Edge states of a Bi$_2$Se$_3$ nanosheet in a perpendicular magnetic field

TL;DR

This work tackles the question of how edge states in Bi$_2$Se$_3$ nanosheets persist when a perpendicular magnetic field is applied. It develops an analytic framework that combines bulk Landau levels with Dirichlet-wall induced states (D-bulk, D-NLS, D-CLS) to describe the edge-state wave function in a semi-infinite geometry, validated by close agreement with numerical solutions. The method yields a precise edge-state dispersion $E(k_y)$ and provides insights into the role of boundary effects and symmetry breaking, showing that edge modes can remain robust even without time-reversal symmetry. The approach enables efficient computation of edge-state properties and local density of states, with potential applications to STM measurements and magneto-transport in topological insulator thin films.

Abstract

Conventional wisdom dictates that the conducting edge states of two-dimensional topological insulators of the Bi$_2$Se$_3$ family are protected by time-reversal symmetry. However, theoretical bulk calculations and a recent experiment show that the edge states persist in the presence of large external magnetic fields. To address this apparent contradiction, we have developed an analytical description for the edge-state wave function of a semi-infinite sample in a perpendicular magnetic field. Our description relies on the usual bulk Landau levels, together with additional states arising due to the presence of the hard wall, which are unnormalizable in the infinite system. The analytical wave functions agree extremely well with numerical calculations and can be used to directly analyze the behavior of the edge states in a magnetic field.

Figures (7)

• Figure 1: Numerical calculation of the spectrum of the system for $\mathcal{B} = 1T$ and $\tau_z={+}1$. For clarity, $\tau_z=-1$ is not shown. The dashed lines are the bulk LLs given by Eq. \ref{['eq:Landau Energies']} for different values of $n$. Notice that there are two LLs for each $n$, except for $n = 0$. The red line corresponds to the edge state, as this is the band that forms the Dirac cone with its $\tau_z = -1$ counterpart (not shown here). It can be seen that every band acquires the bulk LL energy when $k_{y} \rightarrow {-}\infty$.
• Figure 2: Graphical summary of this work, a piecewise construction of an analytical function for the edge state. Each panel shows a dispersion relation of the one-dimensional Bi2Se3 edge state with the wave number $k_y$ on the $x$ axis. The orange (blue) colour represents the $\tau_z =+1$ ($\tau_z = -1$) edge state. In the first panel we show the numerical dispersion relation in faded colors, with wave function $\psi_{\text{num}}$. Full colours represent the fact that we know the analytical wave function corresponding to this $k_y$. Note that we use the numerical solution only for reference, as it is not used in the analytical calculation. In the second and third panel we add $\psi_{n, \text{D-bulk}}$ with $n>0$ and $n=0$ respectively. The dashed lines correspond to the Landau energy. In the fourth panel we add the $\psi_{\text{D-NLS}}$ at their corresponding $k_y$. The D-CLS is not shown here. Finally, in the last panel we interpolate between these known basis vectors as per Sec. \ref{['subsec:Interpolating']} to obtain the full dispersion relation, achieving the goal of this article.
• Figure 3: Dispersion relation of the edge state at $\mathcal{B} = 1T$. The white dots are the numerical solutions while the full lines are obtained via the method described in Sec. \ref{['subsec:Recap']}. The red and blue lines correspond to $\tau_z = {+}1$ and $\tau_z =-1$, respectively. The dashed lines indicate the energies of the different LLs. It can be seen that our method agrees extremely well with the numerical solution. The inset shows the wave function of the D-NLS with $\tau_z=1$ at the point indicated by the black arrow, with $\psi_1$ and $\psi_2$ the first and second component of the wave function, respectively. The black dots show the numerical calculation and the solid lines correspond to our approximation.
• Figure 4: $L^{2}$ difference between the numerical solution and our analytical approximation, i.e., $\int_0^\infty \mathrm{d}x \, \vert\space\vert\psi_{\text{num}}(x) - \psi(x)\vert\space\vert^2$. The red and blue lines correspond to $\tau_z = {+}1$ and $\tau_z =-1$, respectively. We draw attention to the fact that the error is almost exactly zero when $k_y = k_{y, \text{D-NLS}}, k_{y,1}, k_{y,2}$, etc., namely when the energy is equal to that of a LL, shown by the arrows for $\tau_z$. (cf. Fig. \ref{['fig:Dispersion_Edge_state_a_priori_B1']}). The inset shows the comparison at the attached highest-error point and reveals that even in this case the approximation is very accurate.
• Figure 5: Parameters $c_i$ from Eq. \ref{['eq: psi sum general']} for the wave functions with $B=1$ T and $\tau_z =1$. Each colour represents a different part of the wave function. Notice that since this basis is in general not exactly orthonormal, the squared factors at a single $k_y$ do not necessarily add up to one. It can be seen that at $k_y =k_{y, \text{D-NLS}}, k_{y,0}, k_{y,1}$ the full wave function consists solely of a single component. Notice the sizable contribution of $\psi_{-1,\text{D-CLS}}$.