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The work investigates the applicability of the Keating Model to estimation of the crystal lattice dynamics with the atomic coordination different from that of tetrahedral. The general model statement is considered in view of the long-range pattern of Coulomb interaction, software support of estimations with total consistency and accuracy assessment in terms of test systems to be standard for the Keating Model, and calculation results are given with the analysis of the crystal phonon spectrum structured like the mineral salt and fluorite. It is shown that the best choice of the model parameters will result in values for compounds of the structure above that are well consistent with experimental data, including acoustic vibration frequency range, where the Keating Model normally results in the increased values thereof.

The lattice vibrations of the AIIBIII2CVI4 semiconductors with defect-chalcopyrite structure are treated in a simplified version of the Keating model considering only interaction with nearest neighbours and assuming that all anions occupy their ideal lattice sites. It is found that in this model the frequencies of the nonpolar and polar modes with highest energy are determined by the properties of the BIII–CVI sublattice alone. The frequencies of all the other optical modes depend not only on the AII–CVI and BIII–CVI interactions but are also influenced by the presence of the ordered array of vacancies. The results obtained are compared with previous model considerations. Die Gitterschwingungen der AIIBIII2CVI4-Halbleiter mit Defekt-Chalkopyrit-Struktur werden in einer vereinfachten Version des Keating-Modells behandelt, wobei nur die Wechselwirkungen zwischen nachsten Nachbarn berucksichtigt werden und vorausgesetzt wird, das alle Anionen ihre idealen Gitterplatze besetzen. Es wird gefunden, das in diesem Modell die Frequenzen der nichtpolaren und polaren Moden mit der hochsten Energie allein durch die Eigenschaften des BIII–CVI–Untergitters bestimmt werden. Die Frequenzen aller anderen optischen Moden hangen nicht nur von den AII–CVI–und BIII–CVI-Wechselwirkungen ab, sondern werden auch durch das Vorhandensein des geordneten Vakanzensystems beeinflust. Die erhaltenen Ergebnisse werden mit fruheren Modellbetrachtungen verglichen.

The parameters of the modified Keating model deduced for cubic diamond are used to predict the elastic constants and zone-center optic mode frequencies of hexagonal diamond. A geometrical difference between the two structures at the third-neighbor level requires an adjustment to the β* parameter that results in excellent agreement between the model and the three observed Raman frequencies. The lower symmetry and extra degree of freedom associated with the hexagonal unit cell permit exploration of three distinct regimes: quasi-cD (same bond lengths, atomic volume and c/a ratio), equal bondlengths with increased cla ratio, and unequal bondlengths with unchanged c/a ratio. Numerically there is little difference between the predictions in each case. The quasi-cD case implies first an isotropic compressibility consistent with the observed constancy of the cla ratio over a wide range of pressure and second no lifting of the triply degenerate Raman frequency which is consistent with experiment but slightly at variance with a first-principles calculation. All three cases yield a bulk modulus slightly smaller than the fitted value for the cubic allotrope, and similar values for the pressure derivatives of the elastic constants and of the optic mode frequencies.

The experimental data relating to the second- and third-order elasticity and the zone-center optic modes of hexagonal graphite are reviewed and some amendments proposed. A modified Keating model involving three sets of interactions, one planar and two interlayer, has been developed. The harmonic parameters, four planar and seven interlayer, have been fitted by least-squares procedures to five second-order elastic constants, five zone-center optic-mode frequencies and two assumptions relating to internal strain. The anharmonic parameters comprise three planar and three interlayer ones. They have been fitted to the pressure derivatives of the five second-order constants and of three of the optic-mode frequencies. The full spectrum of inner elastic constants and internal strain tensors is given, the composition of the second- and third-order elastic constants is exposed, and the corresponding elastic compliances calculated. A pressure-induced phase transition is correctly predicted at around 16 GPa.

