PHYSICAL REVIEW B
VOLUME 55, NUMBER 16
15 APRIL 1997-II
Theory of the breakdown of the quantum Hall effect
V. Tsemekhman, K. Tsemekhman, C. Wexler, J. H. Han, and D. J. Thouless
Department of Physics, University of Washington, Box 351560, Seattle, Washington 98195
~Received 22 November 1996; revised manuscript received 3 February 1997!
Breakdown of the quantum Hall effect at high values of injected current is explained as a consequence of an
abrupt formation of a metallic conduction path from one edge of the sample to the other. Such a path is formed
when regions of compressible liquid, where the long-range disorder potential is screened, get connected due to
the strong electric field. Our theory explains various features of the breakdown and numerical simulations
based on the theory yield critical currents consistent with experiment. @S0163-1829~97!50420-6#
In this paper we discuss the breakdown of the quantum
Hall effect ~QHE! at high injected currents. Observed1 as a
sudden onset of dissipation when the injected current exceeds some critical value, the breakdown of the integer QHE
has been extensively investigated since then.3 However, the
phenomenon has not been given a satisfactory explanation.
Existing theories4 of this effect include production of hot
electrons,5 inter-Landau level transitions in the high local
electric field ~due to tunneling or emission of phonons!,6,7
increase in the number of the delocalized states in the Landau level ~LL!.8 These theories either produce critical current
orders of magnitude higher than observed experimentally or
need some additional assumptions to be consistent with experiment. Furthermore, none of them can explain a rich variety of phenomena associated with the breakdown.
Experiments on the breakdown of the QHE are performed
on GaAs heterostructures or on high-quality metal-oxidesemiconductor field-effect transistor ~MOSFET! devices,
both characterized by the presence of long-range fluctuations
of the disorder potential. In GaAs systems the disorder potential due to the remote dopants has predominantly long
wavelength fluctuations (l.d.l H , where l is the typical
wavelength of the fluctuations, d is the spacer thickness, and
l H is the magnetic length!. Long-range potential fluctuations
are also present in high-mobility MOSFET devices though
their origin is not evident.9 In such systems, the QHE for
noninteracting electrons can be understood in terms of the
percolation theory.10 The plane of the two-dimensional electron gas ~2DEG! is divided into areas of completely filled
and empty LL’s. The number of completely filled LL’s in the
percolating region determines the value of the Hall conductance.
Coulomb interaction between electrons leads to the
screening of the long-range potential fluctuations, changing
this picture. As the kinetic energy is frozen out in the strong
magnetic field, screening is perfect in some areas of the system where the LL is partially occupied and is absent in the
rest of the system where LL’s are either completely filled or
empty.11 According to Refs. 11 and 12, only in the narrow
range of the filling factors around integers does incompressible region percolate through the sample, leading to the
QHE. While this statement contradicts experimental observations of the wide QH plateaux at low temperatures, it is
justified for temperatures T;1 K, which are high compared
to the energy scales of the short wavelength fluctuations of
0163-1829/97/55~16!/10201~4!/$10.00
55
the disorder potential (l,l H ) that are left unscreened and of
the residual interelectron interaction. As most experiments
on the breakdown of the QHE are performed at temperatures
of this range, one should expect that when the magnetic field
and the density of the 2DEG correspond to a filling factor
close to an integer number, an incompressible region percolates through the sample, leading to the QHE. In the percolating incompressible region electric fields are about the
same as those of the bare disorder potential12
E inc . A^ E 2bare & 5 An 0 /32p (e/ ee 0 d), where n 0 is the density
of the 2DEG. For a typical clean sample this leads to
E inc .0.1\ v c /el H , where v c is the cyclotron frequency.
There are, however, isolated compressible areas of perfectly
screened disorder potential that behave like metallic liquids.
In each of the regions, all states of the highest available
Landau level are partially occupied, and the screened potential fluctuates around the Fermi level e F with an amplitude of
the order of k B T, which is much smaller than the amplitude
of the potential fluctuations in the incompressible region. Below we consider the potential in the compressible regions to
be flat. The main idea of our theory is that at high enough
currents the insulating region separating two adjacent metallic regions suddenly breaks down due to the high electric
field leading to the connection of the regions. When such
connected regions form a metallic path percolating from one
edge of the sample to the other, abrupt onset of the dissipative regime is observed. According to our model, breakdown
of the QHE is totally different from the percolation transition. In complete analogy with the dielectric breakdown of
the insulating media,13 breakdown of the QHE should occur
at any concentration of the compressible liquid in the system,
however small it is. Therefore the predictions of the percolation theory cannot be applied to this phenomenon.
