The Supercell Approach

In the supercell approach, the defect is instead enclosed in a sufficiently large unit cell and periodically repeated throughout space. A common problem with both approaches is the availability of high-level quantum-mechanical periodic solutions, because, as already mentioned, it is difficult to go beyond the one-electron Hamiltonian approximations (HF and DFT), at present. 

The supercell scheme is the most widely adopted approach because it is easily implemented in all periodic ab initio codes. Embedding approaches, on the other hand, may require specific and not widely disseminated softwares, which make their development slow, and their accruacy relatively low. A discussion of limits and merits of the embedding techniques can be found in Pisani.  

In the following pages, we illustrate in more detail the supercell approach and discuss a few examples. 

It is of wide applicability, and it may be adopted, in principle, to model both bulk and surface defects of ionic, covalent, metallic, and molecular systems. 

It is conceptually simple. The size of the supercell depends on an expansion matrix that consists of integers. The matrix is 2 x 2 or 3 x 3 according to the dimensionality of the periodic system. 

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While the supercell approach works well for localized systems, it is typically necessary to consider a very large supercell. This results in a plane-wave basis replicating not only the relevant electronic states but also vacuum regions imposed by the supercell. A much more efficient method to implement for investigating the electronic structures of localized systems is to use real space methods such as the recursion methods [27] and the moments methods [28], These methods do not require symmetry and their cost grows linearly with the number of inequivalent atoms being considered. For these reasons, real space methods are very useful for a description of the electronic properties of complex systems, for which the usual k-space methods are either inapplicable or extremely costly. 

The electronic structure of solids and surfaces is usually described in terms of band structure. To this end, a unit cell containing a given number of atoms is periodically repeated in three dimensions to account for the infinite nature of the crystalline solid, and the Schrodinger equation is solved for the atoms in the unit cell subject to periodic boundary conditions [40]. This approach can also be extended to the study of adsorbates on surfaces or of bulk defects by means of the supercell approach in which an artificial periodic structure is created where the adsorbate is translationally reproduced in correspondence to a given superlattice of the host. This procedure allows the use of efficient computer programs designed for the treatment of periodic systems and has indeed been followed by several authors to study defects using either density functional theory (DFT) and plane waves approaches [41 3] or Hartree-Fock-based (HF) methods with localized atomic orbitals [44,45]. 

In the present review we will discuss the current status of the quantum-chemical treatment of the adsorption of small molecules on oxide surfaces. We will limit our attention to oxide surfaces, because the problems encountered here are quite different from those connected with the treatment of metal surfaces. There are essentially two approaches to deal with a system that consists of a small molecule and an extended solid surface, i.e., a local process on a semi-infinite substrate. One way is the cluster approach described in the following in which a small cluster of atoms is cut out of the surface and the system molecule and cluster is treated as a supermolecule with the methods of molecular quantum chemistry. The alternative way is the supercell approach , in which the adsorbed molecule is repeated periodically on the surface, and the system surface with an ordered overlayer of adsorbed molecules is treated by means of periodic calculations. 

The supercell approach consists of a periodic replica of the defect, which is enclosed in a large nonprimitive unit cell. A pictorial view (in 2-D) of the supercell approach is given in Figure 47, where, by starting from the perfect... 

Within the supercell approach, AE" should tend to a well-defined limit with increasing the supercell size ... 

As a last example, we consider another simple defect the carbon substitution in bulk silicon. " In this case, however, we will not only consider the convergence properties of the supercell approach but also compare the results of the cluster and supercell schemes. Calculations were performed at the HF level with a 6-21G basis set plus polarization fimctions for C and Si and a 2-lG basis set for H (the latter was used in the cluster calculations). 

The defect formation energies for the unrelaxed clusters are similar to those calculated by the supercell approach, even in the case of small clusters. The same screening mechanism, already discussed for the supercell models, is active also in the cluster calculations. 

These examples show that the supercell approach is an accurate and, in many cases, relatively cheap tool for the study of neutral defects in crystalline systems, once properly gauged with respect to supercell size. 

The supercell approach, as implemented in CRYSTAL, has been applied to the study of many different bulk and surface defective systems. These include Ca and Be substitution in bulk MgO, F-center in CaF2, Fe doped NiO, Li doped NiO, V doped Ti02, and Ti substitution in an all-silica Chabazite. An example of reactivity of a surface defect has been... 

Several methods have been developed to calculate the surface electronic structure self-consistently for transition metal systems. All of these involve modeling the surfaces by thin slabs (or by repeated slabs in the case of the supercell approach) and expanding the electron wavefunctions in some basis sets. In conjunction with pseudopotentials, the mixed basis or the LCAO basis are most commonly employed. With basically the surface geometry as input, these calculations yield the work function, surface states, adsorbate states, surface charge densities, densities of states, and often information on preferred sites of adsorption. Surface states are shown to be important in the interpretation of spectroscopic measurements, and chemisorption studies give valuable information concerning the nature of the surface chemical bond. 

The DFT studies represent a natural step towards a more detailed, parameter-free understanding of the properties of the DMS. One possibility is to employ the supercell approach , in which big cells are needed to simulate experimentally observed low concentrations of magnetic atoms and other impurities. Alternatively, one can employ the Green function methods combined with the coherent potential approximation (CP.A) in the framework of the Korringa-Kohn-Rostoker (KKR) method or the tight-binding... 

Plane-wave methods are ideal for periodic systems. For isolated molecules and clusters, however, plane waves are less convenient. Although the supercell approach " discussed above is still able to deliver results for nonperiodic systems, the technique is far from optimum. A large, half-empty supercell must be used to simulate an isolated system, and many plane waves are required to represent it. 

The so-called real-space methods provide a viable alternative to the supercell approach for molecules and clusters. Real-space methods use only the position-space representation (position space is also known as real space ), which implies that molecules and clusters can be dealt with directly, without artificial supercells. The Laplacian operator V, exactly evaluated in momentum space (see equation 97), has to be approximated in real-space methods. The most popular approaches " use a finite-difference approximation for the Laplacian. For example, the second derivative with respect to xofa function y, z) can be approximated by the following finite difference. 

As in any semiconductors, point defects affect the electrical and optical properties of ZnO as well. Point defects include native defects (vacancies, interstitials, and antisites), impurities, and defect complexes. The concentration of point defects depends on their formation energies. Van de WaHe et al. [86,87] calculated formation energies and electronic structure of native point defects and hydrogen in ZnO by using the first-principles, plane-wave pseudopotential technique together with the supercell approach. In this theory, the concentration of a defect in a crystal under thermodynamic equilibrium depends upon its formation energy if in the following form ... 

Figure 2.3 Illustration of the supercell approach to model surfaces the surface is represented by a periodic stack of slabs separated by vacuum.

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