Phonon Dispersion Data Research Articles

A combined helium atom scattering and density-functional theory study of the Nb(100) surface oxide reconstruction: Phonon band structures and vibrational dynamics

Helium atom scattering and density-functional theory (DFT) are used to characterize the phonon band structure of the (3 × 1)-O surface reconstruction of Nb(100). Innovative DFT calculations comparing surface phonons of bare Nb(100) to those of the oxide surface show increased resonances for the oxide, especially at higher energies. Calculated dispersion curves align well with experimental results and yield atomic displacements to characterize polarizations. Inelastic helium time-of-flight measurements show phonons with mixed longitudinal and shear-vertical displacements along both the ⟨1̄00⟩, Γ̄X̄ and ⟨11̄0⟩, Γ̄M̄ symmetry axes over the entire first surface Brillouin zone. Force constants calculated for bulk Nb, Nb(100), and the (3 × 1)-O Nb(100) reconstruction indicate much stronger responses from the oxide surface, particularly for the top few layers of niobium and oxygen atoms. Many of the strengthened bonds at the surface create the characteristic ladder structure, which passivates and stabilizes the surface. These results represent, to our knowledge, the first phonon dispersion data for the oxide surface and the first ab initio calculation of the oxide's surface phonons. This study supplies critical information for the further development of advanced materials for superconducting radiofrequency cavities.

Electro-optical and phonon properties of PbTiO3/CaTiO3/SrTiO3 ferroelectric superlattices: a first-principles calculation

We studied the electronic structure as well as the dielectric, optical, and phonon properties of PbTiO3 / CaTiO3 / SrTiO3 ferroelectric superlattices, based on the density functional theory, using the Perdew–Becke–Johnson-generalized gradient approximation exchange-correlation functional. According to the results, the superlattice under investigation possessed a semiconducting phase with an indirect band gap of 1.84 eV. The results obtained for the partial density of states were indicative of a strong hybridization between the Ti-3d and O-2p states, which affected the amplitude of the Coulomb interaction and ferroelectric property. Regarding the optical properties, we calculated and analyzed the dielectric function and, accordingly, the refractive index, reflectivity, absorption coefficient, energy loss function, and optical conductivity. Thermal vibrations of the lattice were calculated along the high-symmetry path of the first Brillouin zone. Phonon dispersion data revealed stability at point Γ, while ferroelectric and dielectric responses of superlattice could be in unstable modes at other points of the Brillouin zone. Our results comprehensively present the electronic structure, dielectric function, plasmonic features, and dynamical stability based on phonon calculations, which are all essential for novel functional device applications.

First Principles Investigation of Anomalous Pressure-Dependent Thermal Conductivity of Chalcopyrites.

The effect of compression on the thermal conductivity of CuGaS2, CuInS2, CuInTe2, and AgInTe2 chalcopyrites (space group I-42d) was studied at 300 K using phonon Boltzmann transport equation (BTE) calculations. The thermal conductivity was evaluated by solving the BTE with harmonic and third-order interatomic force constants. The thermal conductivity of CuGaS2 increases with pressure, which is a common behavior. Striking differences occur for the other three compounds. CuInTe2 and AgInTe2 exhibit a drop in the thermal conductivity upon increasing pressure, which is anomalous. AgInTe2 reaches a very low thermal conductivity of 0.2 W·m−1·K−1 at 2.6 GPa, being beneficial for many energy devices, such as thermoelectrics. CuInS2 is an intermediate case. Based on the phonon dispersion data, the phonon frequencies of the acoustic modes for CuInTe2 and AgInTe2 decrease with increasing pressure, thereby driving the anomaly, while there is no significant pressure effect for CuGaS2. This leads to the negative Grüneisen parameter for CuInTe2 and AgInTe2, a decreased phonon relaxation time, and a decreased thermal conductivity. This softening of the acoustic modes upon compression is suggested to be due to a rotational motion of the chalcopyrite building blocks rather than a compressive oscillation. The negative Grüneisen parameters and the anomalous phonon behavior yield a negative thermal expansion coefficient at lower temperatures, based on the Grüneisen vibrational theory.

