  As a model for describing finite-size effects in topological insulator thin
films, we study a one-dimensional (1D) effective model of a topological
insulator (TI). Using this effective 1D model, we reveal the precise
correspondence between the spatial profile of the surface wave function, and
the 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 to
show that the Lx-dependence of the size gap is a direct measure of the
amplitude of the surface wave function at the depth of x=Lx+1 [here, the
boundary condition is chosen such that the wave function vanishes at x=0].
Depending on the parameters, the edge state function shows either a damped
oscillation (in the "TI-oscillatory" region of FIG. 2, or becomes overdamped
(ibid., in the "TI-overdamped" phase). In the original 3D bulk TI, an asymmetry
in the spectrum of valence and conduction bands is omnipresent. Here, we
demonstrate by tuning this asymmetry one can drive a crossover from the
TI-oscillatory to the TI-overdamped phase.
