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Microwave spectroscopy of edge and bulk modes of two dimensional electrons in magnetic field

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THE FLORIDA STATE UNIVERSITY
COLLEGE OF ARTS AND SCIENCES
MICROWAVE SPECTROSCOPY OF EDGE AND BULK MODES OF TWO
DIMENSIONAL ELECTRON SYSTEMS IN MAGNETIC FIELD
By
BRENDEN A. MAGILL
A Dissertation submitted to the
Department of Physics
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Degree Awarded:
Spring Semester, 2013
UMI Number: 3564920
All rights reserved
INFORMATION TO ALL USERS
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a note will indicate the deletion.
UMI 3564920
Published by ProQuest LLC (2013). Copyright in the Dissertation held by the Author.
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Brenden A. Magill defended this dissertation on November, 19, 2012.
The members of supervisory committee were:
Lloyd Engel
Professor Directing Dissertation
Geoff Strauss
University Representative
Nick Bonesteel
Professor Co-Directing Dissertation
Peng Xiong
Committee Member
Ingo Wiedenhoever
Committee Member
The Graduate School has verified and approved the above-named committee members,
and certifies that the dissertation has been approved in accordance with university
requirements.
ii
This dissertation is dedicated to my father, who raised me to be persistent and
taught me to be a gentleman.
iii
ACKNOWLEDGMENTS
There were a great many people who helped me become the person who I am today.
I want to start by thanking my parents Michael Magill, Mary Cooney, and my stepmother Lucinda Phobes. They have all been there when I have needed them and I
have learned a great deal from them. From my Dad I learned how to be persistent
and not quit. My mom kindled in me a love of math and taught me to back up my
statements. I remember being ten and her asking me if I could document that statement and having to walk away only to return a week later with a statistic to back up
my claim. Even though I do not recall now what claim I had made, the lesson stuck
with me. I want to thank my stepmother for teaching me how important it is to learn
the rules and work within a bureaucracy instead of just railing against it.
Over the course of my education I have had the pleasure of meeting and learning
from a lot of wonderful teachers. From my undergraduate I want to thank Saul Oseroff
who took me on as his assistant and then as a student He was responsible for me
getting to do research on super-heavy fermion superconductors as an undergraduate
and inculcating in me a love for lab work. From him I will never forget the lesson
that the disorder in a system is never zero and often is of fundamental importance to
the physics of the system. I want to thank Richard Morris who kindled in me a desire
to be an experimentalist and taught me to set up a good experiment and showed
me that I could respond to tough challenges. In graduate school I want to start by
thanking my advisor Lloyd Engel who has gone above and beyond the call of duty in
teaching me and molding me as a physicist. He has worked hard to teach me to not
jump to conclusions and how to evaluate whether an experiment has succeeded or not.
iv
I also want to thank him for standing by me through some tough projects. I want
to thank Nicholas Bonesteel for being my advisor on record with FSU and putting
up with all the administrative hassles that brought. I want to thank the post docs,
Rupert Lewis, Sanbandamurthy Granpathy, Pei-Hsun Jiang, Byoung Hee Moon, and
Anthony Hatke who were wonderful to work with and learn from. Murthy I want to
thank for teaching me what it takes to survive as an experimentalist. Byoung I want
to thank for teaching me about cryogenics and the dilution refrigerator. I want to
give special thanks to Anthony for helping edit my thesis and for helping me finish
up some of the work included here.
I would like to thank the students I have worked with in the Engel group: Yong
Chen, Zhigong Zhang, Han Zhu, and Shantanu Chakraborty. Han I want to thank
for the good times working with him and the company late in the night when we were
the only people in C120. Shanto I want to thank for being an eager collaborator on
the WQW experiment that the second half of this dissertation is about. Without him
I would not have been able to get so much done so quickly and I likely would not be
writing this dissertation this year.
I also want to thank all of my friends and classmates. Aaron Wade who I took
all of my classes with, worked with on many projects, and who I will be pumping
on information about Terahertz measurements and self assembled metal nano rods. I
want to thank Adrienne Serra, Katherine Eastman, and Melissa Hall for helping me
edit and being there to say that paragraph looks really awkward can you try to make
it a little less horrible. Finally, I would like to thank my wife Tadja. She has been
my rock through many years of graduate school, she has supported me through hard
work that had be coming home tired and confused, through technical problems that
had me pulling my hair and gnashing my teeth, and through setback that had me
moping and miserable. Without her I would not be the happy and presumably well
adjusted person I am today.
v
TABLE OF CONTENTS
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
1 Introduction
1.1 Spectroscopy of two regimes . . . . . . . . . . .
1.2 Magnetic edge magnetoplasmons . . . . . . . .
1.3 A new solid phase in “wide” quantum well two
systems . . . . . . . . . . . . . . . . . . . . . .
1.4 Thesis overview . . . . . . . . . . . . . . . . . .
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dimensional
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2 Magnetic edge magnetoplasmons
2.1 Introduction . . . . . . . . . . . . . . . . . . . .
2.2 Plasmons and Magnetoplasmons . . . . . . . . . .
2.3 Edge Magnetoplasmons . . . . . . . . . . . . . . .
2.4 Inter-Edge Magnetoplasmon . . . . . . . . . . . .
2.5 A Semi-Classical Model of Magnetic Confinement
2.6 Magnetic Edge Magnetoplasmon . . . . . . . . . .
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3 Magnetic Edge Magnetoplasmons – Experimental
sults
3.1 Introduction to hybrid ferromagnet 2DES devices . .
3.2 Microwaves connections and mounting . . . . . . . .
3.3 Introduction to gate controlled non-resonant devices .
3.4 Sample characteristics . . . . . . . . . . . . . . . . .
3.5 NRMEMP device fabrication and design . . . . . . .
3.6 NRMEMP devices . . . . . . . . . . . . . . . . . . .
3.7 NRMEMP device results . . . . . . . . . . . . . . . .
3.8 NRMEMP Summary . . . . . . . . . . . . . . . . . .
3.8.1 A simple EMP model with nonzero ∂Bz /∂x .
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Setup and Re.
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4 Wigner crystals and other electron solids
4.1 The Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Wigner Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Pinned Wigner Crystals . . . . . . . . . . . . . . . . . . . . . . . . .
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4.4
4.5
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Integer quantum Hall Wigner crystals . . . . . . . . . . . . . . . . . .
Other electronic solids . . . . . . . . . . . . . . . . . . . . . . . . . .
The reentrant integer quantum Hall effect in 2DES . . . . . . . . . .
5 Wide Quantum Wells
5.1 Introduction . . . . . . . . . . . . . . .
5.2 Two subbands . . . . . . . . . . . . . .
5.3 Measuring the subband separation (Δ)
5.4 Δ in the vicinity of ν = 1 . . . . . . .
5.5 A RIQHE near ν = 1 in WQW . . . .
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34
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6 Microwave spectroscopy of the RIQHE around ν = 1 in WQW 2DES
6.1 Samples and Sample Preparation . . . . . . . . . . . . . . . . . . . .
6.2 Microwave Measurement Technique – CPW . . . . . . . . . . . . . .
6.3 54 nm WQW – microwave spectroscopy of pinning mode resonances
around ν = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Sample 54-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Sample 54-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 54 nm WQW – summary . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 42 nm WQW – pinning mode resonances around ν = 1 . . . . . . . .
6.5.1 42 nm WQW – symmetric charge distribution . . . . . . . . .
6.5.2 42 nm WQW – asymmetric charge distribution . . . . . . . .
6.5.3 42 nm WQW – summary . . . . . . . . . . . . . . . . . . . . .
6.6 Summary and analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
47
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48
7 Significance and future work
7.1 EMPs in magnetic field gradients and MEMPs . . . . . . . . . . . . .
7.1.1 Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.2 Future work, EMPs in magnetic field gradients and MEMP search
7.2 A new electron solid in a WQW . . . . . . . . . . . . . . . . . . . . .
7.2.1 Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.2 Future work in WQW 2DES . . . . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
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Biographical Sketch
93
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vii
51
51
56
61
63
64
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68
71
LIST OF FIGURES
1.1
2.1
2.2
2.3
3.1
3.2
3.3
Example of a pinning mode resonance in an ultra high mobility 65 nm
quantum well. Plotted are spectra showing the real part of the diagonal conductivity (Re[σxx ]) vs frequency, with the density (n) for each
spectrum labeled on the graph. Adapted from Ye et al. [1] . . . . . . .
3
Comparison of the EMP and magnetoplasmon modes for a 10 μm disk
of GaAs semiconductor 2DES with n = 2.8 × 1011 cm−2 . The higher frequency branch on the graph is the magnetoplasmon mode (labeled MP),
the lower frequency branch the EMP (labeled EMP), and the dashed line
the cyclotron frequency (labeled CR). . . . . . . . . . . . . . . . . . .
8
“Skipping” orbit depicted as a series of broken cyclotron orbits confined
to the edge of a sample, n = 0. . . . . . . . . . . . . . . . . . . . . . .
12
a) “snake” orbit depicted as a succession of cyclotron orbits reversing direction where Bz inhomogenously changes from B0 to −B0 . b) “cycloid”
orbit depicted as a series of cyclotron orbits tightening in radius in the
region where Bz is increased by Bm . . . . . . . . . . . . . . . . . . . .
13
Illustration of NRMEMP device the 2DES mesa grey and the contacts
dark grey, all of the features on the are device to scale except for the ends
of the microwave antennas are truncated. Note: the slight overlap of the
tips of the microwave antennas over the edge of the mesa containing the
2DES is not shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
All four device configurations for positive B. For each configuration the
charged gates are colored and the grounded ones are white. The red
dotted line shows the path of the EMP and the blue dashed line shows
the intended path of the MEMP. . . . . . . . . . . . . . . . . . . . . .
20
All four device configurations for negative B. For each configuration the
charged gates are colored and the grounded ones are white. The red
dotted line shows the path of the EMP. . . . . . . . . . . . . . . . . .
21
viii
3.4
3.5
4.1
4.2
4.3
4.4
4.5
4.6
5.1
Transmitted power in dB vs B taken at 4 K. Gate that the EMP is
incident, Py (permalloy) for positive B and Au for negative B on the
graph with the on voltage the gate relative to the 2DES in parentheses
next to it. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
Transmission dip number j (i.e. harmonic) plotted vs B, gate configuration number listed in the caption. The EMP launched from the
transmission antenna is incident on the Au gate for B < 0 and the (Py)
permalloy gate for B > 0 as marked on the graph, lines on the graph
are least squares fits. Configuration numbers and frequencies are shown
on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Density of states for a 2DES showing disorder-broadened LL spectrum.
g the Lande g-factor and μB the Bohr magneton. The Fermi level is as
shown for the LL filling factor ν = 3. . . . . . . . . . . . . . . . . . . .
30
Spectra showing the real part of the diagonal conductivity (Re[σxx ]) vs
frequency, f , at many filling factors ν offset vertically proportional to ν.
