Statistical approach to obtaining vacancy formation energies in high-entropy crystals from first principles calculations: Application to a high-entropy diboride

A reduced pair approximation model for vacancy formation energy in multicomponent materials is proposed as an alternative to the commonly used cluster expansion method. By imposing physical constraints to the interaction coefficients, lower rank models are obtained with improved accuracy as measured by Bayesian information criterion and cross validation. Additionally, reduced models can outperform the full parametrization in high-entropy compounds with as much as 50% less training data. The results are presented for cation vacancies in the high-entropy diboride ${\mathrm{Hf}}_{0.2}{\mathrm{Zr}}_{0.2}{\mathrm{Ti}}_{0.2}{\mathrm{Ta}}_{0.2}{\mathrm{Nb}}_{0.2}{\mathrm{B}}_{2}$ calculated by density functional theory simulations of large cell special quasirandom structures. Further, the calculation of vacancy concentrations from a distribution of energies is considered, wherein the chemical disorder on lattice sites gives rise to non-Arrhenius temperature dependence. Preferential clustering and the possibility of short-range order in the high-entropy lattice are explored through pair affinities derived from model coefficients.