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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.

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 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 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.

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.

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.

The pressure dependences of the second-order elastic constants ij C and the velocity of sound in 3C-SiC and 2H-SiC crystals are calculated in the framework of the Keating model with the additional assumption that the central α and the noncentral β force constants are linear functions of external hydrostatic pressure. Grüneisen parameters for the different acoustic modes of 3C-SiC have been also calculated. The third –order elastic constants ijk C for 3C-SiC are determined from the dependences of ij C on the pressure.

The pressure dependences of the second-order elastic constants Cij and the velocity of sound in 3C-SiC and 2H-SiC crystals are calculated in the framework of the Keating model. The third-order elastic constants Cijk for 3C-SiC are determined from the dependences of the second-order elastic constants Cij on the pressure p.

A model is proposed describing the bond length and bond-bending angle distributions in quaternary A3B5 alloys with ZB structure in quasi-continuum approximation. The model allows to consider reasonably large crystal lattice clusters employing minimum computing resources. Analytical expressions are derived for the radial distribution functions. The developed approach is applied to the analysis of available experimental data on structural disorder in III–V alloys. Employing the valence force field Keating model the deformation energy for several alloys is evaluated.

Raman spectra observed for bulk crystals of defect stannite phases in the Cu–(In, Ga)–Se system are analyzed mainly by the dispersion curve calculations with a superposed lattice model, assuming four fundamental structures. The calculations are performed on the basis of the Keating model, solving dynamical matrix of 24×24. Vacancy induced modifications on force interactions between nearest atoms are also discussed with effective bond overlap population and effective charges determined by the DV-Xα calculations for unit cell clusters. Apparently, invariable Raman profiles with respect to the In to Ga molar ratios through the alloy systems are explained by the superposed lattice model.

We present a theory of electron-phonon interaction and phonon decay in Si nanocrystals based on $s{p}^{3}{d}^{5}{s}^{*}$ empirical tight-binding model and anharmonic Keating model. We demonstrate that the time of optical phonon emission by hot carriers in Si nanocrystal lies in the subpicosecond time range. However, due to the fast phonon recycling, the energy relaxation rate is determined not by the phonon emission but by the phonon decay rate. The decay rate of the optical phonon into two phonons of smaller energy is found to be in the 1--10 ps range. It is this anharmonic phonon decay process that may control the energy relaxation rate of excited carriers in Si nanocrystals.

The incorporation of substitutional C into Ge is studied theoretically with an anharmonic Keating model fitted to recent experimental data and specifically adapted to the computation of the structural properties and the lattice dynamics of ${\mathrm{Ge}}_{1\ensuremath{-}y}{\mathrm{C}}_{y}$ alloys. In the range of physically realistic concentrations $(yl3%),$ the change of lattice parameter due to substitutional carbon is found to agree with Vegard's law of linear interpolation between germanium and diamond with a maximum relative deviation less than 0.03%. This result is obtained in the case of a ${\mathrm{Ge}}_{1\ensuremath{-}y}{\mathrm{C}}_{y}$ alloy either random, diluted, or with C mostly arranged in third nearest neighbors. In the case of a pseudomorphic alloy on germanium (001), the tetragonal distortion of the lattice is well described (error $l0.13%)$ if the elastic constants of the alloy are linearly interpolated between the corresponding parameters of bulk Ge and C in the framework of macroscopic linear elastic theory. The position of the localized vibrational mode of substitutional C in Ge depends on the distribution of carbon, and new phonon modes appear for certain atomic arrangements. Carbon atoms in first nearest neighbor positions give new modes around 425, 480, and $610{\mathrm{cm}}^{\ensuremath{-}1},$ while a new GeC-like mode occurs at $487{\mathrm{cm}}^{\ensuremath{-}1}.$ C in third nearest neighbor (most stable) arrangements induce an additional mode around $550{\mathrm{cm}}^{\ensuremath{-}1}.$ A signature of the fifth nearest neighbor configuration is calculated around $510{\mathrm{cm}}^{\ensuremath{-}1}.$ This study suggests that the content of substitutional carbon in ${\mathrm{Ge}}_{1\ensuremath{-}y}{\mathrm{C}}_{y}$ alloys can be estimated from the lattice constant by x-ray diffraction for $yl3%,$ and that the local order around C can be probed by Raman and infrared absorption spectroscopy.