The inner elastic constants of single-crystal diamond at the second and third order characterize the response of the crystal to the internal displacement of its component sublattices, of which there are two, either alone or coupled to external strain. These constants feature in the decomposition of the macroscopic elastic constants at second and third order, and give rise to one linear and three quadratic independent internal strain parameters. All these constants have been obtained via the implementation of a Keating potential notionally restricted to two-body and three-body interactions between nearest neighbors. Four harmonic parameters have been optimized to reproduce the second-order elastic constants, the frequency of the (triply degenerate) optical mode at the zone center and the internal strain parameter. The resulting fit is excellent and also accounts very well for the frequencies of the TO modes at the X and L critical points. Excessively large frequencies predicted for the TA modes at these points are shown to be due to a particular four-body interaction that cannot be separated elastically from the three-body bond-bending interaction. The assignment of a fifth parameter allows all the critical point frequencies to be well fit, the largest discrepancies being 3% and 6% for the TA modes. One of six anharmonic parameters is shown to be statistically insignificant. The remaining five are fit to the pressure derivatives of the second-order elastic constants and to the various stress derivatives of the frequency of the zone-center modes. These parameters are used to predict the values of all the third-order elastic and inner elastic constants, and of the quadratic internal strain parameters. Finally the Keating strain is redefined so that the parameters of the model no longer depend on the dimensions of the unit cell chosen to describe the structure. New expressions are obtained for all elastic constants and the optimized parameters are appropriately modified.

A generalization of the Keating model is given which treats anharmonic effects in a much improved manner. The dependence of the bond-stretching and bond-bending force constants on the crystal volume was determined by means of ab initio density-functional calculations, revealing simple universal scaling laws. The resulting anharmonic model was used to investigate optical phonons in disordered alloys of Si, Ge, and C. The calculated Raman spectra agree well with experimental results and are analyzed in terms of microscopic and macroscopic strain as well as confinement effects due to mass disorder.

Phonon spectra of (GaAs)n(Ga1−x AlxAs)m superlattices were calculated using the Keating model with allowance made for long-range forces. The calculated summed longitudinal acoustical branches are consistent with observed experimental Raman doublets. The tendency towards a shift to higher frequencies of longitudinal optical modes as the number of GaAs monolayers increases, which is observed both in calculations and in experiments, is caused by localization of these modes within the GaAs layer.

The dynamics of strained (001) ZnSe/ZnS, ZnSe/ZnTe, and ZnS/ZnTe superlattices is calculated in terms of the Keating model, taking into account the long-range Coulomb force. The effect of a plane deformation on the phonon spectra of bulk materials and superlattices is studied. The dependence of a strain-induced shift of the studied modes on the number of monolayers and relative thickness of constituents is analyzed.

The expression for the elastic stridn energy in covalent crystals, proposed by P.N. Keating in 1966, has been used to determine the displacements of near-surface atoms from their ideal positions due to surface reconstructions. The atomic relaxations are compared with those from ab-initio total-energy calculations for Si (111) 3 × 3 - Si by J.E. Northrup and for Si(111)2 × 2-Si by D. Vanderbilt. For the Ge (111) 3 × 3 -Sn , Ge(111) 3 × 3 -Pb and Ge(001)2 × 1 surfaces, the results from the Keating calculations are compared with the atomic positions determined by X-ray diffraction experiments. In general, the agreement is good and it shows that the Keating model can give reliable predictions of the surface relaxations. However, some discrepancies are observed for atoms with heavily distorted bonds.

Lattice dynamics of 36Ar are investigated by Keating model. Owing to the short-range character of the interactions in rare-gas solids, only the nearest neighbour interactions were taken into account. The calculated phonon frequencies and elastic constants are in good agreement with the experimental values. The maximum error for phonon frenquencies is 3.3%, and for the elastic constants is 7%.

The Raman shifts of Si—Si， Ge—Ge and Si—Ge in Si（1-x）Gex alloys are calculated by Keating model. The calculated Raman shifts are 40275，41339 and 38815 cm-1 when the concentrations of Ge are 01， 05 and 09 respectively. These results are consistent with the reported experimental results， which indicates the validity of the Keating model for obtaining the Raman frequency of strained materials by changing the elastic coefficients of stretching and compression and bond-bending interaction. The single-phonon scattering peak at 477029 cm-1 in amorphous silicon is obtained for the first time by Keating model, which is in agreement with the result of 4800 cm-1 from the literature, indicating that the atoms of amorphous silicon as a whole are stretched compared with that of crystalline silicon.