Successive spectra are separated by steps of 0.01 in ν ν for black spectra
are marked at right, in a 30 nm wide quantum well 2DES, and at 30
mK. Adapted from [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
Example of S/fpk plotted vs ν for an IQHEWC pinning mode resonance
in a 30 nm wide quantum well 2DES at 50 mK. The black lines are the
prediction from the sum rule Eq. 4.2 for full quasi-carrier participation.
Apdated from Chen et. al. [2]. . . . . . . . . . . . . . . . . . . . . . .
34
Cartoon of the progression from an IQHEWC to a bubble phase WC and
then a stripe phase WC as ν ∗ increases, with M the number of electrons
per lattice point in the WC. . . . . . . . . . . . . . . . . . . . . . . . .
35
Re[σxx ] vs f for a very high mobility, μ = 24 × 106 cm2 V −1 s−1 , n =
1.0 × 1011 cm−2 , 50 nm QW, and a temperature of 30 mK. Resonance A
shows dispersion with wave vector. Adapted from Chen et. al. [3]. . .
36
Example of the RIQHE observed by Cooper et. al. [4] near ν = 4 as ν
increases with the RIQHE marked by the arrow, a) Rxy in units of h/e2 ,
and b) Rxx in units of Ohms. . . . . . . . . . . . . . . . . . . . . . . .
38
Charge distribution (ρ) on top in red, ground (ΨS in green) and first
excited state (ΨAS in blue) wave functions on bottom. From simultaneous calculations of the Poisson and Schroedinger equations. Difference
in density between the top half and the bottom half of the well (δn)
on top. n in 1011 cm−2 and subband separation (Δ) in K on bottom.
Provided by Yang Liu. . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
ix
5.2
Re[σxx ] in μS plotted vs B at a frequency of 0.5 GHz, from a 54 nm
WQW, a temperature of 30 mK, and an as cooled n = 2.41 × 1011 cm−2 . 43
5.3
B frequency peaks from the Fourier transform of the SdH oscillations
with the B frequencies of the lowest lying state, the first excited state,
the sum of both, and the difference of both marked on the figure. . . .
44
Fan diagram for a 54 nm WQW (left axis) for Δ = 27.6 K. The energies
of the N = 0 Landau level from the lowest lying (S0) and first excited
(A0) subbands are plotted with their spin directions shown. On the right
axis a Re[σxx ] vs B trace taken at 500 MHz is plotted for n = 2.43 × 1011
cm−2 and the same Δ. ν = 1 is labeled on the figure. . . . . . . . . . .
45
Schematic of a coplanar waveguide on the surface of a sample, with
metal in black. The 2DES is a fraction of a micron under the surface of
the sample. An Agilent network analyzer is the source and the receiver
of the microwave signal. . . . . . . . . . . . . . . . . . . . . . . . . . .
48
Gate configurations for the WQW samples. a) back gate only: samples
54-1 and 54-2. b) front and back gate with glass spacer and increased
CPW slot width: sample 42-1. c) front and back gate with etched glass
spacer: sample 54-3. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
Real part of diagonal conductivity (Re[σxx ]) vs B for sample 54-1 taken
at a frequency of 500 MHz. The temperature is 30 mK and the density
is 2.41 × 1011 cm−2 . The sample has no front gate, and there was no
voltage bias on the back gate. A selection of IQHE and FQHE minima
are labeled with their Landau filling factors. . . . . . . . . . . . . . . .
52
Δ vs n for the 54 nm WQW, illustrating how the subband separation
changes as we change n with the back gate. n = 2.41 × 1011 cm−2 is the
as-cooled density with no gate bias. . . . . . . . . . . . . . . . . . . . .
53
Re[σxx ] vs B for n = 1.94 × 1011 cm−2 to n = 3.20 × 1011 cm−2 , each
trace vertically offset for clarity. Taken at ∼ 30 mK and a frequency of
0.5 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
a) Spectra from sample 54-1 showing the real part of the diagonal conductivity (Re[σxx ]) vs frequency taken at many ν. ν of blue traces is
marked on the right, at a temperature of 30 mK, n as-cooled, and normalized to ν = 1 with each spectrum proportionally offset vertically from
the last and the filling factor step between each spectra is ν = 0.005. b)
The peak frequency of the pinning mode resonance (fpk ) vs. ν. c) S/fpk
vs ν. The black lines are the theoretical prediction for full participation
in the resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
5.4
6.1
6.2
6.3
6.4
6.5
6.6
x
6.7
Waterfalls of spectra from sample 54-1 show the real part of the diagonal
conductivity (Re[σxx ]) vs frequency taken at many ν. ν of blue traces
marked on the right, normalized using a ν = 1 spectrum, n marked at
the top of each waterfall, and each is spectrum is proportionally offset
vertically. a) and b) The step in ν between spectra is 0.01. c) The step
in ν between spectra is 0.005. There was no data taken above ν = 1.05
for n = 2.17 × 1011 cm−2 . . . . . . . . . . . . . . . . . . . . . . . . . .
56
Waterfalls of spectra from sample 54-1 showing the real part of the
diagonal conductivity (Re[σxx ]) vs frequency, taken at many ν. ν of
blue traces marked on the right, 30 mK, normalized using a ν = 1
spectrum, n marked at the top of each waterfall, and each spectrum is
proportionally offset vertically. a) and b) The step in ν between spectra
is 0.01. c) The step in ν between spectra is 0.005. . . . . . . . . . . . .
57
6.9
fpk in GHz vs ν as n is increased, from Figs. 6.7 and 6.8. . . . . . . . .
58
6.10
Sample 54-1, S/fpk vs. LL filling factor for all measured n. (See section
4.3.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
a-c) Re[σxx ] vs f at many ν and T = 30 mK. ν of black traces marked
on the right. a) Spectra from ν = 0.8 to 1.2 with a ν step between each
spectrum of ν = 0.0143. b) Spectra from ν = 0.8 to 1.0 with a step of
ν = 0.0111 between spectrum. c) Spectra from ν = 1.0 to 1.2 with a
step of ν = 0.0091 between each spectrum. . . . . . . . . . . . . . . . .
60
Image plots of f vs ν with the color representing Re[σxx ] for a variety
of n. n listed on in image plot in units of 1011 cm−2 . . . . . . . . . . .
61
fpk vs ν, n is listed along the right of the plot in units of 1011 cm−2 .
All fp k taken from spectra taken while maintaining symmetric charge
distribution in the well. . . . . . . . . . . . . . . . . . . . . . . . . . .
62
Location of the transition to a higher frequency range in ν vs n for fpk
vs ν for samples 54-1 and 54-3 for ν < 1. . . . . . . . . . . . . . . . . .
63
Sample 42-1, 50 mK, n = 3.04 × 1011 cm−2 , and no bias on either gate.
a) Many spectra taken at a different ν normalized to ν = 1. ν is marked
on the left of the graph for black spectra, spectra are proportionally
vertically offset, and the step between spectra is ν = 0.00625. b) fpk vs.
ν. c) S/fpk vs. ν. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
6.8
6.11
6.12
6.13
6.14
6.15
xi
6.16
6.17
6.18
6.19
Sample 42-1, waterfalls of spectra taken at many ν. ν of black traces are
shown on the left, offset vertically proportional to ν, taken at 50 mK,
normalized to ν = 1, at similar Δ, with n at the top of each waterfall,
and a step of ν = 0.00625 between spectra. . . . . . . . . . . . . . . . .
66
Sample 42-1 fpk vs. ν, traces are offset vertically by 0.5 GHz per trace
with n = 3.58 not offset, and n is in units of 1011 cm−2 . . . . . . . . . .
67
Sample 42-1, waterfalls of spectra taken at 50 mK, normalized to a
ν = 1 spectrum. ν of black traces is shown on the left, offset vertically
proportional to ν, a step of ν = 0.00625 between spectra, and n 3.25 × 1011 cm−2 . a) Asymmetric charge distribution, with gates set to
give δn = 0.22 × 1011 cm−2 . b) Symmetric charge distribution, gates
balanced. c) Asymmetric charge distribution, gates set to give δn =
−0.22 × 1011 cm−2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
Image plots of f vs. ν with the intensity of the plot representing Re[σxx ]
from sample 42-1. n and δ in units of 1011 cm−2 as marked on each plot;
plots on the left have symmetric charge distribution in the well (δn 0);
plots right of the thick vertical line have asymmetric charge distribution
with postive δn corresponding to more charge added by the front gate.
69
6.20
Sample 42-1, with fpk vs. ν on the left, and S/fpk vs. ν on the right.
Symmetric states are black; non-symmetric with the voltage biased more
on the back gate in blue, non-symmetric with the voltage biased towards
the front gate in red; with the densities on the graph in units of 1011 cm−2 . 70
6.21
Sample 54-1, (Re[σxx ] vs f ) taken at n = 2.41 × 1011 cm−2 , and various
temperatures as marked on graph. a) taken from a region of ν near
ν = 1 where fpk is not enhanced. b) from a region of ν where fpk is
enhanced. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
fpk vs. ν of the pinning mode for samples 54-1 (circles) and 54-2
(squares). Both from spectra taken at n = 2.41 × 1011 cm−2 . . . . . .
75
6.22
xii
ABSTRACT
Edge magnetoplasmons [5] (EMPs) and pinning mode resonances [6] in two dimensional electron systems (2DESs) can both be thought of as lower hybrid modes of
cyclotron and plasma resonances. This dissertation describes low temperature microwave spectroscopy of both of these modes. EMPs have oscillating charge confined
at the 2DES edge by the combination of the perpendicular magnetic field and the
electrostatic potential that produces the edge. Pinning mode resonances are from
electron solids oscillating against confinement provided by disorder in the bulk of the
2DES.
The first part of this dissertation concerns the search for a mode similar to an
EMP but confined solely by a linear magnetic inhomogeneity in the perpendicular
magnetic field (Bz ). While we do not observe such an excitation, we do observe a
marked reduction in the velocity of an EMP in the presence of a Bz -inhomogeneity.
In the second part of this dissertation, we investigate pinning modes in “wide”
quantum well samples, for which the effective electron-electron interaction is softened
at short range due to the vertical extent of the wavefunction. We observe a pinning
mode resonance whose peak frequency (fpk ) vs Landau level filling (ν) shows an
anomalous increase as ν moves away from ν = 1 under roughly the same conditions
as anomalous quantum Hall effects observed previously in DC transport [7]. A region
of ν with enhanced fpk is interpreted as evidence for a new electron solid phase.
xiii
CHAPTER 1
INTRODUCTION
1.1
Spectroscopy of two regimes
This dissertation covers two principal topics, linked by an experimental technique,
low temperature microwave spectroscopy of two-dimensional electron systems (2DES)
in magnetic field. The first topic, concerning the edges of a 2DES, is a study of
charge density (plasma) excitations (edge magnetoplasmons, EMPs) at the edge of
a 2DES in a spatially varying perpendicular magnetic field (Bz ). This arises from a
search for a charge density excitation bound to an inhomogeneity in Bz (a “magnetic”
edge magnetoplasmon or MEMP). The second topic focuses on electron solid phases
found at higher magnetic field and milikelvin temperatures, in the 2DES bulk in
comparatively wide quantum wells. Since the two principal topics in this dissertation
are different, each section will include its own literature review.
