Computation of Defect Energies

An alternative to the static lattice approach is to use quantum mechanical methods in the calculation of defect energies. These methods essentially rely on solution of the Schrodinger equation using the Hartree-Fock approximation. The two main techniques involve calculation on an embedded defect cluster, i.e., the defect and surrounding lattice. Alternatively, calculations may be performed on a defect supercell in which the defect is periodically repeated on a superlattice. Ab initio Haitree-Fock calculations are computationally intensive, and a number of approximations have been made in order to simplify these calculations to obtain reasonable computerprocessing times. Foramore detailed description of ab initio calculations readers are referred to the recent review by Pyper. ... 

Catlow C R A and W C Mackrodt 1982. Theory of Simulation Methods for Lattice and Defect Energy Calculations in Crystals. In Lecture Notes in Physics 166 (Comput. Simul. Solids), pp. 3-20. 

It is important to realize that each of the electronic-structure methods discussed above displays certain shortcomings in reproducing the correct band structure of the host crystal and consequently the positions of defect levels. Hartree-Fock methods severely overestimate the semiconductor band gap, sometimes by several electron volts (Estreicher, 1988). In semi-empirical methods, the situation is usually even worse, and the band structure may not be reliably represented (Deak and Snyder, 1987 Besson et al., 1988). Density-functional theory, on the other hand, provides a quite accurate description of the band structure, except for an underestimation of the band gap (by up to 50%). Indeed, density-functional theory predicts conduction bands and hence conduction-band-derived energy levels to be too low. This problem has been studied in great detail, and its origins are well understood (see, e.g., Hybertsen and Louie, 1986). To solve it, however, requires techniques of many-body theory and carrying out a quasi-particle calculation. Such calculational schemes are presently prohibitively complex and too computationally demanding to apply to defect calculations. 

Recent calculations of hyperfine parameters using pseudopotential-density-functional theory, when combined with the ability to generate accurate total-energy surfaces, establish this technique as a powerful tool for the study of defects in semiconductors. One area in which theory is not yet able to make accurate predictions is for positions of defect levels in the band structure. Methods that go beyond the one-particle description are available but presently too computationally demanding. Increasing computer power and/or the development of simplified schemes will hopefully... 

The inherent problems associated with the computation of the properties of solids have been reduced by a computational technique called Density Functional Theory. This approach to the calculation of the properties of solids again stems from solid-state physics. In Hartree-Fock equations the N electrons need to be specified by 3/V variables, indicating the position of each electron in space. The density functional theory replaces these with just the electron density at a point, specified by just three variables. In the commonest formalism of the theory, due to Kohn and Sham, called the local density approximation (LDA), noninteracting electrons move in an effective potential that is described in terms of a uniform electron gas. Density functional theory is now widely used for many chemical calculations, including the stabilities and bulk properties of solids, as well as defect formation energies and configurations in materials such as silicon, GaN, and Agl. At present, the excited states of solids are not well treated in this way. 

The properties of defects of this type are difficult to determine experimentally, although absorption spectra do give information about electron or hole binding energies. Much information is obtained by calculation, using density functional or other quantum computational methods. In this way, the relative stabilities of defects on plane faces, steps, terraces, and corners can be explored. 

You have now learned about how to use DFT calculations to compute the rates of individual activated processes. This information is extremely useful, but it is still not enough to fully describe many interesting physical problems. In many situations, a system will evolve over time via many individual hops between local minima. For example, creation of catalytic clusters of metal atoms on metal oxide surfaces involves the hopping of multiple individual metal atoms on a surface. These clusters often nucleate at defects on the oxide surface, a process that is the net outcome from both hopping of atoms on the defect-free areas of the surface and in the neighborhood of defects. A characteristic of this problem is that it is the long time behavior of atoms as they move on a complicated energy surface defined by many different local minima that is of interest. 

To our knowledge there have been no reported measurements of equilibrium defect concentrations in soft-sphere models. Similarly, relatively few measurements have been reported of defect free energies in models for real systems. Those that exist rely on integration methods to connect the defective solid to the perfect solid. In ab initio studies the computational cost of this procedure can be high, although results have recently started to appear, most notably for vacancies and interstitial defects in silicon. For a review see Ref. 109. 

Using the calculational method based on DDFT, deviations from the cylinder bulk morphology have been identified as surface reconstructions [58, 62], The constructed structure or phase diagrams allowed surface field and confinement effects to be distinguished [57-59, 107, 145, 186], The comparative analysis of defect types and dynamics disclosed annihilation pathways via temporal phase transitions [36, 111]. Further, a quantitative analysis of defect motion led to an estimate of the interfacial energy between the cylinder and the PL phases [117]. A DDFT-based model was effectively used to simulate a block copolymer film with a free surface and to study the dynamics of terrace development [41,42], We showed how our computational method and an advanced dynamic SFM can be exploited in a synergetic fashion to extend the information about the elementary steps in structural transitions at the mesoscopic level. In particular, the experiments validate the dynamic DDFT method, and the DDFT calculations rationalize the characterization of the film surface in the interior of the film [187],... 

Theoretically, the discussion must turn on the relative energies of complex ordered structures and of defective, random structures. Bertaut (5) (1953) attempted to compute the electrostatic lattice energy of the pyrrhotite phase FeSj.,4 (Fe+2o. 25, Fe+30.25 n + ) (S 2 n ) or (Fe+25, Fe+3, ) (S-28),for several alternative cases a completely disordered structure with higher-valent cations and vacancies randomly distributed over all cation sites and structures with the vacancies ordered into alternate cation sheets, with various hypotheses... 

Catlow, C. R. A., and W. C. Mackrodt (1982). Theory of simulation methods for lattice and defect energy calculations in crystals. In Lecture Notes in Physics, Vol. 166, Computer Simulation of Solids (C. R. A. Catlow and W. C. Mackrodt, eds.) Berlin Springer-Verlag, pp. 3-20. 

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