Observation of a Quantized Hall Resistivity in the Presence of Mesoscopic Fluctuations

Abstract

We present an experimental study of mesoscopic, two-dimensional electronic systems at high magnetic fields. Our samples, prepared from a low-mobility InGaAs/InAlAs wafer, exhibit reproducible, sample specific, resistance fluctuations. Focusing on the lowest Landau level we find that, while the diagonal resistivity displays strong fluctuations, the Hall resistivity is free of fluctuations and remains quantized at its ν = 1 value, h/e 2 . This is true also in the insulating phase that terminates the quantum Hall series. These results extend the validity of the semicircle law of conductivity in the quantum Hall effect to the mesoscopic regime.PACS numbers: 71.30.+h, 72.80.Sk, For small, mesoscopic samples, the conductivity in the Quantum Hall (QH) regime exhibits reproducible, sample specific, fluctuations [1,2,3,4,5,6,7,8,9,10,11,12]. Although quite interesting by themselves, the greatest benefit in studying these fluctuations is that they can provide clues to the nature of the mechanism responsible for conduction in the QH regime.There are several known mechanisms that can lead to such fluctuations. Fluctuations as a result of the modification of the interference between many possible electron paths across the sample are at the heart of the theory of Universal Conductance Fluctuations (UCF) [13]. Although this theory is not expected to be valid at high magnetic fields (B), modified UCF theories have been suggested [14,15] to take into account the influence of B on the electron trajectories, and several experiments were interpreted in terms of UCF at high B [2,3].A different point of view has been suggested by two recent experimental studies: Cobden et al. [4], and Machida et al. [5] have found that in the QH regime fluctuation patterns follow straight lines in the magnetic field-carrier-density plane. These lines appear to be parallel to integer Landau level (LL) filling-factor lines. These results were attributed to charging of electron puddles in the sample [4], or to changes in a compressiblestrip network configuration [5].Another possible source of fluctuations, expected to dominate when B is high enough such that only a small number of LL are occupied, is resonant tunneling [16]. In this process fluctuations arise when an electron scatters from one edge of the sample to the other through a bulk impurity. This model was found to be consistent with observed high-B fluctuations [6], and has been used in measurements of fractional charge [7,8]. Applying this model to the case when only the lowest LL is occupied, Jain and Kivelson [16] argued that the fluctuations will be limited to the diagonal resistivity, ρ xx , leaving the Hall resistivity, ρ xy , quantized. This conjecture was contended by Bütikker [17]. Experiments have so far shown that fluctuations in ρ xx are accompanied by fluctuations in ρ xy [1,2,6], although these experiments were limited to relatively lower B's, where more than one LL participates in the conduction (ν ≥ 2).In this letter we report on an experimental study of the resistivity of meso...

The Quantized Hall Insulator: A "Quantum" Signature of a "Classical" Transport Regime?

“…26,27,28,31 The puddle size is of order l el , and since close to the transition ξ is typically much larger than l el there is an intermediate range where the theory is practically not decisive about the behavior of ρ xy (especially in view of its wide distribution). The experimental data, 34 which mostly accumulate within this intermediate range (where ρ xx increases by at most one order of magnitude), possibly provide evidence for the stability of the QHI behavior in the entire region where ξ > L φ .…”

“…Yet another recent experimental result 34 can possibly be interpreted as a challenge to the theoretical predictions in the quantum coherent limit. Magnetotransport measurements near the transition from a ν = 1 QH state to the insulator were performed on small samples of low-mobility InGaAs/InAlAs (of a few microns in each dimension) at low T .…”

Correlated mesoscopic fluctuations in integer quantum Hall transitions

“…The sample is considered mesoscopic because the characteristic size of its central region between the 4 central terminals, of about 2µm, is comparable to the phase coherent length L φ ≈ 2 − 3µm. 14,15,16 As customary, we denote the various resistances as R ij,mn = V mn /I ij , where V mn = V n − V m is the voltage difference measured between terminals n and m, when the current is injected in the sample through terminal i and extracted through terminal j. For example, to measure the Hall and longitudinal resistances, a small current I is fed into terminal 1 and collected from terminal 4; no net currents are flowing in the other 4 terminals.…”

“…The answer was provided by recent IQHE experiments performed on a mesoscopic sample. 14,15,16 The expected quantum mechanical interference of electronic wave-functions indeed causes reproducible patterns of fluctuations in the resistances near plateau-to-plateau transitions, as the magnetic field is varied (in macroscopic samples all resistances vary smoothly with B). Interestingly, the fluctuations of resistances measured with different combinations of electrical contacts are not independent; instead, various non-trivial correlations were observed.…”

Weakly chiral networks and two-dimensional delocalized states in a weak magnetic field

“…Nodes in the of the CC network also have a transparent meaning: they represent saddle points of the random potential, where equipotentials come as close as magnetic length. Various aspects of the Quantum Hall transition, relevant to experiment [43][44][45][46][47][48][49][50][51] , e.g., divergence of the localization radius (scaling 39,52,68 ), critical statistics of energy levels 53 , mesoscopic conductance fluctuations 54,55 , pointcontact conductance 56 , were studied theoretically using the CC model 57 . Testing the levitation scenario microscopically requires to construct a minimal weakly chiral network model, which captures the physics encoded in the system Eq.…”