How branching can change the conductance of ballistic semiconductor devices

We demonstrate that branching of the electron flow in semiconductor nanostructures can strongly affect macroscopic transport quantities and can significantly change their dependence on external parameters compared to the ideal ballistic case, even when the system size is much smaller than the mean free path. In a corner-shaped ballistic device based on a GaAs/AlGaAs two-dimensional electron gas, we observe a splitting of the commensurability peaks in the magnetoresistance curve. We show that a model which includes a random disorder potential of the two-dimensional electron gas can account for the random splitting of the peaks that result from the collimation of the electron beam. The shape of the splitting depends on the particular realization of the disorder potential. At the same time, magnetic focusing peaks are largely unaffected by the disorder potential.

Figure 1

The first magnetoresistance peaks recorded for two mesoscopic devices fabricated from the same heterostructure. The inset shows the scanning electron microscope picture of the novel magnetic focusing device. $B_0$ is the magnetic field at which the cyclotron radius is commensurate with the wall length of the device.

Figure 2

Branching of a plane-wave front propagating in the $x$ direction. (a) The electron wave flow intensity approximated by classical ray dynamics is plotted in gray scale as the plane wave propagates along the $x$ direction in the disorder landscape shown as a green/white color rendition in the background. The force in the $x$ direction is ignored (quasi-2D). The wave front remains a vertical line in coordinate space. Vertical lines mark the position of the plane-wave front at times $t_1$,$t_2$,$...$,$t_9$. The bottom panel displays the wave fronts at these times in phase space ($y$,$v_y$). Caustics are identified at the turning points (purple dots). (b) Flow density $\rho$ as a function of $y$ at times $t_4$ and $t_7$ [vertical cross sections of the flow density plot in (a)]. Also shown at the bottom are the corresponding wave fronts in phase space. The flow density peaks at the caustics.

Figure 3

Influence of disorder on the flow density $\rho$ (gray scale rendition) emitted from a point source. (a) Gray scale rendition of the flow density $\rho$ in the $(x,y)$ plane when particles are emitted from a point source with a cosinusoidal angular distribution. They are subjected to the disorder potential color-coded in green and white. Forces in both the $x$ and $y$ directions were taken into account and the magnetic field is absent. (b) The formation of caustics in a transverse magnetic focusing geometry. Particles are emitted from the top point contact. The flow density is plotted on a gray scale at a magnetic field where a caustic forms at the collecting point contact. (c) Collimation (left panel) at $B=B_0$ and magnetic focusing at $B=2B_0$ (right panel) in a corner device consisting of two quantum point contacts placed at a 90${}^{\circ }$ angle. Red curves in the lower panels show the flow density hitting the lower wall. (d) Same calculations as in (c) but in the presence of weak disorder. The standard deviation of the amplitude of the disorder potential corresponds to 2$\%E_F$.

Figure 4

Magnetoresistance traces measured in corner devices with different QPC resistances, $R_{\mathrm{QPC}}$. The electron mean free path of the 2DEG is 45 $\mu$m. (a) and (b) Magnetoresistance traces for two devices with a chamfered corner fabricated from the same wafer show different behavior of the resistance $R_{\mathrm{focusing}}$ feature near $B/B_0=1$. In each panel, curves are shown for various values of the resistance of the quantum point contact, $R_{\mathrm{QPC}}$. (c) Magnetoresistance data for a device with a sharp corner design.

Figure 5

Numerical transport simulation and its comparison with experimental results. (a) The number of trajectories which reach the collecting QPC as a function of $B/B_0$ in different realizations of the disorder potential in a corner device with a chamfered corner. Peaks at odd multiples of $B_0$ are strongly affected by branching. The inset shows a contour plot of the potential used to simulate the QPC. (b) and (c) Comparison of experimental data [magnetotransport curves for $R_{\mathrm{QPC}}$ $\approx$ 3 k$\Omega$ in Figs. 4b and 4c] and numerical simulations for different disorder realizations.

Figure 6

Quantification of the influence of disorder. (a) The average distance an electron travels until a caustic forms as a function of the disorder parameters for $B=0$, $B=B_0$, and $B=2B_0$. The dotted lines mark the parameters of the random potential chosen for the simulations in Fig. 5. (b) The numerically calculated inverse participation ratio (IPR) of the flow density along the bottom boundary between $x$ $=$ 0.5$a$ and $x$ $=$ 1.5$a$, where $a$ is the distance from QPC to the corner, as a function of the standard deviation of the amplitude of the disorder potential under collimation and focusing conditions ($B=B_0$ and $B=2B_0$), averaged over 200 realizations of the random potential. The correlation length is identical to the one for the calculations in Fig. 3, and the IPR is normalized by the IPR of the clean system. The rising IPR at $B=B_0$ indicates that increasing disorder produces more peaks in the flow density, while the focusing peak at $B=2B_0$ is broadened by the disorder.