close

Вход

Забыли?

вход по аккаунту

?

Microwave conductivity of magnetic field -induced insulating phase of the two -dimensional hole system in GaAs

код для вставкиСкачать
INFORMATION TO U SERS
This manuscript has been reproduced from the microfilm master. UMI films the
text directly from the original or copy submitted.
Thus, some thesis and
dissertation copies are in typewriter face, while others may be from any type of
computer printer.
The quality o f this reproduction is dependent upon the quality o f the copy
subm itted.
Broken or indistinct print, colored or poor quality illustrations and
photographs, print bleedthrough, substandard margins, and improper alignment
can adversely affect reproduction.
In the unlikely event that the author did not send UMI a complete manuscript and
there are missing pages, these will be noted.
Also, if unauthorized copyright
material had to be removed, a note will indicate the deletion.
Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning
the original, beginning at the upper left-hand comer and continuing from left to
right in equal sections with small overlaps. Each original is also photographed in
one exposure and is included in reduced form at the back of the book.
Photographs included in the original manuscript have been
reproduced
xerographically in this copy. Higher quality 6” x 9” black and white photographic
prints are available for any photographs or illustrations appearing in this copy for
an additional charge. Contact UMI directly to order.
Bell & Howell Information and Learning
300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA
800-521-0600
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
NOTE TO USERS
This reproduction is the best copy available
UM I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
i
I
Microwave Conductivity o f
i
i
Magnetic Field Induced Insulating Phase o f
i
| the Two-Dimensional Hole System in GaAs
Chi-Chun Li
A DISSERTATION
PRESENTED TO THE FACULTY
OF PRINCETON UNIVERSITY
IN CANDIDACY FOR THE DEGREE
OF DOCTOR OF PHILOSOPHY
RECOMMENDED FOR ACCEPTANCE
BY THE DEPARTMENT OF
ELECTRICAL ENGINEERING
NOVEMBER 1999
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
UMI Number:
9944604
UMI Microform 9944604
Copyright 1999, by UMI Company. All rights reserved.
This microform edition is protected against unauthorized
copying under Title 17, United States Code.
UMI
300 North Zeeb Road
Ann Arbor, MI 48103
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
© Copyright by Chi-Chun Li, 1999
All rights reserved
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Abstract
|
This thesis describes the study on the high magnetic field insulating phase o f two-
i
|
dimensional (2D) hole systems, using a novel microwave conductivity measurement
|
technique.
i
While disorder potentials are known to be essential in producing such
l
|
insulating behavior, carrier-carrier interaction effects are expected to be important in this
1
!
phase as well, as it terminates the series o f the fractional quantum Hall states, whose
i
!
existence has long been understood as due to carrier-carrier interactions. In particular, it
i
i
could be the elusive magnetically-induced Wigner crystal (WC), the ground state o f a
j
clean system o f charged carriers,
i
Utilizing a coplanar waveguide to couple microwave power into the 2D system,
j
I
i
!
we measured the power attenuation caused by the 2D holes, from which the conductivity
I
pronounced resonance of conductivity versus frequency in the GHz range.
|
resonance gradually develops from a featureless background near the quantum Hall state-
can be inferred. In the high magnetic field insulator we observed a surprisingly
The
insulator transition, and becomes larger and sharper as the magnetic field is increased.
We studied the evolution o f the resonance with several experimental parameters. The
j
role o f carrier-carrier interactions is established by the density dependence o f the
j
resonance, and the temperature at which the resonance disappears (« 230 mK) is
consistent with the melting o f a WC. The effects of microwave power and a concomitant
DC electric field were also studied. It is evident that the resonance can be related to the
"pinning mode" o f a disordered WC, although so far there is no theory that can explain
all the features and dependences observed in our experiment.
iii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Acknowledgements
i
I
i
Thanks to the wonderful people, my life at Princeton has been a most valuable
j
|
and enjoyable experience. Above all, I am grateful to my thesis advisor, Prof. Daniel
Tsui, for his guidance and encouragement.
A conversation with him was always
I
instructive and inspirational.
I would like to thank Prof. Mansour Shayegan for his advice and the state-of-theart samples that are so crucial for this work.
I thank Prof. Yuan Huei Chang, Prof.
i
j
Stephen Lyon and Prof. Ravin Bhatt for painstakingly reading this thesis and giving me
!
j
|
their valuable comments.
I am especially indebted to Dr. Lloyd Engel.
His vast knowledge and the
i
I
willingness to share it were all that an apprentice could ask for.
I appreciate the
i
opportunity o f working with Dr. Yasunao Katayama, Dr. Dan Shahar and Dr. Jongsoo
!
I
Yoon, who taught me many a trick in the laboratory.
!
graduate students.
I
bizarre hours.
I would not forget my fellow
They were always around when I needed help, even at the most
Finally, I wish to express my deepest gratitude to my beloved wife, Chen-Jung,
|
and my family. This thesis could not have been done without their love and support.
i
i
f
IV
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Contents
Abstract.................................................................................iii
Acknowledgements.............................................................. iv
CHAPTER 1
INTRODUCTION..............................................................1
1.1 Motivation........................................................................................................................ 1
1.2 Wigner Crystal................................................................................................................2
1.3 Existing Experiments on the Magnetically-Induced Insulating P hase.....................4
1.3.1
Reentrant insulating phase................................................................................... 4
1.3.2
Nonlinear I- V and noise generation.................................................................... 5
1.3.3
Giant dielectric constant, inductive anomaly and washboard oscillation
1.3.4
Cyclotron resonance and photoluminescence results....................................... 6
1.3.5
Radio frequency and microwave absorption......................................................7
6
1.4 Structure o f the Thesis................................................................................................... 8
CHAPTER 2
SAMPLES AND EXPERIMENTAL METHOD
9
2.1
The Samples.................................................................................................................... 9
2.2
Coupling Microwave into the 2DHS: the CoplanarW aveguide............................. 11
2.3
Microwave Power M easurement............................................................................... 20
2.4
The Setup inside the Dilution Refrigerator............................................................... 24
2.5
Errors o f Measurement and Conversion o f the Conductivity.................................27
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 3
CONDUCTIVITY RESONANCE IN THE HIGH
MAGNETIC FIELD INSULATING PHASE.............................................28
3.1
Introduction.................................................................................................................. 28
3.2
Microwave frequency conductivity...........................................................................28
3.3
The 5-Dependence o f the Resonance....................................................................... 37
3.4
Interpretation of the resonance: pinning mode o f Wigner crystal.........................43
CHAPTER 4
THE CONDUCTIVITY RESONANCE UNDER
VARIOUS EXPERIMENTAL CONDITIONS..........................................49
4.1
Overview......................................................................................................................49
4.2
Carrier-carrier interaction and the microwave resonance...................................... 49
4.2.1
Introduction.......................................................................................................... 49
4.2.2
Results o f the densitydependence experiment................................................. 50
4.2.3
Discussion............................................................................................................ 60
4.3 Dependence on temperature, microwave power and in-plane DC electric fie ld .. 71
4.3.1
T dependence data............................................................................................... 71
4.3.2
Pav dependence data............................................................................................74
4.3.3
Effect of in-plane DC electric field.................................................................... 78
4.3.4
Discussion............................................................................................................81
CHAPTER 5
CONCLUSIONS
.........................................................90
VI
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
List o f Figures
Figure 2.1
Epitaxial sequence o f the 2DHS samples.........................................................10
Figure 2.2
Top view of the sample for microwave frequency measurement. The black
area indicate evaporated A1 film. Ohmic contacts are made on edges o f the sample,
outside the CPW area...................................................................................................... 12
Figure 2.3
Cross-sectional view of the sample.
The inset lists the thickness o f
substrates and silicon nitride layer o f the samples....................................................... 14
Figure 2.4
Lumped-element equivalent circuit for (a) a unit-length transmission line (b)
a lossless transmission line
(c) the CPW loaded with 2DHS.
rs’ is the sheet
resistance o f the CPW metal, and Yh is the admittance associated with the 2DHS. 16
Figure 2.5Circuit model for a unit length 2DHS coupled to the CPW.................................18
Figure 2.6
Schematic of the microwave detector............................................................. 22
Figure 2.7
Working principle of the bolometric detector. The microwave power P mw
and the dc heater power Poe are modulated at 0.5 Hz and out o f phase from each
other.
A feedback loop adjusted the powers so that the temperature becomes
constant...............................................................................................................................23
Figure 2.8
picture of the bolometer detector, with its housing removed, (a) Top view:
The sapphire evaporated with NiCr film resistor is to the left. The meander line is
the 4kfi DC heater, and the small rectangle resistor next to the white ceramic beam
is the 50Q microwave absorber, (b) Bottom view: The thinned Speer resistor is
attached on the back side o f the sapphire....................................................................... 25
Figure 2.9
The complete measurement setup...................................................................... 26
vii
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 3.1
DC magnetoresistance (fixx) o f the M259B5. ns = 5.4 x 1010 cm'2. The T
dependence indicates insulating behavior for 3.55 T < B < 6.25 T and B > 6.97 T
(0.62 > v > 0.36 and v< 0.32)......................................................................................... 29
Figure 3.2
DC diagonal conductivity (cr**) converted from the 30 mK i?** vs. B in Figure
3.1 and a classical Rxy......................................................................................................31
Figure 3.3
Detector output signal as a function o f B field for two different carrier
densities. The horizontal dotted line is guide to the eye.............................................. 32
Figure 3.4
0.2 GHz Refer**) vs. B for M259B7. The 3.0 x 1010 cm'2.trace is shifted up
by 4 pS................................................................................................................................ 34
!
|
Figure 3.5
Refer**) vs. B for M259B5 measured at several fixed frequencies. T=50mK
.................................................................................................................................35
|
Figure 3.6 (a) Detector output signal v s ./f o r M259B7.at B=13 T and T - 25 mK. (b)
!
Refer**) vs./obtained from the two traces in (a)............................................................36
j
Figure 3.7 Refer**) v s ./ o f M259B5 at several constant B. T= 50 m K............................. 39
|
Figure 3.8 Summary of the resonance features for M259B5. T = 50 mK.
;
(a) f pk (solid
circle) and S (open square) vs. B. (b) apk (solid triangle) and A f (open triangle) vs.
B- (c) Q (diamond) vs. B ................................................................................................. 40
Figure 3.9 Refer**) v s . / o f M230B6 at several B fields, T= 30mK...................................41
Figure 3.10 Summary of the resonance features for M230B6. T = 30 mK. (a) f Pk (solid
;
circle) and S (open square) vs. B. (b) crpk (solid triangle) and A f (open triangle) vs.
B- (c) Q (diamond) vs. B.................................................................................................. 42
!
Figure 3.11 Illustration o f the harmonic oscillator model. L is the domain size, u, the
|
|
displacement.......................................................................................................................44
viii
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 3.12 Pinning frequency (coo) as obtained by (2nfpk coc) from the M259B7 data in
figure 3.8............................................................................................................................46
Figure 4.1
Detector output signal v s ./f o r M259B7. B = 13 T, T = 25 mK Four traces
are shown for Vbg =0, and four for Vbg = 200 V. Among them, two at each V/,g were
measured before, and two after, a high Vbg (400V) was applied to deplete the holes..
................................................................................................................................ 51
Figure 4.2
DC magnetotranport o f M259B7 at different backgate biases.
The inset
shows ns asa function o f Vbg............................................................................................ 52
Figure 4.3
Re(oix) v s ./a t several ns (in 1010 cm'2) o f M259B7. B =13 T, T= 25 mK.. 53
Figure 4.4
Re(crxr) v s ./a t several ns (in 1010 cm'2) o f M230B6. 5 =13 T, 7’ = 25 mK. 55
Figure 4.5 f pk vs. ns for M259B7 (solid symbols) and M230B6 (open symbols), each
with two cooldowns (shown as circles
and squares). B = 13 T,T = 25
mK. The
lines are least square fits to f pk °c ns'r. ..............................................................................56
Figure 4.6
f pk vs. B o f M259B7 at ns = 5.5 and 3.0 x 1010 cm'2..................................... 57
Figure 4.7
Summary of ns dependence o f resonance features forM259B7 (solid
symbols) and M230B6 (open symbols), B =13 T, T= 25 mK. (a) f pk (circles) and Af
(downward triangles) vs. ns
(b) Q (diamonds) vs. ns
(c) S (squares)
and <Jpk
(upward triangles) vs. ns............................................................................................59
Figure 4.8 S / f pk vs. ns for M259B7 and M230B6. 5 =13 T, T = 25 mK. The solid is a
linear fit to the data and its slope is 94% o f the FL prediction......................................64
Figure 4.9 L I a v s. ns computed from the observed f pk data using Eq.(4-4). B = 13 T, T
= 25 mK.......................................................................................................................66
Figure 4.10 Harmonic oscillator fit to the data o f M259B7 (Fig. 4 .3).......................69
ix
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 4.11 Parameters o f harmonic oscillator fit vs. ns.
The symbols represent
experimental data and the lines (solid ones for M259B7, dotted ones for M230B6)
are results o f the fit. In (b), Q vs. ns is plotted against the left axis and r vs. ns is
plotted against the right axis. The dashed line in (c) represents n = ns.....................70
Figure 4.12 Re(t7xc) v s ./a t several T's for M259B7. B =13 T, ns= 5.4 x 1010 cm'2
72
Figure 4.13 Summary of T dependence o f the resonance features for M259B7. ns = 5.4 x
1010 cm'2, B =13 T. (a) f Pk (circles) and Af (downward triangles) vs. T (b) Q vs. T
(c) S (squares) and crpk (upward triangles) vs. T............................................................ 73
Figure 4.14 Summary of resonance features for M259B7, T < 200 mK. ns = 5.4 x 1010
cm'2, 5 = 1 3 T. (a) f pk (circles) and A f (downward triangles) vs. T (b) Q vs. T (c) S
(squares) and apk (upward triangles) vs. T ....................................................................75
Figure 4.15 Re(o*t) v s ./ o f M259B7 measured with different Pav. B = 13 T, T= 25 mK..
...............................................................................................................................76
Figure 4.16 Summary o f Pav dependence o f the resonance for M259B7. B = 13 T, T= 25
mK. (a) f pk (circles) and A f (downward triangles) vs. Pav
(squares) and arpk (upward triangles) vs. Pav
(b) Q VS. Pav
(c) S
The solid lines in (b) and (c) are guide
to the eye............................................................................................................................ 77
Figure 4.17 Pav dependence for M259B7. B = 15 T. Temperatures are 25, 105, 155 and
210 mK for solid black, hollow, solid gray, and dot-centered symbols, respectively.
The solid lines are guide to the eye.................................................................................79
Figure 4.18 Two terminal DC I-V characteristics o f M230B6 at 13 T, 25 mK.
The
arrows indicate the jdc s at which the resonance data are shown in Figure 4.19........ 80
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 4.19 ReCo^) v s . / o f M230B6 measured in the presence o f in-plane DC fields. B
= 13 T, 7 = 2 5 m K .............................................................................................................82
Figure 4.20 Summary o f the effect o f DC current density on the resonance o f M230B6.
