binding method is an effective way to study topological insulators. However, an intrinsic element, i.e., the acoustic sum rule (ASR) in phonon systems, has often been overlooked when constructing the Wannier tight-binding phonon Hamiltonian, which results in edge states of acoustic modes rarely observed in previous works. In the present work, we show that the topological acoustic edge states in graphene need to be assessed by ASR. Interestingly, when the inversion-time-reversal symmetry is broken, the distributions of some edge states will mutate from two edges of the nanoribbon to one edge. Moreover, some other topological phonon modes, which are different from previous reports, have also been observed. Our results would shed light on the method for searching topological phonon states, and provide effective strategies for designing phononic devices.

Compressive sensing lattice dynamics. II. Efficient phonon calculations and long-range interactions

We apply the compressive sensing lattice dynamics method to calculate phonon dispersion for crystalline solids. While existing methods such as frozen phonon, small displacement, and linear response are routinely applied for phonon calculations, they are considerably more expensive or cumbersome to apply to certain solids, including structures with large unit cells or low symmetry, systems that require more expensive electronic structure treatment, and polar semiconductors/insulators. In the latter case, we propose an approach based on a corrected long-range force constant model with proper treatment of the acoustic sum rule and the symmetric on-site force constant matrix. Our approach is demonstrated to be accurate and efficient for these systems through case studies of NaCl, ${\mathrm{CeO}}_{2}$, ${\mathrm{Y}}_{3}{\mathrm{Al}}_{5}{\mathrm{O}}_{12}$, and ${\mathrm{La}}_{2}{\mathrm{Fe}}_{14}\mathrm{B}$.

Low-temperature semiconductor band-gap thermal shifts: T4 shifts from ordinary acoustic and T2 from piezoacoustic coupling

At low temperature T, the experimental gap of silicon decreases as E_g(T)=E_g(0)-AT^4. The main reason is electron-phonon renormalization. The physics behind the T^4-power law is more complex than has been realized. Renormalization by intraband scattering requires a careful non-adiabatic treatment in order to correctly include acoustic phonons and avoid divergences from piezoacoustic phonon interactions. The result is an unexpected low T term E_g(0)+A' T^p with positive coefficient A', and power p=4 for non-piezoelectric materials, and power p=2 for piezoelectric materials. The acoustic phonons in piezoelectric semiconductors generate a piezoelectric field, modifying the electron-phonon coupling. However, at higher T, when thermal acoustic phonons of energy hbar v_s q acquire energies comparable to the electronic intermediate state (higher than the band-edge state by hbar^2 q^2 /2m*), the low q and higher q intraband contributions to T^p rapidly cancel, giving little thermal effect. But there is an additional T-dependence from interband effects of acoustic phonons. This turns out to have power law T^4 for both non-piezoelectric and piezoelectric semiconductors. This term can have either sign, but usually reduces the size of gaps as T increases. It arises after cancellation of the T^2 terms that appear separately in Debye-Waller and Fan parts of the acoustic phonon interband renormalization. The cancellation occurs because of the acoustic sum rule.

Atomic force calculations within the all-electron FLAPW method: Treatment of core states and discontinuities at the muffin-tin sphere boundary

We analyze the accuracy of the atomic force within the all-electron full-potential linearized augmented plane-wave (FLAPW) method using the force formalism of Yu et al. [Phys. Rev. B 43, 6411 (1991)]. A refinement of this formalism is presented that explicitly takes into account the tail of high-lying core states leaking out of the muffin-tin sphere and considers the small discontinuities of LAPW wave function, density, and potential at the muffin-tin sphere boundaries. For MgO and ${\mathrm{EuTiO}}_{3}$ it is demonstrated that these amendments substantially improve the acoustic sum rule and the symmetry of the force constant matrix. Sum rule and symmetry are realized with an accuracy of $\ensuremath{\mu}\mathrm{Htr}/{a}_{\mathrm{B}}$.

An improved version of the Green's function molecular dynamics method

This work presents an improved version of the Green's function molecular dynamics method (Kong et al., 2009; Campana and Muser, 2004 [1,2]), which enables one to study the elastic response of a three-dimensional solid to an external stress field by taking into consideration only atoms near the surface. In the previous implementation, the effective elastic coefficients measured at the Γ-point were altered to reduce finite size effects: their eigenvalues corresponding to the acoustic modes were set to zero. This scheme was found to work well for simple Bravais lattices as long as only atoms within the last layer were treated as Green's function atoms. However, it failed to function as expected in all other cases. It turns out that a violation of the acoustic sum rule for the effective elastic coefficients at Γ (Kong, 2010 [3]) was responsible for this behavior. In the new version, the acoustic sum rule is enforced by adopting an iterative procedure, which is found to be physically more meaningful than the previous one. In addition, the new algorithm allows one to treat lattices with bases and the Green's function slab is no longer confined to one layer.

