Two-dimensional obstructed atomic insulators with fractional corner charge in the MA2Z4 family

According to topological quantum chemistry (TQC), a class of electronic materials has been called obstructed atomic insulators (OAIs), in which a portion of valence electrons necessarily have their centers located on some empty Wyckoff positions (WPs) without atom occupation in the lattice. The obstruction of centering these electrons coinciding with their host atoms is nontrivial and results in metallic boundary states when the boundary is properly cut. Here, on the basis of first-principles calculations in combination with TQC analysis, we propose two-dimensional ${MA}_{2}{Z}_{4}$ $(M=\mathrm{Cr}, \mathrm{Mo}, and \mathrm{W}; A=\mathrm{Si} \mathrm{and} \mathrm{Ge}; Z=\mathrm{N}, \mathrm{P}, \mathrm{and} \mathrm{As})$ monolayers are all OAIs. A typical case is the recently synthesized ${\mathrm{MoSi}}_{2}{\mathrm{N}}_{4}$ (${\ensuremath{\alpha}}_{1}\text{\ensuremath{-}}{\mathrm{MoSi}}_{2}{\mathrm{N}}_{4}$). Although both ${\ensuremath{\alpha}}_{1}$- and ${\ensuremath{\alpha}}_{2}\text{\ensuremath{-}}{\mathrm{MoSi}}_{2}{\mathrm{N}}_{4}$ monolayers are topological trivial insulators, it has valence electrons centering empty WPs and the in-gap corner states at the three vertices of triangle-shaped nanodisk samples respecting ${C}_{3}$ rotation symmetry, which can be explained by the filling anomaly of electrons. In addition, the ${\ensuremath{\alpha}}_{2}\text{\ensuremath{-}}{\mathrm{MoSi}}_{2}{\mathrm{N}}_{4}$ monolayer exhibits unique OAI-induced metallic edge states along the $(1\overline{1})$ edge. The readily synthesized ${\mathrm{MoSi}}_{2}{\mathrm{N}}_{4}$ is quite stable and has a large bulk bandgap of 1.94 eV, which makes the identification of these edge and corner states most possible for experimental clarification.