Topological edge and corner states in bismuth fractal nanostructures

Abstract

Topological materials hosting metallic edges characterized by integer-quantized conductivity in an insulating bulk have revolutionized our understanding of transport in matter. The topological protection of these edge states is based on symmetries and dimensionality. While integer-dimensional effects on topological properties have been studied extensively, the interplay of topology and fractals, which may have a non-integer dimension, remains largely unexplored. Here we demonstrate that topological edge and corner modes arise in fractals formed upon depositing thin layers of bismuth on an indium antimonide substrate. Our scanning tunnelling microscopy results and theoretical calculations reveal the appearance and stability of nearly zero-energy modes at the corners of Sierpiński triangles, as well as the formation of outer and inner edge modes at higher energies. This work opens the perspective to extend electronic device applications in real materials at non-integer dimensions with robust and protected topological states.