Elastic constants of a Si/Ge superlattice and of bulk Si and Ge

Abstract

We report extensive convergence tests of the total energy of Si computed within the local-density approximation with a plane-wave basis and pseudopotentials. These convergence tests indicate that to calculate the elastic constants to about 3% relative error requires the use of 400 plane waves in the electronic structure calculation and 10 special k points to compute the density and energy. Further, we find that using the Ceperley-Alder exchange-correlation form in calculating the elastic constants obtains better agreement with the experimental results than using the Wigner form. We report the calculated lattice constants and elastic constants---bulk modulus, ${\mathit{C}}_{11}$, ${\mathit{C}}_{12}$, and ${\mathit{C}}_{44\mathrm{---}}$of a ``free-standing'' Si/Ge ordered superlattice. For comparison, the results for bulk silicon and germanium are in excellent agreement with existing experiment and other calculations. The calculation for ${\mathit{C}}_{44}$ reported here is unique in the sense that the ``internal'' atom in the diamond unit cell moves while the crystal is sheared, though previously this relaxation has been dealt with differently. The equilibrium position of the sheared crystal cannot be predicted by scaling arguments from the unstrained crystal. An ``averaged elastic theory'' based on bulk Si and bulk Ge predicts our computed Si/Ge lattice constant and bulk modulus surprisingly well.