1.2
Magnetic edge magnetoplasmons
Charge density excitations in 2D electron or hole systems include plasmons, magnetoplasmons, and of most relevance to this dissertation, edge magnetoplasmons
(EMPs). EMPs are charge density excitations (with a frequency below the cyclotron
frequency) that propagate along the edge of a 2DES [5] in a perpendicular magnetic
field (Bz ). The oscillating charge density of an EMP is confined by the combination
of Bz and the static electric potential that causes the electron density in a 2DES to
1
go to zero at its edge. Steps or gradients in Bz (x) can also confine electron orbits
without any electrostatic potential [8]. In this dissertation, we describe a search for
“magnetic edge magnetoplasmons” (MEMPs), charge density excitations analogous
to EMPs, but confined by only such a Bz -inhomogeneity.
To create a magnetic inhomogeneity at the 2DES we pattern ferromagnetic film
onto the surface of the sample. These hybrid ferromagnet-semiconductor devices are
designed to allow us both to launch EMPs and to search for MEMPs. While we found
no evidence of a MEMP, which by definition would be confined solely by a magnetic
inhomogeneity, we observed a marked decrease in the velocity of an EMP moving
along an edge with both vanishing density and a Bz gradient. The magnitude of the
velocity decrease is reasonable when compared to the result of a simple EMP model,
considering what we know about the magnetic properties of the film.
1.3
A new solid phase in “wide” quantum well
two dimensional electron systems
An electron solid, such as a Wigner crystal [9, 10], is a lattice formed by carriers to
minimize their mutual repulsion. Such solids are pinned by disorder and are insulators.
At high magnetic fields the quantum Hall effect series of a 2DES is terminated at high
Bz by an insulating phase [11]. In samples of sufficiently low disorder, the microwave
spectrum of this insulating phase exhibits a resonance that is interpreted as a pinning
mode of an electron solid [12]. The resonance frequency (fpk ) is increased by increasing
the effective strength of the disorder pinning the solid, and decreased by increasing
the stiffness (shear modulus) of the electron solid [13]. Fig. 1.1 shows an example
of the peak frequency of the pinning mode resonance changing as the density of the
2DES is increased. Pinning modes have also been observed for Wigner solids in the
regime of the integer quantum Hall effect [2], for which they have been interpreted as
due to a Wigner crystal in the Landau filling (ν) range of the integer quantum Hall
2
effect, an integer quantum Hall effect Wigner crystal (IQHEWC).
Figure 1.1: Example of a pinning mode resonance in an ultra high mobility 65
nm quantum well. Plotted are spectra showing the real part of the diagonal
conductivity (Re[σxx ]) vs frequency, with the density (n) for each spectrum
labeled on the graph. Adapted from Ye et al. [1]
In “wide” quantum wells (WQW), the increased vertical extent of the electron
wavefunction reduces the effective electron-electron interaction at short range. The
softening of the short range interactions affects the competition between states in
which this interaction plays a crucial role, including electron solids and also the fractional quantum Hall effects (FQHE) [14]. A reentrant integer quantum Hall effect
(RIQHE), in which an integer quantum Hall plateau vanishes then reappears as ν
changes, has been observed by Liu et al. [7] in DC magnetotransport measurements.
They observed the ν of this RIQHE move in toward ν = 1 as they increased the density of the electrons in the 2DES. They proposed that this RIQHE is a manifestation
3
of a new electron solid.
In WQW 2DES we observe a transition in fpk vs ν, with a region of enhanced fpk
appearing at ν and densities similar to those found for the RIQHE in Ref. [7]. We
ascribe the enhanced fpk to a new electron solid distinct from the IQHEWC previously
studied in samples at lower density [2].
We consider two possibilities for the composition of this unknown electron solid.
The first is an electron solid that has some vertical order similar to bilayer electron
solids [15–17]. The second, and more likely possibility is that it is an electron solid
composed of composite fermions (CFs). A CF is an exotic particle that figures importantly into descriptions of the fractional quantum Hall effects and that can be
thought of as an electron bound to an even number of magnetic flux quanta [18–20].
1.4
Thesis overview
Chapter 2 is the introduction to our search for a charge density excitation bound
to an inhomogeneity in the perpendicular magnetic field. In this section, we will
introduce edge magnetoplasmons, also touching on plasmons and magnetoplasmons
in 2DES. We will also mention previous DC transport work done on devices that
combine ferromagnets and 2DES.
Chapter 3 presents the sample design, experimental methods, and the results of
our search for a MEMP. Though we did not observe a MEMP, the same experimental
setup allowed us to characterize EMPs at an edge with a Bz inhomogeneity. We
found a reduction in EMP velocity in a Bz inhomogeneity that is at least plausible
according to classical EMP models.
Chapter 4 introduces pinned solids in systems that show quantum Hall effects.
These solids include the terminal insulating phase at high Bz and solids composed of
quasiparticles under conditions of the integer quantum Hall effect. We also discuss
4
more exotic electron solids that may be relevant to our results. Finally, we introduce
the reentrant quantum Hall effect.
Chapter 5 is an introduction to WQW 2DES. We focus on using the Shubnikov-de
Hass (SdH) oscillations to get information on the vertical charge distribution in the
wells. Also, we describe the specific DC transport measurements that are directly
related to experiments described in the next chapter.
Chapter 6 covers our microwave measurements of WQW 2DES. We give an overview
of our microwave measurement technique and present data taken from samples with
42 nm and 54 nm WQWs. We interpret the data as evidence of a new electron solid
and discuss the possible nature of this solid.
5
CHAPTER 2
MAGNETIC EDGE
MAGNETOPLASMONS
2.1
Introduction
Charge density excitations in 2D electron or hole systems have long been a subject
of considerable theoretical [5, 21–30] and experimental [31–49] interest. Such charge
density excitations include plasmons, magnetoplasmons, and of most relevance to this
dissertation, edge magnetoplasmons (EMPs). EMPs are charge density excitations
the propagate along the edge of a 2DES. In this dissertation, we describe a search
for excitations analogous to EMPs but which are confined by an inhomogeneity in
the magnetic field, rather than by the physical edge of the 2DES. Such magnetically
confined excitations are of interest as novel physical phenomena, and as possible
probes of individual and collective electron physics in low temperature 2DESs.
This chapter, and the next, concern the search for a charge density excitation
bound to a linear inhomogeneity in the perpendicular field Bz . We will refer to
such an excitation as a magnetic edge magnetoplasmon, or MEMP. We will first
give some background on plasmons, magnetoplasmons, and EMPs, followed by a
discussion of a few previous experiments involving 2DESs in the presence of magnetic
inhomogeneities. Lastly we will introduce MEMPs as an analog to EMPs.
6
2.2
Plasmons and Magnetoplasmons
A change in charge density from equilibrium will create an electric field which tends
to restore that equilibrium. If the scattering by disorder is small enough, the inertia
of the electrons will keep them moving past the position where they would cancel
out the original fluctuation, so the charge density fluctuation propagates through the
2DES. Absent nearby metals, such a propagating charge density fluctuation, has been
termed a plasmon [50]. Absent nearby metal surfaces the dispersion of a plasmon in
a 2DES is given by [5]:
ω 2p =
2πne2 k
,
0 m∗
(2.1)
where n is the electron density, m∗ is the effective electron mass of the charge carriers, the relative permittivity of the medium, 0 the permittivity of free space, and
k = 2π/λ with λ the wavelength of the plasmon. In the presence of a finite perpendicular magnetic field the plasmon hybridizes with the cyclotron mode resulting in a
magnetoplasmon whose dispersion is [50]:
ω 2mp = ω p (k)2 + ωc2 ,
(2.2)
where ω c = eB/m∗ is the cyclotron frequency.
2.3
Edge Magnetoplasmons
EMPs [5, 21–23, 25–32, 37, 40, 44, 51] can be thought of as the low frequency
branch of the magnetoplasmon. A 2DES structure can exhibit EMP resonances in its
electromagnetic spectrum when the perimeter of the structure is a integer number of
EMP wavelengths. These resonance frequencies are of lower frequency than the magnetoplasmon. EMP resonances were first observed in semiconductor heterostructure
dots of 4 μm diameter by Stormer et. al. [31]. Fig. 2.1 shows calculated fundamental
EMP and magnetoplasmon resonance frequencies versus Bz for a 10 μm dot. Soon
7
3500
3000
2500
f(GHz)
MP
2000
CR
1500
1000
500
EMP
0
2
4
B(T)
6
8
Figure 2.1: Comparison of the EMP and magnetoplasmon modes for a 10
μm disk of GaAs semiconductor 2DES with n = 2.8 × 1011 cm−2 . The higher
frequency branch on the graph is the magnetoplasmon mode (labeled MP),
the lower frequency branch the EMP (labeled EMP), and the dashed line
the cyclotron frequency (labeled CR).
8
after, Mast et. al. [40] and Glattli et. al. [37] found EMP resonances on the surface
of liquid helium. Grodnensky et. al. [38] in 1991 observed EMPs at radio frequencies
in large (∼ 1 cm square) 2DES GaAs heterostructures. EMPs have low damping in
strong magnetic fields and have been used to probe 2DES in the quantum Hall effect
regime [30, 32, 36, 52].
Experiments involving EMPs have been done in a wide variety of systems. Some
examples are: arrays of quantum dots in semiconductor 2DESs [31, 33, 34], arrays
of anti-dots in semiconductor 2DESs [33], along quantum wires [35], electrons on the
surface of liquid helium [37, 40], and in larger-size semiconductor 2DES structures
[38, 42, 45, 46] at radio frequencies. The first theoretical work on EMPs was done
by Fetter [25–27, 40] and by Volkov and Mikailov [28, 29, 51]. Fetter examined
the problem of EMPs on the surface of liquid helium and obtained a solution using
a classical hydrodynamic approach [25–27, 40]. Volkov and Mikailov addressed the
problem of EMPs on the surface of liquid helium, in GaAs/Alx Ga1−x As semiconductor
heterostructures, and in MOSFETs [28, 29, 51].
Various treatments [21, 25, 29] all obtain
ωEMP = 2 ln
e−γ
2ka
ne2
k,
0 m∗ ωc
(2.3)
where a = 10aB [aB is the Bohr radius and 11.6 nm in GaAs] is the width of the strip
the EMP is confined in at the edge of the sample, γ is the Euler constant (γ = .577),
and the relative permittivity of the medium is often fit from the data as an effective
permittivity to take into account the dielectric environment. The width of the strip
the EMP is confined in (a) is the distance over which the energy of the electronelectron interaction is comparable with the kinetic energy of the individual electrons
and a = aB assumes B such that the Landau filling factor (ν) is less then or equal
to 10 [53]. For low field (and if we assume ω ωc ) a = e2 ν/0 ωc [5]. Then, Eq.
