Fully analytic valence force field model for the elastic and inner elastic properties of diamond and zincblende crystals

Abstract

Using a valence force field model based on that introduced by Martin, we present three related methods through which we analytically determine valence force field parameters. The methods introduced allow easy derivation of valence force field parameters in terms of the Kleinman parameter ζ and bulk properties of zincblende and diamond crystals. We start with a model suited for covalent and weakly ionic materials, where the valence force field parameters are derived in terms of ζ and the bulk elastic constants C11, C12, and C44. We show that this model breaks down as the material becomes more ionic and specifically when the elastic anisotropy factor A = 2C44/(C11 − C12) > 2. The analytic model can be stabilised for ionic materials by including Martin's electrostatic terms with effective cation and anion charges in the valence force field model. Inclusion of effective charges determined via the optical phonon mode splitting provides a stable model for all but two of the materials considered (zincblende GaN and AlN). A stable model is obtained for all materials considered by also utilising the inner elastic constant E11 to determine the magnitude of the effective charges used in the Coulomb interaction. Test calculations show that the models describe well structural relaxation in superlattices and alloys, and reproduce key phonon band structure features.

Impact of stoichiometry and strain on Ge1−x Sn x alloys from first principles calculations

“…To provide a systematic analysis of the incorporation of higher Sn compositions in Ge 1−x Sn x , and to determine the resulting impact on the electronic structure, we undertake first principles electronic structure calculations for pseudomorphically strained Ge 1−x Sn x as a function of both Sn composition and strain. The latter is investigated by pseudomorphically straining Ge 1−x Sn x alloy supercells with respect to Ge, zinc telluride (ZnTe) and cadmium telluride (CdTe) substrates, with ZnTe and CdTe chosen as they respectively possess lattice constants close to that of the fictitious zinc blende IV-IV compound zb-GeSn [40,41], and of α-Sn. To assess the feasibility of thin film growth at different strains and alloy stoichiometries, we also compute the critical thickness of pseudomorphically strained Ge 1−x Sn x , thereby quantifying strain-related limitations to pseudomorphic growth.…”

Raman spectroscopy of group-IV Ge$_{1-x}$Sn$_{x}$ alloys: theory and experiment

“…We further note that the VFF potential presented in this work is broadly applicable to investigate the structural, elastic and vibrational properties of cubic-structured, isovalent multinary alloys (e.g. Ge 1−x C x , Ge 1−x Pb x or Si y Ge 1−x−y Sn x ) and, via inclusion of appropriate electrostatic terms, 55 can be extended to treat cubic-structured polar alloys (e.g. III-V semiconductor alloys).…”

“…( 8), parametrised via Table II For both Ge and α-Sn we note a typical VFF description of the phonon bands: the potential describes well the measured dispersion of the LA, LO and TO bands (solid black lines vs. closed black circles), but fails to capture accurately the full dispersion of the TA bands, tending to overestimate the TA phonon frequency by failing to capture the softening of these modes towards the zone boundaries (dashed grey lines vs. open grey circles). 55 We emphasise that the VFF potential exactly reproduces the relaxed second-order elastic constants C i j , so that the zone-centre LA and TA phonon group velocities are correctly described. 46 We also note that it is in principle possible to soften the dispersion of the TA bands and bring them into quantitative agreement with experiment, by extending the VFF potential to incorporate angular interactions involving four co-planar bonds.…”

“…Recently, we introduced a fully analytic harmonic VFF potential for the relaxation of diamond (non-polar) group-IV and zinc blende (polar) III-V materials and their alloys. 46,55 Here, we are interested solely in the evolution of Raman-active optical phonon modes which, due to the requirement to converse crystal momentum, participate in Raman scattering only when they possess wave vectors in the immediate vicinity of the Brillouin zone centre (Γ-point). 56 We therefore require that the harmonic terms of our VFF potential accurately describe the dispersion of the TO and LO branches close to Γ, while being less concerned with the accuracy with which the dispersion of the transverse and longitudinal acoustic (TA and LA) branches are described.…”

“…In Ref. 55 we showed that it is possible, by expanding Eq. ( 8) to second order in macroscopic and internal strains, to obtain analytical expressions linking the linear elastic properties to the harmonic force constants.…”