Infrared and Raman spectra of oriented CdAl2Se4 single crystals and of as-grown CdAl2S4 single crystals have been measured in the wavenumber range from 600 to 25 cm-1. Both ordered vacancy compounds grown by the chemical vapour transport method crystallize in the thiogallate structure with space group I. All 13 optical phonon modes are Raman active; ten of these are in addition infrared active. We report for the first time the combined infrared and Raman study of the optical phonon modes of both compounds. With the complementary information from infrared and Raman spectroscopy all optical phonon modes have been identified and assigned to their corresponding symmetry classes. The fact that all five modes of E symmetry exhibit a counterpart of B symmetry in the close spectral environment is unambiguous evidence of space group I. The measured frequencies of the nonpolar modes are in a very good agreement with data calculated in the framework of a simplified Keating model. For an oriented CdAl2Se4 single crystal the directional dispersion of the polar phonons has been studied.

The initial stage of growing Ge on single-domain vicinal Si(001) with a large angle of misorientation has been studied theoretically by the modified Keating model and compared with RHEED and STM experiments. The experimentally observed conversion from the DB step configuration, where all dimer rows are normal to the step edges, to a DA-like step configuration, where dimer rows on one terrace are parallel to the step edges, at Ge coverages larger than 1 ML is identified by our calculations. Our results show that this DA-like step is in fact a pair of steps of single atomic height: SA+SB, with a very wide SA terrace and an SB tooth of about 10 A. This new step configuration is energetically favoured over DA and DB step configurations.

Based on the lattice dynamics theories, a size dependence model for the thermal expansion coefficient of [001] oriented Silicon nanowires has been developed. Keating model, as the interactional potential, has been adopted to describe the elastic strain energy of Silicon nanowires. The strained phonon dispersion relations of silicon lattice have been analyzed though the strained Si/Ge superlattices theory. It is found that the thermal expansion coefficient of Silicon nanowires with thickness smaller than about 20nm increase dramatically with decreasing size, and is significantly higher than that of bulk silicon. The value of α is 2.49×10-6 K-1 for bulk silicon, and 2.72×10-6 K-1 for Silicon nanowires with the thickness of 5nm at the room temperature.

The phonon spectra of (GaX)N(Ga1-xAlxX)N(001) superlattices (X=As, N) calculated within a rigid ion model are compared. The elastic forces are characterized by a three-parameter Keating model, while the electric field is fully included using an Ewald summation. The force constants and the effective charges are assumed to be dependent on the chemical environment. The influence of composition and disorder is described within a random-element isodisplacement (REI) model generalized to layered structures with arbitrary wavevectors and composition profiles. Attention is focused on the differences in the spectral and localization behaviour of phonons, which are explained mainly in terms of the different anion masses.

This paper shows that by combining theoretical and experimental results for local Si-C phonon modes in dilute ${\mathrm{Si}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{C}}_{\mathit{x}}$ alloys, information concerning the short-range order can be obtained. Calculations using an anharmonic Keating model predict satellite peaks near the vibrational frequency of an isolated C impurity which are associated with second-, third-, etc., nearest-neighbor C-C pairs. By comparing theoretical spectra with those obtained by Raman and infrared absorption spectroscopy, we conclude that the probability for a third-nearest-neighbor coordination of the C atoms is considerably above that for a purely random alloy. This confirms earlier work predicting an attractive third-nearest-neighbor interaction of the C impurities due to elastic interactions. \textcopyright{} 1996 The American Physical Society.

IR absorption was used to study the vibrational modes of substitutional carbon in strained layers of Si 1-y C y (0≤y≤0.015) alloys, which were grown by molecular beam epitaxy on Si (100) substrates. Satellite peaks that appear in the absorption spectrum close to the vibrational mode of substitutional carbon in the alloy are explained by a recently developed anharmonic Keating model. The interaction of substitutional carbon atoms gives rise to these peaks which can also be observed in Raman spectra. We estimate the carbon concentration and discuss the results. In photocurrent measurements on Si 1-y C y /Si heterostructures we observe a shift of the absorption onset below the fundamental silicon band gap to lower energy with rising carbon concentration.