B = 13 T, T = 25 mK. (a) f pk (circles) and A f (downward triangles) vs. jdc (b) Q vs.
jdc (c) S (squares) and apk (upward triangles) vs .jdc.....................................................83
Figure 4.21 Resonance features as a function o f in-plane EdCfield for M230B6. 5 = 1 3
T, T - 25 mK. (a) f pk (circles) and A f (downward triangles) vs .Jdc (b) Q vs .jdc (c) S
(squares) and crpk (upward triangles) vs .jdc.................................................................... 84
Figure 4.22 Comparison o f two Re(crxt) v s ./tra c e s o f M259B7 at 15 T: one measured
with Pav = 500 pW at T = 25 mK, the other with a smaller Pav (80 pW) but at higher
7 /105 mK).........................................................................................................................87
1
I
t
xi
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Chapter 1 Introduction
i
i
j
i
1.1 Motivation
|
In an intense perpendicular magnetic field, two-dimensional electron systems
;
(2DES) and hole systems (2DHS) display a variety o f remarkable physical phenomena.
The magnetic field, B, drives a low disorder system through several distinct phases,
j
including those exhibiting the spectacular integer [1] and fractional [2] quantum Hall
i
|
effect (QHE). The integer effect (IQHE) can be explained in terms o f single-particle
quantization with disorder-induced localization [3]. The fractional effect (FQHE), on the
j
|
other hand, is the manifestation of a highly correlated, many-body state, in which
Coulomb interaction between the charged carriers is essential [4]. While the IQHE is
j
i
more tolerant o f disorder, the FQHE is only observed in samples o f highest quality [5].
i
Both phases have been extensively studied experimentally and theoretically [6, 7, 8].
|
Less well understood is the magnetic-field-induced insulating phase, which is
observed in all 2D systems to terminate the series of QHE phases at high enough B.
i
For
a clean system, the ground state is predicted to be a Wigner crystal (WC) [9] at
sufficiently small Landau level filling factor v (v = nsh/eB, where ns is the areal density
o f carriers and h the Planck constant) [10].
In this phase, the zero-frequency (DC)
conductivity is expected to vanish as a result o f WC pinning by residual disorder [11].
|
j
!
On the other hand, if the disorder potential overwhelms the Coulomb interaction, carriers
are localized without forming a WC. One would surmise a WC state should be attainable
!
in the extremely high mobility 2D systems available nowadays. However, it is difficult
i
iI
i
i
ji
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
to assess the effect of disorder in real systems, and without conclusive experimental
evidence that WC exists in semiconductor heterostructures (see section 1.3), the nature of
the insulating phase remains unresolved.
Since the insulating behavior, which is a DC phenomenon, can result from vastly
different mechanisms, it is essential to study its frequency response. In this thesis, we
|
present the data o f broadband microwave frequency measurements on the conductivity o f
I
the high mobility 2DHS in GaAs/AlGaAs heterojunctions. At temperatures (7) below
250 mK, a s"harp resonance with substantial conductivity is observed in the high B
!
|
insulating phase. We established the role o f Coulomb interactions in producing such
|
resonance, and compared the results with models o f a pinned WC.
i!
1.2 Wigner Crystal
|
With both disorder and interaction present in the system, the high B insulating
|
phase can be extremely complicated and sample-specific. However, most theoretical and
|
I
|
experimental studies consider such a phase, especially in samples showing FQHE, in
|
specific to the high B, low T conditions appropriate for our experiments, and discuss
!
possible effects of "real-world" disorder on an ideal system.
terms o f a WC perturbed by disorder potential.
Below we describe the WC model
In 1934, Wigner proposed that when the Coulomb energy dominates the kinetic
energy, a degenerate Fermi gas o f interacting electrons would crystallize to minimize the
Coulomb repulsion [9]. For a 2D system, the Fermi energy E f (E/? = ( n h 2/m*)ns, where
|
m is the effective mass) is proportional to ns, while the interaction energy is proportional
I
to nsm , so crystallization occurs at sufficiently low ns.
|
2
j
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The first observation o f an electron solid was actually in a classical system of
electrons on the surface o f liquid 4He, where ksT » £>. In this system, electron ordering
is confirmed by identifying the phonon modes, coupled with the ripplon modes o f the He
|
surface, specific to a WC [12, 13]. In this regime, crystallization is achieved by lowering
|
T to reduce kinetic energy, or by increasing ns to enhance Coulomb interaction, and thus
|
is in a different regime from the original quantum electron solid proposed by Wigner.
The 2DES and 2DHS realized in semiconductor heterostructures are natural
I
i
i
candidate systems to observe a quantum WC because o f their high purity, as manifested
by the observation o f FQHE. However, ns in real samples is generally too high for a WC
!
|
state. It was predicted that crystallization would occur in a strong B field for the ns region
j
|
|
|
I
!
j
where a WC is impossible at zero B field [10], and that a WC terminates the FQHE at
low enough v [14, 15].
Many experiments have been conducted to search for this
magnetic-field-induced WC.
How could a crystalline structure be identified? Observation o f periodicity by x-
j
j
ray diffraction pattern would be direct evidence. Unfortunately, such experiments are
j
|
prohibitively difficult as the light with suitable wavelength (WC lattice constant a =
|
1000A for ns = 2.3 x 1010 cm'2) would be completely attenuated by the GaAs host lattice
before reaching the 2D system.
It would also suffice to observe the rigidity o f the
system, represented by the existence o f a shear mode in the phonon spectrum.
Several
experiments have followed this route, but the results and the comparison to WC phonons
i
have been controversial (see the following section).
i
|
i
3
i
|
1i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A disorder potential, if weak enough so that the WC is not completely destroyed,
modifies the response of the WC to experimental probes. It causes the WC to have finite
correlation length, or domain size (L), and is expected to pin the WC, making the system
insulating against small DC electric field [16, 17, 18, 19, 20]. A large enough electric
field depins the WC, so the DC conduction exhibits threshold behavior. The WC phonon
spectrum is also modified: a gap at zero wavevector (q) is formed, due the finite size o f L.
The data in this thesis is generally compared to the resulting low q "pinning mode", o f
which more details will be discussed in sections 3.4 and 4.2.3.
1.3 Existing Experiments on the Magnetically-Induced Insulating
Phase
In this section we briefly summarize important observations from prior
experiments conducted on the magnetically-induced insulating phase. A thorough review
can be found in ref. [21].
1.3.1
Reentrant insulating phase
In 1990, Jiang et al. reported the observation o f a well-defined v = 1/5 FQHE in
2DES, accompanied by insulating phases at B both above and below the 1/5 state [22]. It
was argued that the existence of the FQHE implies that e-e interactions are also important
in the surrounding insulating phases, especially the one at lower B.
The reentrant
behavior can be expected from the competition o f FQHE and WC as the ground state o f
the system: while the energy o f the WC is a smooth function o f v, the energy o f the
4
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
FQHE shows "cusps" at fractional v's [23], and thus they may intersect both above and
below v = 1/5.
j
i
i
!
Later, the reentrant insulating phase was observed in 2DHS around the v = 1/3
FQHE [24]. The occurrence o f a similar behavior at a much larger v for the holes was
j
|
suggested to result from the much larger effective mass [25, 26]: in GaAs, m* = 0.067 mo
I
for electrons and 0.37 mo for holes [27]. A larger m* means that the Coulomb energy,
e2/4n££o(nnsy ia, becomes relatively more important as compared to the kinetic energy,
A
:
i
E f = (nft !m )ns.
The ratio o f the two energies is captured by the dimensionless
j
parameter rs=(Ttns)/ciB*, where as = 4n££oh2 /m*e2 is the effective Bohr radius. For the
j
same ns = 4 x 1010 cm'2, rs « 3 and 15 for 2DES and 2DHS, respectively. The energy of
|
the WC relative to the FQHE is expected to be lower in the 2DHS than in the 2DES case,
j
so the 1/3 reentrant behavior for 2DHS is consistent with the scenario that a FQHE-WC
transition occurs at a larger v. This was the reason that in our experiments 2DHS were
i
j
chosen over the 2DES.
!
i
i
i
i
I
!
|
1.3.2
Nonlinear I- V and noise generation
Nonlinear DC I-V and accompanying noise generation were observed in
insulating phases surrounding the 1/5 FQHE o f 2DES [28, 29, 30, 31, 32, 33]. As a
threshold field, E j, is exceeded, both the DC current and a low frequency noise start
increasing rapidly.
The nonlinear I-V and the generation o f noise above E t were
attributed to depinning of the WC by the large electric field. At higher T (typically 100 200 mK) both phenomena disappear, consistent with melting o f the WC.
Ii
j
|
However,
5
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
|
different studies report E j differing by two orders o f magnitude. Er is found to be 10 V/m
|
i
|
i
|
in ref. 33 and 0.1 V/m in other references [30, 31,32]. It has been suggested that the
smaller E r comes from generation and separation o f dislocation pair through quantum
tunneling [34].
|
1.3.3
Giant dielectric constant, inductive anomaly and washboard oscillation
j
By measuring both the real part and the imaginary part o f conductivity in the
frequency range 10 - 100 MHz Y. P. Li et al. reported a giant dielectric constant
.1
!
( s > 104) for the insulating phase at v = 0.21 [35]. Similar to the observation in the case
o f pinned charge density wave (CDW) systems, the large s results from a collective mode
weakly pinned by disorder [36]. Experiments at lower frequencies (0.2 —3 MHz) in the
!
presence of a constant DC field [37, 38] revealed an inductive response o f the insulator
!
biased above the conduction threshold, which again found counterpart in the CDW case
[36, 39, 40]. The DC current dependence o f the response can be related to the washboard
i
!
oscillation expected for a sliding electron solid, and is arguably the strongest evidence for
the existence o f a WC in the insulating phase.
1.3.4
Cyclotron resonance and photoluminescence results
Cyclotron resonance experiments conducted on 2DES observed blue shift and
narrowing in the extreme quantum limit (v <1) [41, 42]. The dependences on v and on
substrate bias were found to be consistent with a pinned WC model.
Subsequent
experiments [43, 44] identified an additional peak for v < 1/9, which was originally
j
attributed to either WC or new correlated phases, but was identified later as a spin effect
|
[45]. In photoluminescence experiments on 2DES samples with extra Be-doped layer
!
iii
i
6
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[46, 47], a second luminescence line was observed for v <0.28, and was attributed to a
solid phase.
Experiments on standard single heterojunctions [48] revealed more
complicated structures, whose T dependence and B dependence were utilized to map out
a liquid-solid phase boundary.
1.3.5
Radio frequency and microwave absorption
By measuring the derivative o f absorption in the frequency range o f 0.05-1.4
GHz, Andrei et al. observed multiple resonances for the v < 1/5 insulating phase [49]
which they attributed to the magnetophonon modes o f a WC. These results have been
controversial [50, 51]. Later the sharp resonances were ascribed to instrumental artifact,
and a broader feature was taken as manifestation of a q m mode o f disordered WC [33],
an interpretation again questioned by others [52, 53].
Paalanen et al. determined the conductivity o f 2DES in high mobility
GaAs/AlGaAs heterostructures by measuring the attenuation and velocity o f a surface
acoustic wave (SAW) [54, 55]. A broad resonance was observed at around 1 GHz for v<
1/5, with a well defined critical temperature above which the resonance disappear. These
features were found to be consistent with a pinned WC model.
Our experiment measures the real part o f the longitudinal conductivity, Re(crxx), in
a frequency range similar to the above two experiments. Quantitative ReCcr**), rather than
the derivative-of-absorption obtained in refs. 49 and 33, can be determined from the data.
As compared to the SAW experiment, which measures conductivity at discrete
frequencies limited by the SAW dispersion and the harmonics set by the transducer, we
can sweep frequency continuously from 0.2 to 10 GHz and obtain Re(cr^) in the low q
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
limit. With the zero q pinning mode o f disordered WC predicted to be in this range [56,
|
iI
j
i
j
phase. And indeed, a single conductivity resonance was observed both in 2DES [58] and
I
in 2DHS [59]. The studies of the 2DHS resonance constitute the body o f this thesis.
57], our method is valuable for studying the nature o f the magnetically-induced insulating
I
1.4 Structure of the Thesis
The thesis is organized as follows. Chapter 2 describes the experimental setup,
with the main focus on the novel microwave conductivity measurement method.
In
|
chapter 3 we present conductivity data taken on 2DHS at progressively higher B, and we
I
report the observation of a pronounced conductivity resonance in the insulating phase for
j
v < 0.3. Evolution of the resonance with carrier density ns, temperature T, microwave
I
:
I
|
are compared to models of WC pinning mode.
j
suggests future experiments on the high B insulating phase.
i
electric field Emw and a simultaneous DC field Edc is studied in chapter 4, and the results
Chapter 5 provides a summary and
|
1
8
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Chapter 2 Samples and Experimental Method
2.1 The Samples
The samples are p -type, high mobility GaAs/AlGaAs heterojunctions grown by
molecular beam epitaxy (MBE). Figure 2.1 shows the layered structure of the samples.
Using (311)A GaAs substrates, Si dopants in the 5 doping layers become acceptors [60]
and the 2D holes are realized in the triangular potential well at the GaAs/AlGaAs
interface.
The AlGaAs spacer separates the ionized dopants from the 2DHS, therefore
scattering is reduced and the 2DHS mobility is greatly enhanced.
Nevertheless, the
ionized dopants still present a source of potential fluctuations to the 2DHS.
Other
sources of disorder include unintentional residual impurities (e.g., carbon acceptors) near
the interface, and the roughness o f the GaAs/AlGaAs interfaces.
While the disorder
potential from the remote dopants is usually considered to be a long-ranged, weak
fluctuation, the other two could be short-ranged compared with the inter-carrier spacing.
Various aspects o f our experimental results will be discussed in the context o f the pinned
Wigner crystal models, in which these disorder characteristics play an important role.
The structures of the two MBE wafers, M259 and M230, are quite similar. There
are only slight differences in the doping density o f the Si <S-doping layer, spacer thickness
and the Al concentration. The structural parameters o f wafers are listed in Figure 2.1. The
carrier density, ns, for M259 is around 5.5 x 1010 cm'2, and for M230, 4.2 x 1010 cm'2.
The mobility, ju, for M259 is 3.5 x 105 cm2/V-s, and for M230, 5.0 x 10s cm2/V-s.
9
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
M230
x=0.3
M259
x=0.35
120 A
240 A
2 x 1012 cm'2
1950 A
8 x 1 0 " cm'2
1200 A
GaAs
AlxGai-xAs
Si <5-doping layer
AlxGai-xAs
9 x 1 0 " cm'2
Si (5-doping layer
1150 A
AlxGai.xAs
5 monolayers AlAs
10,000 A
20 periods
GaAs
GaAs/AlAs superlattice
(311)A undoped GaAs substrate
Figure 2.1
Epitaxial sequence o f the 2DHS samples.
10
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Samples from these wafers have been used in the study o f the reentrant insulating phases
j
occurring around the v= 1/3 fractional quantum Hall effect [24], which were interpreted
j
as consistent with Wigner crystal formation.
i
I
I
I
2.2 Coupling Microwave into the 2DHS: the Coplanar Waveguide
Our goal is to obtain the conductivity o f the 2DHS in the microwave frequency
range. At low frequency one can use diffused Ohmic contacts, but at high f the inter­
electrode capacitance, combined with the resistance o f the 2D system, leads to cutoff
frequencies in the MHz range or even lower.
|
j
So we need a coupling scheme to
incorporate the 2D system.
The present method employs a coplanar waveguide (CPW) [61], fabricated on the
front surface o f the sample, to couple the microwave power into the 2D system. Figure
|
2.2 shows the top view o f the sample with the CPW structure. The sample is a 3 mm x 5
|
mm piece with the longer edge along the [233] (high mobility) direction. The CPW
metal is a 3500 A-thick A1 film, patterned by standard photolithography and lift-off
process. It consists of a meandering center conductor with width a = 45 pm, and two side
!
planes. The width of the gaps ( W) between the conductors is 30 pm. Ohmic contacts are
placed on the edges of the sample, outside the CPW pattern.
