Interatomic potentials

Classical modelling assumes that the electrons are integrated out and atoms and ions are treated as point particles. The total energy then becomes a series of interactions between different orders of the total number of atoms [136]. The many-body energy can be written down as an infinite series shown below,

U = N X i=1 Ui+ 1 2 N X i=1 N X j=1 Uij + 1 6 N X i=1 N X j=1 N X k=1 Uijk+ . . . , (3.5)

where the first term in the series is the self-energy of an atom, the second is a pairwise interaction and the third term is a three body term etc. As the number of bodies involved increases the contribution to the total energy diminishes such that it becomes possible to truncate the series without loosing significant accuracy. The trick is to truncate the series in order to still describe a system of interest well whilest decreasing computational complexity. Usually functional forms which are parameterised to experiment are used to represent the N-body terms. There are many of these functional forms to choose from and often it is the case that different functional forms describe different classes of materials. One of the first interatomic potentials created is known as the Lennard- Jones potential which approximates the interaction between atoms as a series of 2-body (pairwise) interactions. The Lennard-Jones model has been shown to work well for gases such as Argon [187–189]. The functional form for the Lennard-Jones model is shown below, VLJ(rij) = 4 "  σ rij 12 − σ rij 6# , (3.6)

where  and σ are parameters which describe the strength and range of the bonding. The general form of this potential is of an attractive r−6 part and a repulsive r−12 part. The repulsive part is based on the Pauli exclusion principle which states that it is impossible for the wavefunctions of two fermions in the same spin state to overlap. The attractive part is more fundamentally based on the Van de Waals force based on the electrostatic dipole-dipole interaction. Although this potential has serious shortcomings when modelling anything other than inert gases [190] this simple method was the starting point for the whole field of materials modelling. Some parameters for the following materials (Ar, Kr, CH4, O2, H2, C2H4) have been determined by Matyushov [191].

Further parameters can be found in Ashcroft-Mermin [64]. 3.1.1.1 Buckingham potentials

In this thesis many simulations of ionic materials are performed so an appropriate in- teratomic potential which describes this bonding type is required. In ionic materials electrons are transferred between atoms, this occurs because this is the most favourable way in which electron shells can be filled. For example in NaCl (or regular salt) sodium is electropositive meaning that it wants to loose electrons, becoming smaller and ionised.

While chlorine is electronegative meaning that it is more favourable to gain electrons and fill its partially filled electron shell becoming larger and an ion. In these materi- als the bonding is predominately electrostatic in nature. Electrostatic bonding means that short range and long range descriptions of the bonding need to be considered. It has been shown that an adjusted Buckingham-Coulomb functional form can be used to model the interactions in metal-oxides [134, 192]. The Buckingham functional form is shown below, Vij(rij) = Aijexp  −rij ρij  −Cij r6 ij + qiqj 4πrij , (3.7)

where A, ρ and C are parameters of the model, qi and qj are the charges of the ions

involved, and  is the permittivity. The variables A, ρ and C are parameterised from experimental values such as the cohesive energy and the lattice constant. The increased number of parameters of the Buckingham potential allows for a larger degree of control on how the potential behaves. There is however a drawback by increasing the number of parameters as it becomes possible to over fit the problem, fitting idiosyncrasies. Buck- ingham potentials have been shown to be effective in bulk, surfaces and GBs. However they are limited in the sense that the charge on each atom is fixed and so is impossible for atoms in these simulation to become polarised or change their charge state. It is possible to extend this model using the shell model potential which allows the ionic charges to po- larise during a simulation [193]. These potentials have been very successful in modelling a range of metal oxides including MgO which is studied in this thesis [194–196].

3.1.1.2 The embedded atom method

The EAM is a classical method used to model the interaction between atoms. It expands on the pairwise type potentials such as the Lennard-Jones and Buckingham potentials described previously by adding many-body terms. The many-body terms consider the sum over many atoms in a locality rather than between pairs. It is beneficial over the use of simple pair potentials for many reasons including describing more accurately the relationship between the coordination and the bond energy of metals which is not linear. The EAM also allows the Cauchy condition2 to be violated which for most metals is positive. At worst the EAM is only twice as computationally expensive as simple pair

2

The Cauchy condition relates two elastic moduli c12= c44, where c44 is the shear modulus and c12

potentials [29, 63]. The total energy of a system of atoms within the EAM is described in the following way,

Etot= 1 2 X i,j V (rij) + X i Fi(ρi) , (3.8) ρi= X j Φ(rij), (3.9)

where V (rij) is a pair potential term which depends on the separation between atoms

i and j and Fi is the embedding energy function for atom i. The latter term accounts

for the many-body aspects of the atomic binding and is expressed in terms of a sum over atom centred functions Φ(rij) which phenomenologically represents the shape of

the electron density around a particular atom. Although the EAM was originally de- signed for use on sp-bonded metals it has been applied to many materials including transition metals. The EAM potentials are usually parameterised by fitting either to experimental data and/or first principles calculations. The EAM offers a good balance between physical accuracy and computational feasibility allowing supercells containing millions of atoms to be simulated in a reasonable time. Extensive research undertaken on transition metals has demonstrated that EAM potentials give an accurate description of many bulk, surface, GB and defect properties [31–34]. It is often the case that many of the potentials do not perform well for situations which are far from the bulk. It is essential that a rigorous testing regime is undertaken to ensure that the correct inter- atomic behaviour is expected. Such tests include reproducing the cohesive energy, the bulk modulus and the lattice constant. Once these have been found more complex tests must be performed, these are tests such as forcing a different geometric structure upon a material. This should have the effect of giving a higher cohesive energy and hence a less stable structure. Although in theory the EAM potential does not contain the details to know which structural phases should be the most energetically favourable a priori [63] EAM potentials are parameterised with the most stable experimental structural phases.