The values of the fourth order elastic moduli (FOEM) for the diamond structure elements, silicon and germanium, have been calculated. The derivation utilizes the Keating model, assuming that both the third and fourth order terms in the interaction potential are composed of a bond‐stretch, an angle‐bend, and a mixed bond‐stretch angle‐bend term. The three force constants in the fourth order interaction potential are evaluated from the measurements of the sound velocity under pressure, hence, numerical values for all 11 FOEM may be obtained. The set of FOEM thus calculated shows markedly different characteristics than the analogous quantities for other materials, e.g., the alkali halides.

In this article, a new semi-continuum model is built to describe the fundamental vibration frequency of the silicon nanowires in orientation. The Keating potential model and the discrete nature in the width and the thickness direction of the silicon nanowires in orientation are applied in the new semi-continuum model. Based on the Keating model and the principle of conservation of energy, the vibration frequency of the silicon nanowires with the triangle, the rhombus, and the hexagon cross sections are derived. It is indicated that the calculation results based on this new model are accordant with the simulation results of the software based on molecular dynamics (MD).

The thermal conductivity of models of amorphous silicon and vitreous silica are discussed. In previous work on amorphous silicon, 216-and 1000-atom models were studied within the Kubo theory and the harmonic approximation using the Stillinger-Weber potential and good agreement with the experimental high-temperature apparent saturation value was obtained. In order to test the model dependence of such results we performed calculations for models based on the Keating potential. The Keating model apparently gives much less force-constant disorder than does the Stillinger-Weber model, as evident from the N dependence and the frequency range of localized states. We also present results of calculations for vitreous silica based on the Feuston-Garofalini model that we relaxed, using their potential, to zero pressure. The results for vitreous silica are in fair agreement with experiment at temperatures above 100K which corresponds to the lowest normal-mode frequency contained within the model.

We present a theoretical study on the relation between the size of the rare earth ions, often known as chemical pressure, and the stability of the coherent Jahn-Teller distortions in undoped perovskite manganites. Using a Keating model expressed in terms of atomic scale symmetry modes, we show that there exists a coupling between the uniform shear distortion and the staggered buckling distortion within the Jahn-Teller energy term. It is found that this coupling provides a mechanism by which the coherent Jahn-Teller distortion is more stabilized by smaller rare earth ions. We analyze the appearance of the uniform shear distortion below the Jahn-Teller ordering temperature, estimate the Jahn-Teller ordering temperature and its variation between NdMnO3 and LaMnO3, and obtain the relations between distortions. We find good agreement between theoretical results and experimental data., Comment: 6 figures

The electronic structure of free-standing (ZnSe${)}_{\mathit{m}}$(ZnS${)}_{\mathit{n}}$ (001) strained-layer superlattices is studied through ab initio non-self-consistent factorized linear combination of atomic orbitals calculations, where the nonrelativistic local-density-functional formalism of Slater is used to represent the crystalline potential. The effects of elastic strain on the crystal superlattice structure are determined using the valence force field or Keating model. The heavy- and light-hole states are found to be localized in the ZnSe sublattice, while the lowest conduction state is delocalized all along the structure. The valence-band offset is determined to be large and increases as the strain on the ZnSe sublattice is increased. The conduction-band offset, though, is determined to be near vanishing, independent of the strain. These results suggest that for all degrees of strain there is a strong confinement of the holes in the ZnSe sublattice and a weak confinement of the electrons. This lack of a strong type-I formation accounts for the weak blue luminescence observed in photoluminescence and absorption measurements at room temperature.

A method is presented for the determination of the deformation of the unit cell of a crystal consisting of atoms bound by a three-body potential energy such as that in the Keating model. This method is based on the idea that the potential energy depends on the unit-cell shape only through the metric tensor of the primitive lattice vectors. The method is applied to obtain the stable structure of strained layer superlattices (GaP${)}_{1}$(InP${)}_{1}$(111) grown on GaAs(001) substrates. The c axis perpendicular to the substrate surface is predicted to be inclined 0.7\ifmmode^\circ\else\textdegree\fi{}.