I
During the measurements the two side conductors are kept grounded, with
|
microwave voltage applied on the central strip. The sole purpose o f the meanders is to
'
i
I
increase the electric length o f the CPW, and hence the measurement sensitivity. The
wide ground planes in-between adjacent meanders reduce the coupling to ~ 35 dB [62].
!
11
|
i
ii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Ohmic contacts
3mm
Figure 2.2
Top view o f the sample for microwave frequency measurement.
The
black area indicate evaporated A1 film. Ohmic contacts are made on edges o f the sample,
outside the CPW area.
12
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
In the analysis below, the CPW will be treated as a straight line with total length (d) equal
to that o f the uncoiled meandering line, which is 28 mm.
Figure 2.3 shows the cross section o f the sample on a plane perpendicular to the
microwave propagation. The 2DHS is 3700 A beneath the sample surface. There are two
samples from the M259 wafer, which will be referred to as M259B5 and M259B7 in the
thesis. M259B5 has no silicon nitride overlay on the sample, and its substrate was not
|
thinned (600 pm thick). Both M259B7 and M230B6, the sample taken from the M230
wafer, are thinned down to 300 pm (± 10%), and with 2400 A (± 5%) silicon nitride
j
!i
|
I
;
I
deposited by plasma enhanced chemical vapor deposition on the front.
The basic idea of the measurement lies in the fact that finite conductivity o f the
2DHS introduces attenuation as microwave propagates through the CPW.
Described
;
below are the analysis of such a transmission line system and the conversion procedure
j
used to obtain ReCcr^) of the 2DHS.
I
First we consider the case when the 2DHS is absent. Because the CPW belongs
!
j
to a family o f so called inhomogeneously filled waveguide, there exists no simple TEM
j
mode. However, for the frequency range up to 10 GHz, the wavelength is much longer
|
I
i
than the CPW width (a+2W « 105 pm). In this case, a quasi-TEM approximation can be
applied [63]. The distributed capacitance along the propagation direction is calculated,
which then determines the propagation constant K and the characteristic impedance Zc of
i
!
the transmission line.
i
j
i
|
!
13
ii
|
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
30 pm
44.7 um
30 um
A1
Si3N4
2DHS
M259B5
M259B7
M230B6
Figure 2.3
h fum)
600
300
300
Si3N4 (A)
0
2400
2400
Cross-sectional view o f the sample.
The inset lists the thickness o f
substrates and silicon nitride layer of the samples.
14
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The CPW thus can be modeled as a lumped-element network o f series impedance
t
i
Z and shunt admittance Y (Figure 2.4 (a)). Suppose microwave propagates in the x!
!
j
direction, the voltage and current in such a network are given by [64]
!
(2-1)
j
V(x) = V+e~Kx + V_eKx;
I(x ) = (F+e~** - F _ e ^ ) / Z c ;
where the propagation constant k and the characteristic impedance Zc are
i
(2-2)
K
=
Z c = 4 z
/ Y
;
Without the 2DHS and other loss mechanism, the differential element circuit is
reduced to a series inductance per unit length L' and a shunt capacitance per unit length C
i
j
(Figure 2.4 (b)).
For a lossless CPW with infinitely extended ground planes and
|
substrate, the characteristic impedance Zo = ^ L '/C ' is determined only by the ratio a/W
j
and the dielectric constant o f the substrate, er. For our CPW geometry, with sr = 13, C -
j
j
1.76 x 10’10 F/m, L' = 4.41 x 10'7 H/m, and Zo = 50Q [61], which is to match the coaxial
j
i
|
cable used in our measurement system. Deviation o f Zo due to finite ground plane width,
!
finite substrate thickness and the existence o f backgate metal, is ~ 2% [62].
Loading the 2DHS introduces an extra admittance, Yfi (Figure 2.4 (c)), which is a
|
|
|
function o f the conductivity o f the hole layer. Since d/W »
;
well-defined E field along the y-direction on the 2DHS plane, only the diagonal
!
I
conductivity cryy (we’ll use c r^ hereafter) is probed. If the frequency is high enough
1, the waveguide imposes a
such that ^J\o’xx\/cL>Cg « W , where Cg is the geometric capacitance per unit area
ii
i
i
i
!
15
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(a)
Z 'd x
Y 'dx
(b)
L 'd x
C ’dx
L 'd x
X
X
rs'd x
V\AAAC 'dx
Figure 2.4
_ L
T
Yh'd x
Lumped-element equivalent circuit for (a) a unit-length transmission line
(b) a lossless transmission line (c) the CPW loaded with 2DHS. rs' is the sheet resistance
o f the CPW metal, and Y is the admittance associated with the 2DHS.
16
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
between the metal and 2DHS, the CPW conductors serve as contacts to the 2DHS, and
the high frequency conduction current is confined under the gaps.
Yh is then simply
2<JxJW, and Y = icoC + 2o^JW. If we further assume that the second term is much
smaller than the first, then we can expand the propagation constant K to first order in
_.
(2-3)
ico L 2cTvv
ico Z c\<t yx
K = — 1+ -----^ ---- > — + u ** ;
vp V icoCW
vp
W
where vp = \/Z qC is the phase velocity.
Assuming zero reflection, Eq.(2-3) leads to a simple formula relating the power
attenuation, A, to the ReCcr**):
(2-4)
Pin
where Pout is the power transmitted through the CPW, and Pin the incident power. In
practice, Pjn is measured as Pout |rr _n , as in the case that the 2DHS is depleted.
O'xx —V
The E field pattern of CPW has been calculated by Gillick, Robertson and Joshi
[65]. Although the in-plane E field is concentrated under the gaps, it is nonzero under the
CPW conductors. The simple expression 7/, = la ^ lW has to be modified to account for
the dissipation occurring under the conductors. To this end we developed a circuit model
which considered the distributed nature o f the capacitive coupling. As shown in Figure
2.5, the 2DHS is modeled as a uniform conductor sheet, and with the CPW metal the
system becomes an R-C network. Yh is evaluated as the admittance value o f a unit length
o f such a network. The result can be expressed as
17
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
a/2
W
center o f CPW
Figure 2.5
Circuit model for a unit length 2DHS coupled to the CPW.
18
s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2-5)
Yh = 2 oJ { W + L W ) ;
with
(2 -6)
AIT = ^ j l + coth(|(l + 0)J;
where £ = y j ^ x x / ^ g
the characteristic decay length o f the R-C line.
In general, Yh is not small compared with icoC, so the expansion used to derive
Eq.(2-3) is not valid. Also, Zc can differ substantially from 50Q, thus reflection will
occur at the CPW/coax interfaces. All these require us to evaluate the scattering matrix
elements o f the system with 50Q interconnections. The corrected expression o f Yh (Eq.
(2-5)) is used, and for completeness, the conductor series resistance is also included in Z.
(zc - l ) s i n h ( ^ )
2 z„
S =
(2-7)
2zc
(zc2 -l)sinh(ATc?)
(zc + l)sinh(ATcf) + 2 zc cosh(AT£/)
where zc = Z c / Z q is the normalized characteristic impedance.
The end reflections at the generator and the detector can also be accounted for,
given
Zg
and
Z j,
their
impedances.
Define
Tg = (Z g - Z o )/(Z g + Z q) , and
frf = (Zd - Z 0)/(Z d + Z q) . The transmission coefficient o f such a system is [66]
(2-8)
T =
1s
>21
1 _ r g5 ll ~ r dS22 ~ r gS2\r dS\2 + r gSUS22r d
19
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
In reality, the generator is matched to the 50Q coax so rg =0. Yd is also taken as
zero in our calculations. The homemade detector is checked for its reflection and
Td < 0. 1was found o f the detectors (see section 2.5 for more details) f o r /< 10 GHz.
In general, the conductivity can have a complex value. Computation o f Ts using
various complex c r^ values shows that Ts is much more sensitive to Re( c r ^ ) than
Im( a ^ ).
For example, at 1.2 GHz the ratio o f power absorbed by a sample with
imaginary conductivity |<x| over the power absorbed by real \a\ is less than 5% for all
|o j . In practice, we have taken c r^ as real for all computational purposes.
, once the attenuation is measured,
Re( axx ) can be determined. This is the principle o f our experiment.
The silicon nitride layer deposited before the CPW serves as insulation between
the CPW metal and the sample.
This layer introduces an extra capacitance, and the
formula for Ts is adjusted accordingly.
2.3 Microwave Power Measurement
To measure the microwave power, we developed a bolometric method, which is
compatible with the high B field and low T environment.
20
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 2.6 shows a schematic diagram o f the components inside the detector.
Three resistors are mounted on a piece o f sapphire so that they are in good thermal
contact with one another.
Two are NiCr film resistors directly evaporated on the
sapphire, the third a thinned Speer 220 resistor attached to the sapphire by epoxy.
The microwave transmission line is terminated by the 50Q NiCr resistor (the
microwave absorber). The microwave power is dissipated, generating heat. The 4kQ
NiCr resistor (the DC heater) is connected to a DC current source (Idc) and works as a
second heat source inside the detector.
The high sensitivity o f the Speer (the
thermometer) at low T is utilized to detect any change o f T o f the sapphire substrate.
All these components are contained in a vacuum chamber.
The only thermal
connection between the sapphire and the 3He-4He mixture outside is through the
microwave path. Effectively the detector is a heat capacitance connected to a reservoir
via a thermal conductance. The thermal time constant was estimated to 0.1 s, but though
extreme care has been taken when the detectors were made, excessive conductive epoxy
could easily dominate the heat capacity, and the time constant would multiply. To ensure
proper operation o f the detector, discussed in the following paragraph, the heat sources
(microwave and Idc) are modulated by a square wave with a 2 s period, which is much
longer than the time constant of the detector.
The working o f the detectors can be described in the following way (See Figure
2.7). First suppose the DC heater is turned off, and the microwave is modulated by the
0.5 Hz square wave. The temperature will vary at the same frequency, as represented by
an oscillating Speer reading. Next, consider that a 0.5 Hz square wave, which is 180° out
of phase with the microwave modulation, is applied on the DC heater.
A lock-in
21
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Vacuum
mw
He - He mixture
^
bath T = 25 -3 0 0 mK
sapphire
50Q NiCr
Figure 2.6
thermistor
(Speer)
4k£2 NiCr
Schematic o f the microwave detector.
!
1
22
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
P
m
wflT LT L
P dc
constant T
“LTLTLr
feedback
=> ^MW = P dC
\ r u \ r - i
Figure 2.7
Working principle o f the bolometric detector. The microwave power Pmw
and the dc heater power P dc are modulated at 0.5 Hz and out o f phase from each other.
A feedback loop adjusted the powers so that the temperature becomes constant.
23
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
amplifier detects the phase o f the temperature oscillation and a feedback loop adjusts the
power level in one o f the heat sources until the oscillation stops, i.e., the temperature
reading is steady. Under this condition the power dissipated in the microwave absorber
equals that generated by the DC heater, so the microwave power is measured by the
current in the DC heater. A good detector could measure 10 pW signals with signal to
noise ratio S/N > 10.
J
!
i
i
A photograph o f the detector is shown in Figure 2.8.
i
I
2.4 The Setup inside the Dilution Refrigerator
i
|
j
J
One difficulty arises from the fact that the sample is sitting in the dilution
|
refrigerator. To reduce the thermal leak, the coaxial cable that connects the microwave
j
source to the sample is made o f poor conductor, stainless steel in our case. Even worse,
i
!
i
|
the coax is more than 3 meters long because o f the size of the fridge. As a result, the
coax has a large frequency dependent attenuation, and the microwave power impinging
on the sample varies accordingly. To alleviate this problem, we have a second detector
i
added near the sample inside the mixing chamber (Figure 2.9). During an experiment,
the modulation amplitude of the current on the DC heater o f the "reference" detector is
maintained constant, and the microwave power is adjusted so that the temperature o f the
|
reference detector is constant. In this way the power at the splitter is fixed, and the effect
l
of the long coax is eliminated. The transmitted power through the CPW is absorbed by
i
I
|
24
j
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I
Figure 2.8
picture o f the bolometer detector, with its housing removed, (a) Top view:
The sapphire evaporated with NiCr film resistor is to the left. The meander line is the
;
4kQ DC heater, and the small rectangle resistor next to the white ceramic beam is the
50Q microwave absorber, (b) Bottom view: The thinned Speer resistor is attached on the
back side o f the sapphire. The smallest scale on the ruler is 0.5 mm.
j
25
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
be
sample
modulator
detector
microwave
generator
splitter
reference detector
level
control
30 mK
300 K
j
!i
Figure 2.9
The complete measurement setup.
j
i
|
26
i
I
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the "sample" detector, and the current on its DC heater is adjusted to match the
microwave power. In this setup the input microwave power can be controlled by
Id c
of
the reference detector: Pav= 4kQ(/oc)2 /2. For most o f the measurements reported here,
Pav ~ 80 pW is used.
2.5 Errors of Measurement and Conversion o f the Conductivity
A major source o f error comes from reflections at the detector, which results in an
j
/-dependent return loss. Extra care has been taken to minimize the discontinuity along
|
the microwave path at various joints. We checked the reflection o f the detector assembly
I
i
j
!
(including the splitter and both detectors) and found that typically the reflection |F[ is
i
much less than 0.1 f o r /< 5 GHz and less than 0.2 for a l l / < 10 GHz. In our experiments,
substantial Re(
) exists only in the < 5 GHz range, and the error due to the neglect of
|
i
j
the reflections is estimated to be 10%.
!
ignored. Without resorting to the RC network model for the 2DHS admittance (section
i
2.2), one can solve the electrostatic problem, given the CPW dimensions and the 2DHS
:
conductivity, and find the formula relating Re(crxx) to power attenuation [67].
In our circuit model for the CPW the edge effect o f the coupling capacitance is
Calculation using these two methods shows a difference less than 1pS for the resonance
conductivity, which is typically between 10 and 30 pS.
27
I
ced with permission of the copyright owner. Further reproduction prohibited without permission.
|
Chapter 3 Conductivity Resonance in the High
Magnetic Field Insulating Phase
j
!
3.1 Introduction
I
DC magnetoresistance,
o f the high B insulator diverges as B increases, so
experimentally.it is impossible to measure R** deep in the insulating phase at very low T.
I
|
Our microwave method, as will be demonstrated below and in the next chapter, measures
|
quantitative Re(<rxv) data at 25 mK and v as low as 1/20, and thus is a valuable tool to
l
|
i
I
i
investigate 2D systems in the extreme quantum regime. The broadband nature o f the
j
j
method helped us identify a well-defined Re(crxx) vs./resonance in the insulator, which is
the subject o f this chapter.
|
3.2 Microwave frequency conductivity
Figure 3.1 shows R^ o f M259B5 as a function o f B field, measured at T= 30, 85
I
and 130 mK. The carrier density ns is 5.4 x 1010 cm'2. The FQHE’s at v = l/3 and 2/3 are
|
well developed, and at 2/5 and 3/5 the FQHE’s appear as dips in R ^ .
Insulating
behavior, as indicated by increasing R ^ with decreasing T, is observed for 3.55 T < B <
1 6.25 T and B > 6.97 T (0.62 > v > 0.36 and i/< 0.32), flanking the 1/3 FQHE. This
reentrant insulating behavior was observed in other samples from the two wafers, M230
and M259, and was reported by Santos et al. in ref. 24.
i
!
i
j
28
I
i
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
80k
30 mK
85 mK
130 mK
M259B5
60k
1/3
2/5i
&
40k
3/5
2/3
20k
B ( T)
Figure 3.1
DC magnetoresistance (/?**) o f the M259B5.