To calculate and design nanoelectronic and nanophotonic devices, in which graphenes and graphene-like 2D nanoallotropes of carbon, silicon, and binary III–V compounds are used, knowledge of the elastic characteristics and piezoelectric, photoelastic, and other properties of 2D materials, which depend on them, is of great significance. In this work, a simple method suitable for engineering calculations of determining the isothermal values of the force constants, elastic rigidities, Young’s modulus, and Poisson coefficient for 2D nanoallotropes of elements of Group IV of the periodic table and binary III–V compounds is proposed. The method is based on the Harrison method of bonding orbitals modified by S.Yu. Davydov and the R. Keating model for describing the elastic properties of such materials. It is shown that the method makes it possible to perform evaluative calculations of the elastic properties for known synthesized 2D crystal structures as well as for structures “constructed” theoretically. It is established that single-layer hexagonal boron nitride and other binary III–V compounds in the form of 2D nanoallotropes of various types of symmetry, which are piezoelectrics in addition, can be promising materials for nanoelectronics along with graphene. This extends the spectrum of the possible practical applications of the studied 2D materials (C, Si, BN, AlN, GaN, AlP, and GaP) both with a graphene-like and more complex structure. The results of this work can be used when developing the acoustic delay lines of the terahertz-frequency range, piezoelectric transducers for exciting and receiving elastic waves in nanoscale 2D acoustic lines, as well as piezoelectric sensors.

The paper presents ab initio density functional calculations of the geometry optimization of BeXAs2 (X = Si, Ge, Sn) crystal structure, primitive cells and phonon spectra in the centre of the Brillouin zone. Interpolation of phonon spectra was performed over the Brillouin zone using the Keating model. Ab initio frequency calculations are used to ascertain the Keating parameters via program optimization. The values of phenomenological parameters are in good agreement with the parameters of the chemical bond obtained by ab initio calculations. The partial phonon spectra are investigated. The structure of acoustic and optical branches of the full phonon spectrum is determined mostly by the ion mass ratio of the crystal compounds.

Polycrystalline AgIn3Te5 has been synthesized by melt-quench technique. Phase homogeneity and crystal structure of AgIn3Te5 have been analyzed using X-ray diffraction (XRD). Vibrational modes and effect of laser irradiation on the surface of AgIn3Te5 have been studied by Raman spectroscopy. XRD analysis has shown that single phase AgIn3Te5 has crystallized in p-type tetragonal structure with P 4 ¯ 2 c space group. Raman spectrum of as-formed sample has shown a strong peak at 125.6 cm−1 which is attributed to A1 mode of chalcopyrite family and substantiates the results obtained from XRD analysis. Tentative mode assignment for the rest of the peaks observed has been done based on simplified Keating model and by comparison with modes reported as well as predicted theoretically for AgInTe2. Raman spectra obtained with laser beam power of 20 mW have revealed the formation of amorphous phase during acquisition. Interaction with 20 mW laser beam has resulted in the formation of In2O3 on the sample surface. The peak at 132.8 cm−1 that appeared after irradiation for 75 s may be attributed to In2O3 or to the silent A2 mode, which probably has become active due to defect formation.

Phonon spectra of hypothetic BeMN2 (M = C, Si, Ge, Sn) crystals with a chalcopyrite lattice are calculated by the ab initio density functonal method in the center of the Brillouin zone and interpolated over the whole Brillouin zone using the phenomenological Keating model. Interaction parameters are found by comparing IR and Raman active frequencies obtained in the phenomenological model with calculations performed by the ab initio method. Numerical values of short-range constants and charges are in accordance with the ab initio calculated characteristics of the chemical bond. These parameters have the obvious physical meaning and the chemical nature and can be further used for both qualitative estimates of any physical and physicochemical values and quantitative calculations of the phonon spectra of isostructural compounds.

