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Magnetic-field-induced transition from an Anderson insulator to a quantum Hall conductor
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surface science
,,::;j I( :)y ../, ,I..,.. ., .,.,.,,, ::::. :.>7.:.+:, :.:,.i.::j:‘,,..,;: ~:‘,‘:~:,iiii;‘:z%i:~~~~~~:.~~:~~~.~,:. .:* ELSEVIER
Surface
Science
305 (1994) 120-123
Magnetic-field-induced transition from an Anderson insulator to a quantum Hall conductor H.W. Jiang
*%“,I. Glozman
‘, C.E. Johnson
‘, S.T. Hannahs
h
‘I Department of Physics, Unil,ersity of‘ Californiu ut Los Angeles, Los Angeles, CA 90024, USA ” Massachusetts Institute qf Technoloa, Francis Bitter National Magnet Lahoratoq, Cambridge, MA 02139, USA (Received 19 April 1993; accepted
for publication
IS June 1993)
Abstract We report the first experimental observation of a magnetic-field induced transition from an Anderson insulator to a quantum Hall conductor in a two-dimensional electron gas (2DEG) system. With sufficiently strong disorder. the 2DEG is an Anderson insulator with a divergent resistance at B = 0. The 2DEG is transformed into a quantum Hall conductor when the lowest Landau level is filled. This magnetic-field-induced delocalization is consistent with a recently proposed global phase diagram for the quantum Hall effect.
1. Introduction It is now generally believed that in the absence of a magnetic field all the states of a 2DEG are localized in the presence of any amount of disorder [l]. The effect of an applied magnetic field, however, is still a subject of much debate and has become central to most discussions of localization in quantum Hall systems. The presence of extended states is, of course, implied by the observation of the quantum Hall effect (QHE). However, experimentally, the localization length in a 2DEG is estimated from 5 = exp(r2h/e2p), here p is the resistivity, to be of the order of a kilometer even in very “dirty” devices. This means that because of its finite size (typically of order 1 mm), a 2DEG system is effectively delocalized at B = 0. It has been suggested that even if the system
* Corresponding
author.
0039.hO2X/94/$07.00 0 1994 Elsevier SSDI 0039.6028(93)EO664-G
Science
were completely localized (with no extended states below the Fermi level E,) in zero magnetic field, it might still be possible to observe the QHE since the system may be subject to magnetic-field-induced delocalization. In this regard, a scaling argument has been developed by Khmelnitskii [2] and a gedanken experiment proposed by Laughlin [3]. The idea is that extended states are assumed to be conserved and, as a consequence of changing disorder or magnetic field, may float or sink relative to localized states and possibly past E,. A schematic illustration of this possibility is suggested in Fig. 1. Recently, a novel global phase diagram by Kivelson, Lee and Zhang [4] predicted a series of direct Anderson insulator (a special type called a Hall insulator [5] which has a non-divergent Hall resistance) to quantum Hall conductor transitions as a consequence of delocalization induced by a change in the magnetic field. Motivated by this theoretical phase diagram, we have conducted an
B.V. All rights reserved
H. W. Jiang et al. /&face
experiment on a system which can serve as an Anderson insulator defined by having all of its electronic states below E, localized due to random potential fluctuations. Experimentally, although the transport properties of the two-dimensional electron gas (2DEG) in the strong localization regime have been explored as early as in the 70’s by utilizing silicon inversion layers [6], the delocalization transition has not been observed.
2. Results The samples used in the present work were modulation-doped GaAs/AlGaAs heterostruc-
Science 305 (1994)120-123
121
tures fabricated by molecular beam epitaxy. The active 2D layer was formed on top of a Si-doped AlGaAs layer without the conventional undoped spacer to ensure a large random fluctuation of the impurity potential. A Hall bar pattern was etched out by standard lithographic techniques and an aluminum gate was evaporated onto its surface. As expected, depletion was achieved by increasing the gate voltage (and thus the disorder) as RX, rose by several orders of magnitude (see inset of Fig. 4). That is, the localization length decreased dramatically as the screening of the random potential was reduced by depletion. The details of tuning the effective disorder by changing the density is described elsewhere [7]. As shown in the figure, at low Vo (high density), the 2DEG is nearly metallic, characterized by a temperature independent resistance as T + 0. At high Vo (low density), R,, shows an exponential divergence R,, as T + 0 at zero magnetic field and in fact follows the well-known Mott law R = exp(To/T)‘/3 as expected for an Anderson insulator, as shown in Fig. 2. From this temperature dependence and Mott’s law [7], ye estimate a localization length of about 1200 A, corresponding to a strongly localized system. To determine how magnetic field affects localization, several R,, versus B traces were taken in the depleted (strongly localized) regime at
n=1.66x10"/cm2 106-
Energy lfiw, I Fig. 1. The density of states versus energy for three different filling factors. For simplicity, spin splitting is assumed to be unresolved. Singular Landau levels are broadened by disorder into bands of extended states (light shading) separated by tails of localized states (dark shading). For v > 2 (low magnetic field) and in the presence of sufficient disorder, there are no extended states below the Fermi level; as a result, no QHE is observed and the system is an insulator. The case v = 2 corresponds to a quantum Hall conductor; dissipation is precluded by an energy gap between E, and the next available extended state into which scattering can take place. For v < 2 (high magnetic field), there are once again no extended states below E, and the system is a regular insulator.
