VOLUME60, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 11 APRIL 1988

Quantum Hall Effect in Quasi One-Dimensional Systems: Resistance Fluctuations and Breakdown

J. K. Jain Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742

and

S. A. Kivelson Department of Physics, State University of New York at Stony Brook, Stony Brook, New York 11794

(Received 17 February 1988)

We propose, and demonstrate with the help of a simple model, that the resistance peaks in the quan­tum Hall experiments on narrow samples result from the phenomenon of "resonant reflection," which in­volves a resonant tunneling of an electron from one extended state to another through localized states. We make a number of predictions which involve the specific nature of the breakdown of the dissipation-less transport.

PACS numbers: 72.20.My, 72.10.Bg, 73.40.Gk, 73.50.Jt

The phenomenon of quantum Hall effect1'2 (QHE) is characterized by plateaus of vanishing (magneto)resis-tance, pXXy and quantized Hall resistance, pxy. Accord­ing to the standard picture,3 the complete absence of dis­sipation (at zero temperature) can be understood by the assumption that in the plateau region the Fermi level lies in the localized states. The changing of the magnetic field moves the Fermi level and when it passes through the extended states, the QHE breaks down and transport becomes dissipative. Recently there have been a number of experiments on quasi one-dimensional (ID) channels4

and the behavior of pxx has been found to be quite different from regular two-dimensional (2D) samples. While in 2D samples pxx is smooth, in quasi ID struc­tures it shows numerous sharp peaks, predominantly near the breakdown. The origin of these peaks as well of the breakdown of QHE in quasi ID systems is the sub­ject of this paper.

The observed structure is reminiscent of that observed in experiments on small solid-state systems with strongly localized electron states5 where finite-size effects give rise to large fluctuations in the low-temperature conduc­tance as some physical parameter, like magnetic field, temperature, chemical potential, etc., is varied. This structure is usually aperiodic, sample specific, and repro­ducible, and consists of sharp and sometimes well-isolated peaks in the conductance. It has been proposed that these large fluctuations arise because of a resonant transmission of the electrons through the localized states of the sample.6,7 Since this is relevant to the later dis­cussion, we consider it in a little more detail. First of all, following Landauer,8 if one looks at transport in terms of a tunneling of electrons through the sample from the filled states in the source lead to the empty states in the drain lead, then the conductance of a ID system is very simply related to the transmission coefficient (T) accord­ing to G = [e2/h]T/(\ -T). This formula, known as the Landauer formula, can also be generalized to the mul­

tichannel case which describes quasi-ID systems. In the strongly localized regime the transport occurs with near­ly zero transmission (T—• 0) so that G = T and the vari­ations in G originate from the variations in T. The transmission coefficient, which is near zero most of the time, is greatly enhanced whenever the energy of the in­cident electron coincides with an eigenenergy of the sys­tem, giving rise to large conductance peaks.

The analogy between the structures seen in quantum Hall experiments and the experiments on strongly local­ized systems is not immediately obvious. The QHE transport on the plateaus occurs with nearly perfect transmission and as a result the peaks occur in the resis­tance rather than the conductance. We propose in this work that the resistance peaks arise because of a "reso­nant reflection" of the current carrying electrons, which is the result of a resonant tunneling of an electron from one extended state into another through localized states. We also identify the complete breakdown as the phe­nomenon in which the "classical" transport of the current-carrying electrons switches from perfect trans­mission to perfect reflection.9 We will demonstrate these ideas within a simple model which permits a semiclassi-cal evaluation of the reflection coefficient. Various quali­tative features of the experiments can be understood on the basis of this picture, and a number of predictions can be made to test these ideas.

We start by considering an ideal potential Vo(y) which is a monotonically increasing function of | y | and therefore confines the electrons to a channel centered about the x axis. In the presence of a transverse magnet­ic field the electrons move on equipotential contours with the velocity determined by the potential gradient along the path.] For a magnetic field in the positive z direction the electrons with y > 0 move towards the right and the electrons with y < 0 move towards the left. If we assume that for positive (negative) y all the states with energy less than n\ (^2) are occupied (assume fi\ > 112), then it

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V O L U M E 60, N U M B E R 15 P H Y S I C A L R E V I E W L E T T E R S 11 A P R I L 1988

can be shown (Halperin10) that the net current in each occupied Landau level (LL) is given by /••(e/AHjii — Hi), which leads to the correct quantization condition for pxy.

