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Review article

Breakdown of the quantum Hall eect G. Nachtwei Max-Planck-Institut fur Festkorperforschung, Heisenbergstr. 1, D-70569 Stuttgart, Germany Received 28 September 1998; accepted 8 October 1998

Abstract The quantum Hall eect (QHE) is one of the most prominent eects in modern solid state physics. Since its discovery, the eect has been attracting interest by a steadily increasing community of researchers. The research activities have been focussed on both application aspects and the basic physics of the eect. The limits of the QHE, in particular its persistence at higher currents, are of crucial importance for the application of the eect as a resistance standard. The observation of the current-induced breakdown of the QHE initiated a variety of experimental and also theoretical work. In this article, the experiments related to the breakdown of the QHE are reviewed. Some current theories are discussed in conjunction with the experiments. Although no comprehensive theory of the breakdown is available yet, an at least qualitatively conclusive picture can be provided on the basis of the present knowledge. ? 1999 Elsevier Science B.V. All rights reserved. Keywords: Quantum Hall eect

1. Introduction and historical remarks Shortly after the discovery of the Quantum Hall Eect (QHE) by von Klitzing et al. [1] in 1980, investigations were made to determine the physical limits of the eect [2,3]. Apart from sample properties as electron density, mobility and sample geometry, the temperature of the sample and the magnitude of the electrical current ow through it were found to be essential for the formation of quantum Hall (QH) plateaus. The QHE is characterized by precisely de ned values of the Hall resistance [1] xy = h=ie2 (h – Planck’s constant, e – electron charge, i – number of occupied Landau levels) in the plateau range, accompanied by a vanishing longitudinal resistance xx → 0 for the temperature approaching zero T → 0. With increasing temperature, a monotonic increase of both xx and the deviation of xy from the plateau

values is observable. In contrast to this rather gradual transition, a sudden breakdown of the QHE occurs if the sample current exceeds a certain limit. At this critical current, the system becomes unstable, and the longitudinal resistance xx changes abruptly by several orders of magnitude (see Fig. 1). This phenomenon naturally attracted attention because of its relevance for the understanding of the very principal basics of the QHE. Moreover, the QHE has been used since 1990 to realize the unit of the electric resistance. For high precision measurements, the sample current should be as high as possible but below the critical current at which the nearly nondissipative current ow breaks down. Therefore, a lot of experiments on samples with dierent material properties and geometries have been performed to understand the physics of the breakdown of the QHE. Due to the principal and applicative aspects of the breakdown of the quantum Hall eect,

1386-9477/99/$ – see front matter ? 1999 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 9 8 ) 0 0 2 5 1 - 3

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Fig. 1. Current–voltage characteristic (plotted as longitudinal electric eld Ex versus current density jx ) at the lling factor = 2 (B = 4:7 T, see upper inset) at T = 1:4 K. Lower inset: sample geometry. Broken curve: Ex -axis magni ed by a factor of 5 × 104 . Data from Ebert et al. [3].

researchers from both National Bureaus of Standards [2,4,5] and from centres dedicated to basic research [3,6–9] were involved in early activities of exploring the breakdown. More recently, the breakdown became a popular topic again for the basic-research community, stimulated by the modern possibilities of submicron patterning. Since all these experimental results are still hardly reconcilable within one conclusive model, dierent mechanisms for the breakdown have been discussed. The proposed models were for example based on assumptions as intra- [10] and interLandau-level transitions [11], the eective reduction of the Landau gap by the electric Hall eld [12] or a phenomenological description on the basis of electron heating [3,8,9]. Some of the models address the electron–phonon interaction as being important for the breakdown [8–10]. Frequently, the breakdown current was found to scale linearly with the sample width. This applies for

samples of lower and medium electron mobilities, for eective sample widths from some ten to hundreds of micrometers [13–16], and for samples with antidot lattices and spacings between the antidots in the submicrometer range [17,18]. Hence, a nearly homogeneous current distribution exists at currents close to the breakdown in the presence of a sucient degree of disorder. In contrast, a sublinear increase of the breakdown current with the sample width was observed in medium- and high-mobility samples [19,20], indicating an inhomogeneous current ow at the breakdown. Earlier, it was shown experimentally that an inhomogeneous current ow leads to a breakdown in distinct local areas of the sample [4,5,21]. Thus, the degree of disorder and the homogeneity of the current distribution are important for the breakdown of the QHE. This could be veri ed by experiments on two-dimensional electron systems (2DES) with arti cial variation of the homogeneity of the current ow by periodic and

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aperiodic antidot arrays [17,18]. The authors found higher breakdown currents for the periodic systems in comparison to the aperiodic ones. Besides the rather simply explainable width dependence, a peculiar length dependence of the QHE breakdown was recently reported [22,23]. Moreover, the breakdown was found fully developed in sample regions which are separated by a relatively large distance from the current-injection into a long constriction of a Hall bar, whereas the nondissipative QH state persisted nearer to this injection position [24]. The authors explained their results by an avalanche heating of the electrons, reaching a steady state after a certain travelling distance. In this review, we will focus on experimental phenomena related to the QHE under conditions far from thermodynamical equilibrium. Theoretical aspects and publications will be mentioned mainly in conjunction with corresponding experiments, to provide conceivable explanation for the measured results. Therefore, this review will be organized as follows. After this introduction, experiments concerning the distribution of current ow and equipotential lines in the quantum Hall regime and their importance for the breakdown will be discussed. Section 3 addresses the role of typical sample dimensions as channel width and length, and the role of disorder and scattering. Some recent results concerning the time- and drift-distancedependent relaxation of hot electrons back to the QH state are also discussed. Typical phenomena of the breakdown of the QHE, as time-dependent uctuations of the sample resistance and the occurrence of a hysteresis in the current–voltage characteristics of the breakdown will be addressed. Finally, we will summarize and attempt to estimate the present state of knowledge and open questions concerning the breakdown of the QHE. 2. Current distribution and breakdown of the quantum Hall eect 2.1. Why is the current distribution crucial? – Edge-current versus bulk-current approach The conductance of two-dimensional systems in high magnetic elds has been discussed theoretically by Ando [25–28] and Gerhardts [29,30] in 1974=1975,

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applying a Kubo-formalism for homogeneous systems with short-range scatterers. Here, the transport was considered in a local picture, assuming the involvement of the entire 2DES in the charge transport. One of the rst theoretical papers to explain the QHE [31] from 1981 was based on the same assumption. In this case, the electric transport properties of the 2DES are determined by locally de ned components of the 2 × 2 conductivity (or resistivity) tensor (or ). If the Hall bar geometry is applied for the measurements, the measured voltages are related to the resistivity tensor. For a homogeneous system, the components xx and xy of the resistivity tensor are independent of the position inside the 2DES and de ne the relation between the electric eld E and the current density j according to E = · j. In the ideal case of the QH regime, i.e. for the temperature and the current approaching zero, the tensor components xx and xy can be written as xx = 0;

xy = h=ie2 :

(1)

For a Hall bar as shown in Fig. 2a and with jk − ex one obtains an electric Hall eld perpendicular to the current ow direction and hence a vanishing power dissipation per unit area, 9PI =9A; in the sample interior: h 0 jx Ex 2 ie = h 0 Ey − 2 0 ie ! 0 9PI = 0: (2) ⇒j ·E = = h 9A − 2 jx ie The current crosses the boundary between metallic contacts (Hall angle H ≈ 0◦ ) and the quantized 2DES (H ≈ 90◦ ), leading to strongly con ned hot spots in two diagonally opposite corners of the 2DES. Thus, the total power dissipation of the Hall bar, Pt , is nonzero (Pt = xy I 2 ) and occurs exclusively at the contacts. For small current densities j¡jc below the breakdown current density jc of the QHE, xx is by several orders of magnitude smaller than xy . Consequently, the dissipation in the sample interior is negligibly small in comparison with the total dissipation. Within this picture, the breakdown can be described as a sudden increase of xx by several orders of magnitude to xx ¿0 and xy ∼ = h=ie2 , if the current density exceeds the critical value, j¿jc . Above the critical current density, an onset of dissipation in the inner

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Ex , pointing in the current ow direction for current densities beyond the breakdown value, j¿jc . As a consequence, the Hall angle deviates from 90◦ , and internal dissipation occurs. This internal dissipation PI over the entire sample length LSD between the source and drain contacts can be approximated by PI = xx jx2 LSD w = xx I 2 LSD =w for xx xy and small contact resistances. PI is usually small in comparison to the total amount of dissipation: Pt = PH + PI = (xx LSD =w + xy )I 2