Ab initio calculations of the electronic spectrum, deformation electron density, and phonon frequencies at the center of the Brillouin zone of the CdAs2 tetragonal compound are carried out in the context of the density-functional method. It is established that the crystal is an indirect-gap semiconductor with a band gap of ~1 eV, which is in good agreement with well-known optical and electrical experimental data. The features of chemical bonding in the crystal are studied. The features are defined by the fact that As atoms form spiral chins of covalent As–As bonds, whereas Cd–As bonds are ionic-covalent. The ab initio calculated phonon frequencies are compared with the frequencies calculated in the Keating model and with experimental data; the partial contributions of Cd and As atoms are analyzed.

The central and noncentral harmonic force constants of the Keating model, proposed previously for describing elastic graphene constant, are obtained for 13 graphene-like IV–IV and III–V compounds. It is shown that the relative role of the noncentral forces decreases with an increase in the compound ionicity. For graphene, the anharmonic central-interaction constant is estimated and the dependences of the secondorder elastic constants on the two-dimensional pressure are calculated. Values of the photoelastic constants are determined.

The concentration dependences of the elastic constants of the two-dimensional SixC1 − x system have been investigated with the use of the Harrison bonding-orbital method and the Keating model. The central and non-central force constants and the Gruneisen parameter have been considered by means of the bonding-orbital method. All quantities under consideration have been shown to exhibit a nonlinear behavior during the transition from graphene to silicene. A nontrivial role of the short-range repulsion has been discussed. The second-order and third-order elastic constants, the pressure dependences of the second-order elastic constants, as well as the Poisson’s ratio and Young’s modulus have been investigated in the Keating model. It has been found that the elastic constants and Young’s modulus change almost linearly upon the transition from graphene to silicene, whereas the other quantities under consideration exhibit nonlinearity.

The phonon spectra of a series of A2B4C25 and A1B3C26 crystals with the chalcopyrite structure are calculated in the Keating model in the basis set of polarization vectors of their sublattices. The dependences of frequency values and partial contributions of sublattices to the polarization vectors on the chemical composition are studied. The effect of the mass ratio and the strengths of covalent bonds between the compound components on the phonon spectrum formation is found.

A model potential for the description of elastic properties of single-layer graphene has been proposed. The potential is reduced to the two-dimensional Keating model at a small strain, but is also applicable to the description of the response of graphene to a finite in-plane strain.

We develop a model for calculating the Raman scattering spectra from phonons confined in for silicon nanocrystals, which is based on the familiar approach taking into account the uncertainty in the quasi-momentum of phonons localized in the nanocrystals. The model is considerably improved by taking into account dispersion of phonons not only in the magnitude of the quasi-momentum, but also in its direction. A significant refinement of the model is also due to the fact that phonon dispersion is calculated using the widely approved Keating model instead of being approximated by empirical expressions as was done in earlier approaches. The calculations based on this model make it possible to determine the sizes of silicon nanocrystals more precisely from analysis of the experimental Raman spectra.

A detailed comparison of continuum and valence force field strain calculations in quantum-dot structures is presented with particular emphasis to boundary conditions, their implementation in the finite-element method, and associated implications for electronic states. The first part of this work provides the equation framework for the elastic continuum model including piezoelectric effects in crystal structures as well as detailing the Keating model equations used in the atomistic valence force field calculations. Given the variety of possible structure shapes, a choice of pyramidal, spherical and cubic-dot shapes is made having in mind their pronounced shape differences and practical relevance. In this part boundary conditions are also considered; in particular the relevance of imposing different types of boundary conditions is highlighted and discussed.

