arXiv:cond-mat/9909314v2 [cond-mat.mes-hall] 22 Sep 1999
Thermodynamic Signature of a Two-Dimensional Metal-Insulator Transition
S. C. Dultz and H. W. Jiang
Department of Physics and Astronomy, University of California at Los Angeles, Los Angeles, CA 90095
(February 1, 2008)
We present a study of the compressibility, κ, of a two-dimensional hole system which exhibits a
metal-insulator phase transition at zero magnetic field. It has been observed thatdκ
the critical density for the metal-insulator transition. Measurements also indicate that the insulating
phase is incompressible for all values of B. Finally, we show how the phase transition evolves as the
magnetic field is varied and construct a phase diagram in the density-magnetic field plane for this
dpchanges sign at
73.40.Hm, 71.30+h, 72.20.My
Recently, we are seeing a growing body of experimen-
tal evidence supporting a metal-insulator quantum phase
transition in a number of two-dimensional electron1and
hole systems2where coulomb interactions are strong
and particle mobility is quite high. These experiments
are of interest because of the prevailing theory of non-
interacting particle systems in two dimensions which
states that only insulating behavior should be seen at
all densities for even the smallest amount of disorder in
the system.3In order to further understand the nature of
the unusual phase transition, it is important to study the
thermodynamic properties near the transition. One par-
ticular question is whether there is any signature for the
phase transition in a thermodynamic measurement. The-
oretically, within the framework of Fermi liquid, one does
not expect any qualitative change in the thermodynamic
properties.4On the other hand, recent theories for strong
interacting systems5,6have predicted that there should
be profound consequences in thermodynamic measure-
In this paper, we address this issue by presenting
a measurement of one of the fundamental thermody-
namic quantities: the thermodynamic density of states
(or equivalently, the compressibility) of a strongly in-
teracting two-dimensional hole system (2DHS). We re-
port evidence that the compressibility measurement in-
deed provides an unambiguous signature for the metal-
insulator transition (MIT). The insulating phase is in-
compressible. Furthermore, we show that the phase tran-
sition at B = 0 is intimately related to the quantum Hall
state to insulator transition for the lowest Landau level
in a finite magnetic field.
Traditionally, to obtain the density of states (DOS) or
the compressibility of a 2D electron system, capacitance
between the 2D electrons and the gate is measured.7–10
This capacitance can be modeled as the geometrical ca-
pacitance in series with a quantum capacitance.
quantum capacitance per unit area cq is related to the
One major drawback of this method is that, in the low
magnetic field limit, the quantum capacitance is much
larger than the geometric capacitance. For two capaci-
tors in series, small uncertainty in the geometric capaci-
?by cq= e2?dn
?, or to κ by cq= n2e2κ where
µ is the chemical potential and n is the carrier density.
tance can lead to a large quantitative error (even a sign
error) in the extracted κ. In a pioneering experiment by
Eisenstein et al.11, the penetration of the electric field
through 2D electrons in one well was detected by the
2D electrons in the other well by using a double quan-
tum well sample. This penetration field measures the
screening ability of the electrons which was shown to be
inversely proportional to κ. For the present study, we ex-
tend the field penetration method to a more conventional
heterostructure with only a single layer of carriers.
The wafer used for the experimental devices was a p-
type MBE grown GaAs/AlxGa1−xAs single heterostruc-
ture. A 400˚ A Al.45Ga.55As undoped spacer was used
to separate the Be-donors (1 × 1018cm−3) from the
2DHS. Just below the 2DHS is a 5000˚ A undoped GaAs
buffer layer and beneath that, a 5000˚ A Al.72Ga.28As
layer which was used as the etching stopping layer for
substrate removal. The mobility of this particular sam-
ple was roughly 123,000 cm2/V-s with a hole density of
p = 2.60 × 1011cm−2. The device for the compress-
ibility measurement was fabricated by sandwiching the
2DHS between two metallic electrodes. To form the top
electrode (2500˚ A from the 2DHS), NiCr was evaporated
on the surface of the sample. To form the bottom elec-
trode, the GaAs substrate was totally removed so that
another NiCr electrode could be placed on the bottom
in close proximity to the 2DHS (10000˚ A). Details of the
substrate removal can be found elsewhere12. To measure
the compressibility, we applied a 10 mV AC excitation
voltage Vacto the bottom electrode (Gate 1), as shown
in Fig. 1a. A DC voltage Vg was superimposed to vary
the carrier density. The 2DHS was grounded to screen
the electric field from the bottom electrode. A lock-in
amplifier was used to detect the penetrating electric field
as current from the top electrode (Gate 2) to ground.
