VOLUME 62, NUMBER 16
PHYSICAL R E V I E W LEITERS
17 APRIL 1989
Comment on the Theories of the Quenching of the Hall Effect
Two recent theoretical Leiters '-2 addressed the
quenching of the Hall effect observed experimentally3 at
low B in intersecting quantum channels in semiconductor heterostructures. Peeters1 calculated the Hall resistance RH of a channel äs measured by weakly coupled Hall probes, and found no quenching. He suggested that the observed quenching is due to strong coupling of the Hall probes to the current-carrying channel in the experi-ments. On the other hand, Beenakker and van Houten2
argued semiclassically that the quenching is due to the suppression of electron edge states in the channel at low B. Their argument appears to be independent of the na-ture of the Hall probes. This Comment attempts to shed some light on these conflicting explanations by means of a more detailed calculation which models strongly cou-pled Hall probes.
In this calculation the current-carrying channel (run-ning along the χ axis) and the Hall probes (a channel along the y axis) are both defined by parabolic confining Potentials. The actual electron potential energy function V(x,y) used was ex2 for \y\ > \ x \ , cy2 for
\x\ > \ y \ . The electron transmission and reflection coefficients T\2, T\-$, T\4, and R\\, äs depicted in Fig. l of Peeters,1 were calculated by expanding the electron
wave function Ψ in terms of the propagating and evanes-cent channel eigenstates and requiring continuity of Ψ and ΫΨ at x — ±y. RH and RL were then found from the Büttiker equations.4 The results for the simplest
case where only l D electron subband is populated are shown in Fig. l äs a function of the normalized cyclotron frequency ω — ω^/ωο, for different normalized electron Fermi energies e —£>Mωό, where ωό ·" (2c/m *)l/2. Note the dip in the longitudinal resistance RL at small <uf which is also seen experimentally. In Fig. l, quench-ing of RH occurs at low <uc provided e is not too small. But at smaller f, when channels are nearing pinchoff, there is no quenching and RH is linear in B near B—0. This is contrary to the edge-state suppression theory2 which holds that quenching should occur for any e when B < Ih/ekpW3, where kp is the electron Fermi wave vec-tor and W is the channel width.5 In fact, the B < 2h/ekpW3 criterion would imply that quenching should occur more easily (and at higher B values) for lower kr (and e), which is the reverse of the trend seen in Fig. 1. Thus, it is clear that strong coupling of the Hall probes can quench RH äs was suggested by Peeters, but edge-state suppression is not the quenching mecha-nism, at least in the present model. The details of V(x,y) are very important and must be included in any
00
FIG. 1. RH (solid line) and RL (dashed line) vs normalized cyclotron frequency ω— £oc/coo. Curves a-e are for e—0.6, 0.8, 1.0, 1.2, and 1.4, respectively. At B— 0, the two lowest l D sub-bands empty at f ~O.5 and 1.5.
valid theory of the quenching. It should be emphasized, however, that no reliable experimental data are currently available for the interesting case close to pinchoff; for multiple occupied bands where good data are available, parabolic confinement potentials are unrealistic and a better strong-coupling model is needed. Finally, the dip in RH near ω—0.52 for e — 1.4 is due to a resonance at the channel junction. Several such resonances occur in this model, and those at higher e and ω are stronger and sharper than that shown. Their energy decreases with decreasing B and they may contribute to the quenching of RH at low B for some f.
George Kirczenow Department of Physics Simon Fräser University
Burnaby, British Columbia, Canada V5A1S6 Received 4 January 1989
PACS numbers: 72.20.My, 73.20.Dx, 73.50.Jt
'F. M. Peeters, Phys. Rev. Lett. 61, 589 (1988).
2C. W. J. Beenakker and H. van Houten, Phys. Rev. Lett. 60, 2406 (1988).
3M. L. Roukes, A. Scherer, S. J. Allen, Jr., H. G. Craighead,
R. M. Ruthen, E. D. Beebe, and J. P. Harbison, Phys. Rev. Lett. 59, 3011 (1988); C. J. B. Ford, T. J. Thornton, R. New-bury, M. Pepper, H. Ahmed, D. C. Peacock, D. A. Ritchie, J. E. F. Frost, and G. A. C. Jones, Phys. Rev. B 38, 8518 (1988).
4M. Büttiker, Phys. Rev. Lett. 57, 1761 (1986). 5Ford et al. (Ref. 3) discuss HP for parabolic channels.
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VOLUME 62, NUMBER 16 P H Y S I C A L R E V I E W LETTERS 17 APRIL 1989
Beenakker and van Houten Reply: The calculation
re-ported by Kirczenow1 demonstrates nicely the
impor-tance of strongly coupled voltage probes for the quench-ing of the Hall resistance. In Ref. 2 we have discussed this issue in relation to our semiclassical theory3 of the
effect (which assumes strong coupling via ballistic leads), and in relation to the perturbation theory of Peeters4 (which assumes weak coupling via a tunneling
barrier).