In the ideal sample the electrons at the upper edge cannot tunnel into the empty states at the lower edge since the edge states are exact eigenstates of the system. However, in the presence of impurities there is a mixing of these states and tunneling becomes possible. Notice that tunneling between the edges leads to backscattering of the electron. Recently a Landauer-type formulation for quantum Hall transport was developed by Streda, Kucera, and MacDonald,11 and by the present authors,9

which relates pxx and pxy to the reflection coefficient, /?(/z), at the chemical potential fi. The simplest case is when only the lowest LL is occupied, since then the problem reduces mathematically to a ID problem. It is found that for small currents 0M—*A*2+) and zero tem­perature pxy remains constant while

pxx = [h/e2]R/(\-R). (1)

Now we are in a position to outline the physics of the resistance fluctuations. Consider some potential scatter­ed (impurities) distributed along the channel. Whenev­er /i is equal to the energy of a bound state on one of the impurities, the electron will experience a resonant tun­neling to the other edge leading to a large enhancement of the reflection coefficient. According to Eq. (1) this will result in a peak in the resistance pxx at this value of

Now we will consider in detail the effect of a single impurity in the channel. (Pokrovsky12 has considered a similar situation in which he studies ^-function impuri­ties at arbitrary locations within the channel.) The typi­cal equipotential contours are shown in Fig. 1(a). There are three types of paths: (i) paths that go from one end of the channel to the other, corresponding to perfect clas­sical transmission; (ii) paths that start and end at the same end, corresponding to perfect classical reflection; and (iii) paths that encircle the impurity. The classical breakdown of the QHE occurs at ^c, when the paths at ii switch from type (i) to type (ii). In the semiclassical picture the bound states on the impurity occur at ener­gies for which the equipotential loop encloses an integral number of flux quanta— the Bohr-Sommerfeld quantiza­tion condition—and it is for these energies that we ex­pect a resonant tunneling. The evaluation of the tunnel­ing coefficients is, in general, rather involved. Here, for simplicity, we will consider only the extreme quantum limit (ho)c—• °°) when only the lowest LL is occupied, which enables us to exploit the simple and accurate semi-classical techniques developed13'14 recently. (The semi-classical approximation is valid when the potential in question is slowly varying over the length scale of /; I2

plays the role of a small parameter analogous to h in the usual WKB approximation.) With reference to Fig. 1(a)

5.80

FIG. 1. (a) Schematic of the three types of classical paths of the model potential in Eq. (4). (b) The calculated zero-temperature reflection coefficient (solid line) as a function of the chemical potential, //. The peak at JJ. =5.783 has zero tem­perature half-width r = 10~4. Also shown is the temperature dependence for this peak: The dashed peak corresponds to khT = 5T and the dotted peak corresponds to k*T = 10T.

our objective is to evaluate the tunneling coefficient from 1 to 2. This tunneling occurs in three steps: First the electron tunnels from 1 to 1', then it proceeds to 2' after making 0,1,2, . . . complete loops around the impurity potential, and finally it tunnels from 2' to 2. With each loop around the impurity the electron picks up an addi­tional Aharonov-Bohm phase of IKQ/QQ where 0 is the flux enclosed in the classical loop and fo^hc/e is the flux quantum. Adding the contributions from all the dif­ferent paths one gets the reflection coefficient

R - I * 1/2/H -/"ir2exp(/2^/0o)] I 2, (2)

where t\ is the amplitude of transmission from 1 to 1', 12 is the transmission amplitude from 2' to 2, and r\ and ri are the corresponding reflection amplitudes. For a semi-classical evaluation of, say, t\ ™(1'| exp(-iHt) | 1), we write it in a coherent-state path-integral form13

f i=J*r f ( r )exp[ -S] ,

S-ifldtl-xy + V(x9y)l (3)

To be specific, we will choose the following model poten­tial:

V(x,y)=V0{y2l2/b2 + a2/[l2(x2+y2))}, (4) where the first term confines the electrons to a channel and the second term denotes an impurity potential at the origin. Here / is the magnetic length, x,y are dimension-less coordinates measured in units of /, b is a measure of the width of the sample, and a is a measure of the size of

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the impurity. In the rest of the paper we will measure energies in units of Vo. Since the impurity is symmetri­cally placed between the two edges, t\=t2 = t, r\=ri = r, and L\=L2=L, where L\ and Li are defined to be the distances of the impurity from the upper and the lower edges, respectively [see Fig. 1(a)]. We have shown in Ref. 13 that the semiclassical calculation of the tunnel­ing coefficient essentially involves an evaluation of the classical action along the equipotential paths in complex space. For the model potential (4) the complex equipo­tential path, V(xc\,yc0

=M> connecting 1 and 1' is such that it has real >>ci and imaginary JCCI. Thus one gets

\t | =roexp(-Re5'ci), rv (5)

ReSci= ~ J j ixc\ dyc\ — aL 2/l2.