(4)

including the dissipation near the current contacts, PH = xy I 2 . This is, because xx xy holds even for the breakdown of the QHE (xx ≈ 1–2 k ; xy ≈ 6:45–25.8 k for lling factors 4¿i¿1). Using xy =xx = tan , one obtains for the relative dissipation of the sample interior: PI =Pt = [tan()w=LSD + 1]−1 ≈

Fig. 2. Schematic view of a Hall bar containing a 2DES under QH conditions. In the sample interior, the Hall angle is 90◦ , and all dissipation occurs near the current contacts. (b) Hall bar for j¿jc (breakdown of the QHE). In the sample interior, the Hall angle is = arctan(Ey =Ex )¡90◦ , and the dissipation occurs partially in the sample interior. Full, dashed and dashed-dotted lines: equipotential lines (the contact zones with a strong bending of the equipotential lines are expanded for clarity).

area L × w is observable. The dissipated power per unit area in the sample interior is then given by Joule heating according to: xx xy jx Ex = Ey −xy xx 0 9PI xx jx = xx jx2 ; ⇒j ·E = = (3) −xy jx 9A assuming that no current ows through the probes ( jk − ex ). The corresponding situation lustrated in Fig. 2b. As visible from Fig. 2b, is a non-vanishing component of the electric

Hall is ilthere eld,

xx LSD xy w

(5)

Eq. (5) provides a useful experimental criterion for the de nition of the breakdown in Hall bars: the definition of the breakdown of the QHE as the exceeding of a certain threshold value, th xx , corresponds to a certain Hall angle or a limit of relative dissipation in the sample interior. As the value for xx increases rather steeply at the breakdown by several orders of magnitude after a slight increase in the pre-breakdown regime (see Fig. 1 and Ref. [3]), th xx should be chosen well above the pre-breakdown values. Typically, the choice of th xx in the range of about 1–10 can provide an appropriate breakdown criterion. All these considerations are, of course, purely phenomenological and do not address the physical process which is responsible for the sudden increase of xx after passing the critical current density. Moreover, Eqs. (1) – (5) were formulated for a homogeneous system, which does not occur in physical reality. However, Eqs. (1) – (3) remain meaningful even for inhomogeneous systems, as long as the tensor components xx and xy can be de ned anywhere in the sample. For this situation, a local model for the quantum transport in 2D systems was developed by Woltjer et al. [32–34], which was successful in explaining early experiments on the areal current distribution under QH conditions. The authors described the inhomogeneous system by tensor components as a

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function of the position (x; y) in the 2DES: (x; y) = ((x; y)). The local value of the lling factor, (x; y), is related to the inhomogeneous carrier density n(x; y) = (x; y) · eB=h. A transport measurement integrates the local quantities within the area L × w as de ned by the positions of the potential probes. The measured Hall resistance Rxy of a Hall bar is then related to the local values of the resistivity tensor (x; y) as follows: Z

w

Uy = Z 0w Rxy = Ix 0

Ey (x; y) dy jx (x; y) dy

with Ey = −xy (x; y)jx (x; y) + xx (x; y)jy (x; y):

(6)

According to this model, slight local gradients of the carrier density result in steep gradients of the local Fermi energy due to the low density of states within the Landau gaps. The local variations of the Fermi energy are of the order of the Landau gap ˜!c and drive a local redistribution of electrons such that a dipolar current path is established where the dissipation disappears, i.e. where the conditions xx (x; y) = 0 and n(x; y) = iN‘ = ieB=h

(7)

(i – QH plateau number, N‘ – Landau level occupancy) hold. In the corresponding sample regions, the Hall resistivity xy = B=en is quantized according to xy = B=(eiN‘ ) = h=ie2 . Due to the inhomogeneity of the 2DES, there is a certain range of magnetic elds where Eq. (7) can be ful lled on a closed path between the current contacts somewhere in the sample, depending on the variaton of n(x; y). This provides a conceivable explanation for some experimental results (see Section 2.2) which can be interpreted as a movement of the nondissipative current path with the magnetic eld. From energy conservation between cyclotron energy and the electrostatic energy of the dipolar current path, Woltjer et al. deduced typical path widths of the order of 100 nm. A variety of experiments at high currents, however, revealed a more or less homogeneous current and potential distribution

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near the breakdown of the QHE. Thus, Woltjer’s picture is either not straightforwardly applicable at high currents, or a rather dense network of current paths develops at current values close to the breakdown (see Section 3.1). In the forthcoming discussion, we will show that the restriction of the validity to small currents applies more strictly for the alternative nonlocal theory of the quantum transport in a 2DES. This theory assumes the current ow in one-dimensional edge channels at the boundary of the 2DES. Although this picture is intriguingly simple and provides nice explanations for experimental results observed at mesoscopic samples (for a review, see Ref. [35]), two remarks should be made in advance: (1) The observation of the QHE is also possible if edge states are absent (results on Corbino devices, see Ref. [36]). (2) Up to now, there is no direct and indisputable evidence that the current (at least at values of some A) indeed ows within edge channels only. Instead, there are principal restrictions for a veri cation of a current ow in edge channels by transport measurements (theoretical work of Thouless [37], see also: van Houten and Beenakker [38], experiments by Nachtwei et al. [39]). In Section 2.2, experiments concerning the current and potential distribution in the QH regime will be reviewed. We will now shortly present the edge-current model and then discuss its applicability to the QHE breakdown. In 1988, Buttiker [40] combined earlier considerations of the electrical resistance on the basis of transmission and re ection of electrons [41,42] and of the appearance of edge states at the boundary of a 2DES [43–45] to provide an explanation of the QHE. Due to the con nement potential V (y) at the boundary, the eigenstates of the 2DES depart from the original Landau levels at E‘ = ˜!c (‘ + 12 ) and cross the Fermi level EF near the boundary. As a consequence, extended states (edge states) develop which are capable of carrying current. The edge states possess a one-dimensional density of states (1D-DOS). Therefore, the current contribution of a single edge channel is proportional to the chemical potential ( = EF at T = 0), as the drift velocity and the 1D-DOS just cancel out in the current integral over energy. If there is

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no backscattering between opposite edges of the 2DES and the dierence of the chemical potentials, , of these edges is small (¡˜!c ), the net current carried by a single edge state is e I‘ = ; h

(8)

irrespective of the Landau level number ‘. Thus, the total current for i completely lled Landau levels is carried by also i edge channels and determined by the dierence of the chemical potentials = eUH (UH – Hall voltage): e I = i eUH : h

(9)

This yields immediately the quantized value for the Hall resistance, Rxy = (1=i)(h=e2 ). More detailed considerations, taking into account scattering and contact resistances, yield the same result as long as the density of scatterers is low enough to prevent eective backscattering [40]. The limits of this model, however, are obvious: for Hall voltages exceeding a limit of eUH ¿˜!c , the tunneling into empty states of the next higher Landau level is strongly enhanced, and the current starts to spread into the 2D-bulk. This results in the breakdown of the QHE and was indeed observed in measurements at certain samples with a constriction (w = 4 m) of the current channel [46] and with a pronounced inhomogeneity [47,48]. For example, an edge-induced breakdown of the QHE at eUH = ˜!c in the second QH plateau at B ≈ 6T would correspond to current values of about 1 A, which are by orders of magnitude smaller than the critical current values which are usually measured for wide Hall bar samples (width w ≈ 100 m) with typical parameters (ns ≈ 3 × 1011 cm−2 , H ¡106 cm2 =V s). This does not mean that the model is inapplicable in general, but an explanation of the breakdown of the QHE on the basis of edge channel transport in Buttiker’s sense is only appropriate in exceptional cases, as, for example, for samples in the ballistic transport regime. Another argument, which also limits the current range of applicability of the edgechannel picture to currents far smaller than the usually measured breakdown values, will be given below in conjunction with the role of phonon emission for the breakdown.