Kirczenow notes that the trend in the parameter dependence of the quench in his calculation is the reverse of what our theory would imply. We want to point out that it is also the reverse of what is found experimental-ly.5·6 In the experiments, increasing the channel width
W rapidly reduces the threshold field /?thres (the
magnet-ic field below whmagnet-ich the Hall resistance vanishes approxi-mately). Kirczenow finds just the opposite.7 The origin
of this atiomalous behavior is most likely the fact that the channel in his calculation is close to pinchoff (only one subband occupied), whereas experimentally several one-dimensional subbands are occupied. As Kirczenow emphasizes in his Comment, the assumption of a para-bolic shape for the lateral confinement potential is un-realistic for channels with multiple occupied subbands. Our explanation3 of the quenching is based on the
correspondence between quantum states and classical trajectories in a square-well confinement potential. Such correspondence becomes meaningless close to pinchoff, so that it is not surprising that Kirczenow's numerical re-sults are not described by our formula for the threshold field,8 which does in fact predict a rapid reduction of
fithres on increasing W.
The anomalous width dependence is not the only way in which Fig. l in Kirczenow's Comment differs qualita-tively from experiment.5'6 As we discussed in Ref. 3, the
experiments show two characteristic field scales, a field Scrit at which the Hall resistance deviates from the clas-sical linear behavior, and a field Äthres below which the Hall resistance is quenched. Typically, 2?thresls an Order
of magnitude smaller than Bcrit, which in turn is much
smaller than the field at which the quantum Hall pla-teaus appear. We have argued3 that B„\t^s2hkF/eW
(with kp the Fermi wave vector) determines the onset of classical size effects when the cyclotron diameter be-comes comparable to the channel width. In Kirczenow's calculation, the Hall resistance has no regime of linear B dependence beyond the quench, but goes directly to the quantum Hall plateau. Again, the reason for this different behavior seems to be that the channel in his cal-culation is close to cutoff.
We are well aware of the fact that experimentally the number of occupied subbands N ~ 3-10 is not » l äs re-quired for a semiclassical treatment, and we stress the need for a numerical calculation to lest our explanation of the experiments. For reasons given above we do not believe that Kirczenow's calculation is in the relevant re-gime.
C. W. J. Beenakker
Philips Research Laboratories 5600 JA Eindhoven, The Netherlands
H. van Houten
Philips Laboratories
Briarcliff Manor, New York 10510 Received 9 February 1989
PACS numbers: 72.20.My, 73.20.Dx, 73.50.Jt
'G. Kirczenow, preceding Comment, Phys. Rev. Lett. 62, 1920 (1989).
2C. W. J. Beenakker, H. van Houten, and B. J. van Wees,
Superlattices Microstruct. 5, 127 (1989).
3C. W. J. Beenakker and H. van Houten, Phys. Rev. Lett.
60, 2406 (1988).
4F. M. Peeters, Phys. Rev. Lett. 61, 589 (1988).
5M. L. Roukes, A. Scherer, S. J. Allen, Jr., H. G. Craighead,
R. M. Ruthen, E. D. Beebe, and J. P. Harbison, Phys. Rev. Lett. 59, 3011 (1987).
6C. J. B. Ford, T. J. Thornton, R. Newbury, M. Pepper, H.
Ahmed, D. C. Peacock, D. A. Ritchie, J. E. F. Frost, and G. A. C. Jones, Phys. Rev. B 38, 8518 (1988).
7The width dependence of the quench is not immediately
ob-vious from Fig. l in Kirczenow's Comment, because of the nor-malization of the parameters. Consider, for example, going from curve d to curve e at constant Fermi energy £> by multi-plying the strength ωό of the parabolic confinement potential with a factor 1.2/1.4. The width Wockp/ωο then increases by 17%. At threshold, uv/iuo^O.S in curve d and «0.5 in curve
e (<oc Ξeßthres/w being the cyclotron frequency at threshold).
h follows that 5thr« increases by a factor (0.5/0.3) x (1.2/1.4)
=s l .4. Alternatively, one can consider going from curve d to curve e at constant electron gas density n, which for one occu-pied subband is /»«(£> — fft<uo)1 / 2. The same conclusion is
reached, that an increase in W leads to an increase in Äthr« (by, respectively, 22% and 30% in this case).
8Note also that this formula is derived using a low-field
ap-proximation for the transverse wavelength of edge states at 5thres, which neglects terms of order (kp W) ~2 [cf. Eq. (3) in Ref. 3l. For the channels considered in Ref. 3 (with multiple occupied subbands, kpWZ, 10) these corrections are negligible — but not so in narrower channels.
