Counterflowing edge current and its equilibration in quantum Hall devices with sharp edge potential: Roles of incompressible strips and contact configuration

We report the observation of counterflowing edge current in InAs quantum wells which leads to the breakdown of quantum Hall (QH) effects at high magnetic fields. Counterflowing edge channels arise from the Fermi-level pinning of InAs and the resultant sharp edge potential with downward bending. By measuring the counterflow conductance for varying edge lengths, we determine the effective number $\langle N_{\text{C}}\rangle$ of counterflowing modes and their equilibration length ${\lambda }_{\text{eq}}$ at a bulk integer filling factor $\nu =1–4$. ${\lambda }_{\text{eq}}$ increased exponentially with magnetic field $B$, reaching $200\mu \mathrm{m}$ for $\nu =4$ at $B\ge 7.6$ T. Our data reveal important roles of the innermost incompressible strip with even filling in determining $\langle N_{\text{C}}\rangle$ and ${\lambda }_{\text{eq}}$ and the impact of the contact configuration on the QH effect breakdown. Our results show that counterflowing edge channels manifest as transport anomalies only at high fields and in short edges. This in turn suggests that, even in the integer QH regime, the actual microscopic structure of the edge states can differ from that anticipated from macroscopic transport measurements, which is relevant to various systems including atomic-layer materials.

Figure 1

Probe-position dependence of $R_{xx}$ vs $B$ of sample A. The insets show the contact configuration for each measurement. Inset of the upper panel: Schematic diagram of the conduction band edge $\left(E_{\text{C}}\right)$ and Landau-level dispersion in the presence of Fermi-level pinning in the conduction band at the edge. Edge channels are formed when the Fermi level $\left(E_{\text{F}}\right)$ crosses the Landau levels.

Figure 2

(a) Schematic of the three-terminal measurement detecting upstream counterflowing current $I_{\text{cntr}}$ in addition to normal forward current $I_{\text{fwd}}$ (shown by the blue and red arrows, respectively). Current $I_{\text{opp}}$ at the opposite side of the Hall bar was also monitored as a measure of bulk conduction. (b) Magnetic field dependence of $I_{\text{fwd}},I_{\text{cntr}}$, and $I_{\text{opp}}$, measured in sample B at $V_{\text{FG}}=0$ V using the configuration shown in (a). (c) $V_{\text{FG}}$ dependence of the normalized counterflow conductance $g_{\text{C}}\left(\propto I_{\text{cntr}}\right)$ for different edge length $\left(L_{\text{edge}}\right)$ at $B=6$ T (see main text for details). The top axis indicates the bulk filling factor estimated from the low-field Shubnikov–de Haas oscillations and Hall measurements at each $V_{\text{FG}}$. (d) $L_{\text{edge}}$ dependence of $N_{\text{C}}{T}_{\text{C}}\left(=g_{\text{C}}\right)$ for $\nu =1$–4 extracted from the data in (c). Solid lines are fitting using a single exponential function.

Figure 3

(a) Equilibration length ${\lambda }_{\text{eq}}$ and (b) effective number of counterflowing modes $\langle N_{\text{C}}\rangle$ for $\nu =1$–4 obtained by fitting the $g_{\text{C}}$ vs $L_{\text{edge}}$ data, plotted as a function of $V_{\text{FG}}$. The data in (a) are replotted vs $B$ in the inset.

Figure 4

Simulated density profile, shown as local filling factor ${\nu }_{\text{local}}$, for a bulk electron density of $3.65\times {10}^{15}{\mathrm{m}}^{-2}$ at magnetic fields corresponding to a bulk filling of (a) $\nu =3$ and (b) 4. The insets are schematics of the top view near the sample edge. The red (blue) arrows represent forward (counterflowing) edge channels. Incompressible (compressible) regions are shown in gray (yellow).

Figure 5

(a) Configurations used for the calculation of $R_{xx}$. Yellow markers at the upper-right and lower-left corners represent hot spots. (b) $R_{xx}$ calculated as a function of $T_{\text{C}}$ for $\nu =2$ (upper panel) and 3 (lower panel). $R_{xx}$ values for different probes, $R_{\alpha }=V_{\alpha }/I_{\text{in}}\left(\alpha =\text{a},...,\text{f}\right)$, are shown. Circles are experimental data in Fig. 1, plotted vs $T_{\text{C}}$ calculated using the ${\lambda }_{\text{eq}}$ values in Fig. 3. Open circles are data for $B<0$, which are included by using the relation $R_{\text{a}}(-B)=R_{\text{c}}\left(B\right)$.