Quantized escape and formation of edge channels at high Landau levels and edge transport mediated zero-differential resistance states

We present nonlocal resistance measurements in an ultrahigh-mobility two-dimensional electron gas. Our experiments show that even at weak magnetic fields, classical guiding along edges leads to a strong nonlocal resistance on macroscopic distances. In this high Landau level regime, the transport along edges is dissipative and can be controlled by the amplitude of the voltage drop along the edge. We report resonances in the nonlocal transport as a function of this voltage that are interpreted as escape and formation of edge channels, and the formation of zero-differential resistance states when the nonlocal voltage is measured on length scales much larger than the mean free path.

Figure 1

Sample geometry in our nonlocal transport experiments. Arrows indicate the geometrical parameters in our experiment, the position of the source and drain electrodes, and the electrodes across which the nonlocal potential drop $V_{\text{nl}}$ is measured. The nonlocal resistance is then defined as $R_{\text{nl}}=V_{\text{nl}}/I$. The closed black contour highlights the geometry of the domain used in our finite-element simulations whose results are displayed in the top panels for $\alpha ={\rho }_{xy}/{\rho }_{xx}=-1$ and $\alpha =-100$, the color (gray-scale) level indicates the potential values [source (drain) potentials are fixed to $\pm 1$]; the potential gradients are concentrated in the center of the sample. The curves on the right represent the dispersion relation ${\epsilon }_n\left(k\right)$ for edge states for a hard wall potential, $k$ is the wave number, and $l_B$ is the magnetic length [37].

Figure 2

Dependence of the nonlocal resistance $R_{\text{nl}}$ (as defined in Fig. 1) and of the longitudinal resistance $R_{xx}\simeq {\rho }_{xx}$ on the magnetic field $B$. The longitudinal resistance $R_{xx}$ is almost a symmetric function of $B$, whereas $R_{\text{nl}}\left(B\right)$ is strongly asymmetric and almost vanishes for $B<0$. The insets illustrate typical classical electron orbits for a capture and an escape event due to the parallel electric field $E_x$ for $B>0$, where electrons propagate along the upper edge in the positive $x$ direction. Capture occurs for $V_{\text{nl}}<0$ and escape occurs for $V_{\text{nl}}>0$.

Figure 3

Dependence of the differential nonlocal resistance $d{V}_{\text{nl}}/dI$ on magnetic field $B$ and on the dc current amplitude $I$ for positive and negative magnetic fields (top and bottom panels, respectively). The data at negative magnetic fields are displayed as a function of $-B$ and $-I$. Temperature was $1.2\mathrm{K}$.

Figure 4

Dependence of the differential nonlocal resistance $d{V}_{\text{nl}}/dI$ on the dimensionless quantity $e{V}_{\text{nl}}/\hslash {\omega }_c$ at a magnetic field of $0.3\mathrm{T}$. The vertical lines on the left and right are equally spaced with respective spacing $0.75$ and $1.3$, and they are a guide to the eye to show the periodicity of the oscillations.

Figure 5

Dependence of the differential nonlocal resistance $d{V}_{\text{nl}}/dI$ (in arbitrary units) on the dimensionless quantity $x=|e{V}_{\text{nl}}|/\hslash {\omega }_c$ at magnetic fields between $0.3$ and $0.9\mathrm{T}$. A voltage offset was applied to fix the position of the first resolved peak at $x=1$. The period of the oscillations is plotted as a function of magnetic field in the inset for positive and negative $V_{\text{nl}}$. It corresponds to the distance between the first resolved peaks at magnetic fields where only a few oscillations could be resolved.

Figure 6

Dependence of the nonlocal differential resistance $d{V}_{\text{nl};F}/dI$ on magnetic field and dc current amplitude; this quantity was measured in a geometry where the separation between voltage probes was $D_x\simeq 500\mu \mathrm{m}$ on the $\mu ={10}^7{\mathrm{cm}}^2/\mathrm{V}\mathrm{s}$ sample from the first two sections. The size of the voltage probe contacts on the Hall probe is still smaller than the mean free path, with a lateral dimension of around $10\mu \mathrm{m}$, as can be seen on the optical image of the sample on Fig. 1. Temperature was $T=1.2\mathrm{K}$.

Figure 7

Dependence of NLDR $d{V}_{\text{nl};L}/dI$ on magnetic field and dc current amplitude for the $\mu =3\times {10}^6{\mathrm{cm}}^2/\mathrm{V}\mathrm{s}$ mobility sample. NLDR was measured in the geometry sketched in the top panel. In this geometry, the distance between the voltage probes and the voltage probe size were both significantly larger than the mean free path with dimensions around $500\mu \mathrm{m}$. Temperature was $T=0.3\mathrm{K}$.

Figure 8

Nonlocal voltage/current characteristics $V_{\text{nl};L}\left(I\right)$ for the $\mu =3\times {10}^6{\mathrm{cm}}^2/\mathrm{V}\mathrm{s}$ mobility sample at several magnetic fields (for resemblance with data from ZDRS experiments in local geometries, we have shown $-V_{\text{nl};L}$ as a function of $-I$ in this figure). The inset shows the voltage as a function of magnetic field for several currents inside the plateau regime. Temperature was $T=0.3\mathrm{K}$.