Unexpected Behavior in a Two-Path Electron Interferometer

We report the observation of an unpredictable behavior of a simple, two-path, electron interferometer. Utilizing an electronic analog of the well-known optical Mach-Zehnder interferometer, with current carrying edge channels in the quantum Hall effect regime, we measured high contrast Aharonov-Bohm (AB) oscillations. Surprisingly, the amplitude of the oscillations varied with energy in a lobe fashion, namely, with distinct maxima and zeros (namely, no AB oscillations) in between. Moreover, the phase of the AB oscillations was constant throughout each lobe period but slipped abruptly by $\pi$ at each zero. The periodicity of the lobes defines a new energy scale, which may be a general characteristic of quantum coherence of interfering electrons.

Figure 1

A top scanning electron microscope micrograph of the Mach-Zehnder interferometer (MZI). The edges of the sample were defined by plasma etching of the $\mathrm{GaAs}\mathrm{\text{−}}{\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ heterostructure, which embeds a high mobility 2D electron gas, some 85 nm below the surface. Edge channels were formed by applying a perpendicular magnetic field (2–5 T), with filling factors $v=1$ or 2 in the bulk. The quantum dot that follows the source S2 (not seen in the picture) was not used, but one of its quantum point contacts (QPC0) served to reflect back the desired edge channel. The incoming edge channel (solid white line) was split by QPC1 to two paths (dotted white lines), of which one moved along the outer edge of the device and the other around the inner drain D1 (under the metallic air bridge). The two paths met again at QPC2, interfered, and resulted in two complementary currents: one in D1 and the other in D2. The modulation gate (MG) changed the contour of one path, thus changing the enclosed area between the two paths and the phase difference between them (via the Aharonov-Bohm effect). Significant changes in path length could also be done by opening gates G1 and G2. The signal at D2 was filtered around a center frequency of $\sim 1\mathrm{MHz}$ with a cold LC resonant circuit and then was amplified by a low noise, preamplifier, cooled to 4.2 K. Note that the centrally located small Ohmic contact ($3\times 3\mu {\mathrm{m}}^2$) served both as D1 and S1. Being grounded, via a long metallic air bridge, it collected the interfering current while injecting edge states at zero chemical potential. Two smaller metallic air bridges applied voltage to the inner gates of QPC1 and QPC2.

Figure 2

Interference oscillations and the visibility. (a) Two-dimensional color plot of the $\sim 1\mathrm{MHz}$ ac signal amplitude measured at D2 as a function of the applied dc bias at the source S2 (that was biased by both 1 MHz ac signal of $\sim 1\mu \mathrm{V}$ and dc) and the modulation gate voltage, at filling factor $\nu =2$ and QPC1 and QPC2 with transmission $T\sim 0.5$. (b) The visibility (defined as $[{\mathrm{V}}_{\mathrm{D}2}(\max )-V_{\mathrm{D}2}(\min )]/[{\mathrm{V}}_{\mathrm{D}2}(\max )+V_{\mathrm{D}2}(\min )]$, with ${\mathrm{V}}_{\mathrm{D}2}$ the measured ac signal at D2) and the phase of the interference pattern at $\nu =2$ as a function of the applied dc bias to S2 [deduced from Fig. 2a]. Five major lobes are visible, each $\sim 14\mu \mathrm{V}$ wide. The phase at each lobe is constant but it slips abruptly by $\pi$ at each node. (c) A similar graph to that of Fig. 2b, but at $\nu =1$, exhibiting only 3 major lobes with similar “stick-slip” phase behavior.

Figure 3

Magnetic field dependence of the visibility plotted as function of the applied ${\mathrm{V}}_{\mathrm{dc}}$. The magnetic field is chosen to be on the conductance plateau of filling factor $\nu =2$. As the magnetic field weakened the periodicity of the lobe structure got smaller and the strength of the oscillations diminished.

Figure 4

The effect of changing the length of one path on the visibility. The length of the path ($L_1$) was increased in two steps by removing the bias from gates G1 and G2 that confine the outer edge of the interferometer. As $L_1$ changed from $8.8$ to $11.2\mu \mathrm{m}$ the lobe periodicity and the phase behavior (not shown) did not change, but the visibility got smaller.