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.

Elasticity of nanometer-sized objects

“…References [15][16][17][18], on the other hand, start from the more general atomic level description and then try to unveil physical properties from the underlying atomic description. It is important to note that these works do not deal directly with nanoscale systems but rather focus on the effects of nanoscale inhomogeneities in bulk systems.…”

“…References [15] and [16] concern the elastic properties of grain boundaries. These works define atomic-level elastic moduli as the second derivative of the energy associated with each atom with respect to strain and then go on to study the behavior of such moduli near grain boundaries.…”

“…In coarse graining from interatomic interactions to mechanical response, some works rely upon the problematic decomposition of the total system energy into a direct sum of atomic energies, 15,16 which is always arbitrary and particularly inconvenient for connection with ab initio electronic structure calculations. The remaining works which attempt to build up overall response from atomic level contributions 17,18 fail to account properly for the Poisson effect.…”

Surface free energy and its effect on the elastic behavior of nano-sized particles, wires and films

“…Wolf and coworkers Kluge et al, 1990;Wolf and Kluge, 1990), who studied superlattices of (0 0 1) twist boundaries, as well as Adams et al (1989), who examined the S ¼ 5ð0 0 1Þ twist boundary in a thin film of copper, have found an increase of the Young's modulus perpendicular to the boundary plane and a substantial decrease of the shear modulus in the boundary plane in the atomic layers adjacent to the boundary. Bassani and co-workers Bassani et al, 1992;Vitek et al, 1994;Marinopoulos et al, 1998) defined the local elastic modulus tensor and determined the values of the local elastic modulus tensor near grain boundaries in several face center cubic metals using molecular dynamic simulations. They, too, found that the local elastic moduli are significantly different for atoms near the grain boundaries.…”

Stress and strain around grain-boundary dislocations measured by high-resolution electron microscopy

“…Compatibility conditions for the stresses and strains across boundaries have been elaborated [9] and used for modelling the strain response of twin boundaries for example [10]. For the moment, however, only theoretical results have been presented, the most relevant to the current study concerning twin boundaries in silicon [11].…”