2.3 gives a ωemp = 2.86 GHz at Bz = 1 T for a 1 cm square of 2DES with the same
9
density (n = 2.8 × 1011 cm−2 ) as the devices for which we will present data for in the
next chapter.
Metal surfaces near the 2DES will change the charge distribution of the EMP. For
a “screened” EMP near a metal surface the dispersion is [42]:
ωEMP =
2ne2 d
k,
0 m∗ ωc a
(2.4)
where d is the distance between the metal surface and the 2DES.
It is useful to mention that the natural log factor in the frequency dependence of
the EMP is absent for different assumptions about the static distribution of charges
near the edge of the 2DES. To summarize this section and prepare the reader for the
rest of this and the next chapter, here are a few useful things to keep in mind about
EMPs.
1. The EMP frequency is proportional to the Hall conductivity σxy , with ω EMP ∝
(1/Bz ) [5].
2. EMPs are chiral, i. e. they propagate in only one direction with their propagation direction dependent on the sign of Bz [5].
3. The velocity of an EMP can be, and generally is, higher than the Fermi velocity
[5].
4. Having metal on or near the surface of the sample will change the dielectric
environment of the EMP, damping the EMP [45], and driving the peak frequency
down.
5. It is an experimental fact that grounded contacts can absorb an EMP [32]; a
sample with a grounded contact along an edge will not show a resonance peak
from circulation of the EMP around the sample.
2.4
Inter-Edge Magnetoplasmon
EMPs have also been observed propagating along a change in charge density in
a 2DES [43, 44, 47]. So-called inter-edge magnetoplasmons [5], IEMPs, travel along
10
a boundary defined by a change in charge carrier density between two different but
non-zero n, instead of along a change between n of some finite value and n = 0 at
the edge of the sample. This IEMP shares some similarity with our theorized MEMP
since it exists in the bulk of the 2D gas not along a physical edge of the sample. The
presence of metal gates, used to create the density step, on the sample will screen the
IEMP. The dispersion of a screened IEMP is predicted to be [5]:
R
σ xy − σ Lxy d
k,
ω IEMP =
2π0
a
(2.5)
L
where σ R
xy and σ xy are the Hall conductivities of the regions on either side of the
boundary on which the IEMP propagates. In typical 2DES in GaAs with evaporated
metal film gates d/a is of order unity. The essential difference between the frequency
dependence of the screened EMP and a screened IEMP is the change from ω EMP ∝
1/B to ω IEMP ∝ (1/BR − 1/BL )
2.5
A Semi-Classical Model of Magnetic
Confinement
There have been many papers involving DC transport through 2DESs decorated
with ferromagnetic materials [8, 54, 55] showing increases in resistance based on the
magnetization of the ferromagnetic material. To give a model of how a magnetic
inhomogeneity can confine an excitation, we will use a semi-classical approach starting
with a model of an EMP confined near a physical edge, and then generalize to a charge
density fluctuation confined by a magnetic inhomogeneity.
In Fig. 2.2, we depict a succession of cyclotron orbits which are broken at the edge
of the sample, forming a chain of semicircular orbits where the path of individual
electrons rebound at the edge of the sample. In the semi-classical model the channel
that confines the charge density fluctuation that composes the EMP is composed of
many of these “skipping” orbits overlapping on top of each other. Then the EMP can
11
n = n₀
n=0
Figure 2.2: “Skipping” orbit depicted as a series of broken cyclotron orbits
confined to the edge of a sample, n = 0.
be thought of as an excitation within these “skipping” orbits. This captures the chiral
nature of the EMP.
To extend this semi-classical model to electrons confined along a change in Bz , we
introduce two simple examples of idealized Bz inhomogeneities, namely linear steps
in Bz . In the first example Bz reverses in sign upon crossing the step, which results
in “snake” orbits like those shown in Fig. 2.3a. If Bz changes in magnitude, but not
in sign, on crossing the step then the electrons travel in “cycloid” orbits along the
magnetic inhomogeneity; as shown in Fig. 2.3b. In both cases, the trajectories of
electrons incident on the step in Bz are bent with the electrons becoming trapped
along the step in Bz , assuming the step is large enough to curve the paths of electrons
incident so that the electrons encounter the step again in a distance on the order of
the mean free path [56].
“Cycloid” orbits at magnetic field steps have been predicted to provide additional
DC magnetoresistance [54, 57]. Ensslin et. al. [8] used a setup in which a ferromagnetic cobalt film was deposited across a narrow Hall bar which was magnetized along
the film using an in plane magnetic field. The result was a large inhomogeneity in Bz
under the edge of the strip in the 2DES. The DC resistance of the device increased,
12
a)
Bz = B0
Bz = -B 0
b)
Bz = B0
Bz = B0 + B m
Figure 2.3: a) “snake” orbit depicted as a succession of cyclotron orbits
reversing direction where Bz inhomogenously changes from B0 to −B0 . b)
“cycloid” orbit depicted as a series of cyclotron orbits tightening in radius in
the region where Bz is increased by Bm .
following the hysteresis curve of the ferromagnetic strip. This magnetoresistance arose
from the charge carrier trajectories being curved under the edge of the magnet, where
the Bz inhomogeneity was located, and being scattered under the edge of the cobalt
film. These previous DC results were part of the inspiration for searching for MEMPs.
2.6
Magnetic Edge Magnetoplasmon
In this dissertation we have hypothesized that there is a charge density excitation
similar to an EMP that travels along a linear magnetic inhomogeneity in Bz instead
of along a change in electron density. To introduce the expected properties of this
13
theorized MEMP, let us consider Eq. 2.5, where, neglecting the quantum Hall effect,
the frequency of an IMEMP is given by
ω emp
L
k σR
xy − σ xy
=
,
2π0
(2.6)
Δne e
,
Bz
(2.7)
where
Δσ xy (IEMP) =
and Δne is the difference of electron densities of the regions in the 2DES on either
side of the boundary. Instead of a density change, if there is a difference in Bz we
can rewrite Eq. 2.7 as:
Δσ xy (IEMP) = ne eΔ
with
Δ
1
Bz
=
1
Bz
1
1
−
,
BzR BzL
,
(2.8)
(2.9)
where BzR and BzL are the perpendicular magnetic fields under either side of the
boundary along which the MEMP is propagating.
To prepare for the next chapter, let us summarize the properties of our hypothesized MEMP:
1. They are confined by a change in Bz .
2. In the IEMP analogy the MEMP frequency is proportional to the change in
Hall conductivity Δσxy between regions of 2DES with different Bz , ω MEMP ∝
Δ(1/Bz ).
3. Like EMPs, MEMPs are expected to have a gapless spectrum with ωMEMP ∝ k
like an IEMP.
14
CHAPTER 3
MAGNETIC EDGE
MAGNETOPLASMONS –
EXPERIMENTAL SETUP AND
RESULTS
3.1
Introduction to hybrid ferromagnet 2DES
devices
This chapter is about hybrid 2DES-ferromagnet devices. These devices use a
ferromagnetic film on the surface of the semiconductor wafer to provide a magnetic
inhomogeneity Bz at the 2DES. We intended this inhomogeneity to support a MEMP.
These devices are intended not to support a resonant mode and we refer to these
devices as non resonant MEMP (NRMEMP) devices.
3.2
Microwaves connections and mounting
We measure the NRMEMP devices immersed in liquid helium at 4K. An Agilent
network analyzer acts as the source and receiver for the microwave signal. Semi-rigid
metallic coax transmit microwaves from room temperature to the sample then back
up to room temperature. We use a pre-amplifier to increase the magnitude of the
returning signal before the receiver.
15
3.3
Introduction to gate controlled non-resonant
devices
For these devices, a ferromagnetic permalloy film is deposited on the surface of the
device to provide a magnetic inhomogeneity in Bz . A diagram of the device is show
in Fig. 3.1. The ferromagnet film can also be biased to deplete the charge carriers in
the area under the it so we often refer to the permalloy as a gate. This fully depleted
region of 2DES creates an edge that EMPs can propagate along though there is a
Bz inhomogeneity as well as a density step at the edge of the biased ferromagnetic
film. The 2DES under a Au gate opposite to the permalloy can also be depleted to
create an edge that will support an EMP. Two grounded contacts block transmission
around the device if the gates are not charged and there is no MEMP along the
permalloy gate. The source antenna excites an EMP traveling along the edge of the
2DES mesa. The direction of the external field controls the propagation direction of
this EMP, directing it towards the permalloy gate (B > 0) and the Au gate (B < 0).
We will refer to these hybrid ferromagnet 2DES devices as non-resonant magnetic
edge magnetoplasmon (NRMEMP) devices. In NRMEMP devices microwaves couple
capacitively to the 2DES in the small region where the antennas overlap the the 2DES
mesa. We optimized the size of this overlap to be large enough for EMP measurements
but not so large that the EMP is damped by the measuring circuits. Transmission
of the signal via an EMP (or EMP-MEMP combination) can be observed as either
an increase or decrease in total transmitted power depending on the phase difference
between the EMP/EMP-MEMP signal and the “crosstalk” which travels through the
2DES bulk or by capacitive coupling between the source and receiver antennas.
From the phase difference we can find the time difference in propagation between
the EMP (EMP-MEMP) and the crosstalk. For there to be dip in the B trace:
τm = τc ± j/2f
16
(3.1)
where j an integer, τm is the edge mode propagation delay, and τc is crosstalk propagation delay. For any series of Np EMP/MEMP on paths p of length lp
Np
lp
τm =
vp
p
(3.2)
If EMP/MEMP velocities are given by αp /B (and for MEMPs they well might not
be) and if we define a Cm then
Np
lp
= BCm .
τm = B
α
p
p
(3.3)
Now the condition for a dip is:
−1
B j = Cm
(τc ± j/2f ),
(3.4)
and if we plot j vs Bj the slope is 2Cm f .
3.4
Sample characteristics
The NRMEMP devices use modulation doped GaAs/Alx Ga1−x As with a quantum
well of width 30 nm that is centered 100 nm below the surface of the wafer. This
wafer was grown by molecular beam epitaxy, MBE, by D. Larouche, M. P. Lilly, and
J. L. Reno at Sandia National Laboratory. Samples from the wafer have have mobility
μ = 9.1 × 105 cm2 V−1 s−1 an as-cooled density n = 2.8 × 1011 cm−2 . This wafer was
used for all the data presented for NRMEMP devices.