2.2. Experiments on the current distribution in the QH regime From the previous section, it is evident that the local (percolation) and nonlocal (edge current) picture imply very dierent distributions of the QH current over the sample. Consequently, the predictions of these models concerning the breakdown of the QHE are very dierent, too. Although some early papers (see, for example, Ref. [46]) suggest an explanation of the experimental data on the basis of edge currents, there are severe arguments against this interpretation. The most serious objection against a breakdown related to edge currents is the absolute amount of the breakdown current and its proportionality to the sample width, as frequently observed (see Section 3.1). The drift velocity of electrons at the Fermi energy, which are assumed to ow within the edge pstates of a typical width of the magnetic length ‘B = ˜=eB, can be estimated for the i-th QH plateau (corresponding to i edge states) by the relation: vD =

I I=i‘B = 4‘B : eN‘ =2 ie

(10)

This drift velocity amounts to vD ∼ I (A) × 4:1 × 105 m=s for a sample with a typical carrier density of ns = 3 × 1011 cm−2 at a magnetic eld of 6 T, corresponding to the 2nd QH plateau. If the drift velocity exceeds the sound velocity (vs ∼ 5 × 103 m=s for GaAs [49]), a spontaneous emission of phonons occurs which results in an onset of dissipation [10]. For our example, this would happen at a current of about I = 12 nA only, irrespective of the sample width. In reality, this constraint is less strict, as the edge channels can be assumed to be wider than the magnetic length due to screening eects [50,51] (formation of so-called compressible strips). However, the sum of the width of all current-carrying compressible strips is smaller than 200 nm for GaAs=AlGaAs samples with etched boundaries [50,51], elevating the maximum of the nondissipative current up to about 120 nA. Thus, the assumption of a wide-spread Hall current near the breakdown appears to be better reconcilable with the experimental observation of typical breakdown current densities of about jc ≈ 0:1–2 A=m (or Ic ≈ 10– 200 A for a current channel being 100 m wide). To clarify the current distribution in the QH regime experimentally, electrical, electro-optical and induc-

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Fig. 3. Distribution of the Hall voltage (total: U48 , partial: U45 , U46 , U47 ) as a function of the magnetic eld, measured with internal contacts (see upper inset) in presence of an illumination-induced gradient of the electron density. Lower diagram: fraction of the total current between internal contacts 5 and 7 as a function of the magnetic eld, as calculated from the voltage distribution. Data from Ebert et al. [52].

tive measurements have been applied. The electrical methods made use of internal metallic contacts, placed into the interior of Hall bars (see upper inset of Fig. 3) [52,53]. At noninteger lling factors, a more or less homogeneous current distribution was observed. Within the QH plateaus, i.e. near integer llings, the current distribution changed drastically with the magnetic eld. A bunching of the current near one edge, a following movement of the current path over the entire sample, and nally a bunching near the other edge was observed while sweeping the magnetic eld from one ank of the QH plateau to the other one (see Fig. 3, Ref. [52]). This result was found in accordance with the calculations of Woltjer [32–34], which explained the observed ‘wandering’ of the current path on the

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basis of local inhomogeneities of the electron density. From Eq. (7) it is evident that a change of the magnetic eld leads to a local shift of the current path. The main objection against this interpretation was based on the argument that an internal contact decouples from the 2DES in the QH regime due to the Corbino eect. However, this argument does not apply for nonideal conditions (xx ¿0), as present in most of the measurements with internal contacts. Investigations of Sichel et al. [54], applying a high-impedance electrometer, yielded maximum impedances of interior contacts in the M -range within the i = 4 plateau at temperatures of 2.2 K. Wolf et al. [55] showed that the onset of the Corbino eect at contacts to the sample interior (and thus the limit of reliable measurements at these contacts) can be detected by a deviation of the sum of all partial voltages at these contacts from the Hall voltage while approaching the QH state (xx → 0). Measurements of Breitlow et al. [56] on samples with contacts attached to several internal boundaries (‘multibridge’ samples) yielded strongly nonlinear current–voltage characteristics at individual contacts, but a perfectly linear sum of all voltages, corresponding to the quantized Hall resistance as measured at the outer boundary of the sample. In general, the measurements at internal contacts are reliable as long as the impedance of the circuitry is large compared with the impedance of these contacts. The presence of contacts itself may, however, change the intrinsic current distribution. Therefore, attempts have been made to measure the current distribution in QH samples with contactless methods. One approach was the application of inductive coupling to Corbino devices [57–59] and Hall bars [60,61]. The recent results of Yahel et al. [60,61] provide a new aspect concerning the ‘current bunching’ as observed earlier: The authors found current bunching if either a top or back gate was closer to the 2DES than a distance corresponding to the sample width. Conductive glue to x the sample can also act as a gate causing current bunching, if the substrate thickness is smaller than the sample width. For samples without any gate, an almost uniform current distribution was found even at integer lling factors. Another experimental approach to measure the current distribution was the application of the linear electro-optical eect (Pockels eect). Due to the

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Fig. 4. Hall potential pro les (2nd QH plateau) measured electro-optically across a 200 m wide stripe, 400 m from the source, for a current value of 23 A (polarity and modulation scheme as indicated). The traces for various magnetic eld values (central column) are shifted vertically for clarity. The value of the reference potential is given by the short at line left of each trace. Dashed lines mark the positions of the channel edges. Data from Knott et al. [65].

in uence of electric elds on the birefringent properties of GaAs, the polarization direction of light changes in dependence on the local potential while passing a 2DES at the GaAs=AlGaAs interface. Fontein et al. [62–64] and later also Knott et al. [65] and Dietsche et al. [66] made use of this eect to monitor the potential and current distribution in the QHE. The results of Knott et al. [65] and Dietsche et al. [66] were in apparent contradiction to those of Fontein et al. [62–64]. Knott and Dietsche et al. [65,66] found rather sharp potential drops, which moved with the lling factor, inside a channel of 200 m width (Fig. 4). In wider samples (w = 2 mm), Fontein et al. [62–64] found most of the Hall voltage drop (ca. 80%) near the sample boundaries at low currents of I = 5 A (Fig. 5). The spatial resolution of the setup (70 m) was about 1=30 of the sample width, so that the extent of the edge zones containing the majority of the current ow could only be estimated. This estimate, however, was found in accordance with the calculation of the Hall potential distribution in the QH regime [62–64] (based on the approach of MacDonald et al. [67]): y − w=2 h ln w ; (11) I ln V (y) = 2 2ie y + w=2

Fig. 5. Hall potential pro les measured electro-optically across a 2 mm wide Hall bar, at a current of 5 A. The scans were taken within the 4th QH plateau, at B = 5:0 T (N) and 5.25 T (+). Full line: calculated pro le using Eq. (11). Data from Fontein et al. [62–64].

for w and with = ‘B2 i=a (a is the eective Bohr radius and is about 10 nm in GaAs). The value for represents the width of the line charge, which is also of the order of 10 nm for typical conditions (B = 5 T, 2nd QH plateau). Eq. (11) is valid in the region between, but outside, the line charges (|y|6w=2 − ). The strongly nonlinear slope of the Hall potential is a consequence of the two dimensionality of the system and the absence of eective screening in the QH regime (xx → 0 for T → 0 and EF = ˜!c =2 – i.e. localized states at the Fermi energy). It should be noted that the current transport occurring preferentially near the sample edges as observed in Fontein’s study is not related to edge channels in the sense of Buttiker [40] (see the previous section). The model of Buttiker predicts the entire net current of electrons owing near one sample edge, within a channel width of the order of some magnetic lengths (some 10 nm). In contrast, both the measurements of Fontein and the calculations of MacDonald yield forward currents at both edges, distributed over macroscopically wide stripes (up to some 10 m, depending on the sample). Another type of scanning optical measurements on Hall bars at integer lling factors was performed by Shashkin et al. [68]. The authors measured the photoresponse of xx with respect to a local optical excitation by scanning a laser spot (exitation energy: 1.96 eV, spot diameter: 20 m) over a Hall bar. At the breakdown of the QHE, they found a more or

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less homogeneous distribution of the photoresponse. At even higher currents beyond the breakdown, the authors observed an increase of xx while illuminating the sample edge with the higher electron potential. Therefore, Shashkin et al. [68] interpreted this result in terms of the edge channel picture [40]. This interpretation, however, appears to be inadequate in view of the experimental conditions. First, the interband excitation of electrons by the laser spot increases the local electron density by about 1% [68]. The corresponding local shift of the lling factor results in a rather drastic change of the QHE breakdown current density (of the order of some 10%) at the position of the laser spot. Second, the applied currents for this experimental regime (up to 300 A) correspond to dierences of the edge potentials of up to 200 × ˜!c . Thus, the observed eect is rather due to a local redistribution of electrons in extremely high electric elds than to a contribution of edge channels. At higher currents (I = 20 A), Fontein et al. [62–64] found a linear distribution of the Hall potential across the sample and attributed this to the breakdown of the QHE. Indeed, a simple argument which was put forward by Thouless [69] can explain a linear spatial variation of the Hall potential for xx 6= 0: if the resistivity tensor components xx and xy are locally de ned and homogeneous throughout the sample, the relation between the potential V (x; y) and the resistivity tensor can be written as: grad V (x; y) = eE = e · j:

(12)

With j = (jx ; 0) for a Hall bar measurement, Eq. (12) yields 9V=9x xx jx : (13) =e −xy jx 9V=9y Assuming xx = const and Ex = 9V=e 9x = const, one obtains for the Hall electric eld Ey : eEy =

Ex 9V = −exy = const: 9y xx

(14)

Thus, in a homogeneous system, the Hall potential has a constant slope across the Hall bar according to Eq. (14). This equation is obviously valid only for nonzero values of xx (xx ). In conclusion, there has been no experimental evidence yet for an edge-current dominated charge transport in the QH regime for higher currents (some A).

