Rich novel zero-dimensional (0D), 1D, and 2D topological elements predicted in the P6₃/m type ternary boride HfIr₃B₄

Topological semimetals, including topological nodal point semimetals (TNPSs), topological nodal line state semimetals (TNLSs), and topological nodal surface semimetals (TNSSs), featuring zero-dimensional (0D), one-dimensional (1D), and two-dimensional (2D) topological elements (TEs), respectively, have attracted widespread attention in recent years. In this work, based on first-principles calculations, we propose for the first time that three different (0D, 1D, and 2D) TEs are simultaneously present in a synthetic compound, HfIr₃B₄, with a P6₃/m type structure. In detail, HfIr₃B₄ hosts a Dirac point (DP) state at the K point, a TNL state in the k_z = 0 plane, and a 2D TNS state in the k_z = π plane, respectively. All sorts of topological elements, 0D, 1D, and 2D TEs, coexisting in the P6₃/m type HfIr₃B₄, provide an ideal platform to study the rich fermionic states and their related physical properties in this type of compound. In addition, because the 0D, 1D, and 2D TEs of HfIr₃B₄ are equally distributed in different energy ranges relative to the Fermi level, an approach is proposed to utilize individual TEs to build on-demand devices.