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

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 ($L_x$) of the film. We solve the boundary problem both in the semi-infinite and slab geometries to show that the $L_x$ dependence of the size gap is a direct measure of the amplitude of the surface wave function $\psi \left(x\right)$ at the depth of $x=L_x+1$ [here, the boundary condition is chosen such that $\psi \left(0\right)=\mathbf{0}$]. Depending on the parameters, the edge state function shows either a damped oscillation (in the “TI-oscillatory” region of Fig. 2), or becomes overdamped (in the “TI-overdamped” phase of Fig. 2). In the original 3D bulk TI, an asymmetry in the spectrum of valence and conduction bands is omnipresent. Here, we demonstrate that by tuning this asymmetry one can drive a crossover from the TI oscillatory to the TI-overdamped phase.

Figure 1

Topological protection, or winding in the 1D model. (a) $argq$ plotted as a function of $k_x$ in the trivial ($m_0/m_{2x}=0.1$, blue curve) and nontrivial ($m_0/m_{2x}=-1.3$, red curve) phases. (b) Locus of the points: $(\mathrm{Re}q,\mathrm{Im}q)$ when $k_x$ sweeps once the entire Brillouin zone: $k_x\in [-\pi ,\pi ]$ [blue: $m_0/m_{2x}=0.1$, (red, dotted): $m_0/m_{2x}=-1.3$]. (c) Global behavior of $argq$ in the $(m_0/m_{2x},k_x)$ plane. When $m_0/m_{2x}\in [-4,0]$, the winding number ${\mathcal{N}}_1$ as defined in Eq. (13) becomes nonzero.

Figure 2

Phase diagram of the 1D topological insulator in the space of mass and hooping parameters [$(m_0/m_{2x},{\tilde{t}}_x/m_{2x})$ plane]. As given in Eq. (45), ${\tilde{t}}_x$ is a function of ${\epsilon }_{2x}$. The latter encodes asymmetry of the valence and conduction bands [see Eq. (3)].

Figure 3

The slab geometry. TI nanofilm of a thickness $L_x$ [the number of (quintuple) layers stacked].

Figure 4

Thickness ($L_x$) dependence of the (half) size energy gap $E_0$ in TI thin films. Comparison of $E_0\left(L_x\right)$ found by numerical diagonalization of the tight-binding Hamiltonian with open boundary conditions at $x=1$ and $L_x$ [for a given set of model parameters, this appears as a series of points with the same symbol and color] and the same dependence predicted by Eq. (66) [shown as a continuous line in the same color]. For the series of filled red points, the model parameters are chosen as given in Eq. (68). In Fig. 2, this corresponds to a point $(m_0/m_{2x},{\tilde{t}}_x/m_{2x})=(-1.3,1.51)$ as indicated by the same symbol and color. The other set of points corresponds to ${\epsilon }_{2x}/m_{2x}=0.3,0.5,0.7,0.9$, i.e., the asymmetry in the spectrum of valence and conduction bands is enhanced from the original Bi${}_2$Se${}_3$ value. These sets of points are all on the (dotted) line $(m_0/m_{2x}=-1.3$ in Fig. 2, while the value of effective hopping ${\tilde{t}}_x$ varies as ${\tilde{t}}_x/m_{2x}=1.57,1.73,2.10,3.44)$. The corresponding points are indicated by the same symbol and color in Fig. 2. Experimental values that have appeared in Refs. [8] and [9] are also shown in the figure in plus and asterisk symbols for comparison.