The lattice vibrations in GaN/Ga 1-x Al x N(001) superlattices are studied within a rigid-ion model. The elastic forces are characterized by a generalized three-parameter Keating model, whereas the microscopic electric field is fully included via an Ewald summation. The composition and the disorder in the barriers are described within a generalized REI model. Cubic symmetry is assumed for each single material layer in the superlattice. In contrast to the GaAs/Ga 1-x Al x As case we observe a confinement of the acoustic phonons in the frequency region between the zone-boundary phonons of GaN and AlN. In the region of optical phonons both propagating and confined modes are found. The results are explained in terms of the light common anion of the compounds.

A simple and accurate model has been developed to study the thermal expansion of crystalline silicon based on the lattice dynamics. It is the first time to calculate thermal expansion of crystalline Si by using the strain phonon spectra. A modified Keating model, as the interactional potential, has been adopted to describe the elastic strain energy of Si lattice. This model includes four interactions and needs four force constants. The phonon dispersion relations with and without strain have been calculated. Combining the modified Keating model with the strain phonon dispersion relations, the analytical expressions for the thermal expansion coefficient a has been obtained. It is found that the value of α is 2.43×10−6 K−1 at the room temperature. The approach could be expected to be used further in nano silicon beam.

An augmented continuum theory, based on lattice dynamics theories, is developed to examine the elasticity of three-dimensional crystalline Si materials. The second-order elastic constants of Si can be expressed as the function of the force constants, with the modified Keating model. The phonon dispersion relations have been calculated by using the density functional perturbation (DFP) theory, from which the force constants can be extracted. Then the elastic modulus in any crystallographic directions can be calculated. The average deviation of Young's modulus from experiment is less than 3.8%. This approach is expected to be used in the design of silicon-based MEMS.

In this work we present a detailed computational study of the structural and elastic properties of cubic Al(x)Ga(y)In(1 - x - y)N alloys in the framework of the Keating valence force field model, for which we perform an accurate parametrization based on state-of-the-art density functional theory calculations. When analysing structural properties, we focus on the concentration dependence of the lattice constant, as well as on the distribution of the nearest and the next nearest neighbour distances. Where possible, we compare our results with experiment and calculations performed within other computational schemes. We also present a detailed study of the elastic constants for Al(x)Ga(y)In(1 - x - y)N alloy over the whole concentration range. Moreover, we include the accurate quadratic parametrization for the dependence of the alloy elastic constants on the composition. Finally, we examine the sensitivity of the obtained results to computational procedures commonly employed in the Keating model for studies of alloys.

Optical, vibrational and structural properties of II‐VI ternary alloys, ZnSxSe1−x and CdSxSe1−x are presented in this work. The estimates of elastic constants and vibrational frequencies are provided following the empirical pseudopotential method. The alloying effects are modeled through the virtual and modified virtual crystal approximation. Elastic constants are computed by combining both the VCA and MVCA with the Harrison bond orbital model. The band gaps for a number of alloys are computed to evaluate the refractive index (n) from Moss formula. We give the longitudinal ωLO(0), transverse ωTO(0) optical frequency, static frequency (e0), high frequency (e∞) dielectric constant and pressure derivatives of e0, e∞, ωLO (0) and ωTO(0). Moreover, photo‐elastic constants are also computed applying the Keating model and prescription of Davydov and Tikhonov.

We present a thorough theoretical study of ordering phenomena in nitride ternary alloys ${\mathrm{Ga}}_{x}{\mathrm{In}}_{1\ensuremath{-}x}\mathrm{N}$, ${\mathrm{Al}}_{x}{\mathrm{In}}_{1\ensuremath{-}x}\mathrm{N}$, and ${\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{N}$. Using the Monte Carlo approach and energetics based on the Keating model, we analyze the influence of various factors on ordering in bulk crystals and epitaxial layers. We characterize the degree of both short-range order (SRO) and long-range order (LRO) for different compositions, temperatures, and for substrates associated with different epitaxial strain. For the description of the SRO, the Warren-Cowley parameters related to the first four coordination shells are used. The LRO is detected by means of the introduced sim-LRO parameter, based on the Bragg-Williams approach. The description of the observed long-range ordering patterns and conditions for their occurrence follows.

