Interplay of Chiral and Helical States in a Quantum Spin Hall Insulator Lateral Junction

We study the electronic transport across an electrostatically gated lateral junction in a HgTe quantum well, a canonical 2D topological insulator, with and without an applied magnetic field. We control the carrier density inside and outside a junction region independently and hence tune the number and nature of 1D edge modes propagating in each of those regions. Outside the bulk gap, the magnetic field drives the system to the quantum Hall regime, and chiral states propagate at the edge. In this regime, we observe fractional plateaus that reflect the equilibration between 1D chiral modes across the junction. As the carrier density approaches zero in the central region and at moderate fields, we observe oscillations in the resistance that we attribute to Fabry-Perot interference in the helical states, enabled by the broken time reversal symmetry. At higher fields, those oscillations disappear, in agreement with the expected absence of helical states when band inversion is lifted.

Figure 1

(a) Electron micrograph of a Hall bar device with a narrow top gate. The geometry of the device is sketched in the inset of panel (c). The width ($W$) of the Hall bar is $10\mu \mathrm{m}$ and the physical length ($L$) of the top gate is $2\mu \mathrm{m}$. The carrier density in the outer and central regions of the junction are denoted by $n$ and $n^{\prime }$, respectively. (b) Four-terminal resistance $R$ across the top-gated region measured as a function of the top-gate and back-gate voltages at a zero applied magnetic field. The diagonal black line marks the estimated position of $n^{\prime }=0$. (c) Selected linecuts extracted from panel (b): linecut 1 (blue) shows the evolution of $R$ as a function only of $V_{\mathrm{TG}}$, that is, as a function of $n^{\prime }$ while the outer region is kept at constant $n={10}^{11}{\mathrm{cm}}^{-2}$. Linecut 2 shows $R$ as function solely of $V_{\mathrm{BG}}$, which changes the density of the whole device, for a value of $V_{\mathrm{TG}}=V_{\mathrm{TG}0}$ such that the device density is homogeneous, that is, $n^{\prime }\simeq n$.

Figure 2

(a),(b) 2D maps of the resistance obtained at $B=3$ and 5 T respectively. The color scale has been chosen to enhance the contrast in the $n\text{−}{n}^{\prime }\text{−}n$ region between 0 and $1h/e^2$. The fractional resistance values match predictions for electron transmission from $N$ QH edge modes in the outer regions of the junction into $N^{\prime }$ modes in the central region in the presence of edge mode equilibration. (c) Horizontal linecuts from panel (a), at $V_{\mathrm{BG}}=3.5\mathrm{V}$ (blue) and 7 V (red), corresponding to $N=1$ and $N=2$. $N^{\prime }$ is tuned by the top-gate voltage $V_{\mathrm{TG}}$. (d) Four-terminal resistance $R$ as a function of density in the junction for $B=0$ (blue), $B=3$ (magenta), and $B=5\mathrm{T}$ (yellow) for a density in the outer region of $n=5\times {10}^{10}$, $5\times {10}^{10}$, and $8\times {10}^{10}{\mathrm{cm}}^{-2}$, respectively. For the cases with a finite applied field, those carrier densities correspond to $N=1$.

Figure 3

(a) Computed band structure of a strained 7.5 nm HgTe quantum well at 0, 3, and 5 T. The critical field $B_c=3.8\mathrm{T}$ in the model. See Ref. [18] for calculation details. (b),(c) Sketch of the band structure and edge states in different scenarios at 3 and 5 T where the outer region hosts one chiral edge state and the inner region hosts either two or one, respectively. (d),(e) Similar sketches for the case $N=1$ and $N^{\prime }=0$ for 5 and 3 T. (d) Above $B_c$ a broad gap opens between the electron and hole Landau levels so no states propagate at the junction. (e) Below $B_c$, band inversion remains and at $\nu =0$ a helical mode propagates in the junction.

Figure 4

(a)–(c) Possible scenarios for edge state matching in the 1-0-1 situation. (d) Enlargement in the $\delta R$ oscillations (obtained by removal of a smoothed background resistance [18]) as a function of the 2D density in the central and outer regions of the junction, $n$, $n^{\prime }$, at an applied field of $B=3\mathrm{T}$. Areas with different numbers of quantum hall modes ($N=1$ and $N=2$), i.e., an integer filling factor, in the outer region are separated by dotted lines.