Physica E 6 (2000) 136–139www.elsevier.nl/locate/physe

QuantumHall e�ect breakdown in two-dimensional hole gases

L. Eaves ∗, S.T. Stoddart, R. Wirtz, A.C. Neumann, B.L. Gallagher, P.C. Main, M. Henini

School of Physics & Astronomy, University of Nottingham, Nottingham NG7 2RD, UK

Abstract

The breakdown of dissipationless current ow in the quantum Hall e�ect is studied for a two-dimensional hole gas at�lling factors i = 1 and 2. At high currents, the magnetoresistance curves at breakdown exhibit a series of steps accompaniedby hysteresis and intermittent noise. These are compared with similar data for electron systems and are discussed in termsof a hydrodynamic model involving inter-Landau level scattering at the sample edge. ? 2000 Elsevier Science B.V. Allrights reserved.

PACS: 73.40.Hm; 73.50.Fq; 73.61.Ey

Keywords: Quantum Hall e�ect; Breakdown; Hole gas; Carrier hydrodynamics

Despite the importance of the quantum Hall e�ect(QHE) [1] in metrology and a wealth of experimen-tal data [2–10], the mechanism for QHE breakdownremains controversial (for a recent review see Ref.[11]). In certain Hall bars used to de�ne the US resis-tance standard, the breakdown of the dissipationlessstate takes the form of a series of up to 20 steps in thelongitudinal magnetoresistance [7–9].Mechanisms proposed for breakdown include

bootstrap electron heating [12,13], percolation dueto an increased number of delocalised states [14],quasi-elastic inter-Landau level scattering (QUILLS)[5,15–18], emission of acoustic phonons due to intra-Landau level scattering [19] and the formation of

∗ Corresponding author. Tel.: +44-115-951-5136; fax:+44-115-951-5180.E-mail address: laurence.eaves@nottingham.ac.uk (L. Eaves)

compressible metallic �laments at high Hall �elds[20]. Recently, it has been suggested that, in somesamples characterised by high values of breakdowncurrent [6–9], breakdown can occur when the Lorentzforce is large enough to push conducting electronsclose to the hard wall potential at the physical edge ofthe Hall bar [21]. Here the current-carrying extendedstates correspond classically to electrons undergoingskipping orbits along the boundary wall. These stateshave a large velocity gradient, i.e. vorticity ∼ !c,where !c is the cyclotron frequency. In addition, theskipping states of the lowest Landau level (n= 0)overlap spatially with bulk states of the un�lled n= 1level so that QUILLS processes can lead to a dissi-pative current ow out of the skipping states. A phe-nomenological similarity between QHE breakdownand breakdown of laminar ow in classical uids hasalso been noted [21].

1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved.PII: S 1386 -9477(99)00080 -6

L. Eaves et al. / Physica E 6 (2000) 136–139 137

Fig. 1. Breakdown of the dissipationless QH state at i = 1 fora 2DHG, showing hysteresis (I = 9:2 �A, T = 300 mK). Arrowsindicate the direction of the �eld sweep and the numbers refer tochronological order of sweeps. The inset shows in higher resolutionthe steps observed in the B-upsweep plots.

Experimental work on QHE breakdown has fo-cussed on n-doped structures and, to our knowledge,there has been no detailed study of breakdown intwo-dimensional hole gases (2DHG) [22]. This issurprising since the hole system has properties whichdi�er signi�cantly from the 2D electron gas (2DEG).In particular, the lowest-energy 2D hole subband hasheavy hole character [23] and hence a cyclotron en-ergy, ˜!ch, several times smaller than for electrons.In addition, the hole coupling to phonons di�ersmarkedly from that of electrons. Since these di�er-ences could provide new insights into breakdown,we have investigated the e�ect for a 2DHG hetero-structure. The Hall bar sample (width w = 200 �mand length l= 400 �m) was prepared from anMBE-grown GaAs=(AlGa)As heterostructure on a(311)A GaAs substrate with Si (acceptor) doping.It has a mobility of 13.4 m2 V−1 s−1 at 300 mK andsheet density of 1:0× 1015 m−2.Fig. 1 shows typical magnetoresistance curves of

the 2DHG (plotted as Vx(B)) at high current values inthe region of the i = 1 QH plateau. By sweeping themagnetic �eld, B (or current I) into and out of the dis-sipationless state, hysteresis is observed together withclearly de�ned step-like structure in Vx(B) on the lowmagnetic �eld side of the i = 1 minimum in Vx(B). Onthe high B side of i = 1 there is a smooth increase ofVx with B. The inset of the �gure shows a detailed plotof the series of small steps observed as B is increased