By modeling the system as a distributed circuit, both
the quantum capacitance Cq and the resistance Rs of
the channel for the 2DHS could be extracted individu-
ally based on the measured values for the in-phase Ix
and 90ophase Iy current components. The model for
the circuit is an extension of the two wire transmission
line problem14. The exact expression for current is as
FIG. 1. (a) Diagram of our experimental setup with an
active sample area of 1 mm2. (b) Typical trace of Cq which is
proportional to compressibility, and (c) Rswhich is a measure
of dissipation as a function of the applied DC voltage at three
different AC frequencies: 43.2 Hz, 100 Hz and 200 Hz. All
data was taken at B = 0 and T = 4.2 K.
C1+ C2+ Cq
C1+ C2+ CqRs;
In this expression, ω is the frequency of our excitation
voltage on gate 1 and Vacis the amplitude. C1(136 pF)
and C2(541 pF) are the geometric capacitances between
gate 1 and gate 2 and the 2DHS, respectively. This is a
complex equation with the real and imaginary parts cou-
pled with respect to Rs and Cq. These values were ob-
tained by solving both equations simultaneously.13In the
low-frequency limit, Ix is directly proportional to −Rs,
the dissipation of the 2DHS, and Iy is proportional to
1/Cq, the inverse compressibility.
Fig. 1b and 1c show typical traces of the Cqand Rsas
a function of the gate voltage at various different frequen-
cies. It is apparent that there is no frequency dependence
over the entire range of gate voltage. Furthermore, we
found that, for frequencies up to 200 Hz, Ixand Iyare in-
deed directly proportional to −Rsand 1/Cqrespectively.
It can then be assumed that the divergence in both chan-
nels is not due to depletion of the channel since this would
produce a noticeable frequency dependence on the high
gate voltage side of the maximum in Cq.
Fig. 2 shows both the inverse compressibility and the
dissipation signals as a function of the density at B = 0
for different temperatures ranging from 0.33 K to 1.28
K. Two main features are immediately noticeable for the
1/κ channel. First, 1/κ is negative for high densities and
becomes more negative as the density is decreased as oth-
ers have seen in diluted electron11and hole systems15.
The negative compressibility is known to be due to the
strong exchange energy contribution to the total chem-
ical potential16. Secondly, a sharp turn-around occurs
at p = 5.5 × 1010cm−2asdκ
dpchanges sign. As the den-
sity is further reduced, 1/κ becomes positive and diverges
rapidly (extraction of Cqshows that κ rapidly approaches
zero from infinity in the low density limit13). Focusing
now on the dissipation channel, one can find a temper-
ature independent crossing point which we believe to be
the critical density for the MIT in this sample (also can
be seen in transport measurements). Although the qual-
FIG. 2. Ix and Iy vs density for five temperatures at an
excitation frequency of 100 Hz: blue- 0.33 K, green- 0.56
K, black- 0.82 K, orange- 1.02 K, red- 1.28 K. The crossing
point of the five dissipation channel curves corresponds to the
metal-insulator phase transition at B = 0. The minimum in
the inverse compressibility channel occurs at the same density
of p = 5.5 × 1010cm−2.
itative shape of Iyas a function of gate voltage was seen
by others11the minimum in this signal was never rec-
ognized as the MIT. Notice the temperature dependence
on the high density side of the crossing point. As the
temperature increases, Ixis getting more negative which
means that the 2DHS is getting more resistive (metallic
behavior). The opposite is seen on the low density side of
this crossing point where the characteristic temperature
dependence is that of an insulator. This crossing point
occurs at the point where∂κ
∂pchanges sign precisely at the
minimum of 1/κ. Therefore, we believe there is a clear
signature of the metal-insulator phase transition at B = 0
in this thermodynamic measurement. Since κ tends to-
ward zero in the insulating phase, the data also suggests
that the insulating phase is incompressible. Theoreti-
cally, it has been argued that the insulating phase is a
Wigner glass phase which is incompressible for strongly
Having identified that the signature of the MIT has
been seen in κ at B = 0, we would like to see how this
critical point evolves as the magnetic field is increased.
We found 1/κ vs. p to be independent of magnetic field
up to about 1.5 T. Fig. 3 shows how 1/κ and Rsevolve
as a function of density in a higher magnetic field where
variations in both Ix and Iy are more dramatic due to
the presence of Landau levels. The data shown was taken
at 3 T and at five different temperatures from 0.33 K
to 1.27 K just as in the B = 0 case. As seen in the
figure, Iy shows a local maximum in an integer filling
factor where the compressibility, which is also propor-
tional to the DOS, tends to zero between two adjacent
Landau levels. The DOS is zero only at T = 0, so the
FIG. 3. Ix and Iy vs density for five temperatures at B = 3
T and excitation frequency, 100 Hz: blue- 0.33 K, green- 0.56
K, black- 0.81 K, orange- 1.05 K, red- 1.27 K. The crossing
point now becomes the DOS peak of the first Landau level.
