Temperature dependence of the electric-field gradient in hcp-Cd from first principles

We determine the temperature dependence of the electric-field gradient in hcp-Cd from first principles. The calculations are based on the ab initio determination of the phonon density of states spectrum of the solid. Using only moderate accuracy requirements, the temperature dependence of the electric-field gradient in hcp-Cd is reasonably well reproduced. The origin of its peculiar $T^{3∕2}$ dependence is discussed.

Figure 1

Different temperature dependence of EFG. Diamonds: experimental data points (Refs. 9, 10, 15). Solid lines: fit through the experimental values, using the relation $V_{zz}\left(T\right)=V_{zz}\left(0\right)(1-B{T}^{\alpha })$.

Figure 2

(a) The unit cell of hcp-Cd used in molecular dynamics calculations. (b) The $2\times 2\times 1$ rhombohedral supercell (24 atoms), representing the hexagonal structure used in the calculation of the dispersion curves. The atom positions along the $z$ axis are indicated in fractional coordinates ($a=b=c=15.78\mathrm{a.u.}$, $\alpha =\beta =\gamma =77.57°$).

Figure 3

Atom positions, in fractional coordinates, as a function of time at $50\mathrm{K}$ and $400\mathrm{K}$ (772 time steps were calculated for $50\mathrm{K}$ and 533 time steps for $400\mathrm{K}$).

Figure 4

(Color online) The five components of the EFG tensor close to the equilibrium in the MD simulation at $400\mathrm{K}$ (the first $4\mathrm{ps}$ of Fig. 3 are not shown): (a) $V_{zz}$; (b) $V_{xx}$; (c) $V_{xy}$, $V_{xz}$, and $V_{yz}$. Inset picture: the complete evolution in time of the five EFG tensor components at $400\mathrm{K}$. In (a) $V_{zz}$ at $50\mathrm{K}$ (the point at $9\mathrm{ps}$ in Fig. 3 is shifted to coincide with $4\mathrm{ps}$ in this picture) is also shown (thin black line).

Figure 5

Ab initio calculated phonon dispersion curves for hcp-Cd (solid line) compared with experiment (dots). The frequencies at $\Gamma$ obtained from MD are indicated by the large open circles.

Figure 6

(a) Calculated values of electric-field gradient for hcp-Cd as a function of temperature: for a fixed lattice constant (diamonds) and for the lattice constant corresponding to that temperature (triangles), compared with experiment. MD results at $50\mathrm{K}$ and $400\mathrm{K}$ (circles) are rescaled so that the $50\mathrm{K}$ result matches the experimental value. (b) The standard deviation of the $9\times 24=216V_{zz}$ values as a function of temperature.