Fig. 2. Logarithmic plot of some of the data in Fig. 1, showingthe wide variation of Vx in the breakdown region.

to a value su�cient to bring the current ow backto a quasi-dissipationless state. Here the step height�Vx ' 1 mV. On the B-downsweep, only larger stepheights (∼6 mV) are observed, indicating a large andsharp transition into a state of dissipative current ow.On the B downsweeps it is impossible to stabilise thedevice in the region of the �Vx ∼ 1 mV steps ob-served on the upsweeps. The data are also displayedlogarithmically in Fig. 2 to show the large variationof Vx over the steps. Although the �eld positions ofthe steps varies slightly in successive sweeps, they arequalitatively similar in all sweeps. If the current andmagnetic �eld are kept constant at certain values, in-termittent switching noise is observed between highand low voltage values, as has been observed previ-ously for 2DEGs [6–9]. The step-like breakdown isdestroyed when the 2DHG is illuminated with lightpulses.Fig. 3 shows the breakdown curves near i = 2. Hys-

teresis is again observed, but in this case the steps oc-cur on the high magnetic �eld side of the minimum.The steps and hysteresis become more pronounced athigher currents and the height �Vx ' 1 mV of the�rst step increases slightly with increasing I from 7 to8.5 �A. At 9 �A, dissipation occurs at all B aroundi = 2. The values of the Hall voltage at breakdown areVH = 0:24 V (i = 1) and VH = 0:11 V (i = 2).The hysteresis steps observed in both upsweep and

downsweep breakdown curves for our 2DHGs at i = 1are qualitatively similar to previously published datafor 2DEG systems at i = 2 [7–9]. For 2DEGs, the

138 L. Eaves et al. / Physica E 6 (2000) 136–139

Fig. 3. Hysteretic breakdown of the dissipationless QH state ati = 2 for a 2DHG at various constant currents. Solid (dotted) linesare sweep up (down).

measured voltage height �Vx of each breakdown stepon the B-upsweeps is typically 6 meV, or equivalently0.3 ˜!ce=e, where ˜!ce is the electron cyclotron en-ergy. For our 2DHG the B-upsweep step height issmaller, typically 1 mV. However, given the large ef-fective mass ratio (∼0.3) for holes and the low sheetdensity (lower B-values at i = 1 and 2) for our sam-ple, the step heights in units of ˜!ch=e are 0.5–1, i.e.similar to those for 2DEGs. The similarity of our dataand those in Refs. [7–9] suggests a common physicalprocess for breakdown in both electron and hole gases.Breakdown due to QUILLS transitions between

sidewall skipping Landau states (n= 0) and bulkLandau states (n= 1) causes a ow of current downthe velocity gradient and leads to a dissipative ef-fect which, in a hydrodynamic model is analogous toviscous drag in classical uids [21]. QUILLS can becharacterised by a kinematic viscosity ∼!c‘2B ∼ ˜=m,which combined with the large velocity gradient,(∼!c), associated with skipping states leads to adissipative drag which scales as ˜!c. This gives qual-itatively the observed scaling of the heights of thevoltage steps for 2DHGs and 2DEGs.Prior to breakdown the ratio VH=Vx is a large num-

ber, around 106 at 4.22T. As can be seen in Fig. 2,it jumps from ∼24 000 to 30 on the step at ∼4.16Tbetween the dissipative and almost dissipationlessstates on the B-downsweeps. A similar large jumpis observed for breakdown in 2DEGs [12,13]. Forthe smallest dissipative step on the B-upsweeps (seeFig. 1 inset at B ' 4:196T);�Vx ∼ 1 mV, giving

