Supercell method

Electronic structure methods for studies of nanostructures can be divided broadly into supercell methods and real-space methods. Supercell methods use standard k-space electronic structure techniques separating periodically repeated nanostructures by distances large enough to neglect their interactions. Direct space methods do not need to use periodic boundary conditions. Various electronic structure methods are developed and applied using both approaches. In this section we will shortly discuss few popular but powerful electronic structure methods the pseudopotential method, linear muffin-tin orbital and related methods, and tight-binding methods. 

All calculations in this study were implemented with the CASTEP package5, which is capable of simulating electronic structures for metals, insulators, or semiconductors. It is based on a supercell method, whereby all studies must be performed on a periodic system. Study of molecules is also possible by assuming that a molecule is put in a box and treated as a periodic system. Forces acting on atoms and stress on the unit cell can be calculated. These can be used to find the equilibrium structure. 

Despite these difficulties, supercell methods have found considerable application in defect studies," and indeed the results compare well with those obtained using the alternative techniques described below. 

In these calculations, the isolated defect or defect cluster is embedded in the crystal, which extends to infinity, and the contrast between this approach and that used in the supercell methods is illustrated diagrammatically in Figure 1. The normal procedure in a Mott-Littleton calculation is to relax all the atoms in a region of crystal surrounding the defect, containing typically 100-300 atoms, until all are at zero force. Newton-Raphson minimization methods are generally used. The relaxation of the remainder of the crystal is then described by more approximate methods in which the polarization, P at a point r, is calculated for crystals that have dielectric isotropy, from the expression ... 

Most calculations of the internal energy of a defect process (hereafter referred to simply as the energy) use classical potential models. Two kinds of approach have been used to calculate defect processes first, that based on the Mott Littleton approximation and second, the supercell method (closely related to the lattice minimization methods discussed in Chapter 3). 

The substitution of divalent cations for Mg2+ ions in MgO has been extensively studied within the ionic model (Mackrodt and Stewart, 1979). Recently these systems have been the subject of a series of systematic studies within the LCAO-HF formalism in which the highly symmetric nature of the system is exploited to lower computational costs and to simplify the geometrical relaxation (Freyria-Fava et al., 1993 Dovesi and Orlando, 1994 Orlando et al., 1994a). These studies contain the only published attempt to demonstrate the convergence of the supercell method for the treatment of defects within a QM formalism in ionic systems. The defect formation energy ( D) of, e.g., substitution of Ca for Mg ions in MgO may be referenced to the energies of the isolated ions ( Mg2< and ca2+)> i.e. 

The most basic data that the Mott-Littleton and supercell methods provide are the energies and entropies of defect formation. Nevertheless, despite the fact that these techniques are essentially static approaches it can also be possible to deduce information on the dynamic processes of diffusion and conductivity. These two processes are related by the Nemst-Einstein relationship ... 

Defects in Ionic Crystals Calculated by the Supercell Method. 

Since the supercell method relies on theory already discussed in forgoing sections, it need not be discussed further. It is referred to in an application to a lattice migration investigation in Section 5.3. 

The various methods used, in fact, belong to two families supercell methods and fixed cluster methods, which are also called force equilibrium methods. Both types of methods have their advantages and disadvantages, with the supercell method possibly being better adapted in the case of solutions with a high concentration of defects, whereas the fixed-cluster method is better suited for the case of dilute defects. The second family of methods is perfiaps a little more accurate, so it is that method which we will describe. 

Theoretical approaches to the DMS can be divided into two groups, namely the model studies and the studies based on the spin-density functional theory (DFT). The model studies mostly employ the kinetic-exchange (KE) model in the connection with a continuum approximation for the distribution of Mn atoms and other defects - that yields a disorder-free problem, although recently this model was refined to include the disorder via the supercell method in the framework of Monte Carlo simulations. ... 

It is possible to apply the plane-wave scheme to nonperiodic systems, such as isolated molecules, by using the supercell method, In the supercell method, identical copies of the molecule in question are used to build up a periodic crystal of molecules. If the distance between neighboring copies is large enough, then a plane-wave treatment of the supercell will give an adequate description of the isolated molecule. Recent results show that the supercell method can reproduce the accuracy of large Gaussian basis calculations at a competitive cost. 

Focusing on the high-temperature cubic phase, the high degree of disorder in the Ag and Bi atoms makes the exact electronic band stmcture calculations impossible. For treating such materials systems by first principles methods, two general methods have been usually carried out, i.e., the supercell method and the virtual crystal approximation (VGA) method [8], but the VGA method is not well suited to simulate the details of this phase transition although it is simpler and more efiftcient. So, in our case, a cubic superceU with 64 atoms is used for the disordered phase in the supercell approximation. 

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