= 5.4 x 1010 cm'2. The T
dependence indicates insulating behavior for 3.55 T < B < 6.25 T and B > 6.97 T (0.62 >
v> 0.36 and v< 0.32).
29
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The diagonal DC conductivity ov^ can be obtained by measuring both DC
and
|
!
. Since our sample is not a Hall bar, a geometric factor is taken to convert Rxx
to Pxx • The resultant c r^ is shown in Figure 3.2. In interpreting this figure, it is useful
i
|
j
to recognize that at very low B field p ^ is greater than Pxy, so cr** ~ 1/ p ^ and is in
i
the mS range. For B greater than a few tenths o f a Tesla, p ^ becomes larger than p ^ ,
I
hence a ^ oc p ^ , which shows minima at the FQHE's. Finally, at very large B (B> 7 T
i
in the present case) the system becomes strongly insulating, and p xx is again larger than
i
P xy. Thus crxv «1 / p ^ vanishes as B is increased.
|
As described in Section 2.3, the transmitted microwave power is measured by the
current applied on the DC heater. The current is limited by a large series resistor (from
i
!
3MQ to 30MQ), so the actual output signal is a DC voltage.
Figure 3.3 shows such a
j
"raw" signal vs. B, measured at fixed frequency, 0.2 GHz, for sample M259B7 at both
i
|
full carrier density (5.6 x 1010 cm'2) and a reduced density (3.0 x 1010 cm'2).
The
I temperature is 25 mK and input microwave power is 80 pW. Note that the microwave
i
! measurements are carried to much higher B than the DC measurements. Since larger
conductivity leads to larger attenuation and smaller output signal, the trace is essentially
I
an upside-down (but not-to-scale) Re(crxc) vs. B.
The full density trace (solid line)
shows all the FQHE "maxima" at the same B fields as the FQHE minima in the DC
magnetoresistance measurements. In the high B region (> 10 T) and at the 1/3 FQHE the
trace coincides with the reduced density dashed trace, which is flat for B > 4.5 T. Clearly
|
the flat region is the zero c r^ signal, and when it is extended down to 1 T (dotted line in
|
the figure), we see that the v —\ IQHE for both densities also lies on this line.
II
30
ii
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10
M259B5, 30 mK
8
(n e )
1/3
6
3/5
2/5
2/3
4
2
0
0
2
4
6
8
B (T)
Figure 3.2
DC diagonal conductivity (cr**) converted from the 30 mK
vs. 5
Figure 3.1 and a classical Rxy.
I
!1
31
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2 .0
1.4
0.8
-O
M259B7,/= 0.2 GHz, T = 25 mK
0 .6
77= 5.6 x 1010cm 2
0.4
3.0xl0'°cm '2
0.2
0.0
0
2
6
4
8
10
12
B ( T)
!
Figure 3.3
Detector output signal as a function o f B field for two different carrier
densities. The horizontal dotted line is guide to the eye.
32
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The 0.2 GHz Re(<x**) vs. B calculated from the full density trace o f Figure 3.3 is
plotted in Figure 3.4. The curve is presented without any filtering or averaging, yet the
I
noise is low enough that even the dip at v =3/5 is resolved. It essentially reproduces all
features observed in the converted DC conductivity trace shown in Figure 3.2: welli
!
i
i
developed v = l, 2/3 and 1/3 QHE's, dips at 2/5 and 3/5, and decreasing conductivity at
high B. The 0.2 GHz 2DHS conductivity, similar to the DC conductivity, vanishes for B
> 10 T.
i
Figure 3.5 shows Refer**) vs. B for M259B5, with ns = 5.4 x 1010 cm'2,
i
■I
|
measured at several fixed frequencies. In the insulator at v > 0.36 (B < 6.2 T), Re (cr**)
j
remains small so that any nonmonotonic / dependence is hidden by the noise. The 1/3
i
FQHE minimum in Re(cr**) vs. B weakens uniformly with increasing/ For v< 0.32 (B
j
i
I
> 7.0 T), where the T dependent DC transport indicates there is an insulator for B above
|
the 1/3 FQHE, Re(cr**) exhibits a strong nonmonotonic / dependence. For 0.2 GHz,
|
Re (cr**) reaches a maximum o f ~ 2 pS at ~7.1 T, and then vanishes with increasing B for
B > 10 T, while even deeper in the insulator the higher/ traces can reach Re (cr**) values
!
|
;
much greater than any observed for v > 1/3. All the curves show dips around v = 2/7,
i
and the 1.41 GHz trace shows a notable dip around B = 11 T ( v = 1/5).
The f dependence is best examined by a continuous / scan at a constant B field.
The detector output signal is plotted v s ./in Figure 3.6 (a) for M259B7 at 13 T. The solid
!
I
trace is measured with carrier density ns =5.6 x 1010 cm'2, and the dashed trace represents
j
the baseline signal, measured with 2D holes depleted by applying 400 V on the backgate.
:
A bump in signal at around 3 GHz, and the small wiggles with 70 to 90 MHz period at/ >
!I
I
j
1
33
!
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
V= 1
M259B7
/ = 0.2 GHz, T ~ 2 5 mK
5 .5 xl O I0cm 2
- - - 3.0 x IQ10 cm 2
%
1/3
v=l
Q
i
0
I i
I\ r
2
I i
I_i
I
i
4
I__i_Ajl i
6
I_i
8
I rV i.-i
10
L__l
12
B(T)
I
|
1
Figure 3.4
0.2 GHz ReCp^) vs. B for M259B7. The 3.0 x 1010 cm '2.trace is shifted
up by 4 pS.
34
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
M259B5, T~ 50 mK
0.20 GHz
(Z)
=L
Figure 3.5
Re(crxc) vs. B for M259B5 measured at several fixed frequencies.
T^SOmK
35
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
i
0
1
2
3
4
5
6
2.0
>
M259B7, B= 13 T, T= 25 mK
8
n = 5.6 x 1010cm'2
baseline (2DHS depleted)
0.5
,¥
0.0
T
T
30
c/3
20
10
0
0
1
2
3
4
5
6
/( G H z )
Figure 3.6
(a) Detector output signal v s ./fo r M259B7.at B= 13 T and T= 25 mK. (b)
Re(Oxr) vs./obtained from the two traces in (a).
I
I
i
!
36
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3 GHz are observed for both traces. We believe these are instrumental artifacts, which
most likely come from imperfect connections between different components in the
j
measurement setup. The baseline signal is always larger, since there are no carriers to
I
|
attenuate the microwave. The two traces coincide at 0.2 GHz, indicating that the 2DHS
|
conductivity is zero, consistent with B scan data in Figure 3.4. For / > 5 GHz the two are
!
again identical, but in-between for the solid line a huge resonance structure exists at
i
|
around 1.2 GHz.
To obtain Re(crxx) vs. f
both traces, with one being the zero
j
conductivity baseline, are needed. The result is shown in Figure 3.6 (b). We see a sharp
j
resonance with substantial conductivity occurring deep inside the insulating phase.
iI
I
We define several quantities to characterize the conductivity resonance, as
I
indicated in Figure 3.6 (b): (1). crpk, the maximum conductivity o f the resonance peak;
|
(2). f pk, the frequency where <jpk occurs; (3). Af the full width at half maximum o f the
i
|
|
i
I
t
|
resonance; (4). Q, a quality factor defined by fpk/Af, and (5). S, the integral o f Retcr^) vs.
/ (i.e., the area under the conductivity curve computed numerically). When the resonance
is well developed, with no high frequency tail outside the experimental frequency range,
S' is a good measure o f the oscillator strength. O f the resonance in Figure 3.6 (b), <Jpk is
j
I
32 ± 3 \iS ,fpk is 1.25 ± 0.02 GHz, A /is 0.23 ± 0.02 GHz, Q is 5.4 ± 0.5, and S is 12 ± 1
pS-GHz.
3.3 The ^-Dependence of the Resonance
J
|
As seen in Figure 3.5, the strong non-monotonic /dependence o f the conductivity
exists only in the high field insulating phase.
This suggests that the conductivity
37
ii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
resonance should be observed only in the insulating phase. Figure 3.7 shows Refcr^) v s . /
for several fixed B's both near the FQHE-insulator transition and deep inside the
insulating phase. At 6.77 T (v = 0.325), Refo**) remains less than 2.5 pS , decreasing
slightly with increasing/ A single peak in ReCoi*) vs./becom es discemable for B > 7.4
I
I
j
I
resonance becomes larger and narrower. /,* also increases with B, but for the curves with
|
the resonance well-developed, the upward shift in/,* with B is slight,
j
I
j
T (v < 0.3). At 7.42 T, the resonance is rounded and broad, but as B is increased, the
The development o f the resonance with B is summarized in Figure 3.8. S is
i
obtained by integrating Re(oi*) over 0.2 GHz to 9.0 GHz. Both f pk and S vs. B are plotted
in Figure 3.8 (a), using left and right axes, which each covers a factor o f 2 range. For B
up to about 10 T (v = 0.22),/,* increases from 1.02 to 1.22 GHz as B is increased. For B
greater than 10 T,f Pk is nearly independent o f B, increasing to only 1.27 GHz by the top B
i
field o f 14.4 T (v=0.154). With B between 10 and 14.4 T the S vs. B curve follows the f pk
vs. B curve well, and is likewise nearly independent o f B.
|
|
I
j
Figure 3.8 (b) shows <rpk and A /o f the resonance. <jpk grows with increasing B
and is 24 pS at the top field. A/decreases throughout the B range over which we observe
the resonance. Figure 3.8 (c) shows the quality factor Q (= f pk / A f ), which is seen to
!
increase linearly with B, even beyond 10 T, where f p^ is approximately independent o f
j
j
!
B. The Q is about 5 at 14.4 T.
Resonances o f M230B6 at several B fields are shown in Figure 3.9. Since the
i
i
!
carrier density (4.2 x 1010 cm'2) is lower, the transition to the insulator occurs at lower B,
\
as does the resonance. Resonance parameters are summarized in Figure 3.10. Similar to
l
38
I
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
25
259B5
r~50m K
«..~5.4 x 1010cm
B(T)
6.8
7.4
8.8
11.6
14.4
C/3
=L
v
(0.325)
(0.297)
(0.250)
(0.190)
(0.153)
JHH
0.0
Figure 3.7
1.0
2.0
/(G H z)
3.0
4.0
Re(c^a:) v s ./ o f M259B5 at several constant B. T= 50 mK.
I
39
i
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.3
0.25
i.o a
V V w
6
O)
4
2
0
8
10
12
14
B (J)
Figure 3.8
Summary of the resonance features for M259B5. r = 5 0 m K . (a) f pk (solid
circle) and S (open square) vs. B. (b) apk (solid triangle) and A/(open triangle) vs. B. (c)
Q (diamond) vs. B.
i
40
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20
M230B6
T - 30 mK
B(T) v
6 T (0.289)
7 (0.248)
8 (0.217)
10 (0.174)
12 (0.145)
j
j
i
|
1
i
I
Figure 3.9
Re(cr**) v s . / o f M230B6 at several 2? fields, r = 30mK.
41
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.3
0.25
V
0.2
0.15
CO
=5.
I
O
co
2.0
20
CO
O
A
0.5
0.0
O)
B ( T)
Figure 3.10
Summary o f the resonance features for M230B6. 7 '= 3 0 m K . (a) f pk (solid
circle) and S (open square) vs. B. (b) apk (solid triangle) and A f (open triangle) vs. B. (c)
Q (diamond) vs. B.
42
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I
i
I
|
the case o f M 259B5,^* saturates at high B, but at a larger frequency, 1.36 GHz. Within
I
range o f the experiment. crpk increases and A/decreases with B. Q vs. B shows a weaker
|
dependence beyond 10 T, which is not observed in the data for M259B5.
the error bars (10% as plotted in Figure 3.10), S vs. B remains constant for the entire B
3.4 Interpretation of the resonance: pinning mode of Wigner crystal
To understand the origin o f the conductivity resonance, we first consider an
j
|
oscillator model, based on the theories o f pinned charge density wave or Wigner crystal
;
pioneered by Fukuyama and Lee (FL) [16,17,57]. As illustrated in Figure 3.11, the 2D
i
I
!
i
system is taken to contain oscillators o f mass M, each moving in a harmonic potential
j
a>o an empirical parameter called the pinning frequency. In the case that the system is a
!
2
*
pinned WC, M is the mass of a domain of linear size Z, so M = L nsm . The pinning
2
2
Vqu 12, where u is a displacement from an equilibrium position, and Vq = M coq , with
potential can include elastic energy of the WC as well as interaction between the domain
|
|
and the pinning impurities [16, 17].
In a large perpendicular B field, the motion of the carriers can be decomposed
into cyclotron motion and guiding center drift.
I
frequencies.
The FL model predicts two mode
One is the analog o f the cyclotron mode, which occurs above the free
♦
electron cyclotron frequency, coc =eB / m . For our 2DHS with hole mass o f 0.37 times
j
I
|I
i
the free electron mass, coc / 27t is 450 GHz for 6 T. This mode frequency is too high to be
43
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
i
pinning potential V0u2!2
Figure 3.11
Illustration of the harmonic oscillator model. L is the domain size, u, the
displacement
|
I
I
44
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
related to the resonance. Assuming coq « coc , the other mode occurs at a much lower
|
frequency,
co_ « coq2 i(oc = VQ/(eBI?ns ) .
|
j
(3-1)
j
This is the so-called pinning mode [16] o f the guiding center in the disorder potential.
'
I
j
We compare the pinning mode predictions to the observed conductivity resonance
in the insulating phase.
First, we consider the resonance frequency. If we assume that
changing the B field does not change Vq and L in (3-1), then co_ oc 1/ B , which is very
i
different from the observed f pk vs. B. The f pk data in Figure 3.8 (a) exhibits two regions in
the insulating phase for v < 0.3: the first region has 7.4 T < B < 12 T, with f p^
increasing with B; and the second has B > 12 T with f p^ approximately constant. The
first region is adjacent to the transition from the 1/3 FQHL, and the increase o f f p^ with
j
i
iI
|
B is probably associated with the transition, possibly reflecting some FQHL correlations
remaining in the insulating regime [58, 68].
A flat
vs. B region may be
|
characteristic o f the insulator in the high B limit. Constant f p^ implies that coq oc
.
We can obtain coo vs. B from the measured f pk, which is plotted in Figure 3.12. The
possibility that B field changes the pinning potential or the domain size will be discussed
in Chapter 4.
A real inconsistency occurs when the B dependences o f the resonance frequency
;
and the oscillator strength are taken together.
I
|
strength associated with the pinning mode, (in units o f the integral o f R e(qit) v s./), given
!
by:
I
II
The FL model predicts the oscillator
45
i
i
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
^
100
B (T)
Figure 3.12
Pinning frequency (a>d) as obtained by (2nfpk a>c) from the M259B7 data in
figure 3.8.
46
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3-2)
S p i = (nse / 4B)a>_ .
