Higher-order topological phase in planar hexacoordinate X2Y monolayers (X = Cu, Ag, or Au; Y = Si, Ge, Sn, or Pb)

Topological insulators (TIs) exhibit insulating bulk behavior with conducting boundary states protected by symmetry and spin–orbit coupling (SOC). Recently, the concept of higher-order topological insulators (HOTIs) has extended this framework, where topological modes appear on lower-dimensional boundaries such as hinges or corners. These states are stabilized by crystalline symmetries and characterized by higher-order invariants, including the Z4 index. In this work, we investigate the two-dimensional X2Y monolayers (X = Cu, Ag, or Au; Y = Si, Ge, Sn, or Pb), a family of materials with planar hexacoordination and hexagonal symmetry. Several compounds, including Cu2Si, Cu2Ge, Ag2Ge, and Au2Ge, have already been synthesized and reported in previous studies, where they were shown to host nodal-line semimetallic features. Using first-principles calculations, we examine the dynamical, electronic, and topological properties of this broader family. Phonon dispersion calculations confirm that ten of the 12 compounds are dynamically stable. Hybrid functional (HSE06) calculations reveal that eight Ag- and Au-based monolayers possess higher-order topology, as indicated by their Z4 numbers. Moreover, Au2Pb undergoes a topological phase transition: without SOC, it displays nodal-line semimetal characteristics, while SOC inclusion opens a gap and induces HOTI behavior with gapped edge states. These results highlight X2Y monolayers as a versatile platform for exploring symmetry-protected topological phases, ranging from nodal-line semimetals to HOTIs. The findings establish this new class of 2D compounds as promising candidates for next-generation topological devices, motivating further theoretical and experimental studies.