3.5
NRMEMP device fabrication and design
We prepare NRMEMP devices by cleaving 2DES wafer into 4 mm x 3 mm rectangles. Then we etch away all the 2DES away except for a mesa of dimensions of 3
mm x 0.5 mm. We evaporate layers of Ge (40 nm), Au (80 nm), Ni (10 nm), and Au
(200 nm), in that order, onto the both ends of the mesa and then anneal the sample
17
Microwave
Antenna
1mm
Contact
Gold
2DES
Microwave
Antenna
Contact
Permalloy
Figure 3.1: Illustration of NRMEMP device the 2DES mesa grey and the
contacts dark grey, all of the features on the are device to scale except for
the ends of the microwave antennas are truncated. Note: the slight overlap
of the tips of the microwave antennas over the edge of the mesa containing
the 2DES is not shown.
18
in a reducing atmosphere (H2 :N2 ) to make ohmic contacts. Next microwave antennas
and the Au gate are evaporated onto the surface with a shallow (10 nm) layer of Cr
under the thick (200 nm) Au layer. Last, a 250 nm thick ferromagnetic permalloy
gate is evaporated onto the surface of the device opposite to the Cr/Au gate. Figure
3.1 depicts the final design configuration.
3.6
NRMEMP devices
We designed these devices to allow EMPs to couple to MEMPs. We work with
four configurations for the NRMEMP device:
1. Both gates grounded.
2. Both gates at -1 V.
3. Au gate grounded, permalloy gate at -1 V.
4. Au gate at -1 V, permalloy gate grounded.
Figs. 3.2 and 3.3 show the expected paths for the EMP and EMP-MEMP through
the device for positive BZ and negative Bz . The antennas are marked in the figures
for what we call forward transmission. For “reverse” transmission the transmitter and
receiver are interchanged.
3.7
NRMEMP device results
Data for this section was taken with the sample at 4 K in an 8 T superconducting magnet, using an Agilent network analyzer as the source and receiver for the
microwaves. In this and the next section of the dissertation we will report traces
where the microwave frequency is held constant and B is swept. The traces show
transmitted power as a ratio with incident power, in dB.
19
Receive
‘ϐ‹‰—”ƒ–‹‘
1
Contact
Gold
B
Contact
2DES
Transmit
Permalloy
Receive
‘ϐ‹‰—”ƒ–‹‘
2
Contact
-1V
Gold
B
Contact
2DES
-1V
Transmit
Permalloy
Receive
‘ϐ‹‰—”ƒ–‹‘
3
Contact
B
Contact
2DES
-1V
Gold
Transmit
Permalloy
Receive
‘ϐ‹‰—”ƒ–‹‘
4
Contact
-1V
Gold
2DES
Transmit
B
Contact
Permalloy
Figure 3.2: All four device configurations for positive B. For each configuration the charged gates are colored and the grounded ones are white. The
red dotted line shows the path of the EMP and the blue dashed line shows
the intended path of the MEMP.
In Figs. 3.2 and 3.3 the EMP is incident on the permalloy gate for B > 0 and the
Au gate for B < 0. Dips are present when the EMP is incident is directed toward a
gate that is biased to deplete the 2DES.
20
Receive
‘ϐ‹‰—”ƒ–‹‘
1
Contact
Gold
Contact
-1V
Gold
Contact
X
B
X
B
X
Contact
2DES
-1V
Transmit
Permalloy
Receive
‘ϐ‹‰—”ƒ–‹‘
3
B
Permalloy
Receive
‘ϐ‹‰—”ƒ–‹‘
2
X
Contact
2DES
Transmit
B
Contact
2DES
-1V
Gold
Transmit
Permalloy
Receive
‘ϐ‹‰—”ƒ–‹‘
4
Contact
-1V
Gold
2DES
Transmit
Contact
Permalloy
Figure 3.3: All four device configurations for negative B. For each configuration the charged gates are colored and the grounded ones are white. The
red dotted line shows the path of the EMP.
The lack of dips on the Au side when both gates are grounded confirms that a
grounded contact absorbs an incident EMP. The lack of dips on the permalloy side
when the permalloy gate is grounded tells us either there is no MEMP traveling
along the permalloy gate or that the power transmitted by MEMP is very small in
21
Configuration 1
2 GHz
4 GHz
-20
-40
Configuration 2
Au (0 V)
Transmitted power (dB)
Transmitted power (dB)
0
Py (0 V)
-60
-80
-100
-3
-2
-1
0
B Z (T)
1
2
3
2 GHz
4 GHz
-40
Au (-1 V)
-120
-160
-3
-2
-1
Au (0 V)
Transmitted power (dB)
Transmitted power (dB)
-60
Py (-1 V)
-80
-100
-120
-3
-2
-1
0
B Z (T)
1
1
-20
2 GHz
4 GHz
-40
0
B Z (T)
2
3
Configuration 4
Configuration 3
-20
Py (-1 V)
-80
2
2 GHz
4 GHz
-40
-60
Au (-1 V)
-80
-100
-120
-140
-3
3
Py (0 V)
-2
-1
0
B Z (T)
1
2
3
Figure 3.4: Transmitted power in dB vs B taken at 4 K. Gate that the EMP
is incident, Py (permalloy) for positive B and Au for negative B on the
graph with the on voltage the gate relative to the 2DES in parentheses next
to it.
comparison to the power transmitted by the crosstalk.
Fig. 3.5 shows the B of the dips from Fig. 3.4 plotted vs the harmonic index (j).
We can fit j vs B in Fig. 3.5 to find the slope and from the slope find an average
velocity for the mode connecting source and receiver. We can rewrite eq. 3.2 as:
τm =
lEM P
lmg
+
vEM P
vmg
(3.5)
where lEM P = 2 mm (the length of the edge the EMP propagates along not near
a gate), vEM P is the velocity of the unscreened EMP, lmg = 0.5 mm (the length of the
Au or permalloy gate), and vmg is the velocity of the mode traveling along the metal
gate (Au or permalloy). The unscreened EMP traveling along the edge of the 2DES
22
Figure 3.5: Transmission dip number j (i.e. harmonic) plotted vs B, gate
configuration number listed in the caption. The EMP launched from the
transmission antenna is incident on the Au gate for B < 0 and the (Py)
permalloy gate for B > 0 as marked on the graph, lines on the graph are
least squares fits. Configuration numbers and frequencies are shown on the
right.
23
mesa has a vB of 1.16 × 107 Tm/s according to Eq. 2.3 using the average relative
permittivity of GaAs (12.9) and vacuum (1) equal to 6.95. We will determine τm for
each observed series of dips from the fit of j vs B using Eq. 3.4, then get vmg from
Eq. 3.5 using the calculated vEM P . Any error in vmg owing to the calculated value of
vEM P will be small because vEM P is approximately 5 times larger than vmg .
The fit for the B < 0 dips in configuration 2 (the EMP along the Au gate) from
Fig. 3.4 give vB = 2.53 × 106 Tm/s for the 2 GHz trace and vB = 2.46 × 106 Tm/s
for the 4 GHz trace. When we fit the dips from configuration 4 for B < 0 (the
EMP along the Au gate) we get vB = 2.52 × 106 Tm/s for the 2 GHz trace and a
vB = 2.56 × 106 Tm/s for the 4 GHz trace. The data for vB for the EMP along the
Au gate can fit Eq. 2.4 which gives vB = 2.71 × 106 Tm/s when is set to 6.95, the
average of the vacuum and GaAs permittivity and the width (a) of the strip on which
the EMP propagates (a) is 10 Bohr radii (aB ) in GaAs, a = 116 nm. The depth of
the 2DES from the top surface in this wafer is d = 100 nm.
When we fit the dips for in Fig. 3.5 for configuration 2 and B > 0 (when the
EMP is incident on the permalloy gate) we get vB = 1.04 × 106 Tm/s for the 2 GHz
trace and vB = 1.07 × 106 Tm/s for the 4 GHz trace. For configuration 3 and B > 0
(when the EMP is incident on the permalloy gate) we get vB = 1.25 × 106 Tm/s at
1 T for the 2 GHz trace and vB = 1.27 × 106 Tm/s for the 4 GHz trace. The main
result for this section is that this velocity is more than a factor of two slower than
velocities observed for the EMPs along the Au gate.
3.8
NRMEMP Summary
The velocities of the modes along the Au gate agree with calculation from Eq. 2.4
for a screened EMP. The modes propagating along the biased permalloy gate have
their velocity reduced compared to the EMP on the biased Au gate. There are no
dips in the traces for transmission along the permalloy gate or the Au gate when
24
they are grounded. The lack of dips from configurations with the permalloy grounded
suggests that the magnetic inhomogeneity along the permalloy gate does not support
a MEMP.
The origin of the increase in delay times for modes along the permalloy gate when
the Au gate is biased is likely from a mode circulating around the whole device when
the Au gate is biased. A small component of the EMP that has made 1.5 revolutions
around the entire device could add an additional component to the detected signal
and change the observed delay time, hence the calculated velocity reported in the
last section. The extra delay from the circulating mode may explain why the velocity
calculated from the slope in Fig. 3.5 is lower for the mode along the permalloy when
both gates are biased.
While we have obtained no evidence for a MEMP, we do see a notable difference
in the EMP velocities along the Au and the permalloy gates. We use a simplified
EMP model to argue that this result can be plausibly interpreted as due to the Bz
inhomogeneity due to the permalloy, though a quantitative analysis will require theoretical development [58], and a more detailed knowledge than is currently available
to us of the magnetic properties of the permalloy film near its edge.
3.8.1
A simple EMP model with nonzero ∂Bz /∂x
An EMP wave equation can be obtained [5, 42, 51, 59] starting from the continuity equation, Ohm’s Law. With ñ the deviation of the density equilibrium from
equilibrium,
ṅ = −∇ · (σE)
(3.6)
where σ is the conductivity tensor, and E = −∇ϕ is the electric field. An integral of
the Green’s function relating the potential (ϕ) to the deviation (ñ) of the local carrier
density from equilibrium, is substituted to give the equation of the EMP. Here we
roughly obtain an expression for the velocity in the local capacitance approximation
25
following ref. [42]. This approximation requires the gate to be much closer to the
2DES than one EMP wavelength. More complete theories for EMPs in Bz gradients
(and for MEMPs) are under development [58], which will define the profile of the strip
of oscillating charge density. For EMPs in the absence of magnetic field gradients,
and fully screened by a gate, the strip width [42] is of order the separation between
the gate and the 2DES.
The potential is then
ϕ=
ñ
,
C
(3.7)
where C is the capacitance per unit area between the 2DES and the gate, ñ is the
time-varying part of the carrier density, and the x-axis perpendicular to the edge. For
x near the maximum of the oscillating potential at the 2DES edge, the EMP velocity
becomes
vp =
∂σxy
.
C∂x
(3.8)
Taking σxy ∝ ne/Bz , and evaluating the derivative, including the x-dependence of n
gives
e
vp =
CBz
n ∂Bz
∂n
−
∂x Bz ∂x
(3.9)
The first term in the parenthesis is due to the density gradient, the second to the Bz
gradient. The ratio of velocities with and without the Bz gradient is
n ∂Bz
η = 1−
Bz ∂x
∂n
,
∂x
(3.10)
in the strip of oscillating charge at the 2DES edge. The measured value of this ratio
is about 0.46.