Here to is an uninteresting prefactor13 roughly of order 1 which contains the contribution to t from the paths near the semiclassical path, and we will assume that *o = l. The quantity a is independent of the magnetic field and depends only on ji and the parameters governing the shape of the potential. The modulus of r can be deduced from /. Notice that the phase of t is irrelevant to the cal­culation of R, while the phase of r (which we will assume to be zero) merely serves to shift the zero of 0. In Fig. 1(b) we show R as a function of // with parameters all =20.0, b/l =7.0. The smallness of / relative to a and b ensures the accuracy of the semiclassical results. The reflection coefficient in Fig. 1 (b) shows huge sharp peaks as fi approaches fic from above. These peaks appear at the values of p for which cos[2^(/i)/0o] ""1.0, which corresponds to a constructive interference of all the paths, and, equivalently, also to the semiclassical ener­gies of the bound states on the impurity potential. No­tice that the R is shown only for fi> i*c ^lalb, which is a result of the way the equations are set up. Similar methods can be used for JJ. < jic as well. However, in this case the relevant process is resonant tunneling along the direction of the current flow, and one expects to see con­ductance fluctuations.

One can estimate the width and height of the peaks for a general impurity at a distance L\ from one edge and Li from the other edge. (Take L\<Li). The width, T, which is also the width of the resonance, is given by the leak rate of an electron on the bound state, and is proportional to exp( — 2L\H2). Similarly, the height of the peak is proportional to exp[(Li2 —L^)//2]. [Here we have assumed a flat enough potential so that the geometrical factors like a in Eq. (5) are unity.] Thus the sharpest peaks are produced by impurities at the center of the channel; for an impurity not at the center of the channel the peak is broader and its height is exponentially smaller than what it would be if the impur­ity were at the center of the channel. Finite temperature also tends to wash out the peaks. This can be easily un­derstood: So long as k^T^T, the temperature is

effectively zero. When k^T^T, the peak broadens, be­cause, unlike at zero temperature, the resonance is acces­sible even when it lies outside the energy range fi\ > E > ^2- At the same time the height is reduced, be­cause at most a fraction T/k^T of the electrons are ever resonant. More formally, it can be shown that the finite-temperature reflection coefficient is given by

R(^T)=fdER(E,T=0)f;(E), (6)

where f^(E) is the Fermi occupation probability. We have plotted R(fi,T) for a peak in Fig. 1(b) for some values of temperature. The numerical results explicitly confirm the intuitive expectations.

Now we enumerate the predictions of our theory, many of which have been implicit in the above discus­sion. These predictions are generally consistent with the available experiments, although a more systematic ex­perimental study is needed for an unambiguous verification, (i) One of the salient features is that the peaks farther from the breakdown are sharper than the peaks closer to the breakdown. Thus these peaks become temperature dependent at much lower temperatures and are the first ones to be washed out as the temperature is raised. In fact, only the peaks with F^knT will be ex­perimentally observable. This, incidentally, clarifies why the peaked structure is observed only in narrow samples; in wide samples the peaks are so narrow that even the smallest experimental temperatures destroy them, (ii) From Eq. (5) it is clear that decreasing / has the same effect as increasing the width of the sample. Thus there are fewer peaks at higher magnetic fields, (iii) For the temperature-dependent peaks (r^ksT), according to Eq. (6) a bound state at EQ yields

R(^T)~f^Eo)-(kBT)-Uxp[\n-Eo\/kBTl

Thus a plot of lnpxx vs /J, will fall off with a slope of (kBT) ~{. For T^>kBT the slope will be temperature in­dependent. (iv) The peaks become more numerous as one approaches the classical breakdown. One reason that we have already discussed is that the peaks far from the breakdown are very sharp and are quickly destroyed by temperature. Another reason is that there is a higher density of resonances near the classical breakdown since then the classical path on the impurity encloses a rela­tively large area so that the enclosed flux changes by one flux quantum with relatively small changes in the mag­netic field or ii. (v) In the Landauer-type treatment, elastic scattering11 mainly affects pxx and causes rela­tively small perturbations in pxy (in the extreme quan­tum limit pxy remains completely unperturbed at its quantized value). This implies that the peaked structure occurs predominantly only in pxx while pxy remains rela­tively structure free.

