Structural stability of phases of black phosphorus

Abstract

The pseudopotential method is used to examine the structural stability of high-pressure orthorhombic, rhombohedral (A-7), and simple cubic (sc) phases of black phosphorus. The calculated ground-state properties are in good agreement with the measured values for each phase. A total-energy study gives the orthorhombic phase as the most stable structure at low pressures. At higher pressures it transforms into the A-7 structure in agreement with experiment. At even higher pressure, the calculations indicate that the A-7 phase is stable with respect to the sc structure as is the case for other group-V elements. Measurements show that the sc phase is stable at 110 kbar at room temperature. This suggests that the calculated crystal energy for the A-7 phase is too low and it requires a finite energy shift. With an a priori energy shift, the transition pressure and volume are in good agreement with experiment. A possible source of this correction is the zero-point energy or temperature renormalization of the phonon frequencies caused by anharmonicity. The changes of the energy band structures and charge densities for the A-7 displacement and selected phonon frequencies are calculated.