Oscillatory tunneling between quantum Hall systems.

Electron tunneling between quantum Hall systems on the same two dimensional plane separated by a narrow barrier is studied. We show that in the limit where inelastic scattering time is much longer than the tunneling time, which can be achieved in practice, electrons can tunnel back and forth through the barrier continously, leading to an oscillating current in the absence of external drives. The oscillatory behavior is dictated by a tunneling gap in the energy spectrum. We shall discuss ways to generate oscillating currents and the phenomenon of natural “dephasing” between the tunneling currents of edge states. The noise spectra of these junctions are also studied. They contain singularites reflecting the existence of tunneling gaps as well as the inherent oscillation in the system. PACS numbers : 1.28, 73.40Lq, 72.s0Dx, 73.40Kp Typeset using REVTEX 1 I. OSCILLATORY TUNNELING IN QUANTUM HALL SYSTEMS In this paper, we study electron tunneling between quantum Hall (QH) systems separated by thin barriers. Examples of these systems are shown in figure 1 to figure 6. The thinness of the barrier allows an electron to tunnel through it many times before scattered away by inelastic effects. Oscillatory tunneling of this kind will occur if the inelastic scattering time τin is much longer than the tunneling time τT , τin >> τT . (1) The existence of oscillatory tunneling can be seen even in the semiclassical (SC) limit, where electron wave-packets moves in circular orbits with cyclotron frequency ωc. When the barrier is infinite, electrons will undergo a sequence of “reflected circular orbits” as shown in fig.1. In the absence of other scattering mechanisms, electrons having collided with the barrier once must collide with it again within the cyclotron period. As a result, they are forever captured by the barrier. (See fig. 6). When the barrier is reduced from infinity to a finite value, the captured electrons on one side of the barrier (say, L) can tunnel to the other side (R). Once tunneled, this electron will collide repeatedly with the barrier and eventually tunnel back to L. When eq.(1) is satisfied, this back and forth tunneling process can proceed without interuption, giving rise to an oscillating current in the absence of external drives. While the SC picture captures the correct physics, it only tells half the story. In a quantum mechanical treatment, we shall see that different edge states tunnel with different frequencies. Thus, even in the absence of inelastic scattering, the tunneling current of different edge states will naturally dephase with each other. As a result, the total tunneling current will decrease in time. However, we show later that despite dephasing effects, there are ways to generate lasting current oscillations (thereby reflecting the oscillatory tunneling near the barrier) without the aid of an a.c. drive. The crucial question is whether eq.(1) can be achieved. We shall argue below that this is possible at least for the case of integer filling. There are two sources of inelastic