Now let us briefly consider the situation when many LL's are occupied,15 which corresponds to a multichan-

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nel problem.l If one assumes that all the LL's have the same n at the edges, which must certainly be true on the plateaus in order to get the correct quantized value of the Hall resistance, then the problem effectively reduces to a single-channel problem. This assumption is justified when the inter-LL scattering on either edge is very large compared to the scattering from one edge to another— so that all the LL's on either edge always equilibrate to the same //. The edges of the LL's are spatially separated with the highest occupied LL corresponding to the inner­most edges. As p, is lowered the transport in the highest LL undergoes a classical breakdown first since the criti­cal chemical potential for the nth LL is given by ficn) —la/b—nhcoc. The transport remains dissipative until n drops below nhcoc and the nth LL becomes unoc­cupied, when pxx again drops to zero and pxy again be­comes quantized—but at a different value.

One can treat the effect of inelastic scattering on the above considerations along the lines of Stone and Lee.7

Their conclusions remain valid here: The peak height is reduced and the peak is broadened because of inelastic scattering. When the inelastic width becomes much greater than the intrinsic width, r , the structure due to resonant tunneling is washed out. Lee16 has shown that in this regime the finite-size effects in Mott's variable-range-hopping conductivity may also give rise to large conductance fluctuations. We have not studied the possi­bility of such processes.

In conclusion, we have proposed that the large fluctua­tions in pxx near the breakdown result from an interplay between the extended (edge) states and the localized (bulk) states. An isolated experimental peak arises from a single bound state on an impurity potential along the channel. From its sharpness, its temperature depen­dence, and its spacing from the nearby peaks, one can learn a lot about the nature of this impurity potential. Thus the transport measurement works as a spectroscop­ic tool for probing the one electron eigenspectrum of the impurities.

We thank Professor V. L. Pokrovsky for an inspiring

conversation which was crucial to our understanding. One of us (J.K.J) was supported by the U.S. Army Research Office. The other (S.A.K.) was supported in part by National Science Foundation Grant No. NSF-DMR-18051. We also thank the University of Mary­land Computing Center for computer time.

'For a review, see The Quantum Hall Effect, edited by R. E. Prange and S. M. Girvin (Springer-Verlag, New York, 1987).

2K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45,494 (1980).

3R. B. Laughlin, Phys. Rev. B 23, 5632 (1981); R. E. Prange, Phys. Rev. B 23, 4802 (1981).

4G. Timp et al, Phys. Rev. Lett. 58, 2814 (1987), and 59, 732 (1987); A. M. Chang et al., to be published.

5A. B. Fowler, G. L. Timp, J. J. Wainer, and R. A. Webb, Phys. Rev. Lett. 57, 138 (1986).

6I. M. Lifshitz and V. Ya. Kirpichenkov, Zh. Eksp. Teor. Fiz. 77, 989 (1979) [Sov. Phys. JETP 50, 499 (1979)1; M. Ya. Azbel, A. Hartstein, and D. P. DiVincenzo, Phys. Rev. Lett. 52, 1641 (1984); J. A. Mclnnes and P. N. Butcher, J. Phys. C 18, L921 (1985).

7A. D. Stone and P. A. Lee, Phys. Rev. Lett. 54 1196 (1985).

8R. Landauer, IBM J. Res. Dev. 1, 223 (1957), and Philos. Mag. 21, 863 (1970); M. Biittiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. B 31, 6207 (1985).

9J. K. Jain and S. A. Kivelson, Phys. Rev. B 37, 4276 (1988).

10B. I. Halperin, Phys. Rev. B 25, 2185 (1982). n P . Streda, J. Kucera, and A. H. MacDonald, Phys. Rev.

Lett. 59, 1973 (1987). 12V. L. Pokrovsky, private communication. 13J. K. Jain and S. Kivelson, Phys. Rev. A 36, 3476 (1987),

and Phys. Rev. B 37, 4111 (1988). 14H. A. Fertig and B. I. Halperin, Phys. Rev. B 36, 7969

(1987). 15X. C. Xie and S. Das Sarma (unpublished) have considered

inter-LL scattering as a possible source of resistance fluctua­tions.

16P. A. Lee, Phys. Rev. Lett. 53, 2042 (1984).

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