z ET II m ", d lo5
:/:
Fig. 2. The logarithmic of the resistance vs. T-‘/3 at a density of 1.66 x 10”/cm2. The data show insulator-like behavior for zero magnetic field as R diverges as T + 0.
n=1.66xlo"/Cm.
10’ - -2.lV
l_._.~____~.
lo6 80nlK
L-. -2.05v
6. -
~
K
-c_-.
_
1
105::
-
.
l
. .
. 1021
’
0.0
’ 8
1
0.2
’
’
0.4
1
’
0.8
0.6
1 .o
T (K) 550rnK
Fig. 1. Semi-logarithmic
w
different
values
proache
1,,,,1,,,,/,,,,,,,,,,,,(
1 2 3 4 MAGNETIC FIELD Fig. 3. Top: Hall resistance. ICYC’I filling
factor
minimum
magnetic
V
ahove and helow,
nearly classical. A well rewlvctd plateau R, ( versus
(T)
is developed
Bottom: the longitudinal
field at three different
R, ~ i\
insulator. Critical
which wparatcs
temperature
b;; and at II=
at
7. I:or
divrrgej
a\ 7
+ 0 as expected
IOI
For Iowcr
value\
ol / I ;, j. R,,
ap
;I quantum
IHall
a5 expected
for
disorder- is expected
to be at
I ;, = ~ 2.tl.i
the two regions. The inset shows a cliwr-
gent R, I it\ 1V,, 1 goes ;I lunction
R, ,
7~1.0 as 7‘+I)
conductor.
For field5 away from the Landau
of I’ = 2. both
I’ = 2 at low-temperature.
5
vci-sus
of the gate voltage
higher vulur~ of 1C;; /. an /\ndcr~m
R,,
plot of
up. implying an increase of diwrtlrr
ii\
of I;,.
around
resistance
temperatures.
is seen at I’ = ? w hich deepens ax the temperature
A is
lowered.
different temperatures. For fields away from v = 2, both above and below, variable-range hopping behavior was observed similar to the zero-field cast and the Hall resistance is approximately classical. K, , = B/net (consistent with the behavior of a Hail insulator). On the other hand. at v = 2 (defined by a well resolved plateau in p,, with the value of (h/e’)/2) a deep minimum was observed which dropped continuously as the temperature was lowered. This minimum is thus consistent with the onset of a quantum Hall conducting phase. It is also interesting to point out that a well defined “crossing point” of the three curves can be seen around B = 2.6 T. This suggests the possibility of a critical field as shown in Fig. 4 by the horizontal transition and certainly warrants further investigation. In a further study of the quantum Hall conductor at v = 2, we were able to map out the
portion of the global phase diagram containing the point of critical disorder on the insulatorquantum Hall conductor phase boundary (Fig. 5).
INSULATING
PHASE
Fig. 5. A simplified
global
Hall effect (adapted
from Ref. [4] and modified).
1x1. relevance insulating
are
the
two
to H quantum
on by a change
phase diagram type\
of
a small
circle
in disorder in the
a transition
Hall conducting
case of the
splitting ix not conaidercd.
Of pnrticufrom
an
phase, one brought
and the other
magnetic field (or filling factor). Critical
for the quantum
hy a change
in
disorder is marked hy
former.
Note
that
spin
H. W. Jiang et al. /Surface
It turns out that a reasonable parametrization of disorder is achieved by the zero field localization length to (estimated from the temperature dependence in the variable-range hopping regime). As we decrease to by increasing VG, the width in B of the quantum Hall conductor decreases continuously until finally the QHE disappears altogether at the critical value V, = -2.05 V (5 = 800 A). For 1V, I> 2.05 V, a pronounced minimum in R,, survives but diverges as T + 0 as shown in Fig. 4. We suggest that this criticality may be described as a percolation threshold. Lakes of mobile 2D electrons (quantum Hall liquids) become intermingled with islands of localized ones. Source-drain conduction becomes likely when the amount of liquid begins to exceed the amount of land as the disorder is reduced. These observed “critical” behaviors are surprisingly similar to the superconductor-insulator transition found in superconducting films [8].