This implies when f pk is independent o f B as observed at high B, S
oc
MB. This is not the
case for the data, which exhibit constant f pk and S a t high B.
i
On the other hand, we can calculate Spp by substituting 2nfpk for co_ in (3-2).
j
For M259B5 at 14.4 T, f pk is 1.27 GHz, so S p i= 12 pS, which is very close to the
|
observed S o f 10 pS. The consistency o f the integrated strength was also reported in
previous work on 2DES [54].
|
An interesting feature o f the FL model that is preserved in a description o f the
present data is the cancellation o f m in the expression for a>_ (Eq. (3-1)). Although the
j
I
|
iI
m o f the holes is 5 times larger than that o f the electrons, our f Pk at high B o f 1.25 GHz is
i
roughly in agreement with those in 2DES exhibiting single resonance [33, 54, 58], which
at low vare between 1 and 2 GHz, for electron densities ranging from 4 to 11 xlO 10 cm'2.
|
I
i
i
i
I
I
One resonance feature that is surprising is the large Q. A resonance originated
from a disorder-induced mode is expected to be broad; conventional theories predicted
that Q « 1 [18, 57]. One may consider that the width o f the resonance results from a
I
j
combination o f damping and inhomogeneous broadening. It is more natural to interpret
I
the increase o f Q with B as due to a decrease in damping, rather than as a reduction in the
j
statistical variance o f oscillator frequencies. As.seen in Figure 3.8 (c), the linear Q vs. B
j
does not saturate, even at our highest experimental B o f 14.4 T, where a Q o f 5 is
|
observed. This suggests that the observed linewidth is predominantly due to damping
j
i
j
rather than to inhomogeneous broadening. However, this raises two difficulties. Firstly,
!
47
\
i
there is no explanation for the linear increase o f Q with B. Theories [69] of damping for
j
I
j
i
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
pinning modes o f CDW in transition metal trichalcogenides are based on the conductivity
o f uncondensed carriers. Such a damping mechanism is unlikely to produce linear Q vs.
|
j
B for the present system, since low temperature DC conductivity decreases more rapidly
than linearly in the insulating phase.
!
Secondly, and more importantly, Q being large and limited by damping implies
i
that the inhomogeneous broadening is extremely small, i.e., all oscillators are alike. This
is surprising for a model with random domains.
In ref. 59 we considered a "strong
pinning" scheme [17, 19], where the pinning potentials result from well-separated
i
charged impurities (e.g., C acceptors) close to the 2DHS, which could produce very
j
j
|
i
!
i
j
i
|
|
similar domains, accounting for the small inhomogeneous broadening. This scheme, as
opposed to the "weak pinning" model, was proposed to explain dielectric constant
measurements for a 2DES at high B in [35], which also connected pinning energy with
reasonable bulk impurity density estimates.
However, subsequent experiments on the resonance show that a weak pinning
i
I
I
model is necessary for the observed ns dependence (section 4.2). Recent theoretical work
i
!
[75, 76, 77, 78] on the conductivity resonance addresses the sharpness issue and point out
j
I
t
that a large Q does not require domains with similar pinning frequencies. Instead, the
j
sharp resonance is the consequence o f two factors: the long range Coulomb interactions
between WC domains, and the mixing o f transverse and longitudinal oscillations by the B
I
|
|
li
field.
We postpone a more detailed comparison between the data and the models until
Ch.4 after we present systematic studies o f the resonance as experimental parameters,
such as carrier density ns and temperature T, are changed.
48
I
i
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Chapter 4 The Conductivity Resonance under Various
Experimental Conditions
4.1 Overview
The characteristics o f the conductivity resonance under different experimental
t
conditions can be used to study the nature o f the high field insulating phase.
The
magnetic field dependence has been presented in the preceding chapter. In this chapter,
!
I
we further investigate the evolution o f the resonance with other parameters, including the
!
2D carrier density (section 4.2), the temperature, the input microwave power, and an in-
i
j
plane "de-pinning" DC electric field (section 4.3). The experimental results are then
!
summarized and compared to more recent theoretical models o f the pinned Wigner
j
crystal.
!
i
!I
I|
!
I
|
4.2 Carrier-carrier interaction and the microwave resonance
4.2.1
Introduction
In this section we directly address the important competition between carrier-
;
carrier and carrier-impurity interactions. Both are expected to play important roles in the
I
2D insulator, and the task is to experimentally separate their effects. This is done by
j
varying the carrier density ns o f the sample through a DC bias (Vig) applied on the
i
backgate, without warming up the sample. Below we demonstrate that this technique
i
I
I
I
49
i
i
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
|
effectively varies the carrier-carrier interactions while leaving the disorder potential
unchanged.
Figure 4.1 shows the detector signals v s . / o f M259B7 at three different Vbg's, OV,
|
i
i
II
200V and 400V. Multiple curves with the same Vbg were taken before and after cycling
Vbg. As can be seen in the figure, these curves are identical, showing no hysteresis on
!
cycling Vbg. This result indicates that the backgate voltage does not alter the disorder
potential, since impurities, if rearranged in position or in charge state by the application
o f a backgate voltage, would not return to their as-cooled condition on removing that
i
1
voltage.
Figure 4.2 shows the low B field, 2-terminal DC magnetotransport o f M259B7 at
|
|
1I
j
|
several Vbg's. For Vbg < 200V, the resistance is dominated by R ^, and quantum Hall
plateaus are visible at v = l, 2, and 3. Beyond 250V Rxx dominates, and only a minimum
at v = l is observed. From these features we extract the ns corresponding to each Vbg, and
j
!
i
the error is estimated to be from 3% for the highest density to 10% for 1.6 x 1010 cm'2
!
(Vbg ~ 280 V). The inset o f Figure 4.2 shows the ns vs. Vbg for M259B7. As can be seen,
ns depends linearly on Vbg, as would be expected if the backgate and the 2DHS form a
parallel-plate capacitor. The slope is consistent with the ratio o f the effective dielectric
I
constant of the backgate insulation (a kapton film and grease) over its thickness being 5.8
j
x 10'7 F/m2.
I
j
4.2.2
Results of the density dependence experiment
We present R e(q^) vs. / data on M259B7 at different ns in Figure 4.3. Unless
i
!
otherwise stated, measurements presented in this section were all done at 13 T and 25
50
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2.4
2.2
Vh„ =+400V
2.0
Q.
oZ3
o
(U 0.8
0)
■a 0.6
0.4
V =+200V
M259B7
B=13 T, T=20 mK
Vu_ = 0V
0.2
0.0
f (GHz)
Figure4.1
Detector output signal v s . / f o r M259B7. B = 13 T, T = 25 mK Four
traces are shown for Vbg =0, and four for Vtg = 200 V. Among them, two at each Vbg
were measured before, and two after, a high Vbg (400V) was applied to deplete the holes.
51
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
30k
+250V
+300V (x 0.1)
V„n = 0V
+200V
a
<
oEu 20k
aCO
+320V (x 0.01)
+100V
’33
e
B
on
ra
E
10k
100
200
^ (V )
0
1
300
40C
.
2
5 (T )
Figure 4.2
DC magnetotranport o f M259B7 at different backgate biases. The inset
shows ns as a function o f Vtg.
52
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
i
M259B7
B = 13 T, T - 2 5 mK
ns
(v)
(grl) (“ojay
-------- 5.42
------- 4.65
......... 3.89
......... 3.13
------- 2.37
......... 1.6
/
^ffiuVi*?*
2
3
(0.172)
(0.148)
(0.124)
(0.100)
(0.075)
(0.051)
V
4
5
f (GHz)
Figure 4.3
Retcr**) v s ./a t several ns (in 1010 cm'2) o f M259B7. B =13 T, T= 25 mK.
|
53
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
mK, with Pav ~ 80 pW. As ns is reduced,^* increases. We can follow this increase up to
l
I
|
a factor o f 3 before Re(oi*) diminishes to such an extent that the resonance maximum
i
t
i
cannot be clearly identified. Decreasing ns also broadens the peak and reduces <jpk. Data
on M230B6 are plotted in Figure 4.4. For ns > 2.7 x 1010 cm'2 the resonance shows
I
j
i
qualitatively similar ns dependence to that o f M259B7. For ns < 2.7 x 1010 cm'2 a second
conductivity peak starts to emerge from the high frequency side o f the original resonance.
This second peak remains a puzzle to us. A possible explanation is that the 2DHS was
I
broken into two regions, each exhibiting a different resonance. We will focus on the
j
!
single resonance regime in this thesis.
I
I
i
Figure 4.5 shows f pk vs. ns on a log-log scale from two different cooldowns of
|
i
both M259B7 (shown as solid circles and squares) and M230B6 (shown as open circles
■
and open squares). For a given sample f Pk varies only 10 percent between cooldowns.
Variation between the two samples is much larger: for example, at ns = 3.0 x 1010 cm'2,
|
|
f pk is around 2.6 GHz for M259B7 and 1.5 GHz for M230B6, as read o ff from the data in
i
Figure 4.5. In both samples the B dependence o ff Pk, even at reduced ns, is much like that
!
]
j
observed in ref. 59: f pk increases with increasing B at small B, but flattens out at large
enough B (Figure 4.6). By 13 T ,fpk vs. B is in this flat regime, so that the 13 T f pk, plotted
in Figure 4.5, can be regarded as an approximate high B limiting value.
M230B6 exhibits f pk uniformly decreasing with ns, with fits o f the data to the
!
I
power law f pk oc ns r that are consistent with y = 1/2, given the errors and range o f
i
i
|
54
|
|
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
]
j
!
20
M 230B6 13T, 25 mK
(Sri) C o ) ^
3.38
2.94
2.72
2.28
;
(0.108)
(0.094)
(0.087)
(0.073)
f (GHz)
i
j
Figure 4.4
Re(oi*) v s ./a t several ns (in 1010 cm'2) o f M230B6. B =13 T, T - 25 mK.
1
55
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
V
4.O r
0.04
I ^ i
0.08
0.12
1---- 1---- 1— i— i—i—|—i—i—r p r n
0.16
| i ii jm
| r r i r rj
3.0 r
3
S 2.0
&
13 T, 25 mK
1.0
1.0
Figure 4.5
* « » i I » i i t I i i i i I i i i i I i i ■t I i li t.I.i.11ill ml mi
2.0
3.0
4.0 5.0 6.0
n (x 1010 cm"2)
f pt vs. ns for M259B7 (solid symbols) and M230B6 (open symbols), each
with two cooldowns (shown as circles and squares). B = 13 T, T= 25 mK. The lines are
least square fits to f pk oc ns'r.
56
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3 .0
—
,
,
,
M259B7
2 .5
■I
•
(
#
1
•
|
•
-T -— r
-
•
1 -
■
•
T
—
I
#
-
ns ~ 3.0 x1010/cm2
•
2 .0
—
%
IsT
X
1.5
o
1.0
'
•
-
-
-
ns ~ 5.5 x1010/cm2
0 .5
0 .0
V _______I_______I ______ I_______1
0
2
4
_
i
.
6
i
8
.
i
10
-
.
i
12
•
14
B(T)
Figure 4.6
f pk vs. B o f M259B7 at ns = 5.5 and 3.0 x 1010
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
measurement. Least squares fits, shown on the graph as solid lines, give y= 0.46 and
j
j
I
|
I
1
0.51 for the two different cooldowns o f M230B6.
The fpk vs. ns traces for M259B7 show three different regions o f ns. In the highest
ns region, (ns > 5.0 x 1010 cm'2) ,fpk changes little with ns. In the intermediate ns region,
*3/O
|
(3.3 x 1010 < ns < 5.0 x 1010 cm"2), the data can be fit well to f pk °c ns
squares fits to f pk ^ n s
-y
. The least
result in the dashed lines shown in the figure, forwhich y= 1.66
and 1.24 for the two cooldowns. In the low ns region, 1.34 x 1010 < ns <3.3 x 1010 cm'2,
i
•i
i
1
the data on M259B7 are, like f Pk vs. rts for M230B6, consistent with f Pk oc ns
-
1 /2
. The least
squares fit lines shown in Figure 4.5 give y= 0.58 for both cooldowns o f M259B7 in the
low ns region.
i
j
i
|
Figure 4.7 summaries the ns dependence o f the resonance features o f M259B7
(solid symbols) and M230B6 (open symbols). Data from one cooldown o f each sample
/
j
(the one represented by circles in Figure 4.5) are plotted.
Generally the behavior of
M259B7 is more complicated than that o f M230B6 because o f the much larger accessible
i
!
s
ns range. Below we'll focus on the case o f M259B7 first, and all the dependences will be
described as relative to a decreasing ns. The three ns regions o f the f pk vs. ns trace, clearly
i
shown on the log-log scale in Figure 4.5, are also visible on the linear scale in Figure 4.7
(a). The Af vs. ns trace also suggests the same demarcations in ns. It remains flat for ns >
I
5.0 x 1010 cm"2, increases at a similar rate, percentagewise, to that o f f pk vs. ns in the
intermediate region, and then increases more rapidly than f Pk for ns < 3.3 x 10 10 cm"2. The
ns dependences o f f pk and Af are reflected in Q vs. ns, which remains between 4.5 and 5
I
58
j
i
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4
1
i
i
i
- t........ i—
'—
3 - (a)
a
N
X
2
1a
0
^
1
0
i
i
.
i
5
« ♦
#
3
i
<>
5 = 13 T
cO°
T =25m K
2
solid symbols: M259B7
open symbols: M230B6 -
0
15
1
:
o
.
00
1
zL
i
1
4 - (b)
O)
.
*
1
i
1
i
.
i
.
i
,
(c )
i
.
i
1
1
1
40
« ■ 1 ■ ■ ■ ■ ■
"
10
11. • •
A *****
c n tP x P ’ A A
b
30
i
co
20
10
CO
i
0
a a a
0
n ( x l 0 l0cm’2)
Figure 4.7
Summary o f ns dependence o f resonance features for M259B7 (solid
symbols) and M230B6 (open symbols), B =13 T, T = 25 mK. (a) f pk (circles) and A /
(downward triangles) vs. ns (b) Q (diamonds) vs. ns
(c) S (squares) and apk (upward
triangles) vs. ns.
59
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
for ns > 3.3 x 1010 cm"2, and decreases rapidly in the low ns region (Figure 4.7 (b)). aPk
decreases with reducing ns, although more slowly in the high ns region (Figure 4.7 (c)).
I
The ns dependence o f S qualitatively agrees with A f x crpk vs. ns. As shown in Figure 4.7
(c) S decreases at high ns, remains approximately constant in the intermediate region, and
decreases again for ns > 3.3 x 1010 cm'2 as apk drops near zero. In the high ns region, S is
!
linearly proportional to ns, as indicated by the dotted line in the figure.
|
It is interesting to compare M230B6 to the low ns region o f M259B7.
As
mentioned previously, the fitting parameter y is close to 1/2 in both cases. Figure 4.7 (a)
J
|
shows that f pk vs. ns and A f vs. ns, taken together, are also similar in that A f increases
|
faster than f pk.
j
difference is in the ns dependence o f S (Figure 4.7 (c)), but its error is also the largest («
j
crpk vs. ns o f M230B6 falls on top o f the M259B7 trace.
The only
i
|
10%).
i
j
i
!]
|
i
4.2.3
Discussion
The evolution o f the peak as ns is reduced leads to the definite conclusion that the
j
resonance cannot be modeled as resulting from noninteracting individual carriers bound
i
:
to impurities in the semiconductor host. If carrier-carrier interactions could be ignored,
j
Re(Oxx) v s ./w o u ld be determined by a simple summation o f the responses o f individual
|
carriers, each trapped in a disorder potential, and removing carriers could only reduce
i
I
Re(crtx). Although the reduction could be /dependent, Re(cr«) could never increase at
j
any / in such a model. On the contrary, Figure 4.3 shows that as ns is reduced, Re(crcx)
1I
;I
!
