Quantum Mechanical Methods

To take explicit account o f the electronic distribution, and the ways in which it is modified by the defect, one must solve the Schroendiger equation for the defective crystal. The “standard” ab-initio HF techniques available in quantum chemistry for the study of molecules and periodic crystals (i.e. the molecular HF and the periodic HF) can be applied to the simulation o f lattice defects. We will focus on these techniques in the following discussion.

In the ‘Cluster Approach’ an isolated molecule is used, containing the defect and reproducing the local environment o f the host crystal (Simonetta, 1986; Colbourn and Mackrodt, 1984; Shangda et. al., 1989).

Although widespread, the method suffers from the obvious implicit shortcomings o f the model: the crystal field is not correctly reproduced and the periodic nature o f the host lattice is not included in the model; surface states associated with dangling bonds introduce unphysical features that can be only partially corrected by terminating the cluster with hydrogen atoms (Surratt and Goddard, 1978; Kenton and Ribarsky, 1981). The convergence with cluster size is usually very slow, and depends not only on the size but also on the shape o f the cluster adopted.

In ionic crystals the problems associated with dangling bonds are less important, and the electrostatic field generated by the rest o f the crystal can be simulated by surrounding the cluster with a lattice o f point ions that generate the correct Madelung field (Almlof and Wahlgren, 1973). A similar method has been applied, for example, to the study o f defects in MgO (Grimes at al., 1989). A different approach is adopted by Barandiaran and Seijo (1988). Their formulation considers the solution for a cluster taking into account its interactions with a "frozen11 crystalline environment within the lattice model potential approximation (Bonifacic and Huzinaga, 1974), so that the multi-electron wavefunction o f the whole system can be factorized in the product o f the cluster wave-function multiplied by the wave-functions o f the external closed-shell ions.

A further improvement o f the cluster method is the inclusion o f the polarizability o f the outer medium, by interfacing the QM cluster with the Mott- Littleton methodology. A successful example o f this approach is the ICECAP code (Vail et al., 1984; Harding et al. 1985), where the HF equations are solved for a molecular cluster embedded in a crystalline lattice o f point ions, whose positions are allowed to relax around the defect. The response o f the lattice outside the cluster is calculated using the Mott-Littleton theory, including ion polarization effects described by the shell model. A self-consistent process guarantees that the matching o f the quantum cluster with the outer region is correctly performed. The localizing potential developed by Kunz and Klein (1987) is adopted in order to avoid the spreading o f the N-electron cluster wave­ function intb the outer lattice (Vail, 1990). Such spreading is undesirable, since there are no electrons in the sites outside the cluster, that are not treated quantum mechanically. ICECAP has been used to characterize electronic defects and impurities in alkali halides and MgO (Vail, 1990), (Shluger et al., 1991).

As an alternative to the cluster approach, periodic boundary conditions can be imposed, to create a “superlattice”: each unit cell contains the defect, and it is large

enough to minimize the interaction between neighbouring defects. This technique is very effective when the perturbation generated by the defect is small and localized as, for instance, in the case o f chemisorbed molecules on a surface. A recent study of impurities in silicon ( Nichols et al, 1989) has shown that even more complicated defects can be accurately treated. This method is less adequate for treating defects that induce long range relaxation processes and charged defects that generate long-range Coulombic fields; it has been suggested that a compensating field corresponding to a uniform charge distribution can be superimposed in order to make charged states treatable by the supercell approach (Bar-Yam and Joannopoulos, 1984).

An alternative is to employ again a partitioned scheme, by dividing the crystal in two regions, both described quantum-mechanically. The problem is then reduced to two simpler ones: the solution for the periodic, defect-free, infinite host lattice and the description o f the finite region surrounding and containing the defect. The aim o f these ‘embedding’ techniques is to obtain the correct solution for the cluster region, including all the interactions with the host lattice (short range exchange and Coulomb interactions, Madelung field, polarization effects etc.).

A first route to the solution o f this problem is the Koster-Slater technique (Koster and Slater, 1954; Baraff and Schluter, 1984), that may be classified as a “perturbed host system” (PHS) embedding scheme; the Green function in the defect region is evaluated in the framework o f the perturbation theory, taking as a reference the host crystal Green function, after assuming that the Hamiltonian matrix remains unchanged outside a localized defect region. As an alternative, a “perturbed cluster” (PC) scheme can be used: in this case the starting point is the solution for a molecular cluster, to which corrections are applied in order to remove boundary and limited size effects and ensuring the correct matching with the surrounding perfect crystal.

An essential prerequisite for the application o f both the PHS and PC techniques, is the availability o f the solution for the perfect host crystal, obtained with the same Hamiltonian and the same computational techniques (basis set, numerical accuracy etc.) as employed for the defect region.

The following discussion will first show and discuss the PHS and the PC embedding equations. We also comment on an alternative scheme proposed by Inglesfield. In section 4.4, we shall concentrate on the latest formulation o f the PC equations, that will be employed in this work; the solution for the perfect host crystal will be provided by CRYSTAL, as discussed in Chapter 3.