In general, incompressible region separating two adjacent
regions can be wide, and the potential in it may have a complicated structure @Fig. 1~a!#. It is clear from physical intuition, and confirmed by our numerical simulations, that the
regions closest to each other get connected first. Let us,
therefore, consider a simple model in which two metallic
regions are separated by a narrow insulating region with one
parabolic potential fluctuation characterized by the rootmean-square electric field E5E inc .0.1\ v c /el H . We will
now estimate the value of the external electric field that
brings these lakes together. In equilibrium, all electrons in
the regions have energies equal to e F ~up to T), and all
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© 1997 The American Physical Society
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V. TSEMEKHMAN et al.
FIG. 1. Distribution of compressible ~gray! metallic regions in
the QHE regime and energy diagrams. In equilibrium all regions are
at the same level ~a!. Application of a small electric field does not
produce significant current between regions, due to incompressibility of the fluid in between, QH regime ~b!. A stronger field creates
an unstable energy-occupation configuration, regions connect forming metallic conduction paths with a sudden increase in the dissipation ~c!.
electrons in the incompressible region separating them have
energies below the Fermi energy @Fig. 1~a!#. Injection of current into the 2DEG leads to the appearance of the Hall electric field throughout the sample. In the regions, as in equilibrium, all the charges are redistributed to screen the external
field. At this stage, however, the two regions are at different
potentials @Fig. 1~b!#. To estimate the critical field we assume that the external electric field is constant between the
two regions. Then, one can easily show that when the external field E 0 5 A3E inc the potential monotonically drops from
one region towards the other @Fig. 1~c!#. At this point, electrons can scatter from the incompressible region into the region, as their energies are higher than those of the partially
filled states in the region. In this simplified picture, E 0 is a
critical field at which the two regions get connected. Possible
existence of fluctuations in the separating region does not
change the estimate for the critical field significantly.
It is easy to see that the local external electric field can be
much larger than the average Hall electric field
E a v 5V H /W, where V H is the measured Hall voltage, and
W is the width of the sample. The potential is constant in the
regions, so the electric field is pushed out into the incompressible region. However, in the large system there will always be an area where two adjacent regions are separated by
the incompressible region whose width is much smaller than
the sizes of the regions. In this region the field is much larger
than the average electric field. Therefore, an average electric
field much smaller than E 0 is needed to connect the regions.
When the two regions merge, the electric field is expelled
from the former incompressible region now covered with
compressible liquid. Therefore, the electric field becomes
even stronger outside the newly formed larger region, facilitating further connections as the injected current is increased.
After the metallic region acquires some critical length in the
direction of the Hall electric field upon the increase of the
injected current, the process of further connection must take
an avalanche form. Finally, when the compressible liquid
forms a path flowing from one edge to the other, dissipation
drastically increases. It is important to notice that just before
the formation of the metallic path, the system is still in the
regime of exponentially small activated longitudinal conductivity: there are many regions on the way of the future path
that are not yet connected, and the distance between them is
as large as in equilibrium. This is the reason why the longitudinal conductivity jumps several orders of magnitude at the
critical current: the mechanism of the conductivity abruptly
switches from the activated to the metallic one.
To see how the breakdown occurs in real systems we
performed numerical simulations ~details will be published
elsewhere14!. Positively charged ions and negatively charged
electrons were assumed to occupy parallel plates of a capacitor separated by the spacer with thickness d. To avoid edge
effects these plates, each 2.5 mm32.5 mm in size, were modeled to be cut from the equally and uniformly charged infinite planes. To simulate the disorder, positive ions were
placed at random on one of these plates with the same average density as in the rest of the plane. Charge distribution on
the negatively charged plate was calculated self-consistently.
Each plate was divided into a grid of elementary squares
with the side of 250 Å. In equilibrium, the free energy of the
interacting 2DEG in the strong magnetic field and in the
smooth disorder potential ~up to an overall factor!
F @ $ n i% # 5
( (i $ k B T @~ 12n ~i N ! ! log~ 12n ~i N ! !
