Finite Temperature Multiscale Methods for Silicon NEMS

In this chapter, we present a finite temperature quasicontinuum method for multiscale analysis of silicon nanostructures at finite temperature. The quasicontinuum method uses the classical continuum mechanics framework, but the constitutive response of the system is determined by employing an atomistic description. For finite temperature solid systems under isothermal conditions, the constitutive response is determined by using the Helmholtz free energy density. The static part of the Helmholtz free energy density is obtained directly from the interatomic potential while the vibrational part is calculated by using the theory of quantum-mechanical lattice dynamics. We describe three quasiharmonic models, namely the real space quasiharmonic model (QHM), the local quasiharmonic model (LQHM), and the reciprocal space quasiharmonic model (QHMK), to compute the vibrational free energy. We also describe a QHMG approach - where the quasiharmonic approximation is combined with the local phonon density of states (LPDOS). The LPDOS is efficiently calculated from the phonon Green's function (GF) by using a recursion method.