Phonon spectra and crystal structure of Zn and Cd using the resonant model-potential approach

Abstract

In the present study, the resonant model potential (RMP) method derived for $d$-band metals is applied to the microscopic calculation of the phonon spectra and to the determination of the crystal structure of lowest energy of zinc and cadmium. The self-consistent RMP is of the form $v(E)=w(E)+\frac{{A}_{d}\mathcal{U}}{(E\ensuremath{-}\mathcal{E})}$, where $w(E)$ is a Heine-Abarenkov-type nonlocal model potential (including the self-consistent screening potential); the second term with $\mathcal{U}=\ensuremath{\Sigma}{m}^{}4\ensuremath{\pi}\ensuremath{\gamma}(r)\ensuremath{\gamma}({r}^{\ensuremath{'}}){Y}_{2m}^{*}(\stackrel{^}{r}){Y}_{2m}({\stackrel{^}{r}}^{\ensuremath{'}})$ is a Heine-type resonant model potential which acts only on $d$ states. The model parameters are determined so that the self-consistent Wigner-Seitz neutral pseudoatom potential is equivalent in the physical energy range to the Wigner-Seitz self-consistent potential calculated in the framework of the Hartree-Slater scheme. The model parameters are then used to calculate the phonon dispersion, bulk modulus, and the structural energy of Zn and Cd. The disperion results and the bulk modulus are found in reasonably good agreement with the experimental data for both metals. Of course the calculation gives the stable hcp structure for the two metals, but the axial ratios are not predicted to be as high as the observed ones. In order to know the effect of the resonant part of the RMP, we performed calculations by neglecting the resonant part. This neglect results (a) in decreasing the value of the bulk modulus by about 20% for Zn and 15% for Cd, (b) in deducing the stable hcp structure at about the ideal axial ratio, and (c) in lowering the phonon frequencies in general by 25-30%. The strong near-resonance hybridization effect in the RMP has also been studied. Our calculations show this effect to be small on the bulk modulus and phonon frequencies of Zn.