N50,1
1n ~i N ! logn ~i N ! # 1 ~ N\ v c 1U i ! 3n ~i N ! %
1
s
(
p l 2H e i, j<i
n in j
u ri2rju
~1!
was minimized with respect to the occupation numbers in
each square at finite temperature. In Eq. ~1! n (N)
is the occui
pation number of the Nth Landau level in the ith elementary
(1)
square; n i 5n (0)
i 1n i , ri is a radius vector for the ith
square; s is the area of the elementary square, and e is the
dielectric constant. The external potential U i is a sum of the
random disorder potential and the constant potential of an
infinite capacitor fixing the chemical potential of the system.
Coulomb interaction between electrons was taken into account in the mean-field approximation. The first two Landau
levels of the spin-degenerate system were included in our
calculations. Figure 2~a! shows the charge distribution for a
clean sample at n 51.93. Regions of compressible liquid
( n ,2, gray areas! are immersed into the percolating incompressible region ( n 52, white areas!. Injection of the Hall
current leads to the appearance of the external electric field.
If the relaxation processes are fast enough, the external field
THEORY OF THE BREAKDOWN OF THE QUANTUM HALL . . .
55
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FIG. 2. Numerical simulation for a 2.5 mm
32.5 mm sample with d525 nm, l H 510 nm,
n51.93, T50.5 K. White areas are the n52 incompressible liquid, gray areas the n,2 partially
filled compressible liquid ~rare regions with n.2
are not shown for clarity!. ~a! Equilibrium, notice
percolating incompressible region characteristic
of the QHE; ~b! Prebreakdown situation, incompressible liquid still percolates; ~c! Breakdown,
paths connect left and right edges.
averaged over a distance much larger that the typical size of
the lakes is constant across the sample and is equal to the
average Hall electric field E a v . 14 On a smaller length scale
the electric field may, of course, fluctuate. Therefore, in our
calculations the external electric field is taken to be uniform
before the response of the charges in the system. Relaxation
current is allowed to flow only in the compressible regions
and through the boundary into the lakes, i.e., only into the
areas where there is empty space for the electrons. The
charge density of the same system with external field of
200 V/cm is shown on Fig. 2~b!, showing no sign of breakdown. At 500 V/cm @Fig. 2~c!#, several regions have merged
into compressible channels parallel to the external electric
field. Evidently, this field is already higher than the critical
one and dissipation becomes important. These values are in
good agreement with experiments.1–4
Tables I and II show results of numerical simulations for
the dependence of the critical field on the filling factor and
temperature. The wide range of values for the same size
sample corresponds to the fluctuations of the breakdown
field in small systems due to different realizations of disorder. When the filling factor deviates from an integer, the size
of the regions grows and the separation between them decreases. This causes stronger enhancement of the electric
field in the incompressible region. Therefore, the critical
electric field is largest at the center of the plateau and decreases when the filling factor is moved away from the
integer.1 The same growth of the metallic regions with increasing temperature leads to lower critical fields at higher
temperatures. This dependence becomes noticeable, however, only at rather high temperatures, and the critical fields
at temperatures in the range 0.5–1 K are basically the same.
Observed localization in space2 of the region where the
dissipation sets in naturally follows from our theory. Only if
the probes measuring the longitudinal voltage appear on different banks of the path will it be possible to observe the
breakdown. As the metallic path is very narrow at the breakdown, situations when the probes are on the same side of the
path and when they are separated by the metallic region are
both likely. Connection of the regions at fields below the
TABLE I. Filling factor dependence of critical electric field at
T51 K. The third column indicates the percolating Landau level.
n
2.08
2.04
2.00
1.96
1.92
E cr ~V/cm!
LL#
330
400
350
350
200
2
1,2
1
1
1
–
–
–
–
–
360
500
400
400
300
critical one causes charge redistribution and energy relaxation. The amount of dissipated energy and the time scale of
the relaxation process vary with the size of the connecting
regions and the distance between them. This energy relaxation shows up as a broadband noise.2
The observed transient switching2 just before the breakdown and switching between different voltage levels after
the breakdown is the result of the formation and immediate
disconnection of the path at the fields just below critical or
switching between metastable states with different number of
metallic paths, respectively. This happens, since after the
path is formed, relaxation is available all over the sample
width. The system can then relax to a new state with incompressible region still percolating, or to the state with other
paths formed. Our numerical simulations show such
connection/disconnection processes.