6o
j
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
increases in a region o f the high frequency wing o f the peak. For example, R e(a^) at
2.55 GHz increases from 1.2 to 13 pS, on reducing ns from 5.42 to 3.13 x 1010 cm'2. This
can only happen when the carrier-carrier interaction plays a role in determining^*.
Interpreting the resonance as the response o f a WC, we extend the discussion o f
l
I
section 3.4 to greater details, which includes the ns dependence o f the mode frequency
co_.
The dynamic properties o f a disorder free, classical 2D WC has been studied by
j
j
Bonsall and Maradudin [70]. At zero B and in the low q limit, the longitudinal phonon,
■|
j
co t,
shows a q m dispersion, and transverse phonon,
go,,
is given by
I
!
(4-1)
q>,=
I ~ i~ q ,
i
j
|
I
where K = a e2n 2a ls is the shear modulus
of the solid, with a ~ 0.02 for a classical
i
\l
|
hexagonal lattice. We note that since a ~ 1
|
cyclotronic mode, whose frequency is too high to be relevant, and one low frequency
/ , K is proportional to n 3a.
In a finite B field, the longitudinal and transverse phonons are mixed into one
i
!
hybrid mode. In the limit that qa « 1 , the hybrid mode is simply
|
(4-2)
|
The mode frequency vanishes with q as q312.
\
eohybrid = com / coc.
We substitute 2 7tfpk from the data into (Ohybrid and obtain the value of q.
With
|
M259B7 at 13 T: (1) f pk - 1.25 GHz for ns = 5.4 x 1010 cm'2, giving q ~ 3.3
|
27t / 1.9 pm, and (2) for ns = 3.1 x 1010 cm'2,f pk - 2.55 GHz, so q ~ 6.9 x104 cm '1 = 2 n /
x 104 cm '1 =
0.9 pm. For M230B6: (3) f pk = 1.42 GHz for ns = 3.2 x 1010 cm'2, so q ~ 4.7 x 104 cm '1 =
i
!
i
61
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2tc / 1.34 pm. With these numbers we find that it is difficult to attribute the resonance to
the hybrid mode of a clean 2D WC. Firstly, the calculated q's are at least 15 times larger
than the q expected from the CPW slot width (27t / 30 pm). Secondly, from Figure 4.7
(c) the crpk s are the same for the two samples at the same ns, hence there is no reason the
q's in case (2) and (3) above should be different [71].
Thirdly, the large difference
between f Pk s o f the two samples at the same ns clearly points to the effects o f disorder.
In the presence of disorder, WC is distorted and the crystalline order is maintained
in domains o f finite size L [16, 18]. This length scale sets a lower limit on q, below which
the hybrid mode frequency approaches a finite value, co_ - (Oq / coc , which is the pinning
mode introduced in section 3.4. Following Normand, Littlewood and Millis [18], the
distortion is mainly transverse because the longitudinal modulus is much larger than the
shear modulus (K) in a 2D solid. As a consequence, the domain size is determined by the
shear and the random potential, and in the q -> 0 limit the pinning frequency can be
written as
(4-3)
co0* =
1
= -)
So the low frequency pinning mode occurs at
(4-4)
K
\ 2 rm \
1/2/1
a>_ = 0)0 = ----*
(
t>
X
(“
)
*
ns
(t) •
eB
CO c
nsm
In view o f Eq. (4-4), the f pk vs. ns data allow us to rule out "strong pinning" by
dilute impurities. In such a model, the pinning potential is strong enough to essentially
immobilize carriers that are bound to impurities, so that the domain size L is fixed by the
distance between pinning sites and is independent o f ns. A s indicated in Eq. (4-4), the ns
62
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I
!
dependence is then determined by the ratio o f the shear modulus K (oc ns3a) over the mass
I
density nsm*, which leads to a>. oc nsm . This is contrary to our observation that f pk
I(
decreases as ns increases.
We interpret the increase o f f pk with decreasing ns as a general consequence of
weak WC pinning.
By definition, in the weak pinning regime, the forces between
carriers in the WC are larger than the forces exerted on carriers by impurities, and the
domain size L is in principle much larger than the WC lattice constant a.
Given a
disorder configuration, increasing ns enhances carrier-carrier interactions and effectively
l
' i
j
makes the system more "crystal-like". In this regime the domain size L is likely to be
|
larger at larger ns, and, according to Eq. (4-4), co. could decrease with increasing ns.
j
!
!
|
|
In the weak pinning model, L is determined by minimizing the sum o f the elastic
energy and the impurity energy. In ref. 18 it is found that
L * K a 2 ln> l2V0 <x«51/2,
|
|
(4-5)
|
where
!
strength . From Eq. (4-4) this leads to eo_ oc ns
|
is the two- dimensional impurity density, and V0 is the impurity potential
I
—1 / 2
, which agrees with the f pk vs. ns data
of M230B6 and that of the low ns region o f M259B7 (Figure 4.5).
As in the B dependence experiment discussed in section 3.4, we take S and f pk
:
together and plot S / f pk vs. ns in Figure 4.8 for comparison with the FL model prediction
th a tS / f pk ~ nsn s /B . While th e S vs. ns is not a simple linear relation (Figure 4.7 (c)), the
]
i------- --1 It has been suggested that this expression o f L is not applicable to a WC, because the original theory was
I
!
extended from a CDW model, in which case the system is only sensitive to the Fourier component o f the
|
disorder potential with period o f a. (R. Chitra and M. Fogler, private communication)
I
63
s
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10.0
13 T, 25 mK
M259B7: solid symbols
M230B6: open symbols
m
0.0
0
1
2
3
4
5
6
/ 10
i n 10 cm-2\)
n (x
I
j
Figure 4.8
|
a linear fit to the data and its slope is 94% o f the FL prediction.
S /fpk vs. ns for M259B7 and M230B6. B =13 T, T = 25 mK. The solid is
i
I
i
i
i
64
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
S I fpk data do appear directly proportional to ns, and the slope o f the fit line drawn in the
figure, is 0.94 of the FL predicted value. This is in contrast to the disagreement with the
FL prediction for the B dependences o f the (5 / f Pk) reported in section 3.4. The measured
value o f the slope is consistent with the resonance oscillator strengths reported earlier
[58, 59].
While we conclude that a weak pinning mechanism is needed to explain the
observed decrease of f pk with ns, conventional FL theories [16, 17, 18, 57] give L / a
comparable to unity for the observed f p k S , contrary to the expectation for weak pinning
conditions. Using Eq. (4-4) and the f pk data we computed L / a v s . ns, as shown in Figure
4.9. While the linear L / a vs. ns is consistent with Eq. (4-5), the values are less than
unity for the measured ns range of both samples. Applying the above analysis to earlier
radio frequency/microwave data [49, 54], Millis and Littlewood [57] found L/a » 1 and
suggested that the system may be better described as a glass [72, 73]. This inconsistency
is addressed later by Ferconi and Vignale [74], who argue that incommensurate pinning,
not considered in previous theories, greatly enhances the pinning frequency. On the other
hand, L is associated only with the commensurate part o f the pinning potential. Deriving
L from the pinning frequency using the likes o f Eq. (4-4) could lead to an underestimate
two orders o f magnitude lower than the actual domain size.
More recent theoretical development [75, 76, 77, 78], motivated by our
experiments, has improved on the model o f pinning mode o f WC.
Chitra et al, [75]
consider a disorder modulated on a length scale much smaller than a, a feature not
captured in earlier theories based on CDW models. Depending on the relative sizes o f
the magnetic length Is and the disorder correlation length rc, the pinning mode frequency
65
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.6
0.5
13 T, 25 mK
M259B7: solid symbols
M230B6: open symbols
0.4
0.2
0.1
0.0
0
4
2
6
ns (x 10 /cm’ )
i
I
Figure 4.9
i
T = 2 5 mK.
L / a vs. ns computed from the observed f pk data using Eq. (4-4). 2? = 13
i
I
66
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
is expected to be oc B2 a t low B and oc B~l at high B. Fertig [7 6 ] points out that, due to the
mixing o f the transverse and longitudinal modes in a B field, long range density
fluctuations are set up as the domains oscillate, and the interactions between the domains
become strong. Even though each domain has its own pinning frequency, the long range
nature o f the coupling between domains locks them together, resulting in a very sharp
absorption corresponding to a frequency which is the average o f the pinning frequencies
o f the individual domains. Fertig's theory can also be cast in a weak pinning picture to
give fp k oc « / 3/2, as observed for M 2 5 9 B 7 for 5 .0 x 1 0 10 > ns >3.3 x 1 0 10 cm'2.
The observed differences between the two samples must be explained by the
differences in their disorder. In these samples, the sources o f disorder are remote Si
acceptors, residual impurities (mainly C acceptors), and heterojunction interface
characteristics.
The smaller spacer and larger doping of M259B7 would increase the
influence o f remote acceptors, and its mobility is less than that o f M230B6. At the same
ns and B, larger disorder would give a larger restoring force in the weak pinning model
and hence a larger f pt, consistent with the f Pk o f M259B7 always exceeding that in
M230B6.
One possible explanation for the two regimes o f decreasing f Pk vs. ns in
M259B7 would be that more than one type o f disorder is playing a role in determining f pk.
Although many aspects of the resonance features are so far not well understood,
we found that phenomenologically most of the lineshape, including the strong
asymmetry, can be captured by an underdamped harmonic oscillator [36].
The WC
domain is treated as a rigid entity, so only the center-of-mass motion is considered. With
67
with permission of the copyright owner. Further reproduction prohibited without permission.
a perpendicular B field and an in-plane ac E field, the equations of motion can be written
as
(4-6)
x = - r x - <oc y - (oh 2x - - ^ E ,
m
y = -r y + a ) c x -(o ho2y ,
where T is a damping constant and co^0 characterizes the restoring potential. Identifying
j = nex - CTxxE, one obtains
'
(4-7)
\
ne 2 '
ico(co2 - o)f,02 - icoT)
if”
m
0(o2 -(o h02 -icoT)2 -(co(oc)2
Using n, T and eOf,0 as parameters, we found that the real part o f (4-7) fits to the
l
|
Re(crxr) v s ./d a ta o f M259B7 remarkably well (See Figure 4.10). The only noticeable
i
|
difference is at the shoulders o f the resonances with high carrier densities, e.g.,
j
ns = 5.42 x 1010 cm'2 and 4.65 x 1010 cm'2. The large shoulders o f the data may have the
|
same origin as the second peak o f M230B6 (which, however, occurs at low densities),
I
and remain an issue to be addressed in future work.
j
2
We plot a)h0 la>c vs. ns in Figure 4.11 (a), which exactly coincides with f pk vs.
ns, as expected for the WC pinning mode.
j
A damping time constant, defined
as r s 2 tt / T , shows the same ns dependence as Q (Figure 4.11 (b)), and the parameter n is
within 10% of ns for the entire density range (Figure 4.11 (c)).
The agreement between the fit and the data suggests that the response o f the
|
2DHS can be cast as dominated by one harmonic oscillator, which contains 90% o f the
I
II
!
68
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
30
M259B7
B = 1 3 T, T ~ 2 5 m K
25
5.42 (0.172) _
4.65 (0.148)
3.89 (0.124) '
3.13 (0.100) 2.37 (0.075)
1.6 (0.051)
CO 2 0
zL
f (GHz)
Figure 4.10
|
Harmonic oscillator fit to the data o f M259B7 (Fig. 4.3)
69
i
ii
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0
1
2
3
4
5
6
4 ----
3 - (a)
2
eg
1
a 0
0
5
I
T
T
-(b )
solid symbols: M259B7
open symbols: M230B6
6
5
' (c)
4
3
2
1
0
0
1
1
2
3
4
5
6
/ , a I° cm-2)x
(xlO
i
Figure 4.11
Parameters of harmonic oscillator fit vs. ns.
The symbols represent
i
experimental data and the lines (solid ones for M259B7, dotted ones for M230B6) are
j
results o f the fit. In (b), Q vs. ns is plotted against the left axis and t vs. ns is plotted
i
i
against the right axis. The dashed line in (c) represents n = ns.
j
70
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
carriers.
This is consistent with the notion that even though the positional order is
maintained only within individual domains, the length scale o f coherent motion is as
large as the sample size [76].
4.3 Dependence on temperature, microwave power and in-plane DC
i
electric field
•j
|
4.3.1
T dependence data
|
Figure 4.12 shows the evolution o f the resonance with T for M259B7 at B = 13 T.
I
j
i
i
The microwave power Pav is 80 pW. As T increases, the peak height diminishes and the
resonance broadens. For the 215 mK trace the Re(cra:) maximum is barely visible, and
for the 240 mK trace ReCcr,*) decreases monotonically with/
I
j
A fit o f Re(o^i;) v s ./a t 240
t
mK to the Drude form Re(crxr)
oc
(l+(ryr) )' results in
100 ps. Flarmonic oscillator
(Eq.(4-6)) does not fit high T traces because it implies zero DC conductivity (pinned at
zero j). A fit to the intermediate temperature 125 mK trace yields a
j
5 ps, although the
discrepancy at low f is clear. The large difference between the two z's is expected from
their different physical origins, with the one of the high T phase being the collision time
in a gaseous system.
Figure 4.13 summarizes the T dependence o f the resonance
features. As T increases,/,* decreases from 1.22 GHz to » 0.9 GHz before the resonance
i
vanishes. A f increases, more rapidly at higher T, from 0.2 GHz to 1.2 GHz at 180 mK.
I
:i
|
ii
i
71
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
30
25
co
=L
M259B7 B= 13 T
25 mK
125
osc. fit for 125 mK
180
215
240
20
JO
0J
a:
2
3
f (GHz)
Figure 4.12
Re(<7a;) v s ./a t several T's for M259B7. B =13 T, ns = 5.4 x 1010 cm'2
72
j
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50
100
150
200
250
2
N
X
o
300
I
o
1
n
0
0
i
6
O)
5
4
3
♦
1
r
M259B7
(b)
B= 13T
2
1
0
j
L
J
i
I
«
L
40
20
50
100
150
200
250
C/3
3.
300
T (mK)
Figure 4.13
Summary o f T dependence o f the resonance features for M259B7. ns =
5.4 x 1010 cm'2, B =13 T. (a) f pk (circles) and A/(downward triangles) vs. T (b) Q vs. T
|
(c) S (squares) and aPk (upward triangles) vs. T.
|
73
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I
I
i
|
1
While S does not show any clear change for the measured T range, both Q and crpk
decrease steadily with increasing T; Q vs. T can be extrapolated to Q - 0 at «200 mK.
At the time when we would like to have a more thorough measurement o f the T
I
dependence o f the width (or Q) in the low T regime, the bolometer detectors became less
|
stable.
Since there is an approximately exponential dependence o f Re(er«) on the
i
I
detector signal (zero output corresponds to infinite conductivity), the error in output
voltage gets amplified when the voltage itself is small, especially in the resonance
|
condition at low T. As shown in Figure 4.14, the scatter o f crpk is large, which leads to
1
j
inaccurate readings o f A f and Q.
|
]
By contrast, f pk is very reproducible, and remains
approximately constant for T < 100 mK.
4.3.2
dependence data
All the data presented so far were measured with Pav = 80 pW, which corresponds
j
!
to a root-mean- square voltage 90 pV and electric field Emw = 3 V/m on the CPW. While
the power and field are small (the cooling power o f the dilution refrigerator is 10 pW at
!