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Thus, a dominant role of edge states for the breakdown of the QHE cannot be expected either. This will be shown in more detail in the discussion of the breakdown experiments below. 3. Breakdown of the quantum Hall eect and sample geometry 3.1. Critical current or critical current density? – width of the current path and breakdown of the quantum Hall eect Soon after the discovery of the QHE [1], investigations of a possible application of the eect as a resistance standard for maintaining the resistance unit ‘Ohm’ had been put forward by various researchers. The essential questions in this respect were the precision of the eect and the physical limits for its steadfastness. In particular for high precision measurements of the Hall resistance, the sample current should be as high as possible but below the critical current where the nearly nondissipative current ow breaks down. Therefore, the rst investigations of the breakdown of the QHE date back to the years shortly after the discovery of the eect. On the conference on electronic properties of two-dimensional systems in 1981, von Klitzing gave a rst estimate of the possible precision level of the QH resistance and its limitation by nite values of xx [70]. In the same conference, Yoshihiro et al. [2] presented measurements of the longitudinal resistance xx of a Si-metal-oxide-semiconductor eld-eect transistor (MOSFET) as a function of the current in the QH regime. These investigations were also aimed at investigating the relation between xx and the precision level of xy . For one of their samples (w = 100 m), they found a pronounced increase of xx at about 15 A for i = 8, whereas no increase of xx was observed for i = 4 up to 25 A. In 1983, the same group found an empirical relation between the deviation of the Hall resistivity from the quantized value, xy , and the corresponding longitudinal resistivity xx which is given by xy = −Sxx and S = 0:1 for the Silicon MOSFETs investigated [71]. For GaAs=GaAlAs heterojunctions, a qualitatively similar behavior was found by Cage et al. [72], but with factors S ranging from 0.015 to 0.507. The values for S can be interpreted as the ratio of the

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Landau level broadening and the Landau gap, =˜!c , applying the theory of Ando et al. [28]. Very recently, Kawaji et al. [73] announced a completely dierent relation between xy and xx . The authors found an abrupt drop of xy by xy =xy ≈ −8 ppm at electric Hall elds which are about 20% smaller than the value for the steep increase of xx . The authors conclude that this drop of xy which they call ‘collapse of the quantum Hall resisistance’ (in distinction from the ‘breakdown of the QHE’ as related to the sudden onset of dissipation) is not due to thermal activation of carriers, but to a delocalization of a part of the localized electrons at a certain electric eld. The physical origin for the obviously very dierent in uence of the assumed delocalization on xy and xx (of course, any change of the number of mobile electrons aects also xx ) can, however, not be understood straightforwardly. The authors also suggest an approach based on the assumption of strip-like inhomogeneities with a strip width smaller than the width of the potential probe. The microscopic origin of the dierent behavior of xy and xx , however, remains an open problem. In 1983, Ebert et al. [3] investigated this currentinduced breakdown of the QHE systematically on GaAs=GaAlAs-heterojunctions, which are preferred worldwide for the QHE-based maintenance of the unit ‘Ohm’ nowadays. The study of Ebert et al. [3] was already aimed at the basic physics of the breakdown. Therefore, the QHE breakdown was investigated as a function of the temperature, the Landau gap and the lling factor. The authors came to the conclusion that it is the interplay between energy gain (from the electric eld at nite values for xx ) and loss (relaxation by inter-Landau-level transitions) of the electrons, which is responsible for the breakdown of the QHE. This picture was later supplemented by Komiyama et al. [8,9] who calculated breakdown characteristics on the basis of a phenomenological model [9]. In their study, Ebert et al. [3] discussed the experimental data on the basis of a local model. Therefore, the current density and the corresponding Hall electric eld were regarded as relevant critical quantities of the breakdown. Critical current densities of jc = 0:5–1.5 A=m for the second QH plateau, increasing superlinearly with the Landau level spacing (or electron density and magnetic eld, respectively), were reported for Hall bars of 380 m width. In other early studies of the breakdown [4,6,7], compa-

Fig. 6. Critical current Icr versus device width w for a wafer with a mobility of H = 2:1 × 105 cm2 =V s. The slope of the full line corresponds to a critical current density of 1.6 A=m. Data from Kawaji et al. [13].

rable values for jc were published (Cage et al. [4]: jc = 0:9 A=m for i = 4 on a Hall bar with w = 380 m, Kuchar et al. [6]: jc = 2:2 A=m for i = 2 on a Hall bar with w = 320 m, Stormer et al. [7]: jc = 1:16 A=m for i = 1; 2 and 4 on a Corbino device with 740 m width of the current-carrying ring). The mobilities of all these samples were at lower or medium values (typical range: H = 0:3–3 × 105 cm2 =V s), and a homogeneous current distribution was assumed. Later, this assumption appeared questionable in view of a variety of theoretical and experimental studies concerning the current distribution under conditions of the QHE (see Section 2). Therefore, an apparently simple approach to clarify the role of the current distribution was to investigate the breakdown current of the QHE as a function of the sample width. Several groups studied this width dependence and obtained rather dierent results. Although these results, and in particular their interpretation by the authors, seem to be hardly reconcilable, two main trends are obvious: for samples with lower and medium mobilities (typically of the order 105 cm2 =V s), a linear increase of Ic with the sample width was found (see, for example, Fig. 6), whereas a sublinear dependence for Ic versus w was occasionally observed on samples of higher mobility (about 106 cm2 =Vs, see Fig. 7). We will discuss these results and the possible conclusions in detail below. Haug et al. [19] reported on the breakdown behavior as a function of the sample width on Hall bars

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Fig. 7. Critical current versus device width for a wafer with a mobility of H = 9:0 × 105 cm2 =V s, at lling factors = 1 ( ) and = 2 ( ). Dashed lines: logarithmic ts. Data from Balaban et al. [74].

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and Corbino devices. While increasing the width of Hall bars from 10 to 80 m, the authors observed a reduction of the critical current density (as de ned by jc = Ic =w, assuming a homogeneous current distribution) from 1.1 toward 0.76 A/m. In Corbino devices, Haug et al. found the critical current density nearly independent of the width. Later investigations showed, also for Hall bars of comparable mobility, a proportionality of the critical current and the sample width [13–16]. Thus, it is likely that the sublinear increase of Ic with w as obtained on Hall bars by Haug et al. [19] is due to slight inhomogeneities of the carrier density and the resulting current distribution. The same explanation is apparently applicable for the results of Balaban et al. [20], obtained on high mobility samples (Fig. 7). In their studies, the authors reported a logarithmic increase of the critical current with the sample width and explained it by applying the theory of MacDonald et al. [67]. But as pointed out by Thouless [69], already small nite values of xx would prove the results of MacDonald et al. inapplicable. As these nite values of xx are always present at currents close to the breakdown, the width dependence observed by Balaban is more likely due to the longer range of inhomogeneities which is usually found in samples with higher mobilities (H ≈ 106 cm2 =V s).

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This was con rmed by a later study of the same authors [74] at samples with lower mobility (H = 1 − 2 × 105 cm2 =V s), yielding a constant critical current density ( jc ≈ 0:15 A=m for sample widths from 10 to 60 m). The frequently observed proportionality of critical current and sample width [13–16,19,74] implies in principle a homogeneous distribution of the current

ow near the breakdown. To be precise, this means a homogeneity on a length scale corresponding to the typical width increments of the dierent samples investigated (ca. 5–10 m). On a shorter scale of some 100 nm (typical width of current path [32–34]), the current distribution may be, however, strongly inhomogeneous. In this case, the proportionality of Ic and w means that the number of (percolative and nondissipative) current paths increases linearly with the sample width ([74], see also the early work of Kazarinov and Luryi from 1982 [75]). In high mobility samples, long-range potential uctuations may cause inhomogeneities in the distribution of the current ow reaching typical length scales of some micrometers. This long-range inhomogeneity of the current ow would result in a nonlinear Ic -versus-w dependence for samples with a width of the order of the typical uctuation length. Such long-range inhomogeneities can be obviously generated or, at least, ampli ed by illumination of the sample [17,18,74]. In a previous work of Nachtwei et al. [18] the breakdown was studied in 2D systems with and without a periodic antidot lattice before and after illumination. The periodic antidot lattices were found to homogenize the current ow close to the breakdown, on length scales large compared with the arti cial lattice period of a0 61 m. In an antidot lattice of a0 = 0:3 m, an enhancement of Ic was observed after illumination. This enhancement is due to the increased electron density, because this increase results in both a higher Landau gap for the integer lling factor (see Ref. [3]) and a reduction of the depletion zones around the antidots. In contrast, the unpatterned reference sample (on the same chip) showed an even slightly reduced breakdown current after illumination. This observation is in accordance with the earlier results of Balaban et al. [74], obtained on samples with a lower mobility. After illumination, the authors observed an increase of Ic with w for samples with w620 m only, whereas no further increase of Ic was observed for wider samples

