Towards an ab initio theory for the temperature dependence of electric-field gradient in solids: application to hexagonal lattices of Zn and Cd

Abstract: Based on ab initio band structure calculations we formulate a general theoretical method for description of the temperature dependence of electric field gradient in solids. The method employs a procedure of averaging multipole electron density component (l \neq 0) inside a sphere vibrating with the nucleus at its center. As a result of averaging each Fourier component (K \neq 0) on the sphere is effectively reduced by the square root of the Debye-Waller factor [exp(-W)]. The electric field gradient related to a sum of K-components most frequently decreases with temperature (T), but under certain conditions because of the interplay between terms of opposite signs it can also increase with T. The method is applied to calculations of the temperature evolution of the electric field gradients of pristine zinc and cadmium crystallized in the hexagonal lattice. For calculations within our model of crucial importance is the temperature dependence of mean-square displacements which can be taken from experiment or obtained from the phonon modes in the harmonic approximation. For the case of Zn we have used data obtained from single crystal x-ray diffraction. In addition, for Zn and Cd we have calculated mean-square displacements with the density functional perturbation treatment of the Quantum Espresso package. With the experimental data for displacements in Zn our calculations reproduce the temperature dependence of the electric field gradient very accurately. Within the harmonic approximation of the Quantum Espresso package the decrease of electric field gradients in Zn and Cd with temperature is overestimated. Our calculations indicate that the anharmonic effects are of considerable importance in the temperature dependence of electric field gradients.