VH=Vx ∼ 240. We can write VH=Vx as wvav=�, wherevav = EHav=B is the average drift velocity of thecurrent-carrying states and EHav = VH=w is the aver-age Hall �eld. The parameter � therefore takes onthe value Vx=B and is analogous to the kinematicviscosity of a classical uid. On the dissipative sideof the �rst B-upsweep step, Vx ' ˜!c=e ' 1 mV, sothat � ' ˜=m∗ ' 3× 10−4 m2 s−1 for the hole gasat i = 1; � has a value about three orders of magni-tude smaller in the almost dissipationless state around4.2 T at the bottom of the step. With the uid dynam-ical analogy we can regard VH=Vx = wvav=� as a pa-rameter related to the Reynolds number, which takeson very high values in the dissipationless region. AsB is decreased away from 4.2 T, � increases graduallyto a critical value of 3× 10−6 m2 s−1 and then jumpsby several orders of magnitude to its dissipative value.Since our breakdown curves on the B-downsweeps

are qualitatively similar to those reported by Komi-yama and coworkers [12,13], the bootstrap carrierheating model that they have proposed for 2DEGs isalso a plausible description for the dissipative transi-tion observed in the B-downsweeps for our 2DHG.However, the bootstrap model cannot easily explainthe large number of steps (up to 20 in all) on theB-upsweeps which have been observed by Cage andcoworkers [3] in the breakdown of the QHE of 2DEGsand the multiple steps observed here for 2DHGs onthe B-upsweeps. A development of the edge break-down model may explain the origin of the steps [24].A better understanding of the hysteresis and intermit-tent noise with strange attractor behaviour observedat breakdown [25] may be achieved by further devel-opment of the hydrodynamic analogy.

Acknowledgements

This work and L.E. were supported by the EPSRC(UK).

References

[1] K. von Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45(1980) 494.

[2] G. Ebert et al., J. Phys. C 16 (1983) 5441.[3] M.E. Cage et al., Phys. Rev. Lett. 51 (1983) 1374.[4] F. Kuchar et al., Surf. Sci. 142 (1984) 196.

L. Eaves et al. / Physica E 6 (2000) 136–139 139

[5] H.L. St�ormer et al., in: J.D. Chadi, W.A. Harrison (Eds.),Proceedings of the 17th International Conference on Physicsof Semiconductors, Springer, Berlin, 1985, pp. 267–270.

[6] V.G. Mokerov et al., JETP Lett. 47 (1988) 71.[7] M.E. Cage, J. Res. Nat. Inst. Stand. Technol. 98 (1993)

361.[8] M.E. Cage, J. Res. Nat. Inst. Stand. Technol. 99 (1994) 757.[9] M.E. Cage, J. Res. Nat. Inst. Stand. Technol. 101 (1996) 175.[10] L. Bliek et al., Surf. Sci. 196 (1988) 156.[11] G. Nachtwei, Physica E 4 (1999) 79.[12] S. Komiyama et al., Solid State Commun. 54 (1985) 479.[13] S. Komiyama et al., Phys. Rev. Lett. 77 (1996) 558.[14] S.A. Trugman, Phys. Rev. B 27 (1983) 7539.

[15] L. Eaves et al., Phys. Rev. Lett. 53 (1984) 608.[16] L. Eaves et al., J. Phys. C 17 (1984) 6177.[17] O. Heinonen, P.L. Taylor, S.M. Girvin, Phys. Rev. 30 (1984)

3016.[18] L. Eaves, F.W. Sheard, Semicond. Sci. Technol. 1 (1986)

346.[19] P. Streda, K. von Klitzing, J. Phys. C 17 (1986) L483.[20] V. Tsemekhman et al., Phys. Rev. B 55 (1997) R10 201.[21] L. Eaves, Physica B 256–258 (1998) 47.[22] M. Henini et al., Appl. Phys. Lett. 65 (1994) 2054.[23] S. Hill et al., Physica B 211 (1995) 440.[24] L. Eaves, submitted for publication.[25] G. Boella et al., Phys. Rev. B 50 (1994) 7608.