Local and Effective Elastic Properties of Grain Boundaries in Silicon

Abstract

When considering the mechanical behaviour of materials an important property is the tensor of elastic moduli. Recently, local elastic moduli of interfaces have been defined and studied for metallic materials [1 to 3]. In these works grain boundaries are regarded as heterogeneous continua composed of ‘phases’ associated with individual atoms which possess elastic moduli identified with the atomic-level moduli evaluated at corresponding atomic positions. From this representation it is possible to define the ‘effective’ moduli of the grain boundary region. In this paper this concept is developed for materials with covalent character of bonding, specifically silicon. Using the Tersoff's potential [4, 5], the atomic-level and effective elastic moduli of the interfacial region have been evaluated for three alternate structures of the Σ = 3 (112-)/[11-0] tilt boundary. These calculations are then compared with the continuum bounds on the effective moduli evaluated using the classical minimum-energy principles of elasticity. The effective moduli calculated in the atomistic framework are generally within the continuum bounds and thus the present study demonstrates that the heterogeneous continuum model of the interfaces is appropriate for the description of the elastic properties of grain boundaries in silicon. An important aspect addressed in this study is the uniqueness of interfacial elastic moduli since their evaluation involves the energy associated with an atom which cannot be defined uniquely. The calculations have been made for two different partitions of the total energy into energies associated with individual atoms. These two partitions lead to almost identical results for the effective moduli and continuum bounds when the tensor of the atomic-level moduli is positive definite. When some atomic-level moduli are not positive definite the results may depend on the chosen energy partition.