Selective equilibration among the current-carrying states in the quantum Hall regime

Abstract

The Hall resistance of a two-dimensional electron gas is measured with gated probes to determine the extent of equilibration among the TV current-carrying states in the quantum Hall regime. After traveling macroscopic distances (-80 /im), current injected into the first state is equilibrated among the TV-1 lowest states but equilibration into the highest state varies strongly across the Hall plateau. This is attributed to a change in the TVth state from being localized within a magnetic length of the edge to substantially extending into the sample. channels. 3 As this current travels along an edge it "equilibrates" by which we mean it tends to redistribute so that an equal fraction of the current is carried by each edge channel. Current equilibration takes place via electron-scattering processes, the required potential being provided by disorder or phonons. By using one nonideal probe as a current injector, and a second nonideal probe as a current detector, the extent of equilibration that occurs between the nonideal probes can be determined. If the spatial separation between the edge channels is not much greater than the magnetic length, equilibration is expected to occur after current travels a distance on the order of the zero-field inelastic and elastic lengths (-10 /im). Recently, van Wees et al. have shown that for high magnetic fields and over a distance of -1 /im no equilibration occurs. 3 Other experiments 4,5 have been interpreted as implying the surprising result that no equilibration occurs for distances of 100 //mor more.Contrary to this interpretation, our experiments indicate that with TV available edge channels the current is always equilibrated among the TV-1 lowest 6 channels after traveling distances -80 /zm. We find further that the fraction of current that redistributes into the TVth channel decreases dramatically as the center of the corresponding Landau level approaches E F . It is the decoupling of this channel, rather than a general effect among all edge channels, that has been observed in previous experiments. 4 ' 5 We believe that this provides a new understanding of the current-carrying states. Current equilibrates among the TV -1 channels existing near the edge of the sample. However, the TVth channel spreads into the sample as the TVth Landau level approaches £>, causing a decrease in scattering between it and the true edge channels.Our device [ Fig. 2(a)] is formed by etching an AlGaAs/GaAs heterostructure (electron density 3.4 xlO 11 cm -2 and mobility 500000 cm 2 /Vsec at 4.2 K) into a Hall bar. Nonideal probes are made by metallic gates with narrow openings of lithographic width 500 nm and length 300 nm on the probe leads. A negative volt-PACS numbers: 72.20.My, 73.40.Cg, 73.40.LqThe recently developed Biittiker formalism 1 has proved useful in understanding two-dimensional electronic systems in the quantum Hall regime. It is based on previous theories 2 in which the conducting properties are determined by states at the Fermi level, £>, located spatially near the edge of the samp...

Quantum Transport in Semiconductor Nanostructures

“…93. The Shubnikov-De Haas maximum at 5 2 T, for example, is found to disappear at gate voltages such that the pomt contact conductance is equal to, or smaller than 2e 2 /h, which means that the pomt contact only transmits two spm-spht edge channels The number of occupied spm-spht Landau levels m the bulk at this magnetic field value is 3 This expenment thus demonstrates that the Shubnikov-De Haas oscillations result from the highest-mdex edge channel only, presumably because that edge channel can penetrate mto the bulk via states in the bulk Landau level with the same mdex that coexist at the Fermi level (cf Section 19 c) Moreover, it is found that this edge channel does not scatter to the lower-mdex edge channels over the distance of 250 μιη from probe p to drain d, consistent with the expenment of Alphenaar et al 429 In Section 19 a we discussed how an "ideal" contact at the 2DEG boundary mduces a local equihbnum by equipartitiomng the outgomg current equally among the edge channels The anomalous Shubnikov-De Haas effect provides a direct way to study this contact-mduced equihbration by means of a second pomt contact between the pomt contact voltage probe p and the current dram d in (Fig 5b), m this case with a l 5-μιη Separation between pomt contact p and the second pomt contact It is found that the Shubnikov-De Haas oscillations m R 3l are suppressed only if the second pomt contact has a conductance of (2e 2 /h)(N vllde . -1) or smaller At larger conductances the oscillations m R 3t return, because this pomt contact can now couple to the highest-mdex edge channel and distnbute the backscattered electrons over the lower-mdex edge channels The pomt contact positioned between contacts p and d thus functions äs a controllable "edge channel mixer "…”

“…This demonstrates that the n = 2 and n = l edge channels have almost fully equilibrated. A quantitative analysis 429 shows that, in fact, 92% of the current originally injected into the n = l edge channel is 677 (1990) redistributed equally over the n = l and n = 2 channels, whereas only 8% is transferred to the n = 3 edge channel. The suppression of scattering between the highest-index n -N edge channel and the group of edge channels with n =ζ N -l was found to exist only if the Fermi level lies in (or near) the Nth bulk Landau level.…”

“…The suppression of scattering between the highest-index n -N edge channel and the group of edge channels with n =ζ N -l was found to exist only if the Fermi level lies in (or near) the Nth bulk Landau level. As a qualitative explanation it was suggested 429 · 476 that the Nth edge channel hybridizes with the Nih bulk Landau level when both types of states coexist at the Fermi level. Such a coexistence does not occur for n ^ N -l if the potential fluctuations are small compared with haf (cf.…”

“…The guiding center energy of the Nth edge channel then approaches zero, and backscattering either by tunneling or by thermally activated processes becomes effective, but for that edge channel only, which remains almost completely decoupled from the other N -l edge channels over distances äs large äs 250 μπι (although on that length scale the edge channels with n ^ N -l have equilibrated to a large extent). 429 …”

“…426 was repeated by Alphenaar et a/. 429 for much larger point contact separations (^ΙΟΟμπι), allowing a study of the length scale for equilibration of edge channels at the 2DEG boundary. Even after such a long distance, no complete equilibration of the edge channels was found, äs manifested by a dependence of the Hall resistance on the gate voltage used to vary the number of edge channels transmitted through the point contact voltage probe (see Fig.…”