Images of Edge Current in $\mathrm{InAs}/\mathrm{GaSb}$ Quantum Wells

Quantum spin Hall devices with edges much longer than several microns do not display ballistic transport; that is, their measured conductances are much less than $e^2/h$ per edge. We imaged edge currents in $\mathrm{InAs}/\mathrm{GaSb}$ quantum wells with long edges and determined an effective edge resistance. Surprisingly, although the effective edge resistance is much greater than $h/e^2$, it is independent of temperature up to 30 K within experimental resolution. Known candidate scattering mechanisms do not explain our observation of an effective edge resistance that is large yet temperature independent.

Figure 1

Flux and current maps in a four-terminal device made from a Si-doped $\mathrm{InAs}/\mathrm{GaSb}$ quantum well. (a) Schematic of the device. Si doping (shown in orange) suppresses residual bulk conductance in the gap. (b) Schematic of the measurement. Alternating current (orange arrows) flows from left to right on the positive part of the cycle. A voltage ($V_g$) applied to the front gate (yellow box) tunes the Fermi level. The SQUID’s pickup loop (red circle) scans across the sample surface, with lock-in detection of the flux through the pickup loop from the out of plane magnetic field produced by the applied current. (c) Four-terminal resistance $R_{14,23}=V_{23}/I_{14}$ as a function of $V_g$, showing both the upwards (black) and downwards (gray) gate sweeps. We measured resistance before and after each image (blue and red circles). $R_{14,23}$ is maximized when the chemical potential is tuned into the gap. (d,e) Flux images for the sample tuned into (d) the bulk gap, $V_g=-2.35\mathrm{V}$, and (e) the n-type regime, $V_g=0\mathrm{V}$. (f–i) Reconstructed horizontal ($j_x$) and vertical ($j_y$) 2D current densities, showing that the current flows on the edges in the bulk gap and uniformly outside the gap. The black bracket in (g) indicates the region of averaging to obtain Fig. 2 and dashed lines in (e) indicate the approximate geometry of the sample. The zero of flux in (e) is not in the center of the device due to the asymmetric geometry of our SQUID’s pickup loop.

Figure 2

Analysis of current flowing along the edges as a function of $V_g$ (a)–(c) and temperature (d)–(f). (a) Selected profiles of the $x$ component of the current density show the evolution from bulk-dominated to edge-dominated transport, offset for clarity. The zero of each profile is indicated by the dashed line. Profiles were averaged over the region between the contacts, as shown in Fig. 1. (b) Resistance vs $V_g$ in a downward gate voltage sweep before imaging the current (gray) and immediately before ($o$) and after ($x$) each image in a subsequent sweep. (c) The fitted percentage of current flowing in the top edge (green circles), bottom edge (blue diamonds) and bulk (purple xs) as a function of $V_g$. (d) Profiles of the $x$ component of the current density at selected temperatures, showing more bulk conductivity at higher temperatures, and the presence of edge states up to 30 K. The profiles are offset for clarity and the zero of each profile is indicated by a dashed line. (e) $R_{14,23}$ of the device as a function of temperature. (f) Fitted percentage of current flowing in the top (green circles), bottom (blue diamonds), and bulk (purple xs). For (a) and (d), the origin in $y$ is defined with respect to the center of the device.

Figure 3

Effective resistance of the bulk and edges as a function of temperature in the quantum spin Hall state. The effective resistance of the bulk varies strongly with temperature. The effective resistance of the top edge is reduced by a factor of two, consistent with shorting of the top contacts by the bulk, rather than a change in the resistance per unit length (see text). The effective resistance of the bottom edge remains constant with temperature, which known scattering theories do not predict.