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up to 100 m. Thus, the short-range inhomogeneities will be replaced by long-range inhomogeneities of the order of some micrometers after illumination. This reduction of short-range inhomogeneities results, however, not necessarily in a nonlinear potential pro le as deduced for the ideal case (homogeneous 2D systems with xx = 0 [67]). It is more likely that just the length scale of the inhomogeneities increases. This leads to an even more pronounced large-scale decomposition of the samples into compressible and incompressible regions and a lesser uniformity in the distribution of the percolative current paths after illumination. To prove the in uence of short- and long-range inhomogeneities of the current ow on the breakdown of the QHE, Nachtwei et al. compared the breakdown behavior in 2D systems with periodic and aperiodic antidot arrays [17,18]. Two arrays of antidots (periodic with the lattice constant a0 and aperiodic with the same average antidot spacing hai) were patterned on the samples. This arrangement permits an immediate comparison of the QHE breakdown in periodic and aperiodic antidot arrays of the same average antidot density (see inset of Fig. 8a). The distribution function of antidots in the aperiodic arrays is of approximately Gaussian shape with a broadening parameter a of about 14 hai for all samples. The antidot distribution corresponds to a large-scale clustering of antidots on length scales comparable with the sample width (w = 50 m). This leads to large-scale inhomogeneities of the current ow, in particular at higher currents, because areas with high local antidot densities are avoided by the current ow [17,18]. Accordingly, the critical currents of the periodic arrays were markedly higher than those of the corresponding aperiodic arrays (Fig. 8a). From these considerations, it appears to be interesting to investigate the breakdown behavior for sample dimensions in the micrometer and submicrometer range. Therefore, investigations of the QHE breakdown at narrow constrictions were performed by several authors [5,46,76,77]. For a constriction of 1 m width and 10 m length, Bliek et al. [5] reported a breakdown current density jc as high as 36 A=m in the center the second QH plateau. This result is in stark contrast to the early measurements at wider samples. The reduction of the in uence of inhomogeneities on the breakdown, which goes along with the reduction of the sample size, cannot account for this drastic en-

Fig. 8. Critical current Ic ( = 2) versus inverse antidot spacing. (a) For samples with square lattices (N) and aperiodic arrays ( ) of antidots. Upper inset in (a) scheme of the corresponding sample geometry (not to scale). Lower inset in (a) critical current Ic as a function of the lling factor for the periodic and aperiodic arrays. Wafer data: ns = 3:0 × 1011 cm−2 , H = 1:6 × 106 cm2 =V s. (b) For samples with square lattices (N) and single lines ( ) of antidots (second wafer). Inset in (b) sample geometry. Wafer data: ns = 2:2 × 1011 cm−2 , H = 9:6 × 105 cm2 =V s. The values for Ic are plotted as normalized values Ic (a)=Ic (∞) (with Ic (∞) as the breakdown current of the unpatterned reference sample) to make the data of both wafers comparable. The linear decrease of Ic with increasing 1=a corresponds to a proportionality of the critical current and the eective width (ranging from 0.23 to 1.5 m) of the current channel. Data from Nachtwei et al. [17,18].

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hancement of jc . Instead, the results can possibly be explained by the avalanche-heating picture as suggested by Komiyama et al. [24]. These authors investigated the dependence of the breakdown on the length of a constriction [22,23] and found higher breakdown

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currents for short drift distances of the electrons (see Section 3.2). If the in uence of the length on the breakdown is eliminated by providing suciently long travelling distances, the critical current densities of submicron constrictions are again in the same range as the values for wide samples (for example jc = 0:4– 1.1 A=m, depending on the wafer properties, measured for eective constriction width from 0.23 to 1.5 m in periodic antidot arrays [17,18]). These values can be assumed close to the ‘intrinsic’ properties, as in uences of the inhomogeneity and of the drift length on the breakdown are minimized in these experiments by a periodic arrangement of the constrictions in the antidot lattices. In the experiments of Nachtwei et al. [17,18], the critical current was measured as a function of the antidot spacing a. Fig. 8 shows the results for two different wafers. As visible, the critical current scales linearly with the inverse distance of the antidots, 1=a0 . This linear dependence of Ic on 1=a0 can be explained by a simple geometrical argument: the breakdown occurs, if the highest local current density jxmax between adjacent antidots reaches the critical current density jc0 known from the unpatterned reference device jc0 = jxmax = Ic (a0 )=(w − Ndel ), where N = w=a0 is the number of antidots across the sample, and del is the electric (eective) antidot diameter. This yields Ic (a0 ) = Ic0 (1 − del =a0 ), with Ic0 = jc0 w as the critical current of the reference region. The equation holds for a constant electric Hall eld (homogeneous current ow) between the antidot lines. The scaling of Ic with 1=a0 is equivalent to a linear increase of the critical current with the eective width of the sample with antidots, N (a0 − del ). This result, obtained on a submicrometer scale, corresponds to the linear dependence of the critical current on the macroscopic sample width as reported earlier [13–16,19,74]. The experimental results agree with the interpretation that the breakdown of the QHE starts where the local current density is at its maximum. The alternative interpretation, that the breakdown would be determined by the current density hji averaged along the current

ow direction or the remaining active area of the array, fails to explain the results in a quantitatively correct manner [17,18]. From the slope of Ic (1=a0 ), values of the electric antidot diameter of del ≈ 400 nm, and of the critical current densities of jc0 = 1:1 and 0.4 A=m (for dier-

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ent wafers [17,18]) can be estimated. The data are consistent with the picture that the current ow at the breakdown occurs exclusively in that region, where an incompressible state of constant local density develops under the conditions of the QHE [50,51,74,78]. This nding eventually rules out any interpretation which relates the breakdown of the QHE to edge channel transport. To conclude, all results reported above can only be explained conclusively if a current ow within the 2D bulk is assumed at current values close to the breakdown of the QHE. The typical length scale of inhomogeneities of the carrier density and their distribution determine the distribution of percolative current paths in the sample. For short-range uctuations (typical length far smaller than the sample width, but larger than the magnetic length), the percolation paths are densely distributed. This leads to a linear dependence of the breakdown current Ic on the sample width w, if the distribution of current paths is homogeneous on larger length scales (comparable with the sample width). If long-range uctuations of the order of the sample width exist, a sublinear increase of Ic with w is observed. This behavior is related to a more eective screening of short-range uctuations and is frequently observable in samples after illumination and in samples of higher mobility. 3.2. Avalanche heating and long-range relaxation of electrons near the breakdown – some relevant length scales In 1986, L. Bliek et al. [5] reported extremely high critical current densities for the breakdown of the QHE, measured on a narrow constriction of 1 m width and 10 m length. The high breakdown current density was explained by the so-called ‘quasielasticinter-Landau-level-scattering’ (QUILLS). This model was proposed by Eaves and Sheard in 1986 [11] and assumed the breakdown as being due to transitions of electrons from the highest populated into the lowest unpopulated Landau level, induced by high electric elds. The QUILLS model yields breakdown current densities which are typically by two orders of magnitude higher than the values measured at wide Hall bars, assuming a homogeneous current distribution. This discrepancy, which is in contrast to the rather good agreement with Blieks measurements,

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Fig. 10. Schematic picture of the nonlocal breakdown of the QHE: (a) avalanche-type multiplication of excited carriers after injection into a constriction, (b) resulting pro le of the spatial evolution of the electron temperature. From Komiyama et al. [24].

