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

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 $\zeta$ 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 $\zeta$ and the bulk elastic constants $C_{11}, C_{12}$, and $C_{44}$. We show that this model breaks down as the material becomes more ionic and specifically when the elastic anisotropy factor $A=2C_{44}/(C_{11}-C_{12})>2$. The analytic model can be stabilized 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 utilizing the inner elastic constant $E_{11}$ 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.

Figure 1

Zincblende primitive cell.

Figure 2

Valence force field interaction terms contributing to Eq. (10). From left to right: bond stretching, $k_r$; bond bending, $k_{\theta }$; bond-bond stretching, $k_{rr}$; bond-stretching angle-bending coupling constant, $k_{r\theta }$.

Figure 3

Phonon band structure of GaAs calculated using different VFF parametrizations: (a) band structure calculated using covalent VFF with effective charge parameter $S=0$, and force constants described by Eqs. (17, 18, 19, 20); (b) band structure calculated using ionic VFF, with effective charge determined via Eq. (23), and force constants determined by Eqs. (24)–(27); (c) band structure calculated using ionic VFF, with effective charges determined via Eq. (29) and force constants given by Eqs. (24)–(27). The filled symbols are experimental frequencies taken from Ref. [65].