The dispersion relations for optical and acoustic phonons are examined in bulk Si and Ge, Si and Ge strained layers grown on (001) and (111) surfaces, and ultrathin Si/Ge superlattices at ambient pressure and under hydrostatic pressure by using a modified Keating model. This model includes four interactions, which involve up to the fifth-nearest-neighbor atom, and force constants that depend on strain. These strain-modified force constants are related to specific cubic anharmonic terms in the potential energy and also to the empirical parameters p, q, and r that have been used to describe zone-center phonon shifts and splittings arising from strain. This model is used to obtain the mode Gr\"uneisen parameters ${\ensuremath{\gamma}}_{\mathit{i}}$ throughout the Brillouin zone for bulk c-Si and c-Ge, and explicit analytic expressions for ${\ensuremath{\gamma}}_{\mathit{i}}$ at the zone center and boundaries. Biaxial strain in the (001) plane is shown to affect phonon dispersion very differently than in previous work that used a much simpler model. For Si and Ge grown on a (111) substrate, the frequency shift due to the biaxial strain for the TO-phonon mode is found to be almost independent of wave vector. The pressure-induced change in the frequency of confined LO phonons in a ${\mathrm{Si}}_{12}$${\mathrm{Ge}}_{4}$ superlattice predicted by the model agrees with the change measured previously by Raman scattering. This modified Keating model is also used to obtain the second- and third-order elastic constants for Si and Ge.

Infrared absorption measurements of heavily carbon doped p-type AlAs have revealed a localized vibrational mode at 631.6 cmhigh-1 which appears as a Fano profile in uncompensated material. Raman scattering measurements show a corresponding line at about 637 cmhigh-1: possible reasons for the difference in frequency are discussed. The line is assigned to CsubAs acceptors and a comparison is made with a two parameter Keating cluster model. Additional weaker lines at 616.7, 632.8, 633.7 and 645.9 cmhigh-1 have been observed in absorption and possible interpretations, including CsubAs-CsubAs pairs, have been investigated. Passivation of a sample with hydrogen led to the observation of the H-CsubAs stretch mode at 2555.4 cmhigh-1, but satellites to this line have not been detected.

Using the Keating model and the Raman polarizability of Alben et al., the phonon Raman spectra of silicon wires are calculated. With the calculation results, the Raman spectra of porous silicon of some published papers are analyzed. Until now different authors have had different views on the luminescence mechanism of porous silicon, which may mainly result from the fact that they had different samples.

A gradient-stage model is proposed to describe the formation of the SiC buffer layer between a diamond film and the silicon substrate during CVD growth. Using the Keating model and minimizing the short-range bond stretching and bending potential, structural parameters were obtained which take into account a uniform strain distribution. Experimentally, X-ray photoelectron spectroscopy (XPS) and Auger electron spectroscopy (AES) were used to characterize the buffer layers during the preparation of diamond films grown on a silicon substrate by hot-filament chemical vapour deposition (HFCVD) and these results were compared to the theoretically derived model.

A simplified version of the Keating model originally derived for defect-chalcopyrite compounds is extended to include all AIIB2IIIC4VI compounds with ordered arrays of vacancies. It is shown that the trends and frequencies of the nonpolar optical modes of these compounds can be described within this model. Simple scaling laws for the interatomic force constants are proposed and compared with corresponding relations for other binary and ternary compounds with tetrahedral coordination. Eine vereinfachte Version des Keating-Modells, das ursprunglich fur die Defekt-Chalkopyrit-Verbindungen hergeleitet worden ist, wird auf alle AIIB2IIIC4VI-Verbindungen mit geordneten Vakanzensystemen erweitert. Es wird gezeigt, das die Trends und Frequenzen der nichtpolaren optischen Moden dieser Verbindungen im Rahmen dieses Modells beschrieben werden konnen. Fur die interatomaren Kraftkonstanten werden einfache Skalierungsgesetze vorgeschlagen und mit entsprechenden Beziehungen fur andere binare und ternare Verbindungen mit tetraedrischer Koordination verglichen.

