The method of homogeneous deformation was combined with first-principles total-energy calculations to provide a general method for determining second- and third-order elastic constants for single crystals of arbitrary symmetry. Lagrangian strain tensors characterized by a single strain parameter are applied to the crystal lattice and the elastic strain energy is calculated from first principles. Second- and third-order elastic constants are obtained from a polynomial fit to the calculated energy-strain relation. Fitting the energy-strain relation, rather than the stress-strain relation, provides a more robust procedure and enables the use of certain first-principles codes where the stress tensor cannot be determined directly. To illustrate the method and to compare with previous work, we calculated the complete set of second- and third-order elastic constants for silicon (cubic lattice). Our results provide better agreement with experimental data than results from previous first-principles calculations. To demonstrate the use of the method for lower-symmetry crystals, second- and third-order elastic constants for alpha-quartz (trigonal lattice) were calculated and reasonable agreement was obtained with experimental results. Our method is general and can be applied to crystals with low symmetry and/or low yield strength where experimental determination of the third-order elastic constants is difficult.
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