Elasticity of si calculated with a lattice dynamics model

The traditional elastic theories based on the continuum assumptions may not be directly applicable for microsystems and nanosystems without any modification. In this paper, an augmented continuum theory, based on lattice dynamics theories, is developed to examine the elasticity of three-dimensional Si materials. The second-order elastic constants of Si can be expressed as the function of the force constants. A modified Keating model, as the interactional potential, includes four interactions, and needs four corresponding force constants. The phonon dispersion relations have been calculated by using the density functional perturbation theory, from which the force constants can be extracted and optimized. The calculated phonon spectra agrees well with experimental results, with the relative error ranging from 1.4% to 6.1%. Combining the modified Keating model with the phonon dispersion relations, the analytic expressions for certain high-symmetry k points phonon frequencies and the elastic constants of Si can be obtained. Then the Young´s modulus in <;100>;, <;110>; and <;111>; crystallographic directions have been calculated, and the average deviation is less than 3.8%. The approach is expected to be used in the nano silicon beam.