Monte Carlo study of temperature-dependent non-diffusive thermal transport in Si nanowires

Non-diffusive thermal transport has gained extensive research interest recently due to its important implications on fundamental understanding of material’s phonon mean free path distributions and many nanoscale energy applications. In this work, we systematically investigate the role of boundary scattering and nanowire length on the non-diffusive thermal transport in thin silicon nanowires by rigorously solving the phonon Boltzmann transport equation (BTE) using a variance reduced Monte Carlo technique across a range of temperatures. The simulations use the complete phonon dispersion and spectral lifetime data obtained from first-principle density function theory calculations as input without any adjustable parameters. Our BTE simulation results show that the nanowire length plays an important role in determining the thermal conductivity of silicon nanowires. In addition, our simulation results suggest significant phonon confinement effect for the previously measured silicon nanowires. These findings are important for a comprehensive understanding of microscopic non-diffusive thermal transport in silicon nanowires.

Comparison of three empirical force fields for phonon calculations in CdSe quantum dots.

Three empirical interatomic force fields are parametrized using structural, elastic, and phonon dispersion data for bulk CdSe and their predictions are then compared for the structures and phonons of CdSe quantum dots having average diameters of ˜2.8 and ˜5.2 nm (˜410 and ˜2630 atoms, respectively). The three force fields include one that contains only two-body interactions (Lennard-Jones plus Coulomb), a Tersoff-type force field that contains both two-body and three-body interactions but no Coulombic terms, and a Stillinger-Weber type force field that contains Coulombic interactions plus two-body and three-body terms. While all three force fields predict nearly identical peak frequencies for the strongly Raman-active "longitudinal optical" phonon in the quantum dots, the predictions for the width of the Raman peak, the peak frequency and width of the infrared absorption peak, and the degree of disorder in the structure are very different. The three force fields also give very different predictions for the variation in phonon frequency with radial position (core versus surface). The Stillinger-Weber plus Coulomb type force field gives the best overall agreement with available experimental data.

A stable, quasi-2D modification of silver: optical, electronic, vibrational and mechanical properties, and first principles calculations

We report the optical, electronic, vibrational and mechanical properties of a stable, anisotropic, hexagonal (4H) form of silver. First principles calculations based on density functional theory were used to simulate the phonon dispersion curves and electronic band structure of 4H-Ag. The phonon dispersion data at 0 K do not contain unstable phonon modes, thereby confirming that it is a locally stable structure. The Fermi surface of the 4H phase differs in a subtle way from that of the cubic phase. Experimental measurements indicate that, when compared to the commonly known face-centered cubic (3C) form of silver, the 4H-Ag form shows a 130-fold higher, strongly anisotropic, in-plane resistivity and a much lower optical reflectance with a pronounced surface plasmon contribution that imparts a distinctive golden hue to the material. Unlike common silver, the lower symmetry of the 4H-Ag structure allows it to be Raman active. Mechanically, 4H-Ag is harder, more brittle and less malleable. Overall, this novel, poorly metallic, anisotropic, darker and harder crystallographic modification of silver bears little resemblance to its conventional counterpart.

Construction of multi-dimensional isotropic kernels for nonlocal elasticity based on phonon dispersion data

Kernels for non-local elasticity are in general obtained from phonon dispersion relations. However, non-local elastic kernels are in the form of three-dimensional (3D) functions, whereas the dispersion relations are always in the form of one-dimensional (1D) frequency versus wave number curves corresponding to a particular wave direction. In this paper, an approach to build 2D and 3D kernels from 1D phonon dispersion data is presented. Our particular focus is on isotropic media where we show that kernels can be obtained using Fourier–Bessel transform, yielding axisymmetric kernel profiles in reciprocal and real spaces. These kernel functions are designed to satisfy the necessary requirements for stable wave propagation, uniformity of nonlocal stress and stress regularization. The proposed concept is demonstrated by developing some physically meaningful 2D and 3D kernels that will find useful applications in nonlocal mechanics. Relative merits of the kernels obtained via proposed methods are explored by fitting 1D kernels to dispersion data for Argon and using the kernel to understand the size effect in non local energy as seen from molecular simulations. A comparison of proposed kernels is made based on their predictions of stress field around a crack tip singularity.