I
20 mK), we found that if the power is further reduced, the conductivity resonance
becomes even larger and sharper. Re(o]cr) v s./m e asu re d at 25 mK and 13 T with Pav
]
;
It is evident that with increasing Pav the
down to 5 pW is plotted in Figure 4.15.
resonance broadens, and the peak height diminishes, reminiscent o f the T dependence
j
result.
The two traces with Pm below 45 pW become noisy but it appears that the
resonance continues to grow. These data are summarized in Figure 4.16. Note that the
j
discussions on the B dependence (section 3.3) and ns dependence (section 4.2) data,
|
j
i
. I
i
74
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0
50
100
150
2
N
200
250
300
1
X
o
«ae
0
0
IU
i
i
8 _
t
6 -
* * iu * A A
▼♦
4 -
O)
1
i
1
i
(b )
, — r ,„
M259B7
-
B = 13T
-
♦
2 —
♦♦♦♦
.1
0
■
.
.
.
, __ .___ 1____
30
60
40
▲▲
20
▲▲
0
50
100
150
200
250
300
T (mK)
Figure 4.14
Summary o f resonance features for M259B7, T < 200 mK. ns = 5.4 x 10 10
(c
cm'2, B =13 T. (a) f pk (circles) and A f (downward
triangles) vs. T (b) Q vs. T (c) 5
(squares) and cj,* (upward triangles) vs. T.
75
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
60
M 2 5 9 B 7 , 13 T, 2 5 m K
2 nW
•■■■ 7 2 0 p W
180 p W
80 pW
45 p W
10 p W
5 pW
*««
f (GHz)
Figure4.15
Re(crxxr) v s . / o f M259B7 measured with different Pav- B = 13 T, T
mK.
j
I
76
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10
100
1000
1.0
0.8
(a)
•
▼
3E
_■ i i 1111_
o.eg
o
•
° '4?
0.2
▼
J
1
i
i ............
0.0
10
8
M 259B7
B = 13 T
6 -
T = 25mK
4 2 -Cb)
o
15
80
60
oo
C/5
40
=s.
20
10
100
1000
P„ (pW)
1
Figure 4.16
Summary of Pav dependence o f the resonance for M259B7. B = 13 T, T =
25 mK. (a) f pk (circles) and A f (downward triangles) vs. Pav
(squares) and upk (upward triangles) vs. Pav
(b) Q vs. Pav
(c) S
The solid lines in (b) and (c) are guide to
the eye.
1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
j
measured with 80 pW, remain unaffected by this Pm dependence. Most conclusions are
|
based on f pk and S, which are independent o f Pav for Pav < 500 pW (Figure 4.16 (a) and
|
(c)).
)
I
j
j
i
discussion on A f and Q can be found in section 4.3.4.
j
mK, Q and crpk vs. Pav are similar to the 13 T data. At higher T, however, both quantities
Descriptions o f crPk, A f and Q should be taken as finite Pav results.
Further
Figure 4.17 shows the Pav dependence for four temperatures at B =15 T. At 25
start to saturate at low Pav4.3.3 Effect o f in-plane DC electric field
|
As mentioned in Chapter 1, several experiments [29, 30, 33] have focused on the
non-linear conduction o f the insulator. In such experiments, DC I-V curves are measured
using Ohmic contacts to the 2D system. The I-V curves in the insulating phases
|
invariably show a threshold voltage, beyond which current begins to increase more
rapidly with voltage. This threshold is interpreted as the signature o f depinning o f the
|
|
i
j
WC.
i
scale and a semi-log scale. The B field is 13 T, and v ~ 0.14. The I-V was taken between
Figure 4.18 shows the two-terminal DC I- V o f M230B6 at 25 mK on both a linear
a pair o f contacts that inject current through the area o f 2DHS that is within the
I
microwave transmission line.
For most part o f the CPW, the DC electric field is
j
perpendicular to the microwave electric field (Figure 2.2). jdc, the current density plotted
on the vertical axis, is the total current divided by the sample width (3 mm), and so is a
rough average value. A clear threshold is observed at 0.17 V.
|
With a nominal DC
electric field defined as EdC= (V I sample length), 0.17 V would correspond to a threshold
i
|
78
i
I
■ii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10
100
1000
2
£
o
1
0
25 mK
M259B7
B = 15T
105 mK O
210 mK. Q
0
X-LJ.
11
50 ------- .
A
40 - ( C)
30 -
4 20
10
0
10
100
1000
p
Figure 4.17
„
(pW)
Pav dependence for M259B7. B = 15 T. Temperatures are 25, 105, 155
and 210 mK for solid black, hollow, solid gray, and dot-centered symbols, respectively.
The solid lines are guide to the eye.
79
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50
40
30
a
B =13 T
T = 25 mK
20
0.0
0.1
0.3
0.2
V ( v o l t)
10'
10'
10'
E
E
10‘
a
’> w '
10
10'
io ;
o .o
0 .3
0.2
0 .4
V (volt)
i
j
Figure 4.18
Two terminal DC I-V characteristics o f M230B6 at 13 T, 25 mK. The
arrows indicate the jdc's at which the resonance data are shown in Figure 4.19.
80
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
j
j
j
field, E t, o f 34 V/m.
Threshold fields for 2DES exhibiting v = 1/5 FQHE in the
aforementioned experiments showed great disparity. 0.02 V/m at 11.8 T (v = 0.19), 44
mK [29] and 0.3 V/m at 12.7 T (v = 0.19), 22 mK [30] were observed in 4-terminal I-V
j
|
i
|
1
measurements. Ref. 33 reported E t = 80 V/m at 19 T (v = 0.15) and 40 mK in a 2terminal I-V measurement.
R e(c^) vs. f traces measured with DC voltages applied across the sample are
presented in Figure 4.19 for M230B6.
The resonance remains prominent for V well
beyond threshold, and over several orders o f magnitude in jdc■The peak broadens and its
height is reduced as jdc increases. f pk and S are plotted against jdc in Figure 4.20 (a) and
|
I
i
(c), respectively. Both remain constant with jdc up to about 300 pA/mm, though they
decrease at high current. The resonance features are plotted against Edc in Figure 4.21.
iI
I
\
All quantities show little or no dependence on Edc for Edc < E t- Above E t, Af Q and apk
|
become strongly dependent on Edc, while f pk and S remains constant across the threshold
|
and start decreasing mildly only at a much higher Edc-
i
i
!
4.3.4
Discussion
i
j
The gradual change o f the resonance features with T (Figure 4.12) is more
consistent with a continuous melting transition as T is raised.
;
The classical melting
temperature, given by Tcm= e2(nns) U2 / 4ns£oks(\30), is 415 mK for 5.4 x 1010 cm"2. The
observed "melting" T of 200 mK for M259B7 at 13 T ( v ~ 0.17), about half o f the Tcm
j
value, agrees with the transition T obtained in SAW experiments (« 200 mK at v ~ 0.17)
j
|
I
!
I
i
i
[54, 55], and in threshold conduction experiments (w 175 mK) [29]. Two characteristic
i
81
i
j
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
25
20
M230B6
13 T, 25 mK
jdc (pA/mm)
6.3
x
x
890
3150
0.0
1.0
2.0
f (GHz)
3.0
4.0
i
I
|
Figure 4.19
!
B = 13 T, T = 2 5 mK.
|
Refer/) v s . / o f M230B6 measured in the presence o f in-plane DC fields.
82
j
|
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.1
%
10
100
1000
~
mil
1.5
g
o
0.5
^
1
o
0
0.0
6
4
ri rini|---
o
> )
1 11111111 1 1
o
o
O)
- B= 13 T
T= 25mK
o
_
o
■ 1 1mill--- 1 L
- l. I J . llll_____
10
30
20
C/3
ZL
i.o
C/3
5
zL
I
O
CO
0
0.1
1
100
1000
j , (p A /m m )
Figure 4.20
Summary o f the effect o f DC current density on the resonance of
:
M230B6. B = 13 T, T = 25 mK. (a) f pk (circles) and A/(downward triangles) vs. jdc (b)
j
j
Q vs-jdc (c) S (squares) and crPk (upward triangles) vs .jdc-
ji
j
|
83
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
10
2
ooooo
5 1-0
1
o
0.5
0.0
0
6
O)
5 = 13T
T= 25mK
10
30
20
C/3
ZL
C/3
=L
5
:£
B
CO
0
100
10
s * (V/m)
Figure 4.21
Resonance features as a function of in-plane Edc field for M230B6. B =
13 T, T = 25 mK. (a)
(circles) and A f (downward triangles) vs .jdc (b) Q vs .jdc (c) S
(squares) and apk (upward triangles) vs.y*.
84
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
temperatures were reported in the dielectric constant measurement (To * 1 2 0 and 7 / « 220
mK for v * 0.18), which were found to agree, respectively, with the temperature above
j
ii
j
which the low f noise vanishes, and the temperature above which the DC conduction
nonlinearity vanishes [79]. It was argued that To corresponds to the loss o f positional
i
j
i
order and T/, orientational order. The agreement between Tj and our melting T may
suggest that the pinning mode does not have a stringent requirement on the positional
order o f the carriers.
The ever-increasing Q o f the 80 pW data with reducing T (Figure 4.13, 4.14)
indicates that the resonance is thermally broadened to our lowest temperature. Similarly,
I
it is broadened by Pm, as small as 5 pW at 25 mK (Figure 4.16). The fact that Q has not
'
I
saturated at such low temperature and low power is very intriguing, which possibly
j
reflects the nature o f the pinning potential, e.g., large anharmonicity. On the other hand,
j
|
that Q is very large in the T
0, Pav
I
the resonance does not come from inhomogeneous broadening. In Figure 4.17 (b), the
1
i
|
higher temperature traces of Q vs. Pav branch out o f the 25 mK trace, and then flatten at
low Pav. Extrapolating the data to Pav
;
for the 155 mK trace. The 25 mK data show logarithmic Pav dependence and can be
I
I
extrapolated to Q = 12.5 at 1 pW.
0 limit provides strong evidence that the width o f
0 gives Q = 4 for the 105 mK trace, and Q =2
At higher T, the resonance features become independent o f Pav when Pav is small.
1
For example, we see in Figure 4.17 that for T > 100 mK, apk and Q (in fact, the whole
Re(crxx) vs. / trace) do not depend on the microwave power for Pav < 300 pW. The T
dependence also becomes less significant with progressively larger Pav85
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The observations made in the previous two paragraphs indicate that T and Pm
have similar effect on the resonance. This argument is further supported by the data
!
|
j
Ii
!
ii
i
presented in Figure 4.22, where the trace measured with Pav = 500 pW at 25 mK is very
similar to the trace with 80 pW at 105 mK. It can be suggested that the effect o f high
electric field is simply heating the carriers to a temperature higher than T o f the bath.
Below is an attempt to visualize such a comparison.
Either .an increase in microwave electric field or in T can be thought as to cause
|
increase in a "vibrational amplitude" o f the charged carriers about their equilibrium
positions, so in that sense Pav and T are equivalent. For the T part, an estimate of this
|
I)
!
!
amplitude, u, can be obtained by setting (m*a>o2u2)/2 « kaT.
Using the GaAs hole
effective mass (0.37 m0) and the pinning frequency coo = 2.5 x 1011 rad/s at 13 T and 25
mK (from Figure 3.12), m«60A. Introduction o f such a length scale is motivated by
|
I
recent theories [75, 76], which proposed that the magnetic length Ib is essential in
|
understanding the anomalous B dependence o ff pk (section 3.3). Ib represents the spread
o f electric charge, and the disorder was modeled to be potential "pits" with a very short
|
range, rc «
a, and a pit density larger than ns. When Ib > rc, the energy gain from the
disorder potential decreases as Ib increases, since the charge density spreads further out of
the potential pit. Applying the conventional Fukuyama-Lee-Rice argument, i.e.,
balancing the disorder energy gain and elastic energy cost to obtain the optimal domain
i
size and the pinning frequency, it was found that co. decreases with increasing Ib when Ib
> rc. In our case, u replaces Ib to be the spread o f charge in the model, which predicts
|
I
j
that co. drops as T or Pav increases (u > rc assumed), consistent with the data. About Af: it
was suggested that the broadening of the resonance results from anharmonicity in the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
30
25
M259B7, B=15 T
• • • 25 mK, 500 pW
— 105 mK, 80 pW
X
X
*•
f (GHz)
Figure 4.22
Comparison o f two ReCcr**) v s./tra c e s o f M259B7 at 15 T: one measured
with Pm = 500 pW at T = 25 mK, the other with a smaller Pav (80 pW) but at higher T
(105 mK).
87
i
i
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
pinning potential [80], so the length scale argument may again be relevant because,
naively, anharmonicity is manifest at large amplitudes. An interesting implication is that
i
the largest length scale (among Ib,
|
u
from T, and u from Pav) would dominate and
i
|
determine the resonance characteristics, an idea that requires a lot more data to falsify.
i
|
About the jdc dependence data (Figure 4.19-Figure 4.21), the first question would
be: How would a pinning mode survive in a "depinned" condition? It is possible that the
,
DC current flow is filamentary, leaving the bulk o f the 2DHS unaffected by the DC
i
electric field. In this case, broadening o f the resonance with increasing jdc is consistent
!
with heating o f the sample by the DC current. On the other hand, our data may not rule
out a spatially uniform flow of the WC.
In the case o f the CDW, it has been
i
|
demonstrated by x-ray experiments [81] that CDW is not destroyed by electric field
,
larger than the threshold field, but is driven to slide as a whole, as originally suggested by
|
Frohlich [82]. The current exhibits the washboard oscillation, with a fundamental
|
frequency given by the ratio o f CDW drift velocity over the CDW wavelength. For the
WC case, one may define analogously a washboard frequency f , = Vd / a = jdc / nsea,
\
where Vd is the average drift velocity of the WC.
It has been observed that in the
presence o f a DC current, radio frequency Imfo**) v s . / shows minimum at / close to f w
[37, 38]. In our experiment, f w is 0.96 MHz for the largest jdc in Figure 4.18, which is
i
three orders o f magnitude less than/,*. Thus it is possible that the pinning mode survives
because the sliding is much slower than the pinning mode motion, and the energy gain for
the carriers from the DC field is too slight to modify the pinning energy.
i
j
j
|
!
!
88
ii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The threshold field E j is often compared with the pinning frequency
coq
by
assuming a sinusoidal pinning potential with period equal to carrier spacing [30, 36, 33,
79]. Depinning occurs when the external field exceeds the maximum pinning force, so
eET = m*coo2a/2iz [79]. With E r = 34 V/m for M230B6 at 13 T, a*2/ 2ncoc « 53 MHz,
much less than f pk.
This large discrepancy may originate from the simplistic model o f
the potential; on the other hand, the electric field inside the sample may be very different
from the nominal £ * . First, our sample geometry is far from ideal for applying a uniform
electric field across the entire sample. Secondly, the contact resistance in our 2 terminal
measurement could be large, and the real E j is most certainly smaller (This factor would
worsen the discrepancy, though). In order to make a meaningful comparison between
DC and microwave results, both 2-terminal and 4-terminal DC measurements have to be
done on a sample with improved geometry and, together with the microwave
experiments, at different B and T.
89
with permission of the copyright owner. Further reproduction prohibited without permission.
Chapter 5 Conclusions
]
I
j
The objective of this thesis is to study high quality 2DHS in the high magnetic
|
field limit, where an insulating phase terminates the FQHE. We measured the microwave
(
t
frequency conductivity o f such an insulator and observed a surprisingly pronounced
conductivity resonance in the GHz range. We studied the evolution o f the resonance with
several experimental parameters: magnetic field, carrier density, temperature, microwave
power and a simultaneous in-plane DC electric field. The results are compared to models
|
of the WC pinning mode. Although at this stage our understanding is far from complete,
i
the data suggests that the resonance comes from a disordered system o f interacting
i
particles.
