Understanding the\nFinite Size and Surface Relaxation\nEffects on the Surface States of Bi<sub>2</sub>Se<sub>3</sub> Family\nTopological Insulators

Topological insulators (TIs) of the Bi<sub>2</sub>Se<sub>3</sub> family exhibit a topological phase transition from three-dimensional\n(3D) to two-dimensional (2D) TIs in thin films with decreasing thickness.\nUnderstanding the driving force of this transition is critical for\nthe applications of TIs in nanodevices. Herein, we investigate the\nfinite-size effects on bulk band inversion and the structural relaxation\neffects on the surface states within the Bi<sub>2</sub>Se<sub>3</sub> family via first-principles calculations. Thin films exposing the\nthree lowest-energy surfaces are modeled by 2D slabs with tunable\nthicknesses. We propose that the thickness dependence of the topological\nphase originates from electron confinement created by surface cuts.\nThe increase in film thickness then counteracts these confinement\neffects, resulting in a monotonically decreasing band gap evaluated\nat the spin–orbit decoupled level. This dependence of the bulk\ngap on the thickness is found to be consistent for various surface\nslabs. We utilize this relationship to predict the required thickness\nfor maintaining the 3D TI phase and 2D surface states. Our findings\nunderscore the importance of electron delocalization in determining\nthe topological phase of TI thin films. In addition, the actual manifestation\nof topological surface states on the side surfaces is affected significantly\nby the coexisting dangling bonds produced by surface cuts. Therefore,\nsurface relaxation plays a crucial role in disentangling the trivial\nand nontrivial surface states.