3. Conclusion The ability to tune disorder has of late become a powerful experimental tool in the study of localization phenomena in 2DEG systems. In particular, we have demonstrated a magneticfield-induced transition from an Anderson insulator (Hall insulator) to a quantum Hall conductor in a disordered 2DEG system. Consistent with
Science 305 (1994) 120-123
123
the global phase diagram, two types of transitions were observed, one brought on by a change in the magnetic field and the other by a change in the disorder. Furthermore, above a critical disorder no transition appears to be possible and the insulating phase persists, irrespective of the magnetic field.
4. Acknowledgements
We would like to think S. Kivelson and S. Feng for useful discussions. H.W.J. acknowledges support from the Alfred Sloan Foundation.
5. References [ll E. Abrahams,
P.W. Anderson, D.C. Licciardello and T.V. Ramarkrishnan, Phys. Rev. Lett. 42 (1979) 673. 121D.E. Khmelnitskii, Phys. Lett. 106 (1984) 182. [31 R.B. Laughlin, Phys. Rev. Lett. 52 (1984) 2304. (41 S. Kivelson, D.H. Lee and SC. Zhang, Phys. Rev. B 46 (1992) 2223. and K.B. Efetov, Phys. Rev. B 44 (1991) [51 0. Viehweger 1168. [61 For early experimental works see, for example, D.C. Tsui and S.J. Allen, Phys. Rev. Lett. 32 (1974) 1200, A.B. Fowler, Phys. Rev. Lett. 34 (1975) 15, M. Pepper, Philos. Mag. B 37 (1978) 83. [71 H.W. Jiang, C.E. Johnson and K.L. Wang, Phys. Rev. B 46 (1992) 12830. A.F. Hebard and R.R. [81 See, for example, M.A. Paalanen, Ruel, Phys. Rev. Lett. 69 (1992) 1604.
,:,,.
i ~
..,.,
::
. .. ..
. .. . ,. ::;
y:
.:,:
..:>
5
;,.
..:.
:. j . .
.:j ..:
surface science
,,::;j I( :)y ../, ,I..,.. ., .,.,.,,, ::::. :.>7.:.+:, :.:,.i.::j:‘,,..,;: ~:‘,‘:~:,iiii;‘:z%i:~~~~~~:.~~:~~~.~,:. .:* ELSEVIER
Surface
Science
305 (1994) 120-123
Magnetic-field-induced transition from an Anderson insulator to a quantum Hall conductor H.W. Jiang
*%“,I. Glozman
‘, C.E. Johnson
‘, S.T. Hannahs
h
‘I Department of Physics, Unil,ersity of‘ Californiu ut Los Angeles, Los Angeles, CA 90024, USA ” Massachusetts Institute qf Technoloa, Francis Bitter National Magnet Lahoratoq, Cambridge, MA 02139, USA (Received 19 April 1993; accepted
for publication
IS June 1993)
Abstract We report the first experimental observation of a magnetic-field induced transition from an Anderson insulator to a quantum Hall conductor in a two-dimensional electron gas (2DEG) system. With sufficiently strong disorder. the 2DEG is an Anderson insulator with a divergent resistance at B = 0. The 2DEG is transformed into a quantum Hall conductor when the lowest Landau level is filled. This magnetic-field-induced delocalization is consistent with a recently proposed global phase diagram for the quantum Hall effect.
1. Introduction It is now generally believed that in the absence of a magnetic field all the states of a 2DEG are localized in the presence of any amount of disorder [l]. The effect of an applied magnetic field, however, is still a subject of much debate and has become central to most discussions of localization in quantum Hall systems. The presence of extended states is, of course, implied by the observation of the quantum Hall effect (QHE). However, experimentally, the localization length in a 2DEG is estimated from 5 = exp(r2h/e2p), here p is the resistivity, to be of the order of a kilometer even in very “dirty” devices. This means that because of its finite size (typically of order 1 mm), a 2DEG system is effectively delocalized at B = 0. It has been suggested that even if the system
* Corresponding
author.
