  Bismuth (Bi) atomic layers are known as 2D topological materials with variety
of the electronic structures and topological orders depending on the number of
stacking layers. Recently, it is reported that few layers of Bi grown on
semiconductor substrate InSb(111)B exhibit the Sierpi\'{n}ski-triangle (ST)
fractal patterns on the surface. In this work, we have grown Bi layers on
InSb(111)B and traced the evolution of the atomic and electronic structures of
Bi. The surface atomic structures and growth modes were monitored by using
reflective high-energy electron diffraction and core-level photoelectron
spectroscopy. It is suggested that the single layer of the ST-phase Bi grows on
InSb(111)B and the following Bi deposition causes layer-by-layer growth up to
nominally 4 atomic layers. Diffuse band dispersion and quantum well states
observed by angle-resolved photoelectron spectroscopy are consistent with the
small surface domains and variation of the thickness even during the
layer-by-layer growth region. The further Bi evaporation changes the growth
mode to the 3D island formation. The unveiled growth behavior of Bi on
InSb(111)B would provide a new interesting playground to study 2D topological
electronic structure of quasi-periodic 2D atomic layers.
