Molecular-Dynamics Simulations and Interatomic Potentials

Atomistic simulations are used to study the structure, energetics, and dynamics of a collection of interacting atoms following classical mechanics. For a system containing N atoms with positions {ri}_i^N₌₁and velocities {!i = dri / dt}_i^N₌₁, the total Hamiltonian is given by

H({r_i}_i=1^N )= T+ U = 1

2m_i!_i²+ U({r_i}_i=1^N )

i=1

"

where T and U represent kinetic and potential energy, respectively, of the system and mi

is the mass of atom i. A molecular dynamics (MD) simulation is based on the integration of Newton’s equations of motion for all N interacting atoms and generates a trajectory of the atomic positions of all the atoms. Most of the thermodynamic properties of the material system can be calculated as time averages along the atomic trajectories and the corresponding transport properties can be calculated from computation of time correlation functions (Allen & Tildesley, 1990). MD simulations, however, are most useful in identifying the dynamical response of a material system to external forces; the structural evolution of the material can be monitored easily over the course of the simulation and atomic-scale information that is not easily accessible in experiments can be extracted from MD simulations. Hence, MD simulations are very useful for the fundamental understanding of structure-property relationships.

As can be seen from the expression for Hamiltonian, Eq. (2.1), the accuracy of the potential energy function, U({r_i}_i^N₌₁) , in describing the interatomic interactions is vital to obtaining reasonably accurate predictions for the behavior of real materials. In the MD simulation studies of metallic systems conducted in this thesis, semi-empirical interatomic potentials that are based on the embedded-atom method (EAM) in order to describe the interatomic interactions in metals were employed (Foiles et al., 1986; Oh &

Johnson, 1988; Angelo et al., 1995; Mishin et al., 1999, 2001). In the EAM formalism, the total potential energy U of an elemental system is represented as

U = 1

2 u(r_ij)

!

!

where u(r_ij) is the pair potential as a function of the distance between atoms i and j, rij, and ! is the embedding energy that is a function of the host electron density, given by

!_i^' = !(r_ij)

i" j

#

functional form of the EAM potential is based on certain concepts and principles of solid-state physics, the comprising potential functions used in MD simulation practice are essentially empirical and their parameterization is based on fitting procedures to reproduce known (mainly from experimental measurements) material properties.

In almost all EAM parameterizations, and certainly all the EAM parameterizations used in the studies within this thesis, the total energy of the fcc lattice under uniform compression and dilation is required to obey the universal equation of state proposed by Rose and coworkers (Rose et al., 1984). As a result, the EAM potentials account properly for the lattice anharmonicity (Foiles & Daw, 1988) and provide a good modeling framework for simulating high-pressure and shock-loading experiments (Oh & Johnson, 1988; Johnson, 1988; Foiles & Daw, 1988). These potentials have been tested extensively regarding their predictions of structural and mechanical properties. Specifically, the potentials predict accurately point-defect properties and surface energies and favor energetically the fcc phase over the hexagonal close-packed (hcp) and body-centered cubic (bcc) phases. Furthermore, the computed unstable, generalized, and intrinsic stacking-fault energies are in good agreement with experimental results (Oh & Johnson, 1988; Angelo et al., 1995; Mishin et al., 1999, 2001).

Figure 2.1. Stress-strain curves for shearing deformation of copper in the [111] plane along the ^[112] direction. Shear strain is defined as the shear displacement divided by a/!6. The black and red curves correspond to the potential parameterizations used in the present work. The blue and green curves correspond to ab initio and EAM-Ackland calculations as reported by Zhu and co-workers (Zhu et al., 2004a).

In addition to their good predictions of single-value properties, the EAM potentials used in the studies within this thesis predict accurately the elastic deformation of Cu. Figure 2.1 compares the stress-strain response for a simple shearing deformation in the (111) plane along the [112] direction predicted by the Oh & Johnson potential (Oh & Johnson, 1988) and the EAM potential parameterization by Mishin and coworkers (Mishin et al., 2001) with those obtained from ab initio density functional theory (DFT) calculations and from another EAM parameterization developed by Ackland and co-workers (Ackland et al., 1997) that was not used in the studies within this thesis. The stress-strain response from the ab initio, Ackland, and EAM-Mishin potentials shown in Fig. 2.1 were calculated by Zhu and coworkers (Zhu et al.,

2004a). It is demonstrated clearly in Fig. 2.1 that the predictions of the Oh & Johnson potential and Mishin potentials are in good agreement with the ab initio DFT calculations. Zhu and coworkers verified further the ability of the EAM-Mishin potential to predict accurately the characteristics of the first nucleated dislocation in nanoindentation experiments; the EAM-Mishin potential accurately predicted the mixed shear mode of the Shockley partial in nanoindentation simulations (Zhu et al., 2004a).

In order to model silicon-germanium heterostructures (Si1-xGex, 0 ! x !1), an empirical many-body interaction potential proposed by Tersoff was used (Tersoff, 1989). In Tersoff’s formulation, the potential energy U is expressed as

U = 1

where the indices i, j, and k are used to label the atoms in the system, rij denotes the distance between atoms i and j, and !ijk is the bond formed between distance vectors rij

and rik at vertex i. Single subscripted parameters, such as "i and ni depend only the type of the atom. The Tersoff potential has been used quite successfully for modeling the solid-state mechanics and surface physics of semiconductor materials.

2.2. Atomistic Simulations of the Mechanical Behavior of