Fig. 9. (a) Shubnikov–de Haas oscillations, measured in a narrow constrictions (we = 2:4 m) of Hall bars (H = 8:0 × 105 cm2 =V s), for dierent lengths LC of the constriction (LC = Lv + 10 m, Lv is the distance between the potential probes). (b) Current–voltage characteristics of the breakdown (2nd QH plateau) for dierent constriction lengths. Data from Kawaguchi et al. [22,23].

was attributed to inhomogeneities of the current ow in wide samples [5,76]. About ten years later, these results appeared to be explainable in a completely dierent manner. In 1995 and 1996, studies on the QHE breakdown in narrow constrictions of dierent length were published by Kawaguchi et al. [22,23]. The authors investigated constrictions of an eective width of 1.8 and 2.4 m, with constriction lengths ranging from 20 to 440 m (Fig. 9). Whereas the breakdown was observed at current values of 2 A (corresponding to jc ≈ 1 A=m) for longer constrictions with l¿210 m, about 10 times higher currents were necessary to induce the breakdown in the shortest constriction (l = 20 m) of the same width. Thus, there is a certain travelling length for the electrons necessary

to develop the complete breakdown of the QHE. This length was observed to depend on the mobility, with a trend of smaller lengths for smaller mobilities. In a subsequent paper by Komiyama et al. [24], this was con rmed by investigating the breakdown at dierent positions inside the constriction. The breakdown was found at higher currents for contacts closer to the position were the electrons are injected into the constriction. These experiments were explained by an avalanche-like heating process for the electrons which need a certain length to achieve a stationary state (Fig. 10). This length was estimated to be of the order of 10 m and of 20 m for the lling factor = 4 and mobilities of 1 and of 8 × 105 cm2 =V s, respectively. For the high mobility sample, even higher avalanche lengths up to 130 m were found at = 2. As these length scales are dependent on the scattering properties of the samples and also on extrinsic sample parameters (sample geometry and current), the values given above for the avalanche length cannot be taken as typical or even intrinsic parameters. However, the trend of shorter avalanche lengths with decreasing mobility is evident. The authors have chosen the term ‘nonlocal breakdown’, because the long-range development of the dissipative state cannot be described by a local relation of the critical Hall eld EH (r) and xx (r), but by a balance equation between energy

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gain of the electrons from the electric eld, xx j 2 , and the energy loss by relaxation processes of heated electrons: xx (Tel )j 2 =

(Tel ) − (TL ) ; 0

(15)

where (T ) is the energy of the electron system (per unit area) at the electron [lattice] temperature Tel [TL ], and 0 is the energy relaxation time of the heated electrons. This balance is very sensible against uctuations, and its (local) violation leads to a runaway of the electron temperature. If the typical time scales for this runaway are long enough, this results in a spatially resolvable electron temperature pro le after the ignition of the breakdown. The ignition, however, has a completely local character and can dominate the breakdown, if the ignition condition is periodically repeated or is present over a drift distance longer than the avalanche length, respectively. This was shown by the investigation of the breakdown in antidot lattices [17,18]. The correlation of the QHE breakdown with the highest local current density as observed in this study does not contradict the nonlocal electron heating approach [24], as the antidot array extends over a length of more than 50 m. The electrons are subsequently heated while passing the array line by line. This could be con rmed by the results obtained on samples with single antidot lines. Fig. 8b shows the results, obtained on samples containing a single line of antidots and an antidot lattice with the same antidot spacing. The breakdown currents of the sample areas containing a single line are usually larger (the exceptional case for a0 = 1200 nm is attributed to a local carrier density uctuation) than those of the corresponding square lattice. Although a single line causes the same local enhancement of the current density as the lattice, the extension of this constriction along the current ow direction of some hundred nanometers is obviously far too small to eectively heat the electrons. In the lattice, the travelling distance of the electrons is large enough to develop a stationary state of elevated electron temperature. These results show that the electron heating is governed both by the local enhancement of the current density and the extent of the current path containing the constrictions. From the observation of macroscopic avalancheheating lengths for the breakdown, naturally the question of the length scale for the opposite process arises.

Fig. 11. (a) Resistivity xx of a sample as depicted in the inset (schematic view of the sample geometry. ‘W is the length of the wires, w is the width of the quantum Hall conductor, and ‘ is the distance between the potential probes) near the second QH plateau, measured at dierent pairs of adjacent potential probes with current ow of electrons towards (left: I = +35 A) and from (right: I = −35 A) the wire array. The resistivity xx is shown as grey scale plot (black: xx 64 white: xx ¿100 ) in dependence on the magnetic eld B and probing position x (taken in the middle between the adjacent potential probes). Only for the current ow from the wires into the 2DES, a signi cant distance dependence of xx occurs. (b) Dependence of the xx minimum on the distance of the probing contacts from the end of the wire array at dierent currents: I = 34:0 A (square), 34.2, 34.4, 34.6, 34.8, 35.0, 35.2, 35.4, and 35.6 A (dash). At the breakdown current (I = 35:6 A), the value of the xx minimum is independent of the probing position. Data from Kaya et al. [79,80].

Therefore, Kaya et al. have analyzed the energy relaxation of hot electrons in the pre-breakdown regime of the quantum Hall eect by measurements of the resistivity as a function of the distance from the position of hot-electron injection [79]. The electrons are heated in arrays of parallel narrow channels (wires, see inset of Fig. 11a). These wires form a series of parallel constrictions with equivalent properties, providing a homogeneous temperature front of heated electrons

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which enter the unpatterned two-dimensional electron system (2DES). Within a small range of currents below the critical value of the unpatterned region, the resistivity was found to depend strongly on the position of the potential probes used for the measurement (Fig. 11). The resistivity decreased with increasing distance of the potential probes from the injection front of hot electrons. The characteristic decay length of the resistivity was observed to strongly increase with current. Above a certain current level, the dissipation persists over the entire sample length (Fig. 11b). For electrons injected from a wide metallic contact (opposite current direction), the QHE breakdown was found at higher currents and more or less uniformly distributed over the sample (Fig. 11a). A remarkable in uence of the current and of the electron mobility on the measured resistivity pro les was observed. The pre-breakdown current range I , in which a decay of the resistivity was observable, was found narrower with decreasing mobility (I=IC ≈ 5–25% for mobilities from H = 4:0 × 104 to 5:3 × 105 cm2 =V s). The results were compared with calculations on the basis of a nonequilibrium between the excitation and relaxation of hot electrons. These calculations yielded an exponential decay of the resistivity with increasing distance x from the injection front of hot electrons. According to the model, the resistivity pro le xx (x) in dependence on the current I (I 6Ic ) can be written as " ( 2 #) I x − x 0 1− (16) xx (x) = 0xx exp − ‘D Ic with 0xx as the saturation value of the resistivity in a region 0¡x¡x0 close to the injection front of hot electrons (see Fig. 11b), and ‘D = vD 0 as the average drifting length of electrons between two inelastic scattering events [79,80]. From tting the experimental resistivity pro les to Eq. (16), the relaxation lengths ‘D and the inelastic scattering times 0 could be determined. A remarkable coincidence of the relaxation lengths ‘D with the mean free path ‘mfp was observed for samples with mobilities from 4:0 × 104 to 5:3 × 105 cm2 =V s. As the latter values are mainly determined by elastic Coulomb scattering at ionized donors, this means that the inelastic scattering (responsible for the dissipation near the QHE

breakdown) is also related to Coulomb scattering. The energy relaxation times, 0 , were found proportional to the Hall mobility of the samples. The order of magnitude of the energy relaxation times (0.24–2.6 ns) is in good agreement with earlier estimates of Komiyama et al. [8,9], which were explained by phonon-assisted dissipation. However, the observed proportionality between energy relaxation time and Hall mobility means that the inelastic scattering rate in our samples is dominated by the scattering at impurities instead of intrinsic electron–phonon coupling properties of the GaAs-bulk. In conjunction with these relaxation experiments, a remark concerning the ‘bootstrap’-type breakdown [24] should be made. Also this avalanche-like development of the QHE breakdown may in principle be related to similar excitation and relaxation mechanisms as suggested above. In this case, the energy gain locally dominates over the loss until a stationary equilibrium develops due to the increase of the electron temperature along the drift direction of electrons. This is because of the increase of the energy relaxation rate with temperature [8,9], stabilizing the electron temperature at a higher value corresponding to an equilibrium of energy gain and loss. Indeed, spatially resolved measurements of the evolution of the hot-electron avalanche in an antidot array by Kaya et al. con rm qualitatively this assumption, as depicted in Fig. 12 (from Ref. [79]). However, in contrast to the relaxation of hot electrons under subcritical conditions, the avalanche multiplication of hot electrons cannot be adequately described by a linear dierential equation. A highly nonlinear process is in general very sensitive to even slight changes of the boundary conditions. Thus, the individual density uctuations of a given sample are of crucial importance for the development of the hot-electron avalanche. Already some years before the investigations of the avalanche-type breakdown in GaAs samples [22–24,79,80], the in uence of the con guration of current-injecting and voltage-probing contacts on the breakdown of the QHE was observed in Si-MOSFETs [81–83]. Van Son et al. found the trend that the breakdown occurs rst close to the current-injecting contact and then spreads towards the opposite dissipative corner. This trend is just opposite to the behaviour as reported for the GaAs samples [22–24]. The results obtained on Si-MOSFETs are