The first term in the parenthesis in (3.9) involves only the density of the 2DES
at the edge. Chklovskii and coworkers [53] calculated the density profile without
considering the depth, d = 100 nm, of the 2DES below the surface, but as discussed
in ref. [60] that should be a reasonable approximation for the the gate voltage we
26
applied, which is about 2.5 times that required to deplete the 2DES well under the
gate. Ref. [53] calculated the horizontal distance L between edges of the gate and
edge of the 2DES (at which the equilibrium density n(x) goes to 0, as
L=
2Vg 0
πn0 e
(3.11)
where n0 is the density far from the edge, and for GaAs = 12.8. For Vg = 1 V
and n0 = 2.8 × 1011 cm
−2
, L ≈ 160 nm. For the equilibrium density the reference
obtained
neq (x) =
L
1−
x
1/2
n0 ,
(3.12)
where the x-axis is perpendicular to the edge, and x = 0 at the edge of the gate.
The second term involves the magnetic field gradient. With the external field
perpendicular to the permalloy film, according to ref. [61] the permalloy magnetization M vs H is roughly linear up to μ0 H = 0.96 T. The susceptibility χ in that
direction is about 1. From continuity of the normal B, the film magnetization is
μ0 M = χ/(χ + 1)B0 ∼ B0 /2, where B0 is the applied field. Neglecting edge effects
and taking M to be uniform, (which will tend to overestimate fields near the gate
edge) we then have [62]
Bz = B0
2χhx
1−
(χ + 1)(x2 + d2 )
,
(3.13)
where x = 0 just under the gate edge and h is the height of the magnet. Grain size
in permalloy films of thickness around ∼ 100 nm can be as large as 1 μm [61, 63], so
when the film is not fully saturated, variations in Bz of around that spatial extent,
at around that distance from the edge, may exist.
Simply using equations (3.11),(3.12) and (3.13) to evaluate η, we find that the
magnetic term is much too large for the observed velocity change due to the Bz
gradient. The most natural explanation is that (3.13) is overestimating ∂Bz /∂x,
which is expected since the uniform magnetization model would be expected to be
27
incorrect for B0 below saturation. Setting the field in equation (3.13) 1 μm further
away from the 2DES gives a result in which the two terms are comparable, and would
give the measured η for x = 700 nm.
28
CHAPTER 4
WIGNER CRYSTALS AND OTHER
ELECTRON SOLIDS
4.1
The Quantum Hall Effect
At low temperatures, sufficiently low disorder 2DESs exhibit the quantum Hall
effect (QHE) [64, 65]. The QHE is characterized by vanishing diagonal resistance,
Rxx , concomitant with plateaus in Hall resistance Rxy with integer [65] or rational
fractional [14] multiples of h/e2 in certain ranges of magnetic fields. The integer
quantum Hall effect (IQHE) is an effect of a resolved Landau level (LL) spectrum,
and occurs when the Fermi energy lies between the LLs, as shown in the Fig. 4.1. We
will refer to this gap in the density of states between LLs as a “Landau vacuum”. This
Landau vacuum is responsible for the zero observed in Rxx , and the plateaus observed
in Rxy . The these LLs are separated by the cyclotron energy ωc = eB/m∗ , where
m∗ is the band mass of the charge carriers [66], or by the Zeeman splitting gμB B,
which in GaAs is generally much less than ωc . The QHE plateau centered at ν = νc
has a quantized Rxy = h/e2 νc . Each of the LLs has a degeneracy of the density of
flux quanta, nφ = 2eB/h, and the LL filling factor is:
ν=
nh
n
.
=
nφ
2eB
(4.1)
As depicted in Fig. 4.1, in the presence of disorder each of these LLs is broadened.
The IQHE plateaus are centered at integers of ν and have a Rxy = h/e2 .
29
Figure 4.1: Density of states for a 2DES showing disorder-broadened LL
spectrum. g the Lande g-factor and μB the Bohr magneton. The Fermi
level is as shown for the LL filling factor ν = 3.
30
The fractional quantum Hall effect (FQHE) is a manifestation of a many-body
quantum liquid which has fractionally charged quasi-particles, and was first described
by Laughlin [67]. The composite fermion (CF) model [18–20] encompasses most of
the physics of the FQHE. A CF can be thought of as an electron bound to an even
number of magnetic flux quanta. In the CF description, the FQHE is an analog of
the IQHE with LLs of CFs instead of electrons; here the simplest FQHE states arise
at odd denominator rational fractional ν for which the CF Fermi level lies between
CF LL.
4.2
Wigner Crystals
In 1934, Eugene Wigner [9] theorized that the ground state of a low density
electron gas in a uniform neutralizing background was a regular lattice, or crystal,
of electrons. He pointed out that an electron gas in a neutralizing background has
two principal energy scales: the Coulomb energy, EC , and the kinetic energy, Ek .
The Coulomb energy scales as EC ∝ 1/r where r is the inter-electron spacing. As T
goes to 0, the EK is taken to be the the Fermi energy, EF , which is proportional to
kF2 where kF is the Fermi wavevector. In 2DES, KF2 is proportional to the density
n. So EF goes as 1/r2 Hence at sufficiently low density, EC will be larger than EK .
Under such conditions electrons arrange themselves into a crystal lattice to minimize
EC . The lattice is expected to be triangular in 2D [68]. This regular lattice of
electrons is referred to as a Wigner crystal (WC). While 2D Wigner crystals hosted
in semiconductor systems at B = 0 have been considered theoretically [69, 70], they
have yet to be definitively observed.
The addition of a perpendicular magnetic field (B) changes the shape of free
electron wave functions from plane waves to Landau orbitals. Experimental evidence
of a WC stabilized by the presence of a perpendicular magnetic field in semiconductor
2DES does exist, for examples see reviews in [17, 68]. As B is increased, the area that
31
an electron in a Landau orbital takes up decreases. We can think of ν as a measure
of the overlap between electron wavefunctions. This is because ν = 2(lB /r)2 where
lB = /eB is the magnetic length and is the size of the lowest LL single particle
wavefunction and r = 1/πn the average electron separation. Landau quantization
allows for WCs of arbitrarily high n as long as sufficient B is available. The WC is
the predicted ground state of a 2DES, with no disorder, for ν 1/7 [11, 71, 72]. The
existence of WCs in this “ low ν” regime at the high magnetic field termination of
the QHE series is corroborated by microwave measurements as described in the next
section [3, 12, 73–76].
4.3
Pinned Wigner Crystals
In the presence of weak disorder, the crystalline order of the WC will have a finite
correlation length, or equivalently the disorder breaks the crystal up into domains.
This correlation length is a consequence of the electrons moving to reduce their total
energy, the sum of the electron-electron energy, and the electron-disorder energy [77,
78]. Also in the presence of disorder, the WC is pinned since it can no longer slide
as a whole, and thus is an insulator [6]. The disordered WC can oscillate collectively
within the disorder potential. This collective oscillation in 2DES in high B turns out
to have a frequency in the microwave range or RF range, and it is referred to as a
pinning mode resonance.
The frequency of a pinning mode resonance is a function of the collective restoring
force from disorder on the pinned WCs. If the strength of the pinning potential
increases the peak frequency of the pinning mode also increases. For example, if the
electron density (n) is increased, then the individual electrons will not be as closely
associated with the disorder pinning potentials, and the frequency of the pinning
mode decreases [13].
Fukuyama and Lee [13] derived a sum rule for the integrated intensity of a pinning
32
Figure 4.2: Spectra showing the real part of the diagonal conductivity
(Re[σxx ]) vs frequency, f , at many filling factors ν offset vertically proportional to ν. Successive spectra are separated by steps of 0.01 in ν ν for
black spectra are marked at right, in a 30 nm wide quantum well 2DES, and
at 30 mK. Adapted from [2].
mode resonance if all available charge carriers are participating in the resonance:
S/fpk = neπ/2B,
where S is the integrated pinning mode resonance conductivity S(Re[σxx ]) =
(4.2)
Re[σxx ]df .
We can use Eq, 4.2 to evaluate the carrier density in a pinning mode resonance.
33
4.4
Integer quantum Hall Wigner crystals
In ultra high mobility 2DESs pinning mode resonances have been observed near
IQHE plateaus [2]. As ν moves away from the central filling, νc , of some IQHE
plateau, a population of quasi-carriers is created, quasi-holes for ν < νc , and quasielectrons for ν > νc . These quasi-carriers interact with each other. The population
of these quasi-carriers is n∗ = n|ν − νc |/ν. This dilute population of quasi-carriers
crystallizes, similar to the low ν WC. fpk , the peak frequency of the pinning mode
decreases as n∗ increases [2, 79], as discussed in the previous section.
Figure 4.3: Example of S/fpk plotted vs ν for an IQHEWC pinning mode
resonance in a 30 nm wide quantum well 2DES at 50 mK. The black lines are
the prediction from the sum rule Eq. 4.2 for full quasi-carrier participation.
Apdated from Chen et. al. [2].
Using the sum rule, Eq. 4.2, the IQHEWC pinning mode appears to involve nearly
all the quasi-carriers available near the integer filling as shown in Fig. 4.3.
34
4.5
Other electronic solids
Electronic solids can have more complex carrier arrangements than the single
carrier-per-site arrangement of the WC we introduced earlier. A WC can be regarded
as a particular example of an electron solid. We will refer to any of these other “exotic”
crystals as an electronic solid, electron solid. In general electron solids have a regular
crystal lattice as a consequence of Coulomb repulsion, and in the presence of disorder
have a finite correlation length and are pinned.
v*
M=1
M=2
IQHEWC
Wigner Crystal
Bubble
Phases
Stripes
Figure 4.4: Cartoon of the progression from an IQHEWC to a bubble phase
WC and then a stripe phase WC as ν ∗ increases, with M the number of
electrons per lattice point in the WC.
Now we will give a few examples of previously observed electron solids. One
example of an electron solid is a so-called “bubble phase” in certain ν ranges in the
N ≥ 2 LL in ultra high mobility 2DESs [79, 80] that are ascribed to bubble phase
electron solids. These bubble phase electron solids are thought to be clusters of M
electrons, or holes, in a triangular lattice [12]. They are also predicted to be pinned
by disorder, and pinning mode resonances of these modes have been studied [79]
35
Figure 4.5: Re[σxx ] vs f for a very high mobility, μ = 24 × 106 cm2 V −1 s−1 ,
n = 1.0 × 1011 cm−2 , 50 nm QW, and a temperature of 30 mK. Resonance A
shows dispersion with wave vector. Adapted from Chen et. al. [3].
Another example is a crystal composed of skyrmions. Skyrmions are particles
composed of a single charge in a spin texture containing multiple flipped spins, spread
out in space to reduce exchange energy. For sufficiently low Zeeman energy, skyrmions
(anti-skyrmions) are predicted to be the quasi-carriers around ν = 1 [81]. The pinning
mode frequency is shown experimentally to shift upward with the addition of an inplane magnetic field, consistent with the expectation that increased Zeeman energy
due to the in-plane field would cause the skyrmions to shrink into regular Landau
quasi-carriers [82–84].
