A temperature-related boundary Cauchy–Born method for multi-scale modeling of silicon nano-structures

•Multi-scale method is used for thermal properties of silicon at finite temperature.•The multi-scale model is performed based on the molecular dynamics and FEM method.•The Cauchy–Born hypothesis is used to relate the atomic position to continuum media.•The thermo-mechanical finite element model is applied in continuum domain.•The Boundary Cauchy–Born model is applied for the surface, edge and corner effects.

The surface, edge and corner effects have significant influences in the electrical and optical properties of silicon nano-structures. In this paper, a novel hierarchical temperature-related multi-scale model is presented based on the boundary Cauchy–Born method to investigate not only the surface but also the edge and corner effects in thermal properties of diamond-like structures such as silicon nano-structures at finite temperature. A combined finite element method and molecular dynamics are respectively employed in macro- and micro-scale levels. The temperature-related Cauchy–Born rule is applied using the Helmholtz free energy, as the energy density of equivalent continua relating to the Tersoff inter-atomic potential. The model employs radial quadratures at the surface, edge and corner elements as an indicator of material behavior. The capability of computational algorithm is illustrated by numerical simulation of a nano-scale cube at finite temperature and the results are compared with the atomistic model.