Computer Simulation of Interfaces in Ceramics
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for a given plane and then add the contributions of the planes to make a crystal. The calculation is divided into two regions; an inner region close to the interface where the ions are relaxed to positions of zero force and an outer region where the ions move as a block. The outer region ensures that the inner region is properly embedded in the lattice, and also permits the interface to dilate. Periodic boundary conditions are imposed in the interfacial plane. Many calculations, particularly electronic structure calculations, are performed with three-dimensional periodicity. If standard periodic boundary conditions are used. this enforces one of two choices. Either the calculations are performed using a slab geometry or two boundaries must be considered within the unit cell, the second undoing the geometrical effects of the first. (It is possible in principle to achieve this result without using two boundaries in the cell by using more complex boundary conditions but the programs are awkward to write and difficult to check [11]). A more recent approach, based on considering the grain boundary as an array of dislocations, has been developed by Parker and co-workers [12], resulting in the METADISE code. In this case one must consider the electrostatic sums for (one-dimensional) strings of ions. This has particular advantages if one wishes to consider dislocations at interfaces. The calculations done by Tasker and Duffv were with simple pair potential models, on svmmetric interfaces. Indeed. all calculations have been performed on pure tilt or twist boundaries with a high density of coincidence lattice sites. This work has been reviewed by Duffy [13]. Features similar to those calculated by Duffy and Tasker have been seen by high-resolution microscopy [14] for the highly symmetric (310) twin boundary in nickel oxide. The holes in the structure suggest whv it is easier for migration to occur down grain boundaries than in the bulk. However, they are not large enough for the ion to pass down the boundary as though it were an open channel. The methods have been extended to hetero-interfaces. Here the near-coincidence site lattice theory has been used to pick suitable systems for study. Almost all calculations have artificially compressed one or both lattices so that the calculations can be performed in periodic boundary conditions in the interfacial plane. Hetero-interfaces present problems at a variety of length scales. On the atomic scale, there is the question of the detailed structure of the interface. On the macro scale, there are the problems of misfit and the generation of strain. These have been much discussed. particularly with reference to the attempts to grow epitaxial layers of one semiconductor on dnother. Here relevant questions are the stability of the system against insertion of misfit dislocations (and the determination of the critical thickness of the laver when this happens) and the links between dislocation nucleation, motion and strain relaxation. These issues have recently been reviewed bv Jain and c
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