Some issues that arise when modeling elastic energy for binary alloys are discussed within the context of a Keating model and density-functional calculations. The Keating model is a simplified atomistic formulation based on modeling elastic interactions of a binary alloy with harmonic springs whose equilibrium length is species dependent. It is demonstrated that the continuum limit for the strain field are the usual equations of linear elasticity for alloys and that they correctly capture the coarse-grained behavior of the displacement field. In addition, it is established that Euler-Lagrange equation of the continuum limit of the elastic energy will yield the same strain field equation. This is the same energy functional that is often used to model elastic effects in binary alloys. However, a direct calculation of the elastic energy atomistic model reveals that the continuum expression for the elastic energy is both qualitatively and quantitatively incorrect. This is because it does not take atomistic scale compositional nonuniformity into account. Importantly, this result also shows that finely mixed alloys tend to have more elastic energy than segregated systems, which is the exact opposite of predictions made by some continuum theories. It is also shown that for strained thin films the traditionally used effective misfit for alloys systematically underestimate the strain energy. In some models, this drawback is handled by including an elastic contribution to the enthalpy of mixing, which is characterized in terms of the continuum concentration. The direct calculation of the atomistic model reveals that this approach suffers serious difficulties. It is demonstrated that elastic contribution to the enthalpy of mixing is nonisotropic and scale dependent. It is also shown that such effects are present in density-functional theory calculations for the Si-Ge system. This work demonstrates that it is critical to include the microscopic arrangements in any elastic model to achieve even qualitatively correct behavior.

A semi-continuum approach is developed for mechanical analysis of a silicon nanowire, which captures the atomistic physics and retains the efficiency of continuum models. By using the Keating model, the strain energy of the nanowire required in the semi-continuum approach is obtained. Young’s modulus of the silicon (001) nanowire along [100] direction is obtained by the developed semi-continuum approach. Young’s modulus decreases dramatically as the size of a silicon nanowire width and thickness scaling down, especially at several nanometers, which is different from its bulk counterpart. The semi-continuum approach is extended to perform a mechanical analysis of the silicon nanowire at finite temperature. Taking into account the variations of the lattice parameter and the bond length with the temperature, the strain energy of the system is computed by using Keating anharmonic model. The dependence of young’s modulus of the nanowire on temperature is predicted, and it exhibits a negative temperature coefficient.

The theoretical calculation of the electronic structure of any constituent materials is the first step toward the interpretation and understanding of experimental data and reliable device design. This is essentially true for nanoscale devices where both the atomistic granularity of the underlying materials and the quantum-mechanical nature of charge carriers play critical roles in determining the overall device performance. In this paper, within a fully atomistic and quantum-mechanical framework, we investigate the electronic structure of wurtzite InN quantum dots (QDs) self-assembled on GaN substrates. The main objectives are threefold: 1) to explore the nature and the role of crystal atomicity, strain field, and piezoelectric and pyroelectric potentials in determining the energy spectrum and the wave functions; 2) to address the redshift in the ground state, the symmetry lowering and the nondegeneracy in the first excited state, and the strong band mixing in the overall conduction-band electronic states, which is a group of interrelated phenomena that has been revealed in recent spectroscopic analyses; and 3) to study the size dependence of the internal fields and its impact on the electronic structure as a whole. We also demonstrate the importance of 3-D atomistic material representation and the need for using realistically extended substrate and cap layers (multimillion-atom modeling) in studying the built-in structural and electric fields in these reduced dimensional QDs. The models used in this study are as follows: 1) valence-force-field Keating model for atomistic strain relaxation; 2) 20-band nearest neighbor sp 3 d 5 s* tight-binding model for the calculation of single-particle energy states; and 3) microscopically determined polarization constants in conjunction with an atomistic 3-D Poisson solver for the calculation of piezo- and pyroelectric contributions.

For an effective use of Raman scattering as strain characterization tool in SiGe nanostructures a precise knowledge of the phonon deformation potentials (DPs) is strictly necessary. The optical phonon DPs can be determined by means of Raman scattering measurements from the cleaved edge of a biaxially strained SiGe alloy layer grown pseudomorphically on silicon and subsequently capped. Due to uncertainties in the literature values of the unstrained phonon frequencies it turns out that the desired degree of accuracy is only attained by complementing the Raman measurements from the edge with that of the hydrostatic pressure coefficient of the optical phonons. For that purpose we have grown by molecular beam epitaxy up to seven partially strained Si 1-x Ge x alloys on Si spanning the entire compositional range and measured the dependence on hydrostatic pressure of the frequency of the Ge-like, Si-like and mixed Si-Ge optical modes. After correcting for the pressure dependent biaxial stress induced by the Si substrate on the alloy layer and taking into account the dependence on alloy composition of the bulk modulus we obtain a fairly constant value of the Gruneisen parameter around 1.0 for all three optical modes in the whole range of Ge contents. We also determined the strain shift coefficients for the three modes, which are essentially independent of Ge content between 0.4 and 1. Our results are in very good agreement with recentcalculations of the SiGe phonon deformation potentials using a modified Keating model, which settles the longstanding issue about the large discrepancies between results from different experiments.