36
In very high mobility, μ = 24 × 106 cm2 V −1 s−1 , 50 nm wide quantum well 2DES at
low ν just above ν = 1/5 a pinning mode resonance was observed [3], and as was done
by Chen et. al. we will refer to it as the A-phase. This A-phase initially coexists
with the low ν pinning mode resonance (B-phase), with the A-phase vanishing by
ν ∝ 0.12 and only the B-phase (low ν WC) remaining. This A-phase pinning mode
resonance shows dispersion with respect to the width of the microwave transmission
line used to measure it (see Fig. 4.5, indicating that it has a larger correlation length
compared to previously measured electron solids). The FQHE has been suggested to
play a major role in the A-phase electron solid, and it has been proposed that this
electron solid is composed of CFs.
4.6
The reentrant integer quantum Hall effect in
2DES
The reentrant integer quantum Hall effect, RIQHE, is characterized by the vanishing and then appearance of an IQHE, both the plateau in Rxy and the zero (or
minimum) in Rxx [4, 74, 79, 85]. The RIQHE was first observed by R. R. Du et. al.
and M. P. Lilly et. al. 1998 [86, 87]. Both groups observed this reentrance of the
quantum Hall effect in a GaAs/Alx Ga1−x As semiconductor 2DESs between ν = 4 and
ν = 5 at low temperature (∼ 20 mK), as shown in Fig. 4.6.
Fig. 4.6a shows Rxy and Fig 4.6b shows Rxx , the three regions marked on them
are the IQHEWC near ν = 4 on the left, the bubble phase electron solid on the right,
and the region where they compete with each other in the middle. Concurrent with
the RQHE observed between ν = 4 and ν = 5, two different pinning mode resonances
have also been observed for the same range in ν [79]. These regions were identified
by a combination of DC measurements [4, 79] and microwave spectroscopy [74]. The
pinning mode resonance observed near the IQHE plateaus grows weaker as the filling
factor moves away from the plateau, where a second pinning mode resonance appears
37
Bubble
Phase Competing
Phases IQHEWC
WC
a)
0.250
R xy
0.249
RIQHE
b)
20
RIQHE
R xx
0
2.6
B (T)
2.7
Figure 4.6: Example of the RIQHE observed by Cooper et. al. [4] near
ν = 4 as ν increases with the RIQHE marked by the arrow, a) Rxy in units
of h/e2 , and b) Rxx in units of Ohms.
38
in the same region of ν where in DC the IQHE zero in Rxx vanishes. The zero in
Rxx then returns as the first resonance vanishes, leaving only the second resonance.
The first resonance, near the integer value of ν, has been identified as an IQHEWC,
and the second resonance as a bubble phase Wigner solid [79]. As the “domains” of
the bubble phase WC increase in size there will be a critical ν where the individual
domains link up and their walls merge, forming a path across the 2DES and allowing
electrons to move again, short circuiting the ν = 4 IQHE plateau.
Another example of the RIQHE occurs at very low T, < 15 mK, in ultra high
mobility 30 nm quantum well 2DESs. Multiple RQHE have been observed in the first
excited Landau level, between ν = 2 and ν = 3 with both ν = 2 and ν = 3, reentering
multiple times [85, 88]. This RIQHE is likely related to multiple transitions between
bubble and stripe phases.
A RIQHE has been observed in 42nm wide quantum well samples on either side
of ν = 1 as ν moves away from ν = 1 [7]. The location of this RIQHE in ν has been
observed to move in toward ν = 1 from either side of ν = 1 and eventually merge
with the ν = 1 plateau, as n is increased. This RIQHE has also been observed to
vanish below a critical n. The DC experiment that observed this specific example of
the RIQHE was one of the principle inspirations for the experiments described in Ch.
6.
39
CHAPTER 5
WIDE QUANTUM WELLS
5.1
Introduction
Mansour Shayegan‘s group in Princeton has recently been investigating 2DES QW
samples with wider wells [7, 89–94]. In comparison with narrower QW 2DESs, “wide”
quantum well (WQW) 2DESs have reduced electron-electron interaction at short
range from the larger extent of the electron wavefunction in the vertical direction
(in the growth direction, perpendicular to the 2D plane) and they have an electron
distribution that can be tuned from a single layer to a bilayer like distribution [95].
These WQW 2DES exhibit a wide variety of phenomena due to the addition of
the subband degree of freedom. Stabilization and enhancement of the q/2 FQHE
states [90, 91, 93, 96] (where q is an integer) and enhancement or weakening of the
q/3 FQHE states [92, 94] has been observed based on which specific subband-split
LL is below the Fermi energy. While level crossings between the subbands, have
been observed to cause the IQHE plateaus to vanish [97]. These WQW 2DES also
display quantum Hall ferromagnetism [95, 98, 99] when there are crossings between
subbands with the same spin orientation. A possible bilayer electron solid [15], where
the positions of individual charges are correlated between the top and bottom layers
of charge in the well, has also been observed in WQW 2DES. Finally, an RIQHE has
been observed around ν = 1 by Liu et al. [7] in 31, 42, and 44 nm WQWs. This
instance of the RIQHE is notable as the RIQHE has never before been observed in
40
the lowest LL (ν < 2) in high mobility 2DESs, though disordered 2DES can show an
RIQHE in the lowest LL [100, 101].
5.2
Two subbands
Figure 5.1: Charge distribution (ρ) on top in red, ground (ΨS in green) and
first excited state (ΨAS in blue) wave functions on bottom. From simultaneous calculations of the Poisson and Schroedinger equations. Difference in
density between the top half and the bottom half of the well (δn) on top.
n in 1011 cm−2 and subband separation (Δ) in K on bottom. Provided by
Yang Liu.
41
If we consider a 1D quantum well that is symmetric about the origin and solve
for the wavefunction of a single electron in that well, the ground state is symmetric
about the origin and the first excited state is antisymmetric. We will refer to these
states as ΨS and ΨAS . In semiconductor QWs, the addition of carriers will change
the potential of the well and we will need to simultaneously solve the Poisson and
the Schroedinger equations to calculate ΨS and ΨAS . As we increase the number of
carriers in the well, the ground state (ΨS ) will fill up and we will begin to populate
the first excited state (ΨAS ). The electron charge distribution in the well then is
ρ = e(nS |ΨS |2 + nAS |ΨAS |2 ) where nS is the population in the ground state and
nAS is the population in the first excited state. As show on the left in Fig. 5.1 ρ
is symmetric about the center of the well and double peaked. We can visualize this
charge distribution as a pseudo-bilayer, each peak representing a layer of 2DES.
If we apply a bias to a front and back gate on the sample we can tune the symmetry
of the electron distribution as well as the overall density (n). Increasing n while
keeping the charge distribution symmetric decreases the gap (Δ) between the ground
and the first excited states, and moves the peaks in the charge distribution toward
the walls of the well [102, 103]. We can also operate the gates to move charge from
one side of the well to the other, as shown in the center and on the right of Fig.
5.1. The uneven charge distribution in the well then increases Δ [102]. The well and
its ground state are then no longer symmetric. We will keep referring to the lowest
lying state with the subscript S and the first excited state with the subscript AS one
even though the ground state is no longer symmetric and the first excited state is no
longer antisymmetric. As can be seen in the simulations shown in Fig. 5.1, there is a
relationship between δn, the difference between nt (n of the top half of the well) and
nb (n of the bottom half of the well), and the subband separation Δ [103].
42
Re[σxx(μS)]
30
20
10
0
0.1
0.2
0.3
B(T)
0.4
0.5
0.6
Figure 5.2: Re[σxx ] in μS plotted vs B at a frequency of 0.5 GHz, from a 54
nm WQW, a temperature of 30 mK, and an as cooled n = 2.41 × 1011 cm−2 .
5.3
Measuring the subband separation (Δ)
Δ can be experimentally determined from Fourier analysis of the Shubnikov de
Haas (SdH) oscillations. An example of SdH oscillations (Re[σxx ] vs B) in a WQW
is shown in Fig. 5.2. We find a “B spectrum” of the SdH from the “Fast” Fourier
transform (FFT) of (Re[σxx ] vs 1/B) in the SdH regime. The FFT has peaks whose
position is (in Tesla) depends on the subband populations.
Fig. 5.3 shows the B spectrum from the data in Fig. 5.2. There are four peaks
corresponding to the total density(n), the density of electrons in the lowest lying
subband (ns ), the first excited subband (nAS ), and the difference between the two
subband densities (nS − nAS ). The population density of the lowest lying and first
excited subbands is given by:
e
nS = 2BS ,
h
(5.1)
e
nAS = 2BAS ,
h
(5.2)
and
43
Fourier transform intensity (a.u.)
BAS
BS - BAS
BS + BAS
BS
1
2
3
frequency (T)
4
5
6
Figure 5.3: B frequency peaks from the Fourier transform of the SdH oscillations with the B frequencies of the lowest lying state, the first excited
state, the sum of both, and the difference of both marked on the figure.
where BS and BAS are the SdH frequencies of the the lowest lying and first excited
subbands. Once we have the subband densities we can calculate the subband separation as a temperature:
1 (nS − nAS )π2
Δ=
kB
m∗
(5.3)
where kB is the Boltzmann constant. For GaAs, with m∗ = 0.067 free electron masses,
Eq. 5.3 becomes:
Δ = 20.03|BS − BAS |,
(5.4)
using BS and BAS in Tesla and producing Δ in units of K.
5.4
Δ in the vicinity of ν = 1
Fig. 5.4 shows a fan diagram of the LL energies vs B in a 54 nm WQW in the
lowest LL and a Δ = 27.6 K superimposed over a Re[σxx ] vs B trace from the same
44
Figure 5.4: Fan diagram for a 54 nm WQW (left axis) for Δ = 27.6 K.
The energies of the N = 0 Landau level from the lowest lying (S0) and first
excited (A0) subbands are plotted with their spin directions shown. On the
right axis a Re[σxx ] vs B trace taken at 500 MHz is plotted for n = 2.43×1011
cm−2 and the same Δ. ν = 1 is labeled on the figure.
well with n = 2.43 × 1011 cm−2 . The energies of the LLs of each of the subbands is:
ES = (N + 1)eB/m∗ ± gμB B,
(5.5)
EAS = Δ + (N + 1)eB/m∗ ± gμB B,
(5.6)
ES is for the lowest lying subband, EAS is for the first excited subband, and N is 0
for the lowest LL.
At ν = 1 there is one filled LL under the Fermi energy. For the samples we studied,
there are no level crossings between the symmetric and antisymmetric LLs around
ν = 1.
