Crystal stability ofα- andβ-boron

Abstract

The crystal stabilities of $\ensuremath{\alpha}$- and $\ensuremath{\beta}$-boron are studied theoretically by the density-functional calculations. The ground-state properties and thermodynamic properties are calculated by the pseudopotential method. These calculated thermodynamic properties include the effect of atomic disorder, observed experimentally, as well as the effect of phonons. The pressure dependence of the free energy is also studied. At zero temperature, it is found that $\ensuremath{\alpha}$-boron is more stable than $\ensuremath{\beta}$-boron. This does not change even if the zero-point energy and atomic disorders are considered. The contribution of these effects to the energy is small at $T=0\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. However, these effects eventually cause a phase transition to $\ensuremath{\beta}$-boron at high temperatures. By considering the phonon contribution as the chief source of the temperature dependence of the free energy, $970\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ is obtained as the transition temperature, which is in qualitative agreement with the experimental value of $1400\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. The difference between these values could be attributed to anharmonic effects. The effect of thermal expansion on the transition temperature is insignificant. At finite pressures, the stability of various polymorphs can be determined mainly using the atom density. The basic feature underlying all the above properties is that $\ensuremath{\alpha}$-boron is dense, while $\ensuremath{\beta}$-boron is dilute. For $\ensuremath{\beta}$-boron, an energetic consideration shows that the disorder in the atom arrangement is inherent. The present calculations reveal a small change in bond length for specific intericosahedral bonds, which is caused by an atomic disorder.