Recent years have seen great advances in our ability to predict crystal structures from first principles. However, previous algorithms have focussed on the prediction of bulk crystal structures, where the global minimum is the target. Here, we present a general atomistic approach to simulate in multicomponent systems the structures and free energies of grain boundaries and heterophase interfaces with fixed stoichiometric and non-stoichiometric compositions. The approach combines a novel genetic algorithm using empirical interatomic potentials to explore the configurational phase space of boundaries, and thereafter refining structures and free energies with first principles electronic structure methods. We introduce a structural order parameter to bias the genetic algorithm search away from the global minimum (which would be bulk crystal), while not favouring any particular structure types, unless they lower the energy. We demonstrate the power and efficiency of the algorithm by considering nonstoichiometric grain boundaries in a ternary oxide, SrTiO 3 .
The technological importance of polycrystalline strontium titanate (SrTiO3) is directly linked to its interfacial and grain boundary properties, which are at present poorly understood. A complete understanding (including links with experiment) requires information from many length scales, including electronic and atomistic, up to microstructural and macroscopic. In addition, the size and complexity of many general grain boundaries makes first-principles simulations prohibitively expensive. We have tested the ability of a number of interatomic potentials from the literature to accurately describe at least the structures of some simple grain boundaries in SrTiO3. The potentials we have tested are of three types: rigid ion model with either fixed formal or partial charges and shell model. We have also performed a detailed density functional theory (DFT) study of the same boundaries and used this data (interface structures and energies) to validate the interatomic potentials. Our conclusion is that none of the potentials can reproduce the energy ordering of the boundaries predicted by the DFT calculations. The boundary structures produced by some of the potentials do however agree reasonably well with the DFT structures. We discuss the implications of our findings for ionic oxide grain boundary research and critically examine the rigid ion and shell model approximations
Exact Langevin equations are derived for the height fluctuations of surfaces driven by the deposition of material from a molecular beam. We consider two types of model: deposition models, where growth proceeds by the deposition and instantaneous local relaxation of particles, with no subsequent movement, and models with concurrent random deposition and surface diffusion. Starting from a Chapman-Kolmogorov equation the deposition, relaxation, and hopping rules of these models are first expressed as transition rates within a master equation for the joint height probability density function. The Kramers-Moyal-van Kampen expansion of the master equation in terms of an appropriate "largeness" parameter yields, according to a limit theorem due to Kurtz [Stoch. Proc. Appl. 6, 223 (1978)], a Fokker-Planck equation that embodies the statistical properties of the original lattice model. The statistical equivalence of this Fokker-Planck equation, solved in terms of the associated Langevin equation, and solutions of the Chapman-Kolmogorov equation, as determined by kinetic Monte Carlo (KMC) simulations of the lattice transition rules, is demonstrated by comparing the surface roughness and the lateral height correlations obtained from the two formulations for the Edwards-Wilkinson [Proc. R. Soc. London Ser. A 381, 17 (1982)] and Wolf-Villain [Europhys. Lett. 13, 389 (1990)] deposition models, and for a model with random deposition and surface diffusion. In each case, as the largeness parameter is increased, the Langevin equation converges to the surface roughness and lateral height correlations produced by KMC simulations for all times, including the crossover between different scaling regimes. We conclude by examining some of the wider implications of these results, including applications to heteroepitaxial systems and the passage to the continuum limit.
We review a combination of theoretical and experimental techniques that have been applied to the study of grain boundaries in SrTiO3, with particular attention to Σ3 and ( 100 )-oriented grain boundaries. Electron microscopy, which includes high-resolution transmission and high-angle annular dark-field methods, is discussed, with successful applications to mapping atomic columns and testing theoretical models. Then, we compare and contrast different techniques of electron holography that may be used to map electrostatic potentials. Problems with the current methods of interpretation in holography and impedance spectroscopy are highlighted in an attempt to reconcile their respective estimates of electrostatic potentials at grain boundaries. Then, standard theoretical tools for the atomistic simulation of boundary structures are critically reviewed, which include classical potentials and density functional theory. A promising genetic algorithm for discovering low-energy grain boundary structures is described and tested. Finally, the synergy of experiment, theory, and simulation that is required to understand boundaries is demonstrated, and we identify major challenges to understanding multicomponent systems.
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