Configurational thermodynamics of alloys from first principles: effective cluster interactions

Phase equilibria in alloys to a great extent are governed by the ordering behavior of alloy species. One of the important goals of alloy theory is therefore to be able to simulate these kinds of phenomena on the basis of first principles. Unfortunately, it is impossible, even with present day total energy software, to calculate entirely from first principles the changes in the internal energy caused by changes of the atomic configurations in systems with several thousand atoms at the rate required by statistical thermodynamics simulations. The time-honored solution to this problem that we shall review in this paper is to obtain the configurational energy needed in the simulations from an Ising-type Hamiltonian with so-called effective cluster interactions associated with specific changes in the local atomic configuration. Finding accurate and reliable effective cluster interactions, which take into consideration all relevant thermal excitations, on the basis of first-principles methods is a formidable task. However, it pays off by opening new exciting perspectives and possibilities for materials science as well as for physics itself. In this paper we outline the basic principles and methods for calculating effective cluster interactions in metallic alloys. Special attention is paid to the source of errors in different computational schemes. We briefly review first-principles methods concentrating on approximations used in density functional theory calculations, Green's function method and methods for random alloys based on the coherent potential approximation. We formulate criteria for the validity of the supercell approach in the calculations of properties of random alloys. The generalized perturbation method, which is an effective and accurate tool for obtaining cluster interactions, is described in more detail. Concentrating mostly on the methodological side we give only a few examples of applications to the real systems. In particular, we show that the ground state structure of Au₃Pd alloys should be a complex long-period superstructure, which is neither DO₂₂ nor DO₂₃ as has been recently predicted.