Based on the Keating model, a semi-continuum approach is developed in which the strain energy of silicon nanoplates is presented. Using the quasiharmonic approximation, the temperature dependence of the lattice parameter of silicon has been coupled into the semi-continuum approach. By considering (2 × 1) surface reconstruction of the silicon nanoplate, Young's moduli at finite temperature are modeled and the surface effects on the mechanical properties of the silicon nanoplate are predicted. As the nanoplate thickness is scaled down to 100 nm, Young's moduli begin to deviate from that of the bulk silicon. It is interesting to note that Young's moduli exhibits opposite behavior with and without surface reconstruction. Without surface reconstruction, Young's modulus of the nanoplate decreases dramatically as the nanoplate is scaled down to several tens of nanometer, which means that the nanoplate is elastically softer than bulk. The surface reconstruction leads to stronger bonds and hence an increase in the Young's modulus of the material as it is scaled down, which makes the nanoplate stiffer along the [1 0 0] direction. Young's modulus of the nanoplate exhibits a negative temperature coefficient.

A detailed microstructural analysis of amorphous silicon is performed via numerical modeling technique. Nanoporous paracrystalline models have been proposed. Intermixed nanocrystallites and nanovoids of various sizes and concentrations have been introduced into a continuous random network that was generated with a vacancy model. Using the conjugate gradient method, the structures have been relaxed by minimizing their total strain energy described by the anharmonic Keating model. The obtained nanoporous structures are energetically competitive with the voidless paracrystalline networks. Nanoporous models with large voids are energetically more favorable than those with small voids. The nanoporous paracrystalline model is less dense than the crystalline phase, contrary to the paracrystalline model. This density decreases with increasing the void size for a fixed void volume fraction. Nanoporous paracrystalline structures reproduce the experimental structure factor better than the paracrystalline network. They account for, in particular, the intense small-angle scattering observed for some a-Si samples. The paracrystallites form amorphous zones but with local and topological ordering neatly better than the surrounding matrix. Such structural heterogeneity gives so a satisfactory explanation of the nanoscale fluctuation electron microscopy data reported recently by Treacy and Gibson.

A detailed microstructural analysis of amorphous silicon is performed by means of a numerical modeling technique. Paracrystalline models of amorphous silicon, first proposed by Treacy, Gibson and Keblinski, have been generated. Nanocrystallites of various sizes and concentrations have been introduced into a continuous random network that was generated with a vacancy model. Using the conjugate gradient method, the structures have been relaxed by minimizing their total strain energy described by the anharmonic Keating model. The computed pair correlation functions of these structural models bring to the fore a unique behavior of the paracrystalline networks in the context of diffraction experiments; they appear amorphous as the continuous random network model. The paracrystalline model remains denser than the crystalline phase, contrary to experimental observations. However, the former is found to be less homogenous than the CRN model, thus giving a satisfactory explanation of the nanoscale fluctuation electron microscopy data reported recently by Treacy and coworkers.

A theoretical approach to the lattice dynamics of Si 1 - x Ge x alloys within a valence force field framework is discussed. A modified Keating model, the anharmonic Keating, was employed to perform supercell calculations in order to investigate accurately the three Raman active optical phonon modes. Theoretical results on both relaxed and strained SiGe samples are shown. A detailed study of the biaxial strain-induced shift of phonon modes of epitaxial SiGe alloys has been performed in all the composition range. Furthermore, a comparison of experimental data with the model results on phonon strain shift coefficients will be also discussed.