The traditional elastic theories based on the continuum assumptions may not be directly applicable for microsystems and nanosystems without any modification. In this paper, an augmented continuum theory, based on lattice dynamics theories, is developed to examine the elasticity of three-dimensional Si materials. The second-order elastic constants of Si can be expressed as the function of the force constants. A modified Keating model, as the interactional potential, includes four interactions, and needs four corresponding force constants. The phonon dispersion relations have been calculated by using the density functional perturbation theory, from which the force constants can be extracted and optimized. The calculated phonon spectra agrees well with experimental results, with the relative error ranging from 1.4% to 6.1%. Combining the modified Keating model with the phonon dispersion relations, the analytic expressions for certain high-symmetry k points phonon frequencies and the elastic constants of Si can be obtained. Then the Young's modulus in , and crystallographic directions have been calculated, and the average deviation is less than 3.8%. The approach is expected to be used in the nano silicon beam.

Monte Carlo simulations with the Keating model have been performed to predict the lattice constant and bond length variations with composition for pseudo-binary semiconducting alloys. In general, it is observed that the deviations of the lattice constants from Vegard's law predictions are larger as the lattice mismatch between the constituent binaries increases. Further, it is noted that these alloys have partial virtual crystal model characteristics and tend to be more towards the flexible (floppy) crystal limit as compared to the rigid crystal limit. The topological rigidity parameters are bond-type dependent. The angular deviations from perfect tetrahedral structure are also measured.

A direct method for calculating local vibrational mode (LVM) frequencies from defect complexes in zinc blende structure semiconductors is discussed. Interatomic interactions are described by an extension of the Keating model. Sets of force constants or eleven pure host crystals, including homopolar and III-V semiconductors, are found by fitting to available phonon dispersion relations. Simulations of isolated impurities in III-V hosts are reported and, from these, some trends are deduced for the changes in force constants needed for reproduction of experimental measurements. The use of the method in the interpretation of experimental results from defect complexes is illustrated by detailed calculations of (i) isotopic fine structures of the CAs line and CAs-(As)interstitial lines in GaAs and (ii) line patterns associated with carbon in AlxGa1-xAs.

The geometrical structures and phonon spectra of strained α-Sn/Ge superlattices grown pseudomorphically on (001)-oriented substrates are studied using a generalized Keating model. It is found that symmetrically strained α-Sn/Ge superlattices grown on properly chosen substrates can be approximately achieved, which renders the growth of superlattices with large thickness possible. The general features of phonon spectra are discussed

The local electronic structure of intrinsic defects, including antisites, interstitials, vacancies and complexes with themselves or with Si impurities, in a (GaP)1/(lnP)1 (001) strained-layer superlattice determined by the Keating model, is calculated using a developed recursion method. Besides the localized states obtained, attention is paid to the electronic occupancy on various defects. A general consequence of describing the influence of isolated defects on the electron distribution of their neighbours is given. It is found that the charged state of an atom will be changed when it becomes an antisite. A closely bound (I(p)-P(III)) (III - Ga or In) pair, formed by coulombic interaction, will induce some states in energy gap. No evidence is found anywhere of Et-2-like behaviour in this system. Some novel diffusion mechanisms, based on I(p) and V(p) complex formation and dissociation, are presented.

A strain k.p model is used to investigate lattice temperature dependence of the piezoresistivity of p-doped silicon nanobeam sensor in the range of 100-600k based on the self-consistent solution of the Schodinger and Poisson equations. Based on quasiharmonic approximation, an analytical semi-continuum model is presented to describe the effect of size and temperature on elastic constants of the silicon nanobeam sensor by using Keating model. A quantitative comparison of the piezoresistive coefficient calculated with and without considering elastic constant correction indicates it is crucial to incorporate elastic constant correction in order to quantify the piezoresistivity of the silicon nanobeam. Our calculations demonstrate that size-dependent elastic constant correction is more important than temperature-dependent elastic constant correction, however, lattice temperature can lead to considerable changes in the piezoresistive coefficient due to the large redistribution of the carriers between the heavy hole and light hole subbands, which may impact the application of the nanobeam as sensors. Fortunately, the interaction of a piezoresistive effect with hole quantization effect results in a remarkable enhanced electromechanical properties, exhibiting a promising application in mechanical sensors.