0039.hO2X/94/$07.00 0 1994 Elsevier SSDI 0039.6028(93)EO664-G
Science
were completely localized (with no extended states below the Fermi level E,) in zero magnetic field, it might still be possible to observe the QHE since the system may be subject to magnetic-field-induced delocalization. In this regard, a scaling argument has been developed by Khmelnitskii [2] and a gedanken experiment proposed by Laughlin [3]. The idea is that extended states are assumed to be conserved and, as a consequence of changing disorder or magnetic field, may float or sink relative to localized states and possibly past E,. A schematic illustration of this possibility is suggested in Fig. 1. Recently, a novel global phase diagram by Kivelson, Lee and Zhang [4] predicted a series of direct Anderson insulator (a special type called a Hall insulator [5] which has a non-divergent Hall resistance) to quantum Hall conductor transitions as a consequence of delocalization induced by a change in the magnetic field. Motivated by this theoretical phase diagram, we have conducted an
B.V. All rights reserved
H. W. Jiang et al. /&face
experiment on a system which can serve as an Anderson insulator defined by having all of its electronic states below E, localized due to random potential fluctuations. Experimentally, although the transport properties of the two-dimensional electron gas (2DEG) in the strong localization regime have been explored as early as in the 70’s by utilizing silicon inversion layers [6], the delocalization transition has not been observed.
2. Results The samples used in the present work were modulation-doped GaAs/AlGaAs heterostruc-
Science 305 (1994)120-123
121
tures fabricated by molecular beam epitaxy. The active 2D layer was formed on top of a Si-doped AlGaAs layer without the conventional undoped spacer to ensure a large random fluctuation of the impurity potential. A Hall bar pattern was etched out by standard lithographic techniques and an aluminum gate was evaporated onto its surface. As expected, depletion was achieved by increasing the gate voltage (and thus the disorder) as RX, rose by several orders of magnitude (see inset of Fig. 4). That is, the localization length decreased dramatically as the screening of the random potential was reduced by depletion. The details of tuning the effective disorder by changing the density is described elsewhere [7]. As shown in the figure, at low Vo (high density), the 2DEG is nearly metallic, characterized by a temperature independent resistance as T + 0. At high Vo (low density), R,, shows an exponential divergence R,, as T + 0 at zero magnetic field and in fact follows the well-known Mott law R = exp(To/T)‘/3 as expected for an Anderson insulator, as shown in Fig. 2. From this temperature dependence and Mott’s law [7], ye estimate a localization length of about 1200 A, corresponding to a strongly localized system. To determine how magnetic field affects localization, several R,, versus B traces were taken in the depleted (strongly localized) regime at
n=1.66x10"/cm2 106-
Energy lfiw, I Fig. 1. The density of states versus energy for three different filling factors. For simplicity, spin splitting is assumed to be unresolved. Singular Landau levels are broadened by disorder into bands of extended states (light shading) separated by tails of localized states (dark shading). For v > 2 (low magnetic field) and in the presence of sufficient disorder, there are no extended states below the Fermi level; as a result, no QHE is observed and the system is an insulator. The case v = 2 corresponds to a quantum Hall conductor; dissipation is precluded by an energy gap between E, and the next available extended state into which scattering can take place. For v < 2 (high magnetic field), there are once again no extended states below E, and the system is a regular insulator.
z ET II m ", d lo5
:/:
Fig. 2. The logarithmic of the resistance vs. T-‘/3 at a density of 1.66 x 10”/cm2. The data show insulator-like behavior for zero magnetic field as R diverges as T + 0.
n=1.66xlo"/Cm.
10’ - -2.lV
l_._.~____~.
lo6 80nlK
L-. -2.05v
6. -
~
K
-c_-.
_
1
105::
-
.
l
. .
. 1021
’
0.0
’ 8
1
0.2
’
’
0.4
1
’
0.8
0.6
1 .o
T (K) 550rnK
Fig. 1. Semi-logarithmic
w
different
values
proache
1,,,,1,,,,/,,,,,,,,,,,,(
1 2 3 4 MAGNETIC FIELD Fig. 3. Top: Hall resistance. ICYC’I filling
factor
minimum
magnetic
V
ahove and helow,
nearly classical. A well rewlvctd plateau R, ( versus
(T)
is developed
Bottom: the longitudinal
field at three different
R, ~ i\
insulator. Critical
which wparatcs
temperature
b;; and at II=
at
7. I:or
divrrgej
a\ 7
+ 0 as expected
IOI
For Iowcr
value\
ol / I ;, j. R,,
ap
;I quantum
IHall
a5 expected
for
disorder- is expected
to be at
I ;, = ~ 2.tl.i
the two regions. The inset shows a cliwr-
gent R, I it\ 1V,, 1 goes ;I lunction
R, ,
7~1.0 as 7‘+I)
conductor.