Peaks in the Iy channel represent regions where the DOS is
going to zero. The Ix channel corresponds to the set of lower
peaks get more pronounced as one goes to lower tem-
peratures. Conversely, Iy reaches a minimum when the
DOS reaches a peak in a Landau level center where the
delocalized states (states which exist at the Landau level
centers where the electronic wavefunction is expected to
extend spatially throughout the sample) reside. Mean-
while, Ixalso undergoes oscillations. It is important to
note here, Ixin the high field is not proportional to ρxx
but rather to a constant term plus a term that goes like
1/σxxin the limit of high conductivity7. Because σxxis
also going to zero in the quantum Hall liquid regime, this
gives us a peak in Ixwhen the DOS tends to zero. At 3
T, we also see a temperature independent point in both
channels and in the same place where the inverse com-
pressibility reaches a minimum. In this case however, the
temperature independent point marks the phase bound-
ary between the insulator and the ν = 1 quantum Hall
state. On the insulator side, 1/κ diverges just as in the
B = 0 case and so one cannot distinguish between the
insulating phase at B = 0 versus the insulator at finite
As we vary the magnetic field from 0 to 12 T, we keep
track of where the 1/κ minimums (i.e. phase boundaries)
are occurring in density. Fig. 4 is a phase diagram in the
density-magnetic field plane. We can see how the DOS
FIG. 4. A map of the phase diagram in the p − B plane
for the 2DHS at T = 0.3 K. The dots are data taken from
graphs like figure 3 where the local minimums exist in Iy(peak
in DOS) and the temperature independent crossing points
exist in Ix. The uncertainty in peak position is approximately
5 × 109cm−2in density. The lines are calculated positions
for Landau level centers in the density-magnetic field plane
from standard non-interacting particle picture.
peaks are evolving as one goes from the high field to the
low field regime. There are number of interesting fea-
tures. First, the phase boundary for the lowest Landau
level flattens out as B is reduced. If we can assume that
the delocalized states are occurring in the DOS maxi-
mum, we can draw the conclusion that the delocalized
states do not float up as B tends toward zero for the
strongly interacting 2DHS. The rsvalue which is the ra-
tio of the Coulomb energy to the Fermi energy, reaches
about 20 near the MIT. For non-interacting electron sys-
tems with low rsvalues, the lowest delocalized states are
found to “float up” in energy17as B → 0. This energy di-
vergence means that the Fermi energy can only be tuned
through localized states which do not contribute to cur-
rent but give only insulating behavior at B = 0 (i.e.-
resistivity diverges as temperature decreases). The cur-
rent observation is consistent with the studies that have
been done in the past for the 2DHS through transport
measurements.18,19This implies that for the 2DHS there
is indeed a metallic regime (as shown) in the thermody-
namic limit which does not exist in the phase diagrams
for the 2DES. We would like to note here the data in
Fig. 4 was taken from another 2DHS sample which was
cut from the same wafer but had slightly lower density
and we see the lowest delocalized state terminating at a
density of roughly 4.0×1010cm−2at B = 0 rather than
a density of 5.5×1010cm−2as seen in Fig. 2. Although
we have only clearly seen the phase boundary between
the metallic and insulating states in low magnetic field
as we cross the critical density, we can see from our data
that the metallic phase exists all the way to our highest
density of 2.6 × 1011cm−2at low magnetic fields. We
have also seen that this metallic phase is preserved to at
least 1 T in our sample. The shaded vertical band marks
an ill-defined region between the metallic phase and the
quantum Hall phases. This region needs to be explored
in more detail at lower temperatures when the quantum
Hall plateaus are better resolved. The insulating regime
is also shaded in the diagram for clarity. Secondly, the
data suggests that the insulator to ν = 1 transition is re-
lated to the B = 0 MIT based upon the similar behavior
of the compressibility. In fact, the same argument has
been made based on the tracking of temperature inde-
pendent crossing points in a transport experiment.19
In summary we have found, using an improved elec-
tric field penetration technique, the compressibility of
a strongly interacting hole system undergoes a qualita-
tive change traversing the MIT. More importantly, we
have observed that the local maximum in compressibil-
ity occurs precisely at the critical density for the MIT at
B = 0. The divergence of the compressibility shows the
insulating phase is incompressible for all values of mag-
netic field. The phase transition at B = 0, in fact, evolves
into the quantum Hall to insulator transition at high B.
We believe these observations reported here cannot be
explained by simple non-interacting models. Theoretical
analysis for this strongly interacting system is called for
to understand the thermodynamic properties presented
The authors would like to thank S. Chakravarty, Q.
Shi, J. Simmons, S. Sondhi, and C. Varma for helpful
discussions, and B. Alavi for technical assistance. This
work is supported by NSF under grant # DMR 9705439.
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