As in any dielectric breakdown, hysteresis3 is the result of
the irreversible change of the system properties ~namely the
pattern of paths and regions! after the current was allowed to
flow through the system. Finally, the peculiar steps observed
in the magnetic field dependence of the longitudinal voltage
in the critical current regime3 show the evidence of the opening of new metallic channels when the system is moved
away from the center of the plateau. As mentioned above, in
equilibrium, when the filling factor is shifted from integer,
the size of the regions of compressible liquid increases and
their separation becomes smaller. Therefore, in the critical
regime the number of percolating narrow metallic channels
also grows, each path giving its own discrete contribution in
the voltage drop. Further study is needed to see if the individual contributions in the longitudinal voltage from these
paths are quantized.
We would like to emphasize that the nature of the phenomenon does not depend on the system size. However, the
value of the breakdown electric field is not a universal quantity and is determined by the system size and by the disorder
configuration. Any large system can be divided into smaller
pieces such that each of them still breaks down through the
avalanche process. Once an avalanche starts in any one of
them it spreads across the system. Thus the breakdown field
TABLE II. Temperature dependence of the critical electric field
at n52.00.
T ~K!
E cr ~V/cm!
LL#
1.0
2.0
3.0
3.8
350 – 400
200 – 250
100 – 200
; 120
1
1
1
1
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V. TSEMEKHMAN et al.
in the larger system is determined by the weakest piece, and
not by the average over all pieces comprising the sample.
The larger the system the higher is the probability to find a
piece with the lower critical electric field and, hence, the
smaller is, on average, the breakdown field for the whole
sample.15 Therefore, the numbers from Table I give the upper bounds for the critical field in the macroscopic samples.
This, as well as possible deviations of the external electric
field from the uniform one explain the fact that our values of
the critical field are about two times higher than the experimental results. Quantitative conclusions about the breakdown electric field dependence on the size, filling factor,
temperature, etc., as well as about its fluctuations require
1
G. Ebert, K. von Klitzing, K. Ploog, and G. Weimann, J. Phys. C
16, 5441 ~1983!.
2
M. E. Cage et al., Phys. Rev. Lett. 51, 1374 ~1983!.
3
M. Cage, J. Res. Natl. Inst. Stand. Technol. 98, 361 ~1993!, and
references therein.
4
The Quantum Hall Effect, edited by R. E. Prange and S. M.
Girvin ~Springer-Verlag, New York, 1990!, and references
therein.
5
S. Komiyama, T. Takamasu, S. Hiyamizu, and S. Sasa, Solid
State Commun. 54, 479 ~1985!.
6
O. Heinonen, P. L. Taylor, and S. M. Girvin, Phys. Rev. B 30,
3016 ~1984!.
7
L. Eaves and F. W. Sheard, Semicond. Sci. Technol. 1, L346
~1986!.
8
S. A. Trugman, Phys. Rev. B 27, 7539 ~1983!.
55
averaging over a large number of samples. This work is currently under way.
In conclusion, we propose a theory of the breakdown of
the QHE based on the existence of the compressible regions
in the inhomogeneous 2DEG. Calculations based on our
theory produce values of the critical fields that are in good
agreement with the results of measurements. We are also
able to explain the observed dependence of the critical fields
on the filling factor and temperature, as well as qualitatively
explain some of the rich features observed in the experiments.
This work was supported by the NSF Grant No. DMR9528345.
9
K. von Klitzing, in Proceedings of the 17th International Conference on the Physics of Semiconductors, San Francisco, CA,
1984, edited by James W. Chadi ~Springer-Verlag, New York,
1985!.
10
R. F. Kazarinov and S. Luryi, Phys. Rev. B 25, 7626 ~1982!.
11
A. Efros, Solid State Commun. 65, 1281 ~1988!; 67, 1019 ~1988!.
12
N. R. Cooper and J. T. Chalker, Phys. Rev. B 48, 4530 ~1993!.
13
Non-linearity and Breakdown in Soft Condensed Matter, edited
by K. K. Bardhan, B. K. Chakrabarti, and A. Hansen ~SpringerVerlag, Berlin, 1994!.
14
K. Tsemekhman, V. Tsemekhman, and C. Wexler ~unpublished!.
15
R. J. Haug, K. von Klitzing, and K. Ploog, in High Magnetic
Fields in Semiconductor Physics, edited by G. Landwehr
~Springer-Verlag, Berlin, 1989!, p. 185.