Non-diffusive relaxation of a transient thermal grating analyzed with the Boltzmann transport equation

The relaxation of an one-dimensional transient thermal grating (TTG) in a medium with phonon-mediated thermal transport is analyzed within the framework of the Boltzmann transport equation (BTE), with the goal of extracting phonon mean free path (MFP) information from TTG measurements of non-diffusive phonon transport. Both gray-medium (constant MFP) and spectrally dependent MFP models are considered. In the gray-medium approximation, an analytical solution is derived. For large TTG periods compared to the MFP, the model yields an exponential decay of grating amplitude with time in agreement with Fourier's heat diffusion equation, and at shorter periods, phonon transport transitions to the ballistic regime, with the decay becoming strongly non-exponential. Spectral solutions are obtained for Si and PbSe at 300 K using phonon dispersion and lifetime data from density functional theory calculations. The spectral decay behaviors are compared to several approximate models: a single MFP solution, a frequency-integrated gray-medium model, and a “two-fluid” BTE solution. We investigate the utility of using the approximate models for the reconstruction of phonon MFP distributions from non-diffusive TTG measurements.

Non-local modeling of epoxy using an atomistically-informed kernel

An integral type non-local continuum model for epoxy, used as the matrix material in aerospace structural composites, is developed from phonon dispersion data. Non-local continuum models can be used to regularize the stress fields at crack tips and molecular defect cores (e.g. disclinations in epoxies) where local (classical) elasticity theories fail to give bounded solutions. Integral type non-local elastic models phenomenologically incorporate microstructure information through a weighting function known as a kernel function. The kernel functions are typically obtained by matching dispersion curves computed using lattice dynamics. However, the use of lattice dynamics for amorphous polymers that do not have an underlying lattice structure is computationally prohibitive. In this paper, a molecular dynamics based approach is used for the computation of the non-local kernel, for amorphous epoxy. Dispersion relations calculated from the reciprocal-space velocity–velocity autocorrelation function are used to build the kernels. The computed atomistic kernel is used to predict stress solutions for some example problems where classical elasticity predicts singularities.

Thermal conductivity of indium arsenide nanowires with wurtzite and zinc blende phases

The thermal conductivity of wurtzite and zinc blende indium arsenide nanowires was measured using a microfabricated device, with the crystal structure of each sample controlled during growth and determined by transmission electron microscopy. Nanowires of both phases showed a reduction of the thermal conductivity by a factor of 2 or more compared to values reported for zinc blende indium arsenide bulk crystals within the measured temperature range. Theoretical models were developed to analyze the measurement results and determine the effect of phase on phonon transport. Branch-specific phonon dispersion data within the discretized first Brillouin zone were calculated from first principles and used in numerical models of volumetric heat capacity and thermal conductivity. The combined results of the experimental and theoretical studies suggest that wurtzite indium arsenide possesses similar volumetric heat capacity, weighted average group velocity, weighted average phonon-phonon scattering mean free path, and anharmonic scattering-limited thermal conductivity as the zinc blende phase. Hence, we attribute the differing thermal conductivity values observed in the indium arsenide nanowires of different phases to differences in the surface scattering mean free paths between the nanowire samples.

Structural, electronic, mechanical, and thermodynamic properties of UN 2: Systematic density functional calculations

We have performed a comparative study of UN 2 and β-U 2N 3using the generalized gradient approximation (GGA) and the GGA + U approaches based on the density functional theory (DFT). The lattice parameters obtained from the GGA + U calculations can be ameliorated appreciably, however, the density of states (DOS) is insensitive to Hubbard U for both compounds. Our Bader analysis shows that the effective charges increase with decreasing U:N ratio (from UN to UN 2). The f electrons in UN 2 and β-U 2N 3 seem less localized than that in UN phase. The effects of Hubbard U on mechanical properties is evident, while the phonon dispersion depends weakly on the changes of Hubbard U. Based on our phonon dispersion data, the lattice vibration energy, thermal expansion, and specific heat are obtained by utilizing the quasiharmonic approximation (QHA).

Electronic structures, mechanical and thermodynamic properties of ThN from first-principles calculations

Lattice parameter, electronic structure, mechanical and thermodynamic properties of ThN are systematically studied using the projector-augmented-wave method and the generalized gradient approximation based on the density functional theory. The calculated electronic structure indicates the important contributions of Th 6 d and 5 f states to the Fermi-level electron occupation. Through Bader analysis it is found that the effective valencies in ThN can be represented as Th +1.82 N −1.82. Elastic constant calculations show that ThN is mechanically stable and elastically anisotropic. Furthermore, the melting curve of ThN is presented up to 120 GPa. Based on the phonon dispersion data, our calculated specific heat capacities including both lattice and conduction electron contributions agree well with experimental results in a wide range of temperature.