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Fig. 12. Resistivity xx as a function of the magnetic eld B and probing position x, shown as grey scale plot (black: xx 640 white: xx ¿1000 ), throughout a Hall bar of a width w = 90 m (as depicted schematically on top of the plot, with an antidot array embedded) around the second QH plateau. The current is I = 47 A (electron ow direction as indicated in the sample scheme). Both the nite lengths for electron heating after the entrance of the current into the antidot lattice and the hot-electron relaxation after leaving the array are clearly visible. Data from Kaya et al. [79].

attributed to a dissipation of the current source contact which dominates over the dissipation near the drain contact [81,82,84]. This assumption could not be con rmed by phonon emission experiments on Si-MOSFETs, performed by Russell et al. [85,86]. The authors found equal shares of dissipation on both contacts over a wide range of total power input (0.3– 2000 W). However, experiments on GaAs=GaAlAs heterojunctions, applying a similar technique [87] and an imaging technique based on the fountain-pressure eect of super uid helium [88], indeed yielded a predominant dissipation near the current-injecting contact at higher currents. It should be noted here that the behavior of contacts is not the only essential dierence between QH samples based either on Silicon or on GaAs. Due to the presence of a gate in Si devices, the bunching eect of the sample current as described by Yahel et al. [60,61] is unavoidable, in particular at higher currents. Accordingly, a direct comparison of the current distribution in Si and GaAs samples using an internal contact [89] revealed pronounced dierences for both material systems. Further, the scattering rate in 2DES based on Si is usually higher than in GaAs-based systems. At the present state of knowledge, it is impossible to decide which of the dierences between Si and GaAs (concerning the contacts, the current distribution and

the scattering) is dominant for the apparently different spatial evolution of the breakdown in both materials. 3.3. Metastability near the breakdown: bistable switching and time-dependent resistance uctuations One of the most intriguing features of the breakdown of the QHE is the steepness of the transition from a low-dissipative towards a high-dissipative conduction when the critical current is reached. This abruptness attracted the interest of many researchers from the early 1980s on. The challenge to nd a physical explanation for this behavior led to questions beyond application aspects of the QHE. A variety of mechanisms for the breakdown, as for example the sudden delocalization of electrons (proposed by Trugman in 1983 [12]), the establishment of a compressible path between the sample edges (suggested in 1997 by Tsemekhman et al. [90]), the spontaneous emission of phonons [10], the hotelectron picture [3,8,9], the Zener tunneling [91] or inter-Landau-level transitions induced by high electric elds [11] were proposed to explain the jump-like increase of xx or xx , respectively, at the breakdown.

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Fig. 14. Time-dependent uctuations of the longitudinal voltage Vx , measured on a Hall bar (w = 380 m, H ¿1 × 105 cm2 =V s) near the breakdown of the 4th QH plateau (B = 5:65 T, TL = 1:1 K), at a source-drain current of I = 325 A. Data from Cage et al. [4].

Fig. 13. Current–voltage characteristics (2nd QH plateau, B = 6 T) of a Hall bar (w = 380 m, H = 3:8 × 104 cm2 =V s) at dierent lattice temperatures TL (inset: critical current density versus TL ). Several jump-like transitions are visible for TL 65:87 K. Data from Ebert et al. [3].

In most of the experimental studies, several steps were observed in the transition from the lowdissipative state towards higher dissipation (see, for example, Fig. 13 and Refs. [3–5,19,47,48,92,93]). To explain this particular behavior, a novel type of quantization in narrow constrictions [11,92], the population of more than one Landau levels above the equilibrium Fermi energy [4,47,48], or simply macroscopic inhomogeneities [93,94] were suggested. In some studies, just one sharp step (Fig. 1) was observed in the transition from quantum Hall conduction to dissipative transport [8,9,95,96]. This type of transition was readily explainable by the a hot-electron relaxation model, which will be explained in some more detail below. For both the single- and the multi-step transitions, time resolved mesurements revealed that timedependent uctuations occur between the lowdissipative quantum Hall conduction and one or more resistive states (see, for example, Fig. 14 and Refs. [4,47,48,95]). A comprehensive study of the statistics of these uctuations was made by Scherer [97]. The author concluded that the uctuations represent a type of nonchaotic telegraph noise. The time scale of the

uctuations was found to be strongly dependent on the lling factor, the current and the sample width. The lowest switching frequencies were observed at lling factors slightly below the integer value [97]. At currents below, but close to the breakdown, the system was found in the low-dissipative state most of the time [95]. At slightly higher current values, the probabilitiy of observation of a dissipative state increased drastically. No signi cant correlation between the uctuation amplitude and the cyclotron energy was found, in contrast to earlier observations [46–48]. Ahlers et al. explained their results with time-dependent uctuations of macroscopic hotelectron domains, in accordance with the scenario suggested by Ebert et al. [3] and Komiyama et al. [8,9]. The hot-electron picture indeed provides an explanation of the steep jumps. Moreover, the frequently observed hysteresis eects in the breakdown characteristics while either sweeping the current at a xed lling factor, or vice versa, can also be explained. The essential reason for the development of the hysteresis is the dierent functional dependence of the energy gain and the energy loss on the electron temperature [3,8,9]. We will exemplarily show this by comparing a typical I–V characteristic, measured on a Hall bar at the lling factor = 2 (B = 6 T), with a calculation of the electron temperature as a function of the current for the corresponding system. The measured curve for the sample with an electron density of ns = 3:1 × 1015 m−2 and a Hall mobility of

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Fig. 15. (a) Measured I –Vx -characteristic for a typical sample (ns = 3:1 × 1015 m−2 and H = 6:0 × 105 cm2 =V s) at a lling factor of = 2. A pronounced hysteresis between the current upand down-sweep is observable. (b) Electron temperatures versus current for the corresponding sample, calculated using the energy relaxation parameter of Cep = 1:2 × 107 K−2 s−1 and the density of states of DBG = 2 × 109 cm−2 meV−1 (see Section 3.3). The hysteresis is depicted by arrows and agrees well with the experimental observation. Data from Nachtwei et al. [18].

H = 6:0 × 105 cm2 =V s is shown in Fig. 15a. A clear hysteresis between up- and down-sweep of the current is present. To explain these experimental results, we now calculate the dependence of the electron temperature Tel on the sample current. We adopt the relation between the inelastic scattering time, 0 , and the electron temperature of the form 1=0 = Cep Tel2 as suggested by Komiyama [8,9]. The constant Cep is an empirical constant determined by electron-phonon interaction (Cep = 1:2 × 107 K−2 s−1 for = 2 [8,9]). The value for Cep was experimentally found to be only weakly dependent on the sample properties and the magnetic eld [8,9]. The energy of the electron system, (T ),

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can be easily calculated for EF in the middle of the Landau gap, if a constant density of states (DOS), DBG , is assumed [17,18,98,99]. For our example, we use DBG = 2 × 109 cm−2 meV−1 (as reported in Ref. [98] for a sample with comparable mobility) and a Landau gap of ˜!c = 10:4 meV at B = 6 T. The balance between energy gain and energy loss (Eq. (15)) yields an s-shaped Tel versus jx -dependence. Since xx is a monotonic function of Tel , this corresponds to a s-shaped Vx –I -characteristic. As the parts of the curve, where 9Tel =9jx ¡0 (and 9Vx =9jx ¡0, respectively) holds, are instable, a hysteresis in the corresponding experimental curve develops. The hysteresis is con ned between two limiting values of jx ; jc1 and jc2 (jc1 corresponds to lower and jc2 to higher electron temperatures, jc1 ¿jc2 ). In Fig. 15b, the calculated electron temperature as a function of the current (de ned by I = jw) shows a corresponding hysteresis between Ic1 and Ic2 . For current densities jc1 ¡j¡jc2 , spontaneous resistance uctuations can be expected. Both values, jc1 and jc2 , are not only dependent on the parameters DBG and Cep , but also on the temperature dependence of xx (Tel ). To calculate jc1 and jc2 of the hysteresis, we used the relation xx = 0 exp{−=kTel } + BG for the resistivity as a function of temperature [98], with = ˜!c =2 (EF in the middle of the Landau gap). The rst term describes the resistivity contribution due to thermal activation over the Landau gap. The prefactor 0 was chosen as 0 = h=4e2 (see Ref. [18]). The additional contribution to the resistivity, BG , is of crucial in uence on the breakdown current density jc1 . If purely activated behavior of xx (T ) is assumed, the electric power gain xx jx2 is very small at low temperatures, even for rather large values of jx . Thus, the values for jc1 become unrealistically high. To obtain values of jc1 closer to the experiment, nite values of BG have to be assumed. A possible origin of BG is variable range hopping (VRH) [100,101]. Different temperature dependences and parameters for BG were proposed by Ono [100] and Polyakov and Shklovskii [101]. However, the value of jc2 , which marks the lower limit of the hysteresis at higher electron temperatures, is nearly unaected by the choice of BG . This is, because for the determination of jc2 the activated conduction provides the dominating contribution. Using the parameters Cep , DBG , and