I
:
j
Central to our study is the microwave measurement technique, which enables us
to quantitatively measure Re(cr^) over a wide / span, 0.2 - 10 GHz.
The coplanar
i
j
waveguide effectively couples microwaves into the 2DHS in the insulating phase, where
the DC resistance is extremely large.
The dual bolometer scheme ensures the
compatibility with the cryogenic system and good control on input microwave power. A
|
backgate on the sample serves as a means to change ns, and to deplete the carriers to
obtain a zero conductivity baseline.
The resonance only exists in the high B insulating phase, so a natural candidate
would be the pinning mode o f a disordered WC. The most surprising aspect o f our first
!
observation was the sharpness o f the resonance, with Q increasing linearly with B to
j
around 5 at 15 T. Also puzzling were the B dependences o f f Pk and S, both remain
|
constant at high B in our experiment. Recent theories suggest that due to the long range
j
90
i
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
|
Coulomb interaction between the oscillating domains in the B field, the inhomogeneous
|
broadening is eliminated; and that the unexpected B dependences come from length
|
|
scales unique to magnetically induced WC, such as disorder correlation length and
|
|
!
magnetic length.
Our density dependence study firmly established that carrier-carrier interactions
i
!
are essential in producing such a resonance, as would be in the case o f a WC.
The
difference in the details of the dependence between the two samples, on the other hand,
implies that disorder plays a crucial role. The upward shift in f Pk with reducing ns is more
i
j
consistent with a weak pinning model, in which domain size grows as ns increases.
j
When temperature is raised, the resonance weakens gradually, suggestive o f a
i
|
continuous "melting" transition, and above about 200 mK (at 13 T for M259B7) the
!
conductivity becomes Drude-like. At 25 mK, Q and crpk continues to increase as Pm is
reduced down to 5 pW (Emw « 0.8 V/m), indicating that the response is not in the linear
I
regime even at such a low power. Another intriguing aspect o f the resonance is that it
iI
|
survives in a large DC field.
The threshold field does leave its mark on Q and apk,
though: below threshold both quantities are essentially constant, but beyond threshold,
both decrease with increasing jdc■ Whether the broadened resonance comes from a
1
"pinning" mode o f a moving lattice would require further experiments, e.g., a MHz range
measurement to detect washboard oscillations [37, 38].
Given the vast, multi-dimensional parameter space, apparently there is much more
;
to be done even with the present setup. For example: (1) An experiment on low mobility
|
2D systems with no FQHE, or even with no IQHE. The insulating behavior o f such
|
samples is expected from single particle localization in random potentials, and the
!
91
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
microwave conductivity should be qualitatively different from the observed resonance in
high quality 2DHS with FQHE. (2) More samples with quality comparable to the ones
currently used are needed to study the sample dependence, such as the different f pk vs. ns
between M259 and M230. In addition, although the results from M259B5 and M259B7
are almost identical, the twin-peak observed at low ns in M230B6 might be sample
specific. (3) "Melting" transition driven by B, or by nSt should be further studied. (4)
"Melting" transition driven by T can be done at different ns, which would further
demonstrate the interaction effect. Preliminary data show that at a reduced density, the
resonance persists to T > 300 mK. (5) In relation to the recent theories o f weak pinning
[75, 76], a set o f experiments with different (B, T) could be done to test if the larger o f the
length scales corresponding to B and T would determine the WC characteristics, as
suggested in such models. (6) The jdc experiment can be repeated with a substantially
larger microwave power and observe the combined effect o f microwave and DC field,
e.g., if the effect o f dc threshold on Q changes from the case o f small Pav (section 4.3.3).
It is very desirable to improve the detection below 10 pW and hopefully one may
reach the linear response regime at low T. This could be done by using more sensitive
bolometric element, reducing thermal time constant and adopting a double modulation
scheme, i.e., adding additional modulating on ns to obtain the baseline simultaneously.
Complementary to the microwave measurement, detailed study o f the DC I-V on
the 2DHS samples would also be helpful for understanding o f the high B insulator.
Careful 4-terminal and 2-terminal DC measurements, as have been done in various
experiments on 2DES, can be taken together with the microwave measurement to study
the underlying pinning-depinning mechanism.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Bibliography
j
iI
|
[1]
K. v. Klitzing, G. Dorda and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).
j
[2]
D. C. Tsui, H. L. Stormer and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982).
[3]
R. B. Laughlin, Phys. Rev. B 23, 5632 (1981).
[4]
R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).
[5]
M. A. Paalanen, D. C. Tsui, A. C. Gossard
and J. C. M. Huang, Solid State
Commun. 50, 841 (1984).
|
l
|
[6]
|
[7]
R. E. Prange and S. M. Girvin (ed.), The Quantum Hall Effect (Springer-Verlag,
New York, 1990).
|
T. Chakraborty and P. Pietilainen, The Quantum Hall Effects (Springer-Verlag,
Berlin, 1995).
|
[8]
j
S. Das Sarma and A. Pinczuk (ed.), Perspectives in Quantum Hall Effects (John
Wiley & Sons, New York, 1997).
i
[9]
E. Wigner, Phys. Rev. 46, 1002 (1934).
[10]
Y. E. Lozovik and V. I. Yudson, JETP Lett 22, 11 (1975).
[11]
P. A. Lee, T. M. Rice, P. W. Anderson, Solid State Commun. 14, 703 (1974).
[12]
C. C. Grimes and G. Adams, Phys. Rev. Lett. 42,795 (1979).
[13]
D. S. Fisher, B. I.Halperin and P. M. Platzman, Phys. Rev. Lett.42, 798 (1979).
[14]
P. K. Lam and S. M. Girvin, Phys. Rev. B 30, 473 (1984).
j
[15]
D. Levesque, J. J. Weis and A. H. MacDonald, Phys. Rev. B 30, 1056 (1984).
j
[16]
H. Fukuyama and P. A. Lee, Phys. Rev. B 17, 535 (1978); 18, 6245 (1978).
|
|
93
l
I
Ii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[17] P. A. Lee and T. M. Rice, Phys. Rev. B 19, 3970 (1979).
[18] B. G. A. Normand, P. B. Littlewood and A. J. Millis, Phys. Rev.
B 46, 3920
(1992).
[19]
I. M. Ruzin, S. Marianer and B. I. Shklovskii, Phys. Rev. B 46, 3999 (1992).
[20]
S. T. Chui, J. Phys. Condens. Matter 5, L405 (1993).
[21]
M. Shayegan, Case fo r the Magnetic-Field-Induced Two-Dimensional Wigner
Crystal, in ref. [8].
[22]
H. W. Jiang, R. L. Willett, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer and K. W.
West, Phys. Rev. Lett. 65, 633 (1990).
[23]
B. I. Halperin, Phys. Rev. Lett. 52, 1583 (1984).
[24]
M. B. Santos, Y. W. Suen, M. Shayegan, Y. P. Li, L. W. Engel and D.C. Tsui,
Phys. Rev. Lett. 68, 1188 (1992); M. B. Santos, J. Jo, Y. W. Suen, L. W. Engel
and M. Shayegan, Phys. Rev. B 46, 13639 (1992).
[25]
X. Zhu and S. G. Louie, Phys. Rev. Lett 70, 335 (1993).
[26]
R. Price, P. M. Platzman and Song He, Phys. Rev. Lett. 70, 339 (1993)
[27]
K. Hirakawa, Y. Zhao, M. B. Santos, M. Shayegan and D. C. Tsui, Phys. Rev. B
47, 4076(1993).
[28]
R. L. Willett, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. West, M. Shayegan,
M. B. Santos and T. Sajoto, Phys. Rev. B 40, 6432 (1989).
[29]
V. J. Goldman, M. B. Santos, M. Shayegan and J. E. Cunningham, Phys. Rev.
Lett. 65,2189(1990).
i
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[30]
|
Y. P. Li, T. Sajoto, L. W. Engel, D. C. Tsui and M. Shayegan, Phys. Rev. Lett.
67,1630(1991).
j
[31]
T. Sajoto, Ph. D. thesis, Princeton University (1993).
[32]
H. W. Jiang, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer and K. W. West, Phys. Rev.
j
B 44, 8107 (1991).
[33]
F. I. B. Williams, P. A. Wright, R. G. Clark, E. Y.Andrei, G.Deville, D. C.
Glattli, O. Probst, B. Etienne, C. Dorin, C. T.Foxon and J. J. Harris, Phys. Rev.
i
I
I
i
|
[34] S. T. Chui, Phys. Lett. A 180, 149 (1993).
|
[35]
Lett. 66, 3285 (1991).
|
|
f
j
!
iI
I
i
I
Y. P. Li, D. C. Tsui, T. Sajoto, L. W. Engel, M. Santos and M. Shayegan, Solid
State Commun. 95, 619 (1995).
[36] G. Griiner, Rev. Mod. Phys. 60, 1129 (1988).
[37]
Y. P. Li, D. C. Tsui, L. W. Engel, M. B. Santos, T. Sajoto and M. Shayegan, Solid
State Commun. 96, 379 (1995).
[38] Y. P. Li, D. C. Tsui, L. W. Engel, M. B. Santos and M. Shayegan, Solid State
Commun. 99, 255 (1996).
[39] S. Bhattacharya, J. P. Stokes and M. J. Higgins, Phys. Rev. B 43, 1835 (1991).
l
[40]
S. N. Coppersmith and P. B. Littlewood, Phys .Rev. B 31, 4049 ( 1985).
[41]
T. A. Kennedy, R. J. Wagner, B. D. McCombe and D. C. Tsui, Solid State
Commun. 21, 459 (1977).
|
[42] B. A. Wilson, S. J. Allen and D. C. Tsui, Phys. Rev. B 24, 5887 (1981).
I
j
95
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
j
[43]
M. Besson, E. Gomik, C. M. Engelhardt and G. Weimann, Semicond. Sci.
Technol. 7, 1274 (1992).
i
!
[44]
!
I
G. M. Summers, R. J. Warburton, J. G. Michels, R. J. Nicholas, J. J. Harris and C.
T. Foxon, Phys. Rev. Lett. 70, 2150 (1993).
i
[45]
Chalker, Phys. Rev. Lett. 72, 2057 (1994).
[46]
H. Buhmann, W. Joss, K. von Klitzing, I. V. Kukushkin, A. S. Plaut, G. Martinez,
K. Ploog and V. B. Timofeev, Phys. Rev. Lett. 66, 926 (1991).
|
[47]
i
I. V. Kukushkin, N. J. Pulsford, K. von Klitzing, K. Ploog, R. J. Haug and V. B.
Timofeev, Phys. Rev. B 45,4532 (1992).
I
[48]
E. M. Goldys, S. A. Brown, R. B. Dunford, A. G. Davis, R. Newbury, R. G.
|
Clark, P. E. Simmonds, J. J. Harris and C. T. Foxon, Phys. Rev. B 46, 7957
\
(1992).
!
[49]
!
|
E. Y. Andrei, G. Deville, D. C. Glattli, F. I. B. Williams, E. Paris and B. Etienne,
Phys. Rev. Lett. 60, 2765 (1988).
[50]
H. L. Stormer and R. L. Willett, Phys. Rev. Lett. 62, 972 (1989).
[51]
E. Y. Andrei, G. Deville, D. C. Glattli, F. I. B. Williams, E. Paris and B. Etienne,
i
i
j
j
!
Phys. Rev. Lett. 62, 973 and 1926 (1989).
[52]
H. L. Stormer and R. L. Willett, Phys. Rev. Lett. 68, 2104 (1992).
[53]
F. I. B. Williams, G. Deville, D. C. Glattli, E. Y. Andrei, B. Etienne and E. Paris,
|
|
|
j
Phys. Rev. Lett. 68, 2105 (1992).
[54]
M. A. Paalanen, R. L. Willett, P. B. Littlewood, R. R. Ruel, K. W. West, L. N.
Pfeiffer and D. J. Bishop, Phys. Rev. B 45, 11342 (1992).
96
j
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
i
|
|
[55]
M. A. Paalanen, R. L. Willett, R. R. Ruel, P. B. Littlewood, K. W. W est and L. N.
1
[56]
S. T. Chui, J. Phys. Condens. Matter 5, L405 (1993).
[57]
A. J. Millis and P. B. Littlewood, Phys. Rev. B 50, 17632 (1994).
Pfeiffer, Phys. Rev. B 45, 13784 (1992).
i
[58] L. W. Engel, C.-C. Li, D. Shahar, D. C. Tsui and M. Shayegan, Solid State
I
Commun. 104, 167(1997).
[59] C.-C. Li, L. W. Engel, D. Shahar, D. C. Tsui and M. Shayegan, Phys. Rev. Lett.
I
79,1353 (1997).
[60]
W. I. Wong, E. E. Mendez and I. Iye, J. Appl. Phys. 60, 1834 (1986).
[61]
C. P. Wen, IEEE Trans. Microwave Theory and Tech., M TT-17, 1087 (1969).
[62]
G. Ghione and C. U. Naldi, Electronics Letters 20, 179 (1983); IEEE Trans.
Microwave Theory and Tech., M TT-35, 260 (1987).
[63]
i
|
T. Itoh (ed.), Planar Transmission Line Structures
(IEEE Press, New York,
1987).
[64]
S. Y. Liao, Microwave Devices and Circuits (Prentice Hall, Englewood Cliffs,
1990).
[65]
M. Gillick, I. D. Robertson and J. S. Joshi, IEEE Trans. Microwave Theory and
Tech., MTT-41, 1606 (1993).
[66]
S. F. Adam, Microwave Theory and Applications (Prentice Hall, Englewood
Cliffs, 1969).
|
[67]
M. M. Fogler, private communications.
[68]
S. Kivelson, D.-H. Lee and S. C. Zhang, Phys. Rev. B 46, 2223 (1992).
I
|
|
97
*
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[69]
See, for example, P. B. Littlewood, Phys. Rev. B 36, 3108 (1987).
[70]
L. Bonsall and A. A. Maradudin, Phys. Rev. B 15, 1959 (1977).
[71]
M. M. Fogler, private communication. The conductivity o f the 2DHS affects the
q o f the electric field.
j
[72]
H. Aoki, J. Phys. C, 12, 633 (1979).
[73]
R. N. Bhatt and T. V. Ramakrishnan, J. Phys. C 17, L639 (1984).
[74]
M. Ferconi and G. Vignale, Phys. Rev. B 48,2831 (1993).
[75]
R. Chitra, T. Giamarchi and P. Le Doussal, Phys. Rev. Lett.80, 3827 (1998).
[76]
H. A. Fertig, Phys. Rev. B 59, 2120 (1999).
[77]
U. Wulf, Phys. Rev. B 59, 6700 (1999).
[78]
M. M. Fogler and D. A. Huse, cond-mat/9904245 (1999).
[79]
Y. P. Li, Ph. D. thesis, Princeton University (1993).
[80]
H. Yi and H. A. Fertig, cond-mat/9905350 (1999).
[81]
R. M. Fleming, D. E. Moncton and D. B. Me Whan, Phys. Rev. B 18, 5560 (1978)
[82]
H. Frohlich, Proc. R. Soc. London A 223, 296 (1953).
98
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Документ
Категория
Без категории
Просмотров
0
Размер файла
3 476 Кб
Теги
sdewsdweddes
1/--страниц
Пожаловаться на содержимое документа