Towards an ab initio theory for the temperature dependence of electric field gradients in solids: Application to hexagonal lattices of Zn and Cd
Based on ab initio band-structure calculations we formulate a general theoretical method for description of the temperature dependence of an electric-field gradient in solids. The method employs a procedure of averaging multipole electron-density component (l ≠0 ) inside a sphere vibrating with the nucleus at its center. As a result of averaging, each Fourier component (K ≠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 uc(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 uc(quantum espresso) package, the decrease in 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.