Exploring the configurational space of binary alloys: Different sampling for different cell shapes

Abstract

In many areas of alloy theory, such as determination of the $T=0$ ground state structures or calculation of finite-$T$ alloy thermodynamics, one needs to enumerate and evaluate the $\ensuremath{\sim}{2}^{N}$ configurations $\ensuremath{\sigma}$ created by different substitutions of atoms A and B on the $N$ sites of a unit cell. These configurations consist of ${M}_{\mathrm{ICS}}$ ``inequivalent cell shapes'' (ICS's), each having ${M}_{\mathrm{SSS}}$ ``same-shape structures'' (SSS's). Exhaustive evaluation approaches attempt to compute the physical properties $P(\ensuremath{\sigma})$ of all SSS's belonging to all ICS's. ``Inverse band structure'' approaches sample the physical properties of all SSS's belonging to a single inequivalent cell shape. We show that the number ${M}_{\mathrm{ICS}}$ of ICS's rises only as $B{N}^{\ensuremath{\alpha}}$, whereas the total number of SSS's scales as $A{e}^{\ensuremath{\gamma}N}$. Thus, one can enumerate the former (i.e., calculate all) and only sample the latter (i.e., calculate but a few). Indeed, we show here that it is possible to span the full configurational space efficiently by sampling all SSS's (using a genetic algorithm) and repeating this by explicit evaluation for all ICS's. This is demonstrated for the problem of ground state search of a generalized cluster expansion for the Au-Pd and Mo-Ta alloys constructed from first-principles total-energy calculations. This approach enables the search of much larger spaces than hitherto possible. This is illustrated here for the ${2}^{32}$ alloy configurations relative to the previously possible ${2}^{20}$.