# Modelling the Defect Processes of Materials for Energy Applications
## 2 Methodology
### Cluster Expansion and Special Quasirandom Structures  
Modelling crystal structures using DFT is straightforward from a structural point of view as all that is required is the construction of supercells with periodic boundary conditions. Complications arise when there is structural disorder (i.e., non-stoichiometry and/or atomic species able to occupy multiple crystal sites). Modelling such a system using DFT with the use of large supercells where atoms are randomly inserted in sites can be computationally very expensive, and reproduction of the statistics of the random system may even be practically intractable. Conversely, the SQS approach effectively enables the use of DFT in disordered systems by extracting the physical properties of the random alloy from a small designed cell. The properties of random alloys (for example, the band gap, total energy and point defect energies) are strongly dependent upon the arrangement of the atoms on the underlying lattice, which is known as configuration \(\sigma\). In the cluster expansion (CE) method, the dependence of a given property \(f\) on \(\sigma\) can be described via [52; 53; 54; 55; 56; 57; 58]:  
\[f(\sigma)=\sum_{\alpha}D_{\alpha}J_{\alpha}\overline{\Pi}_{\alpha}(\sigma) \tag{12}\]  
where \(\alpha\) is a figure (cluster) that comprises a group of \(k\) lattice sites (\(k=1,2,3\) indicates single site, pair, and triplet, and so on), \(D_{\alpha}\) is the degeneracy factor that expresses the number of symmetrically equivalent figures of type \(\alpha\) per lattice site, \(\overline{\Pi}_{\alpha}(\sigma)\) is the correlation function and \(J_{\alpha}\) is the effective cluster interaction for figure \(\alpha\). When the effective cluster interactions are figured, Equation (12) may be used to extract the properties of any lattice configuration. Although CE can be deemed as a robust way to characterise the ordered and disordered states within a single framework, the SQS methodology is far more computationally efficient when considering the physical properties of the random alloy. In practice, SQSs are designed small-unit-cell ordered structures that possess correlation functions comparable to the correlation functions of the random alloy [59; 60; 31]. Since in random alloys there is no pattern in the occupation of the available sites, this results in the alloy correlation functions being expressed as the product of the lattice-averaged spin variables. In a random binary \(A_{1-x}B_{x}\) alloy [58],  
\[\left\langle\overline{\Pi}_{\alpha}\right\rangle_{R}=(2x-1)^{k_{\alpha}} \tag{13}\]  
where \(k_{\alpha}\) is the number of vertices in figure \(\alpha\). One of the major advantages of the SQS approach in comparison to the CE method is that it can be employed to resolve non-scalar properties such as elastic constants. Employing periodically repeating supercells may induce erroneous correlations that are not appropriate when modelling a random alloy. The error due to the SQS approximation is [58]:  
\[f_{\text{SQS}}-\left\langle f(\sigma)\right\rangle_{R}=\sum_{\alpha}D_{\alpha }J_{\alpha}\big{[}\big{(}\overline{\Pi}_{\alpha}\big{)}_{\text{SQS}}-\left\langle \overline{\Pi}_{\alpha}\right\rangle_{R}\big{]} \tag{14}\]  
The basic assumption at the heart of the SQS method is that interatomic interactions at large distances are weaker than interactions at smaller distances. This means that the cluster expansion of a property is strongly dependent upon the cluster interactions of the short-ranged clusters. Consequently, it is feasible to form SQSs cells that reproduce the pair and multisite correlation functions of the random alloy between the first few nearest neighbours. Arguably, more extended SQSs cells can be better because they have smaller periodicity errors and in essence serve as better approximations of the real random alloy. This in turn implies that the predicted physical properties for more extended SQS cells will more accurately match the properties of the random alloy.