Quantum Hall edge states in topological insulator nanoribbons

We present a microscopic theory of the chiral one-dimensional electron gas system localized on the sidewalls of magnetically doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$-family topological insulator nanoribbons in the quantum anomalous Hall effect (QAHE) regime. Our theory is based on a simple continuum model of sidewall states whose parameters are extracted from detailed ribbon and film geometry tight-binding model calculations. In contrast to the familiar case of the quantum Hall effect in semiconductor quantum wells, the number of microscopic chiral channels depends simply and systematically on the ribbon thickness and on the position of the Fermi level within the surface state gap. We use our theory to interpret recent transport experiments that exhibit nonzero longitudinal resistance in samples with accurately quantized Hall conductances.

Figure 1

(a) A simplified model for states localized on a $y\text{−}z$ plane sidewall. (b) Quasi-1D energy bands calculated at $\mu =8/\pi$ with energies in $\hslash \pi v_{\mathrm{D}z}/T$ units. (c) Positive nonchiral band energies at $\chi =0$ as a function of $\mu$. The nonchiral bands are particle-hole symmetric. For $\epsilon >\mu$ all states extend across the top and bottom surfaces. In the limit $\mu \to \infty$ the dimensionless energies of nonchiral states approach integers. For $\epsilon <\mu$, the $\chi =0$ band energy lines separate regions labeled by the number of nonchiral channels $N_{\mathrm{NC}}$ that are present. (d) Wave functions of the chiral state (black dashed line) and the first three negative energy nonchiral states at positive (dimensionless) momentum $\chi =5/\pi$. The eigenvalues associated with the nonchiral wave functions are marked by correspondingly colored dots in (b).

Figure 2

(a) ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ nanoribbon with thickness $T$ and width $W$. The thick red arrows represent currents carried by ballistic chiral edge states, while the thin blue arrows represent currents carried by ballistic nonchiral edge states. The sidewall localization property of chiral states is reflected by the illustrated ${\left|\mathrm{\Psi }\right|}^2$ probability density distribution across the ribbon cross section. Top views of the $\left(\overline{1}10\right)$ sidewall (b) and the $\left(111\right)$ top and bottom surface layers (c), with black arrows for 2D crystal unit vectors.

Figure 3

Band structure of a ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ nanoribbon with $T=5\mathrm{QL}$ and $W=208\mathrm{nm}$ for $B_{\mathrm{ex}}=0$ (a) and $B_{\mathrm{ex}}=0.16\mathrm{eV}$ (b). Chiral edge states are shown in red in panel (b). Spatial distribution of the wave function across the ribbon cross section for a right-going state ${\mathrm{\Sigma }}_R$ (c) and a left-going state ${\mathrm{\Sigma }}_L$ (d) with energy $\approx 0.11\mathrm{eV}$ [dashed line in (b)], and a nonchiral state $\mathrm{\Lambda }$ (e) with energy $\approx 0.23\mathrm{eV}$ [white circle in (b)]. The dashed lines in (d) mark the positions of the outermost Se layers in each QL. In (c) and (d) only the ribbon edges are shown.

Figure 4

Band structure of a ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ nanoribbon with $T=3\mathrm{QL}$ for $W=21\mathrm{nm}$ and $B_{\mathrm{ex}}=0.16\mathrm{eV}$ (a), $W=21\mathrm{nm}$ and $B_{\mathrm{ex}}=0.4\mathrm{eV}$ (b), and $W=125\mathrm{nm}$ and $B_{\mathrm{ex}}=0.16\mathrm{eV}$ (c). Chiral edge states are shown in red. (d) Chiral state avoided crossing gap for $T=3\mathrm{QL}$ (circles) and $T=5\mathrm{QL}$ (squares) as a function of $W$ for $B_{\mathrm{ex}}=0.16\mathrm{eV}$. The inset shows the logarithm of the gap for 3QL films over a larger range of $W$'s. (e) Gap as a function of $B_{\mathrm{ex}}$ for a fixed $W=21\mathrm{nm}$ in the same two cases. The vertical dashed line marks the critical exchange field ($B_{\mathrm{ex}}^{\mathrm{cr}}\approx 0.09\mathrm{eV}$) for 3QL.