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As a model for describing finite-size effects in topological insulator thinfilms, we study a one-dimensional (1D) effective model of a topologicalinsulator (TI). Using this effective 1D model, we reveal the precisecorrespondence between the spatial profile of the surface wave function, andthe dependence of the finite-size energy gap on the thickness (Lx) of the film.We solve the boundary problem both in the semi-infinite and slab geometries toshow that the Lx-dependence of the size gap is a direct measure of theamplitude of the surface wave function at the depth of x=Lx+1 [here, theboundary condition is chosen such that the wave function vanishes at x=0].Depending on the parameters, the edge state function shows either a dampedoscillation (in the "TI-oscillatory" region of FIG. 2, or becomes overdamped(ibid., in the "TI-overdamped" phase). In the original 3D bulk TI, an asymmetryin the spectrum of valence and conduction bands is omnipresent. Here, wedemonstrate by tuning this asymmetry one can drive a crossover from theTI-oscillatory to the TI-overdamped phase.

One-dimensional topological insulator: a model for studying finite-size effects in topological insulator thin films