Elastic properties of Si, Ge and GaAs

Calculations of the full stress tensor is a method ideally suited to the derivation of elastic constants, since it contains up to six independent pieces of information that otherwise would require extensive calculations of total energy. The c- and c. 2 elastic constants can be found from the stress-strain relation with the application of an ei-strain. (The Voigt notation is used, see e.g. (Nye, 1957), i.e. 11- -1, 22- 2, 33- 3, 23 4, 13 5, 12- -6 thus 

With a strain of =-0.004 we obtain the c and c 2 given in Table 1. The differences from experiment are -6 for Si, and up to -9/5 for Ge and GaAs where where only two special k-points are used, resulting in a lower accuracy as estimated above. These numbers agree well with the independently calculated bulk moduli B 

The force constant matrix for atomic displacements is denoted (t,t ), T( t) is the third-rank internal strain tensor, c is a fourth rank elastic constant tensor. Tensor contraction is understood throughout. Thus the restoring force on an atom is 

The force constants equals Pj, where p is the reduced mass 

Thus the two force calculations determine without necessarily calculating the stress. The strains and displacements are illustrated in Fig. 2. 

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