Optimized Tersoff and Brenner empirical potential parameters for lattice dynamics and phonon thermal transport in carbon nanotubes and graphene

We have examined the commonly used Tersoff and Brenner empirical interatomic potentials in the context of the phonon dispersions in graphene. We have found a parameter set for each empirical potential that provides improved fits to some structural data and to the in-plane phonon dispersion data for graphite. These optimized parameter sets yield values of the acoustic phonon velocities that are in better agreement with measured data. They also provide lattice thermal conductivity values in single-walled carbon nanotubes that are considerably improved compared to those obtained from the original parameter sets.

Crystal dynamics ofδfcc Pu-Ga alloy by high-resolution inelastic x-ray scattering

We have used a microbeam on large grain sample concept to carry out inelastic x-ray scattering experiments to measure the phonon dispersion curves of a fcc $\ensuremath{\delta}$-phase Pu-Ga alloy along the main symmetry directions of the cubic lattice. This approach obviates experimental difficulties with conventional inelastic neutron scattering due to the high absorption cross section of the common $^{239}\mathrm{Pu}$ isotope and the nonavailability of large (millimeter size) single crystal materials for Pu and its alloys. A classical Born-von K\'arm\'an force constant model was used to model the experimental results, and up to fourth nearest neighbor interactions had to be included to obtain sufficient agreement. Several unusual features including, a large elastic anisotropy, a small shear elastic modulus $({C}_{11}\ensuremath{-}{C}_{12})∕2$, a positive kink in the ${T}_{1}[0\ensuremath{\xi}\ensuremath{\xi}]$ branch, and a pronounced bending (toward lower energy) of the $T[\ensuremath{\xi}\ensuremath{\xi}\ensuremath{\xi}]$ branch near the $L$ point in the Brillouin zone are found. These features are discussed in light of the various phase transformations of $\ensuremath{\delta}$ plutonium. The phonon dispersion data also provide a critical test and benchmark for theoretical treatments of highly correlated $5f$ electron systems.

Effect of pressure on the Raman anomaly of zinc-blende CuBr and Raman spectra of high-pressure phases

Raman spectra of isotopically pure CuBr ${(}^{63}{\mathrm{Cu}}^{81}\mathrm{Br})$ were measured under hydrostatic pressures up to 10 GPa at 10 K. The anomalous line shape of the longitudinal-optic (LO) scattering in the zinc-blende phase (CuBr-III, 0--4 GPa), consisting of a broad structure between 155 and $177{\mathrm{cm}}^{\ensuremath{-}1},$ exhibits a continuous change with pressure and develops into a narrow LO phonon peak near 4 GPa. The disappearance of the LO Raman anomaly as well as the pronounced broadening of the TO mode with pressure are explained in terms of pressure-dependent third-order anharmonic interactions with two-phonon states. A Fermi resonance model, which is based on a shell model fit of available phonon dispersion data, fully accounts for the changes in Raman line shapes under pressure. Low-temperature Raman spectra of the tetragonal CuBr-IV and cubic CuBr-V high-pressure phases are also reported. An assignment of the observed Raman modes of these phases is proposed.

Landau free energy for a bcc-hcp reconstructive phase transformation

We study the bcc-hcp phase transition in Ti and Zr with the use of first-principles calculations. We have determined the complete energy surface from the bcc to hcp structure. The results are used to find an appropriate Landau free energy density for describing this transformation. The proposed Landau free energy density has two relevant order parameters: shear and shuffle. Through first-principles calculations, we show that the bcc structure is unstable with respect to the shuffle of atoms ${(TA}_{1}$ N-point phonon) rather than the shear. Therefore, we reduce the two order parameter Landau free energy to an effective one order parameter (shuffle) potential, which is a reasonable approximation. In general, the effective Landau free energy is a triple-well potential. From the phonon dispersion data and the change in entropy at the transition temperature we find the free energy coefficients for Ti and Zr.

