First-Principles Study of Electronic Structure and Bulk Modulus of High-Entropy Transition Metal Carbides

High-entropy transition metal carbides combine ultrahigh hardness, excellent thermal stability, and intrinsic structural disorder, making them attractive for extreme-environment applications. Using density functional theory in the generalized gradient approximation (GGA-PBE) as implemented in the ABINIT package, we systematically calculate the electronic structure and bulk modulus B of a series of (TiZrHf-X)C compositions (X = Sc, V, Nb, Ta, Mo, W) with varying average d-electron count per metal site (ndest). A 24-atom rock-salt (B1) supercell with numerical-annealing relaxation of atomic positions is employed. The calculated DOS for group IV carbides TiC, ZrC, and HfC reveals a strikingly similar electronic structure: in all three cases the Fermi level is located within a wide pseudogap—responsible for the wide carbon nonstoichiometry range—and falls precisely on a small local peak resembling a Van Hove singularity, which promotes vacancy formation even at low temperatures. Qualitatively similar DOS profiles are found for all HECs studied, indicating that this electronic stabilization mechanism persists in multi-component systems. The bulk modulus increases monotonically with ndest from 209±1 GPa for (TiZrHfSc)C to 269±2 GPa for (TiZrHfW)C. At fixed ndest, heavier homologue metals (Ta > Nb > V; W > Mo) yield higher B due to greater core-electron Pauli repulsion. A single metal vacancy reduces B by approximately 21–35 GPa and simultaneously increases configurational entropy, suggesting that metal vacancies function as an additional thermodynamic stabilizing component of the high-entropy compound.