For field5 away from the Landau
of I’ = 2. both
I’ = 2 at low-temperature.
5
vci-sus
of the gate voltage
higher vulur~ of 1C;; /. an /\ndcr~m
R,,
plot of
up. implying an increase of diwrtlrr
ii\
of I;,.
around
resistance
temperatures.
is seen at I’ = ? w hich deepens ax the temperature
A is
lowered.
different temperatures. For fields away from v = 2, both above and below, variable-range hopping behavior was observed similar to the zero-field cast and the Hall resistance is approximately classical. K, , = B/net (consistent with the behavior of a Hail insulator). On the other hand. at v = 2 (defined by a well resolved plateau in p,, with the value of (h/e’)/2) a deep minimum was observed which dropped continuously as the temperature was lowered. This minimum is thus consistent with the onset of a quantum Hall conducting phase. It is also interesting to point out that a well defined “crossing point” of the three curves can be seen around B = 2.6 T. This suggests the possibility of a critical field as shown in Fig. 4 by the horizontal transition and certainly warrants further investigation. In a further study of the quantum Hall conductor at v = 2, we were able to map out the
portion of the global phase diagram containing the point of critical disorder on the insulatorquantum Hall conductor phase boundary (Fig. 5).
INSULATING
PHASE
Fig. 5. A simplified
global
Hall effect (adapted
from Ref. [4] and modified).
1x1. relevance insulating
are
the
two
to H quantum
on by a change
phase diagram type\
of
a small
circle
in disorder in the
a transition
Hall conducting
case of the
splitting ix not conaidercd.
Of pnrticufrom
an
phase, one brought
and the other
magnetic field (or filling factor). Critical
for the quantum
hy a change
in
disorder is marked hy
former.
Note
that
spin
H. W. Jiang et al. /Surface
It turns out that a reasonable parametrization of disorder is achieved by the zero field localization length to (estimated from the temperature dependence in the variable-range hopping regime). As we decrease to by increasing VG, the width in B of the quantum Hall conductor decreases continuously until finally the QHE disappears altogether at the critical value V, = -2.05 V (5 = 800 A). For 1V, I> 2.05 V, a pronounced minimum in R,, survives but diverges as T + 0 as shown in Fig. 4. We suggest that this criticality may be described as a percolation threshold. Lakes of mobile 2D electrons (quantum Hall liquids) become intermingled with islands of localized ones. Source-drain conduction becomes likely when the amount of liquid begins to exceed the amount of land as the disorder is reduced. These observed “critical” behaviors are surprisingly similar to the superconductor-insulator transition found in superconducting films [8].
3. Conclusion The ability to tune disorder has of late become a powerful experimental tool in the study of localization phenomena in 2DEG systems. In particular, we have demonstrated a magneticfield-induced transition from an Anderson insulator (Hall insulator) to a quantum Hall conductor in a disordered 2DEG system. Consistent with
Science 305 (1994) 120-123
123
the global phase diagram, two types of transitions were observed, one brought on by a change in the magnetic field and the other by a change in the disorder. Furthermore, above a critical disorder no transition appears to be possible and the insulating phase persists, irrespective of the magnetic field.
4. Acknowledgements
We would like to think S. Kivelson and S. Feng for useful discussions. H.W.J. acknowledges support from the Alfred Sloan Foundation.
5. References [ll E. Abrahams,
P.W. Anderson, D.C. Licciardello and T.V. Ramarkrishnan, Phys. Rev. Lett. 42 (1979) 673. 121D.E. Khmelnitskii, Phys. Lett. 106 (1984) 182. [31 R.B. Laughlin, Phys. Rev. Lett. 52 (1984) 2304. (41 S. Kivelson, D.H. Lee and SC. Zhang, Phys. Rev. B 46 (1992) 2223. and K.B. Efetov, Phys. Rev. B 44 (1991) [51 0. Viehweger 1168. [61 For early experimental works see, for example, D.C. Tsui and S.J. Allen, Phys. Rev. Lett. 32 (1974) 1200, A.B. Fowler, Phys. Rev. Lett. 34 (1975) 15, M. Pepper, Philos. Mag. B 37 (1978) 83. [71 H.W. Jiang, C.E. Johnson and K.L. Wang, Phys. Rev. B 46 (1992) 12830. A.F. Hebard and R.R. [81 See, for example, M.A. Paalanen, Ruel, Phys. Rev. Lett. 69 (1992) 1604.