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the activation energy = ˜!c =2 as given above, we obtain the lower limit of the hysteresis (Ic2 = jc2 w) for the I –V -characteristics of the sample in excellent agreement with the experiment (Fig. 15a and b). To reproduce the upper limit of the observed hysteresis, values of BG approaching BG ≈ 1 at electron temperatures near the breakdown are necessary. The essential conclusion from this nding is that the breakdown of the QHE can be explained consistently within the hot electron model only, if xx is locally enhanced by additional transport mechanisms (e.g. by hopping or by inter-Landau-level tunneling) in the pre-breakdown regime. However, the hot-electron model as described above does not provide a complete microscopic picture of the breakdown. The essential physics of excitation and relaxation of hot electrons within the hot-electron model is contained in the temperature functions of xx (Tel ) and 0 (Tel ), respectively. Whereas a variety of theories was proposed for xx (Tel ) (see above), the excitation and relaxation processes have not yet been fully understood. Apart from the hot-electron model, a variety of alternative theoretical approaches to the breakdown of the QHE have been given (see above). A more microscopic approach to the bistable behavior near the breakdown was suggested by Riess [102,103] on the basis of a solution of the time-dependent Schrodinger equation, leading to a shift of the mobility edge towards the Fermi energy near the breakdown. The results of these considerations are akin to the earlier work of Trugman [12]. Chaubet et al. [104,105] explained the breakdown by a combination of local enhancements of the Hall electric eld, inter-Landaulevel transitions (see also Ref. [11]) and cyclotron emission. However, further work is needed to gain a detailed microscopic understanding of the full variety of experimental results concerning the breakdown of the QHE. 4. Summary: what do we know and what questions remain open? In this review, we have summarized experimental investigations of the breakdown of the QHE. For the breakdown behavior of any sample, the current distribution and the sample geometry are essen-

tial. The current distribution in QH samples near the breakdown is re ected in the dependence of the critical current Ic on the sample width w. Two dierent models for the current distribution under QH conditions were proposed: the local picture (current ow in the 2D-interior) and the nonlocal picture (current

ow in quasi-one-dimensional states close to the sample edge). In apparent accordance with these pictures, both a linear and a sublinear increase of the critical current Ic with w were found, depending on the sample properties as e.g. the mobility. However, recent experiments clearly demonstrate that even the sublinear Ic (w) behavior of some samples can be only conclusively explained if a current ow within the 2D bulk is assumed at current values close to the breakdown of the QHE. There has been no convincing experimental evidence for an edge-current dominated charge transport in the QH regime at higher currents (some ten A). Thus, a dominant role of edge states for the breakdown of the QHE can be excluded for samples which are large in comparison with the scattering length (i.e. for nonballistic transport). Instead, the existence of a network of percolative current paths near the breakdown is probable. For the distribution of percolative current paths in the sample, the typical length scale of inhomogeneities of the carrier density is essential. For short-range uctuations (typical length far smaller than the sample width), the percolation paths are densely distributed. This leads to a linear dependence of the breakdown current Ic on the sample width w. If long-range uctuations of the order of the sample width exist, a sublinear increase of Ic with w is observed. This behavior is related to a more eective screening of short-range uctuations, which is usually present in samples after illumination and in samples of higher mobility. The evolution of the dissipation inside the sample can be related to rather large drifting distances of the electrons. From this, it was concluded that an avalanche of excited electrons is generated close to the breakdown. As the breakdown at any location inside the sample is then in uenced by the energy gain and loss of electrons along their previous traveling, the breakdown is nonlocal on a certain length scale. This length scale was found the shorter, the higher the density of scatterers (or the lower the mobility) of the sample was. The ‘ignition’ of the breakdown, however, is determined by local enhancements of the

G. Nachtwei = Physica E 4 (1999) 79–101

energy gain by the electrons. Thus, the question as to whether the breakdown of the QHE is a local or a nonlocal phenomenon can be answered by relating the sample dimensions to the corresponding length scales of the energy excitation and relaxation processes of non-equilibrium electrons. This means, that lowmobility samples tend to a more local behavior, whereas high-mobility samples tend to behave nonlocal. Concerning the spatial evolution of the breakdown, a dierence between Silicon- and GaAs-based QH samples was found. In 2DES on GaAs, the breakdown occurred at currents being the lower, the farther the distance between electron injection and voltage probing was (avalanche heating of electrons). In contrast, a qualitatively opposite behavior was observed in SiMOSFETs. At the present state of knowledge, it is impossible to decide which of the dierences between Si and GaAs (concerning the contacts, the current distribution and the scattering) is dominant for the apparently dierent spatial evolution of the breakdown in both materials. Some theories were discussed in conjunction with the corresponding experiments. Typical features of the breakdown, as for example the steepness, the hysteresis and the bistability of the transition between quantum Hall and dissipative conduction, are well explainable in the hot-electron picture. A coexistence of dissipative (‘hot’) and nondissipative (‘cold’) regions within the sample explains the instability of the QH regime near its breakdown. It should be noted here that the hot-electron model yields reasonable results only, if the resistivity xx is locally determined not only by the thermal excitation of electrons over the Landau gap, but also by additional transport mechanisms (e.g. hopping conductivity as intra-Landau level process and inter-Landau level tunneling) in the pre-breakdown regime. Such additional contributions have been found in measurements of the resistivity xx as a function of temperature and current and are meanwhile also theoretically understood. Although most of the theories are succesfull in explaining certain aspects of the breakdown of the QHE, there is no theory available yet which were able to conclusively explain the full variety of experimental observations. A promising approach may be the assumption of a sudden delocalization of electrons in strong electric elds, as discussed by several authors

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over more than a decade. However, further work is needed to gain in particular a detailed understanding of the microscopic processes of the breakdown of the QHE. Acknowledgements The author would like to thank W. Dietsche, R.R. Gerhardts and K. von Klitzing for critical reading of the manuscript and valuable discussions. The author also acknowledges the kind permission for the presentation of data of other groups. References [1] K. von Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45 (1980) 494. [2] K. Yoshihiro, J. Kinoshita, K. Inagaki, C. Yamanouchi, J. Moriyama, S. Kawaji, Surf. Sci. 113 (1982) 16. [3] G. Ebert, K. von Klitzing, K. Ploog, G. Weimann, J. Phys. C. 16 (1983) 5441. [4] M.E. Cage, R.F. Dziuba, B.F. Field, E.R. Williams, S.M. Girvin, A.C. Gossard, D.C. Tsui, R.J. Wagner, Phys. Rev. Lett. 51 (1983) 1374. [5] L. Bliek, E. Braun, G. Hein, V. Kose, J. Niemeyer, G. Weimann, W. Schlapp, Semicond. Sci. Technol. 1 (1986) 110. [6] F. Kuchar, G. Bauer, G. Weimann, H. Burkhard, Surf. Sci. 142 (1984) 196. [7] H.L. Stormer, A.M. Chang, D.C. Tsui, J.C.M. Hwang, Proc. 17th Int. Conf. on the Physics of Semiconductors, San Francisco, 1984, Springer, Berlin, 1985, pp. 267 – 270. [8] S. Komiyama, T. Takamasu, S. Hiyamizu, S. Sasa, Solid State Commun. 54 (1985) 479. [9] T. Takamasu, S. Komiyama, S. Hiyamizu, S. Sasa, Surf. Sci. 170 (1986) 202. [10] P. Streda, K. von Klitzing, J. Phys. C. 17 (1984) L483. [11] L. Eaves, F.W. Sheard, Semicond. Sci. Technol. 1 (1986) 346. [12] S.A. Trugman, Phys. Rev. B 27 (1983) 7539. [13] S. Kawaji, H. Hirakawa, M. Nagata, Physica B 184 (1993) 17. [14] S. Kawaji, H. Hirakawa, M. Nagata, T. Okamoto, T. Fukase, T. Goto, Surf. Sci. 305 (1994) 161. [15] S. Kawaji, H. Hirakawa, M. Nagata, T. Okamoto, T. Fukase, T. Goto, J. Phys. Soc. Japan 63 (1994) 2303. [16] A. Boisen, P. BHggild, A. Kristensen, P.E. Lindelof, Phys. Rev. B 50 (1994) 1957. [17] G. Nachtwei, G. Lutjering, D. Weiss, Z.H. Liu, K. von Klitzing, C.T. Foxon, Phys. Rev. B 55 (1997) 6731. [18] G. Nachtwei, Z.H. Liu, G. Lutjering, R.R. Gerhardts, D. Weiss, K. von Klitzing, K. Eberl, Rev. B 57 (1998) 9937.

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