Wannier functions and Born charge tensors of brookiteTiO2

The dynamical Born charge tensor of brookite $\mathrm{Ti}{\mathrm{O}}_{2}$ has been calculated and analyzed in terms of the Wannier functions associated with the O $2s∕2p$ valence states. It is shown that the fundamental $\mathrm{O}{\mathrm{Ti}}_{3}$ triangular complex, whose opposite distortion in rutile and anatase explains the anisotropy of the Born charge in these polymorphs, is also the relevant building block for understanding this quantity in brookite. This compound consists of an equal number of distorted rutilelike and anataselike $\mathrm{O}{\mathrm{Ti}}_{3}$ structural units, which are nonplanar due to the low ${C}_{1}$ site symmetry of oxygen atoms. Born charge tensors associated with the two nonequivalent O sites correspond in terms of values and anisotropies to those for rutile and anatase, and, consistent with the acoustic sum rule, the Ti charge tensor is an ``average'' of those for the two O types. The intermediate character of brookite, between rutile and anatase, should be expected also for other physical properties.

Phonon Dispersions of a Single-Wall (8,0) Carbon Nanotube: Effects of the Rotational Acoustic Sum Rule and of Surface Attachment

ABSTRACTThe lattice dynamics of single-walled carbon nanotubes (SWCNT) is studied from first-principles using density-functional perturbation theory (DFPT) at the GGA-PBE level. The phonon dispersions of a pristine, infinite zigzag (8,0) SWCNT are obtained and the effect of applying the rotational acoustic sum rule on vibrational properties is discussed. Finally we study the effects of covalent functionalizations on the SWCNT phonon frequencies by selectively increasing the effective mass of the carbon atoms that would link to the functional groups.

Approximation of Local‐Field Effects in the Dielectric Formalism of Lattice Dynamics

AbstractThe phonon dispersion is investigated of a crystal where local field effects are taken into account using a first‐order perturbation expansion with respect to the off‐diagonal of the dielectric matrix. Such an expansion is used in other applications in the literature. It is shown that a straightforward application of this approach is not sufficient for a proper description of acoustic phonons in semiconductors and insulators in the limit of long wavelength because the acoustic sum rule is violated. A general expression for the correction term being of second order is given. Furthermore, the relations are illustrated with the phonon dispersion and certain phonon induced polarization charge densities in Si using a suitable model for the dielectric function.

Symmetry properties of the polarizability matrix of an inhomogeneous electron gas

The properties of the polarizability matrix of an inhomogeneous electron gas in nonconducting crystals, are subjected to group-theoretical analysis. The analytical properties of the eigenvalues and eigenvectors of the polarization operator (intrinsic polarizabilities and intrinsic densities) are discussed. It is shown that intrinsic densities transformed with respect to the vector representation of the point symmetry group of the crystal are excited in an external homogeneous electrical field. The acoustic sum rule for a dielectric matrix is obtained as a consequence of invariance relative to an infinitely small shift of space.

Valence charge density and effective charges within the density-response theory.

In many applications of the electron-response theory a decomposition of the electronic charge distribution \ensuremath{\rho}(x) into contributions ${\ensuremath{\rho}}_{\ensuremath{\alpha}}$(x) of the different sublattices (ion types \ensuremath{\alpha}) is of interest. It is shown that such a decomposition is indeed possible. Starting with the acoustic sum rule the electronic charge density can directly be expressed by the density-response function and the bare-ion pseudopotential, and this representation then allows for a decomposition in a unique way. The integrated partial densities define partial charges ${Z}_{\ensuremath{\alpha}}^{e}$, which are shown to be the electronic contribution to the longitudinal (Callen) charge. All the statements are numerically illustrated with Si as an example. Furthermore, the local-field effect in chemical binding is discussed using the partial densities ${\ensuremath{\rho}}_{\ensuremath{\alpha}}$(x). In the Appendix the relation between the acoustic and Keating's sum rule is investigated with regard to the partial charge densities and the long-wavelength limit of the acoustic-phonon modes.

Effective-charge tensor and acoustic sum rule for an insulating slab

Explicit expressions of the effective-charge tensor for a thin slab of an insulating material are presented. To obtain them we use a Ward identity for the surface which determines the behavior of the microscopic response function $\ensuremath{\chi}$ for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}\ensuremath{\rightarrow}0$, where $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$ is a two-dimensional wave vector. The effective-charge tensor is shown to obey the acoustic sum rule.

The Transverse Effective Charge of the IV-VI Compound Semiconductors

Littlewood's assumption in the calculation of the transverse effective charge (TEC) of IV-VI compound semiconductors is studied by the simple calculation at W -point on the Jones zone surfaces based on the Heine-Jones model. It is shown that the main peak of optical absorption spectrum cannot be assigned only to (111) component of the effective charge in contrast with Littlewood's assumption. The TEC is directly calculated by executing the band calculation with the potential obtained by the EPM in the framework of Vogl's expression. The calculated result gives better agreement with experiment than Littlewood's. The acoustic sum rule is also studied by using the similar formulation given by Vogl.

Comments on the phonon spectra of covalent semiconductors

The results of a calculation of the static dielectric function and the phonon frequencies of the covalent semiconductors C, Si, Ge are discussed and compared with other calculations. The present article gives evidence of the inadequacy of the extreme tight-binding model to calculate phonon dispersion curves. In this model consistency between the electron-ion and crystal pseudopotential cannot be achieved and consequently the generalised acoustic sum rule is not satisfied. Therefore the model is not appropriate to explain phonon properties of crystals.

