PHYSICAL REVIEW B VOLUME 50, NUMBER 7 15 AUGUST 1994-I Resistance nuctuations in the quantum Hall regime P. C. Main, A. K. Geim, H. A. Carmona, C. V. Brown, and T. J. Foster Department of Physics, University of Nottingham, Nottingham, NG7 2RD, United Kingdom R. Taboryski and P. E. Lindelof Niels Bohr Institute, The Oersted Laboratory, DK-2100 Copenhagen, Denmark (Received 22 February 1994; revised manuscript received 19 April 1994) We have measured the low-temperature magnetoresistance of long () 50 pm), narrow {-1 pm) quantum wires fabricated from a high-mobility two-dimensional electron gas. In the quantum Hall re- gime at high magnetic fields we observe that the Shubnikov-de Haas oscillations split into a series of aperiodic sharp peaks. Both the amplitude and sharpness of the peaks are strongly dependent on tem- perature below 1 K. We attribute the peaks to resonant tunneling between edge states through localized states in the bulk. The results are in quantitative agreement with the theoretical model of Jain and Kivelson. I. INTRODUCTION The existence of aperiodic conductance or resistance fluctuations (RF) as the magnetic field is varied is a com- mon feature of mesoscopic systems in which the dimen- sions are comparable with the phase-coherence length of electrons in the material. In low magnetic fields, cu, «1 where co, is the cyclotron frequency, the fluctuations are well described by the standard theory of universal con- ductance fluctuations' (UCF). Various authors have also studied RF in high fields, both in diffusive conductors and in the quantum Hall regime of a high-mobility, two- dimensional electron gas (2DEG), which is of principal concern here. In the latter case, RF have been observed in the Shubnikov — de Haas oscillations (SdHO) which separate the zeros in the longitudinal resistance. Al- though many experimental observations are qualitatively similar there has been a variety of explanations of the behavior. In broad terms the explanations may be split into two categories: (a) interference between difFerent electron trajectories, ' ' ' which is the basic mechanism for UCF at low magnetic fields, and (b) resonant back- scattering of electrons which, in general, involves scatter- ing between edge states on opposite sides of the conduc- tor. The scattering process may itself rely on quantum interference, either by itself '" or in conjunction with Coulomb blockade. ' To understand these mechanisms we need to consider the electron trajectories when the magnetoresistance ex- hibits a SdHO between the zeros of the quantum Hall effect. At this magnetic field, the chemical potential lies within the extended states of the highest-energy occupied Landau level. A resistance is measured in this regime due to the difference in chemical potential between the extended states of the highest-energy occupied level and the edge states of the other levels. This in turn arises from the backscattering of the electrons between opposite sides of the wire. ' ' The physica1 processes which modify the backscattering therefore control the measured resistance. The first category of process involves some sort of elec- tron interference. Diffusive electron trajectories cross the wire. In general, these would give rise to a measured resistance but the trajectories may form closed loops, in- cluding the trajectories which follow the edges of the wire, and hence interference can occur. Alternatively, a model which considers only trajectories within the bulk of the wire has been proposed by Xiong and Stone but this is unlikely to be valid in wires which are narrow compared to the phase coherence length 1&, as has al- ready been demonstrated by Geim et al. in the diffusive regime. In either case, the period of the RF is deter- mined by I& which determines the maximum loop size over which coherence is maintained. Figure 1 illustrates schematically the second sort of process which can give rise to RF at high magnetic fields. Again the chemical potential lies within the bulk states as shown in Fig. 1(a). In Fig. 1(b) we show the correspond- ing equipotentials at the Fermi energy. %e assume the existence of a localized state within the wire. At certain values of the magnetic field, where the energy of the lo- b) FIG. 1. Schematic illustration of the variation of the Landau level energies across the wire (a) and the corresponding equipo- tentials (b). When the chemical potential is as shown by the dashed line, there is a SdHO peak in the longitudinal resistance. 0163-1829/94/50(7)/4450(6)/$06. 00 50 4450 1994 The American Physical Society