Application of generalized gradient approximations: The diamond–β-tin phase transition in Si and Ge

Generalized gradient approximations (GGA) for the exchange-correlation energy have been suggested to improve upon the local-density approximation (LDA) in density-functional theory. It has been demonstrated by various authors that the gradient correction suggested by Becke for the exchange functional and Perdew for the correlation energy, as well as the recent GGA by Perdew and Wang (1991), yield improved cohesive energies for solids, correcting for the overbinding in LDA. In this paper we focus on a structural phase transition of Si, which under pressure transforms from the diamond structure to the -tin structure. The coexistence pressure calculated within the LDA is 80 kbar, smaller than the experimental values (103--125 kbar). Both gradient-corrected schemes yield a coexistence pressure which is larger than the LDA value (146 kbar for the Becke-Perdew and 122 kbar for the Perdew-Wang exchange-correlation functional). Ge undergoes a similar transition at around 105 kbar. The LDA result for the coexistence pressure is 73 kbar, it amounts to 118 kbar when calculated with the Becke-Perdew GGA, and 98 kbar within the Perdew-Wang GGA.