  We use high-throughput first-principles sampling to investigate competitive
factors that determine the crystal structure of high-entropy alloys (HEAs) and
the energetics dependence of the stable phase on the atomic configuration of
fully ordered L1$_2$, D0$_{22}$, and random solid solution (RSS) phases of
equiatomic quaternary alloys comprising four of the six constituent elements
(Cr, Mn, Fe, Co, Ni, and Cu). Considering the configurational entropy, we
demonstrate that valence electron concentration (VEC) and temperature are
crucial to determine the phase stability of HEAs at finite temperatures,
wherein the ordered phases are energetically more favorable than RSS phases.
Some D0$_{22}$ phases with high VEC are energetically more stable than L1$_2$
phases, though both phases are metastable. Further, we explore magnetic
configurations to identify the origin of the enthalpy term. The calculations
reveal that ordered phases comprising antiferromagnetic atoms surrounded by
ferromagnetic atoms are energetically stable. The quantitative
structure--property relationship is also discussed.