Lattice dynamical Debye-Waller factor for silicon

The mean-square amplitude of vibration of silicon at 293 K has been determined by a lattice-dynamical procedure. A Born\char21{}von K\'arm\'an model has been used to fit phonon dispersion data from inelastic neutron-scattering measurements. The force-constant model included the first six nearest neighbors in the diamond-type lattice. The least-squares results from the fitting of the force constants were used to carry out variance analyses of properties dependent on the harmonic model. The density of phonon states was determined by sampling an even mesh of 5.7 billion points in the unique part of the Brillouin zone. Moments of the frequency distribution up to eighth order are tabulated. The frequency distribution function was used to calculate a $〈{u}_{\mathrm{LD}}^{2}〉$ for silicon. The result is $0.005941\ifmmode\pm\else\textpm\fi{}0.000021{\AA{}}^{2}.$ An Einstein-type potential of Dawson and Willis was used to extract an anharmonic force constant from the temperature dependence of neutron-diffraction measurements of silicon carried out by Batterman and co-workers. These measurements were restricted to the weak reflections from the (222), (442), and (622) diffracting planes. With the use of the lattice-dynamical value for the vibrational amplitude of silicon the result for the ${\ensuremath{\beta}}_{\mathrm{DW}}$ anharmonic constant is $18.58\ifmmode\pm\else\textpm\fi{}0.27{\mathrm{N}\mathrm{}\mathrm{m}}^{\mathrm{\ensuremath{-}}1}{\AA{}}^{\ensuremath{-}1}.$

The phonon density of states and low-temperature specific heat: the blue bronze and the platinum chain compound KCP

The quasi-one-dimensional charge-density-wave (CDW) compounds , the blue bronze and the platinum chain compound KCP(Br) show a strong deviation from a Debye behaviour of the specific heat at temperatures of several degrees Kelvin. To determine the origin of this specific heat `anomaly' in the case of , we have performed time-of-flight neutron scattering measurements and extracted a generalized phonon density of states (PDOS). The excess over a Debye behaviour at low energies in the generalized PDOS lies at the origin of the observed anomalous specific heat. We have used a simplified dispersion model based on experimental phonon dispersion curves to calculate the low-energy PDOS of the system. The model succeeds in reproducing the observed specific heat anomaly and establishes its lattice phonon origin. Similar calculations of the low-energy PDOS based on available phonon dispersion data for KCP allow us to state that the anomalous behaviour of the specific heat for this compound also has a phonon origin.

Vibrational dynamics of a stepped metallic surface: Step-edge phonons and terrace softening on Ni(977)

Inelastic helium atom scattering has been used to measure the surface and step localized phonons on a stepped metallic surface, Ni(977). These time-of-flight measurements were carried out both perpendicular and parallel to the step direction. Surface phonon dispersion data collected across the steps show backfolding of the surface Rayleigh mode, and, most importantly, dramatic softening as compared to the forces present at the smooth Ni(111) surface. This softening suggests significant relaxation perpendicular to the step edge. Single-phonon scattering data collected along the step direction reveals the presence of two new step-edge localized modes, as well as the Rayleigh mode for this direction of the crystal. The Rayleigh mode here does not exhibit the notable softening that was found for the other direction. Novel in- and out-of-phase scattering measurements, with respect to the terraces, lead us to assign the new step induced modes as the two transversely polarized vibrations which propagate along the direction of the step edge. An analytic one-dimensional lattice model is proposed which well represents the dispersion data for these two step modes; its use allows us to determine the effective local force field in the two transverse directions with respect to the step edge. The findings reported herein shed new light on such topics as interface stability, crystal growth, and charge redistribution in the vicinity of well-characterized extended surface defects.

Lattice dynamics of Nd2CuO4.

We present a lattice-dynamical treatment of ${\mathrm{Nd}}_{2}$${\mathrm{CuO}}_{4}$ (T' structure) in the framework of a shell model that is based on short-range overlap and long-range Coulomb potentials, and includes ionic polarizabilities. We have formulated the static equilibrium conditions for the T' structure under this interaction model, and use the equilibrium conditions to determine the potential parameters for some of the unknown short-range interactions (e.g., Nd-O). Two versions of the shell model are used in the calculation: model I (``generic'' model), which was developed without reference to any experimental phonon-dispersion data of ${\mathrm{Nd}}_{2}$${\mathrm{CuO}}_{4}$, and model II, which involved some fitting to the experimental data. In addition to the phonon-dispersion curves, we present results for the phonon density of states F(\ensuremath{\omega}) and the phonon specific heat ${\mathit{c}}_{\mathit{v}}$(T). We discuss the low-temperature behavior of ${\mathit{c}}_{\mathit{v}}$(T) by examining ${\mathit{c}}_{\mathit{v}}$(T)/${\mathit{T}}^{3}$ and FTHETA(T) in the range between 0 and 40 K.