Dielectric screening and zone-center phonons in Si

We present a microscopic calculation of the zone-center phonon frequencies of silicon. The electronic contributions to the phonons are fully incorporated into the dynamical matrix via the dielectric matrix ${\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{G}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{G}}}$. The acoustic sum rule is shown to be satisfied.

Dynamical effective charges in semiconductors: A pseudopotential approach

A method for an ab-initio calculation of transverse dynamical effective charges Z* in partially covalent semiconductors within the framework of pseudopotential theory is presented. It is based on previously established expressions for Z* in terms of the inverse microscopic dielectric function of the crystal but circumvents the direct inversion of the latter. In contrast to earlier calculations the results are definitely boundary insensitive. Moreover, the acoustic sum rule expressing charge neutrality is inherently guaranteed. Within the local empirical pseudopotential method a simple analytical model calculation, which is compared with the bond orbital model, as well as a full numerical computation is performed. For III-V compounds the resulting Z* underestimate the experimental effective charges by approximately 30% while good agreement for II-VI materials is obtained. Reasons for these findings are discussed and the effects of several refinements are estimated.

Local Field Correction and Acoustic Sum Rule for Si and Ge

The microscopic inverse dielectric matrix in the limit of a long wavelength is calculated in the random phase approximation for Si and Ge by rushing the realistic band models. It is shown that, if a proper form of the pseudopotential is employed, the acoustic sum rule can be substantially satisfied without any adjustable parameter, and the local field correction to the static dielectric constant can not be neglected.

Microscopic theory of electron-phonon interaction in insulators or semiconductors

A unified treatment of the electron-phonon interaction matrix $M$ in insulators (or semiconductors at low temperatures) is given in terms of the complete linear electron response within the adiabatic approximation. The effective electron-one-phonon (el-ph) potential $M$ which is constructed, analogously to metals, as a screened vertex-corrected renormalized (bare) ion potential is defined, including all many-body corrections. We derive the small-wave-vector ($\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$) limit of $M$ leading to multipole as well as short-range contributions by separating out the electron response for fixed macroscopic electric field and using the analytical properties of the inverse dielectric tensor ${\ensuremath{\epsilon}}^{\ensuremath{-}1}$. We obtain explicit expressions for all standard types of el-ph couplings in terms of microscopic quantities in principle calculable from band structure. The main results of Lawaetz are rederived and new---indirectly induced---multipole terms are obtained. A new quadrupole sum rule for $\ensuremath{\epsilon}$ is proved which guarantees, together with the already known acoustic sum rule, the well-known behavior of the acoustic scattering potential as a function of $q$ in the elastic limit. An effective el-two-ph scattering is investigated; further sum rules are proved representing constraints on a nonlinear response function.

General theory of the effect of ionic motion on charge density

By analysing the first-order change in the total charge density of a system when one ion moves, a unique definition of a 'generalized' pseudoatom, involving both a rigid and a deformation part, is given. The dipole moment of this pseudoatom is related to the 'effective charge', which is shown to be the sum of two tensors, an isotropic one coming from the rigid part and an antisymmetric one coming from the deformation part of the pseudoatom. A generalized 'acoustic sum rule' is obtained. Force constants, pair-potential functions, bond-stretching and bond-bending effects are discussed in terms of the pseudoatom. The application of the theory to the perfect solid and its connection with standard theories involving the dielectric constant is made. The application of the theory to nearly-free-electron metals transition metals and covalently bonded semiconductors is discussed and the phonon spectra of diamond-type semiconductors is qualitatively explained.

Dielectric-screening matrix and lattice dynamics of Si

The dynamical matrix of a semiconductor is set up by evaluating the full dielectric matrix $\ensuremath{\epsilon}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}+\stackrel{\ensuremath{\rightarrow}}{\mathrm{G}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}+{\stackrel{\ensuremath{\rightarrow}}{\mathrm{G}}}^{\ensuremath{'}})$ to the lowest order in a Brillouin-Wigner perturbation scheme. The introduction in the dynamical matrix of the nondiagonal elements of the dielectric matrix, which account for the lattice effects on the electron screening, gives rise to noncentral forces between the atoms and restores the correct $q\ensuremath{\rightarrow}0$ limit for the longitudinal acoustic frequencies through the fulfillment of the acoustic sum rule. A calculation of the phonon frequencies of Si is presented. The results, which are in reasonable agreement with experimental data, show that the off-diagonal elements are responsible for the stability of the crystal against shear and give real values for transverse acoustic frequencies. It is also shown that a third-order Rayleigh-Schr\"odinger perturbation theory, with some correction to include higher-order effects, can provide a good description of phonon spectrum of Si, but poorly reproduces the experimental elastic constants.

On dielectric screening matrix and localized potentials in diamomd type semiconductors

A necessary condition for a dielectric screening matrix to describe long range effects of a phonon-induced charge distortion is pointed out. It is also shown that the translational invariance of the system imposes an infinite set of sum rules on the dielectric matrix, which imply the acoustic sum rule as a special case.