Transition pressures and enthalpy barriers for the cubic diamond$\to \beta$-tin transition in Si and Ge under nonhydrostatic conditions

We present an ab initio study of the phase transition cubic diamond$\to \beta$-tin in Si and Ge under hydrostatic and nonhydrostatic pressure. For this purpose we have developed a method to calculate the influence of nonhydrostatic pressure components not only on the transition pressure but also on the enthalpy barriers between the phases. The calculations were performed using the plane-wave pseudopotential approach to the density-functional theory within the local-density and the generalized-gradient approximation implemented in VASP. We find good agreement with available experimental and other theoretical data.

Figure 1

Contour plot of the total energy $E(V,c∕a)$ (solid lines) and of the average pressure $p_0(V,c∕a)$ [dashed lines, see Eq. (6)] for Si with GGA. The bold-dashed line corresponds to the value $p_0=0$. The interval of the contour lines is $50\mathrm{meV}$ for the energy and $20\mathrm{kbar}$ for the pressure surfaces. The black dots mark the equilibrium positions of the cd $(c∕a=\sqrt{2})$ and the $\beta$-tin phase $(c∕a=0.55)$. The dotted line marks the parameters under hydrostatic condition.

Figure 2

Volume $V$, reduced enthalpy (see text), and enthalpy barrier $\mathrm{\Delta }H$ as a function of the hydrostatic pressure for Si within GGA. The crossing of the solid and dashed lines determines the transition pressure.

Figure 3

Contour plot of the total energy $E(V,c∕a)$ (solid lines) for Ge (GGA) with an interval of $0.2\mathrm{eV}$ between contour lines. Besides the hydrostatic condition (bold solid line) nonhydrostatic conditions $(p_z-p_x=-15,-10,\dots ,20)$ are shown.

Figure 4

Equation of state $H\left(p_0\right)$ for nonhydrostatic conditions as a function of the average pressure $p_0$. The difference $p_z-p_x$ of two neighboring lines is $5\mathrm{kbar}$. The black dots mark the transition pressures $p_0^{\mathrm{t}}$.

Figure 5

Enthalpy barriers at the average transition pressures $p_0^{\mathrm{t}}$ and transition pressures (average transition pressure $p_0^{\mathrm{t}}$ and the corresponding components $p_x^{\mathrm{t}}$ and $p_z^{\mathrm{t}}$) as a function of $p_z-p_x$. The dashed line marks the boundary $p_x=0$ and $p_z=0$.

Figure 6

Enthalpy barriers, as in Fig. 2, for fixed nonhydrostatic conditions $p_z-p_x=-20,-15,\dots$, $35\mathrm{kbar}$. The solid horizontal line at $25\mathrm{meV}$ marks the thermal excitation energy at RT.