45
5.5
A RIQHE near ν = 1 in WQW
Shayegan’s group observed an RIQHE for fillings both above and below ν = 1 in
31, 42, and 44 nm WQWs [7]. In the 42 nm WQW, as n is increased while keeping the
well symmetric, the RIQHE appears at ν ≈ 0.81 as n reaches 2.05 × 1011 cm−2 . The
filling factor of the RIQHE then increases toward the main ν = 1 plateau region as n
is increased further, eventually merging with the main plateau around n = 3.14 × 1011
cm−2 . At n = 2.87 × 1011 cm−2 another RIQHE appears, this time at ν ≈ 1.18. As n
is increased, this RIQHE, like the one for ν < 1 moves in toward, then merges with,
the plateau around ν = 1. In Yang Liu et al. [7] it was proposed that both of these
RIQHEs were due to electron solids composed of CFs. Our experimental data to be
presented in the next chapter addresses the nature of these RIQHE states and will
discussed in comparison with data of Ref. [7].
46
CHAPTER 6
MICROWAVE SPECTROSCOPY OF
THE RIQHE AROUND ν = 1 IN WQW
2DES
The origin of the RIQHE around ν = 1 in WQW 2DES is not currently clear [7]. In
this chapter we present microwave measurements of WQW 2DESs. We will start by
presenting our measurement techniques. Next, we will present data from microwave
transmission measurements from 54 and 42 nm WQW 2DES. Finally, we will interpret
this data in the context of multiple competing electron solids. All data for the samples
in this chapter was taken with a bottom loading dilution refrigerator with a base
temperature of 30 mK.
6.1
Samples and Sample Preparation
The microwave measurements reported in this chapter are all of modulation doped
GaAs/Alx Ga1−x As samples. The specifics of each individual sample are shown in
Table 6.1. These samples are grown by molecular beam epitaxy (MBE) by Loren
Pfeiffer and Ken West at Princeton University. To prepare the samples we first
cleaved a rectangle of wafer approximately 4 mm x 3 mm in size. We then fabricated
contacts of thermally evaporated Ge/Au/Ni, which we then annealed at 440 C in a
reducing atmosphere of H2 : N2 for 10 minutes. After that we define a microwave
waveguide on the surface of the sample using a 200 nm thick Au layer on top of a
47
Table 6.1: Sample properties
sample #
ID #
well width
54-1
8-17-10.1
54 nm
54-2
8-17-10.1
54 nm
54-3
8-17-10.1
54 nm
42-1
2-9-11.1
42 nm
mobility (μ)
4 × 106 cm2 V−1
4 × 106 cm2 V−1
4 × 106 cm2 V−1
9 × 106 cm2 V−1
−1
s
s−1
s−1
s−1
CPW
30 μm,
80 μm,
30 μm,
40 μm,
meander
slot
meander
meander
thin 5 nm Cr layer.
6.2
Microwave Measurement Technique – CPW
Ohmic Contacts
Sample
50 :
Microwave
Source
Detector
3 mm
Figure 6.1: Schematic of a coplanar waveguide on the surface of a sample,
with metal in black. The 2DES is a fraction of a micron under the surface
of the sample. An Agilent network analyzer is the source and the receiver
of the microwave signal.
To obtain the diagonal ac conductivity of the 2DES we measure the microwave
attenuation through a coplanar waveguide (CPW transmission line) patterned on top
of the sample as shown in Fig. 6.1. CPW, invented by C. P. Wen in 1969 [104], is
used widely in planar integrated circuit devices.
In our setup, an Agilent network analyzer acts as both the source and receiver of
the microwave signal The signal from the network analyzer goes through attenuators
48
before traveling down coaxial cable (coax) to the sample in the cryostat. From the
sample, the signal is carried by coax to the top of the cryostat and pre-amplified
before being returned to the network analyzer. The center conductor of the CPW is
connected to the center conductor of the coax, and the planes on either side of the
center conductor are grounded. The electric field from the driven center conductor to
the ground planes interacts mainly with the 2DEG lying under the slots separating
the ground planes from the center conductor. For the samples considered here, Table
6.1 shows the slot width (W ) of the CPW, and whether the CPW is straight or if it
meanders. The meander CPW, like that shown in Fig. 6.1, increases the sensitivity
of the transmission line and is not intended to couple to wavevectors related to the
meander pitch.
The characteristic impedance of the CPW on the sample with σxx set to 0, Z0 ,
is determined by the ratio of the width of the center conductor of the CPW to the
width of the slot of the CPW. We designed the CPW so that the Z0 = 50 Ω, matching
the impedance of the coax in the experiment. We model the 2DES conductivity as a
perturbation of the transmission line, assuming the following [105]:
1. The measuring system has no reflections at the end of the lines.
2. The per-unit-area coupling capacitance (CC ) between the 2DES and CPW film,
is sufficiently large that the RF current is well confined in the 2DES under the
slots |σxx |/πf CC W .
3. The measurement is done at a sufficiently high frequency that |σxx |v0 Z0 /πf W , where v0 is the propagation velocity of the RF signal through the 2DES.
When the above assumptions are satisfied, the diagonal conductivity is approximated
by:
W
Re[σxx (f )] =
ln
2lZ0
P
P0
,
(6.1)
where P0 is the transmitted power with zero 2DES conductivity (usually found using a trace taken on a quantum Hall plateau where Re[σxx ] is vanishing), P is the
49
Figure 6.2: Gate configurations for the WQW samples. a) back gate only:
samples 54-1 and 54-2. b) front and back gate with glass spacer and increased
CPW slot width: sample 42-1. c) front and back gate with etched glass
spacer: sample 54-3.
50
transmitted power, and l the total length of the CPW on the sample. We verify the
result of Eq. 6.1 using more complicated calculations [106] that include higher loss,
greater conductivity through the 2DES, and the distributed nature of the CPW-2DES
coupling.
Fig. 6.2 shows the gate configurations for all the WQW samples. To keep the
top gate from capacitively shorting out the CPW, a spacer is needed between the top
gate and the CPW. All of our samples with top gates have a spacer made out of glass.
For sample 42-1, the CPW has a wider slot width to take into account the increase in
the effective dielectric constant from the glass spacer. Sample 54-3 has a glass slide
with the area over the CPW etched away so that the effective dielectric constant near
the CPW is less affected by the spacer, and strain due to differential contraction is
reduced.
6.3
6.3.1
54 nm WQW – microwave spectroscopy of
pinning mode resonances around ν = 1
Sample 54-1
Sample 54-1 has a back gate, and no front gate. The lack of a front gate on
sample 54-1 prevents us from controlling Δ when we vary n. Δ vs n for sample 54-1,
as assessed from SdH measurements, is shown in Fig. 6.4. Fig. 6.3 shows Re[σxx ] vs
B for n = 2.41 × 1011 cm−2 with a selection of IQHE and FQHE minima marked on
the graph. Fig. 6.5 shows traces of Re[σxx ] vs B, taken at a frequency of 0.5 GHz, a
temperature of 30 mK, and several n which are shown on the right in units of 1011
cm−2 . We calculated the densities for the spectra we present in the section from the
traces in Fig. 6.5.
Fig. 6.6 shows data from sample 54-1 in its as-cooled state (i.e. with no gate
voltage). Fig. 6.6a displays spectra, Re[σxx ] vs f , of 54-1 for many ν. These spectra
show a pinning mode resonance for ν on either side of ν = 1. The small (∼300
51
10PS
Re[ σ xx ](μS)
3
1
2
8/3 6/3
8/5
6/5
4/5
5/3 4/3
2
4
6
8
10
12
14
B(T)
Figure 6.3: Real part of diagonal conductivity (Re[σxx ]) vs B for sample
54-1 taken at a frequency of 500 MHz. The temperature is 30 mK and the
density is 2.41 × 1011 cm−2 . The sample has no front gate, and there was no
voltage bias on the back gate. A selection of IQHE and FQHE minima are
labeled with their Landau filling factors.
MHz period) peaks on the spectra are from reflections arising from the increased
conductivity of the pinning mode resonance, causing an impedance mismatch. The
resonance is absent at ν = 1, and vanishes for ν just above 6/5 and just below 4/5.
Fig 6.6b shows fpk vs ν from the data in Fig 6.6a. There is a prominent kink in
the curve between ν = 0.91 and 0.87 Fig. 6.6c displays S/fpk vs ν from the spectra in
Fig. 6.6a, with the black lines showing the theoretical prediction for full participation
of the quasicarriers at density n∗ = nν ∗ /ν, as explained in Section 4.3 (see Eq. 4.2).
The drop off in participation seen in Fig. 6.6c near ν = 4/5 and ν = 6/5 is from the
pinning mode resonance dying off as ν approaches the 4/5 and 6/5 FQHE states. In
Fig. 6.6c, we do not observe any changes in the number of quasicarriers participating
in the resonance for 0.87 ≤ ν ≤ 0.91, where where there is a kink in fpk vs ν in
Fig. 6.6b. The kink is of central importance; the rest of this section will detail its
dependence on n.
52
Figure 6.4: Δ vs n for the 54 nm WQW, illustrating how the subband
separation changes as we change n with the back gate. n = 2.41 × 1011 cm−2
is the as-cooled density with no gate bias.
Fig. 6.7 shows spectra taken for many ν (0.8 < ν < 1.2) for n = 2.83 × 1011 cm−2 ,
3.00 × 1011 cm−2 , and 3.20 × 1011 cm−2 . The smaller range in ν for Figs. 6.7b and
6.7c is due to the 14 Tesla limit of the superconducting magnet that was used to take
these spectra. Relative to the n = 2.41×1011 cm−2 spectra of 6.6a the n = 2.83×1011
cm−2 spectra in Fig. 6.7b are shifted to higher frequency for ν between 1.1 and 1.15.
The transition to a mode with enhanced frequency moves in towards ν = 1 as n is
increased as shown in Figs. 6.7b and 6.7c. At n = 3.20 × 1011 cm−2 the resonance in
almost its whole range for ν > 1 has moved to higher frequency.
The effect of reducing n on the pinning mode resonance is not as dramatic as
53
Figure 6.5: Re[σxx ] vs B for n = 1.94 × 1011 cm−2 to n = 3.20 × 1011 cm−2 ,
each trace vertically offset for clarity. Taken at ∼ 30 mK and a frequency
of 0.5 GHz.
increasing n. Fig. 6.8 shows spectra for many ν (0.8 < ν < 1.2) for n = 2.17 × 1011
cm−2 , 2.08 × 1011 cm−2 , and 1.94 × 1011 cm−2 . We do not observe any changes
in the pinning mode for ν > 1 as n is decreased. The transition observed in the
n = 2.43 × 1011 cm−2 spectra for ν < 1 weakens and moves away from ν = 1 as n is
decreased and has almost completely vanished by n = 1.94 × 1011 cm−2 .
Fig. ?? shows how fpk vs. ν evolves as n is changed. For ν > 1, if n < 2.83 × 1011
cm−2 fpk fpk smoothly decreases as ν moves away from 1, but a region of enhanced
fpk appears for n ≥ 2.83 × 1011 cm−2 . The ν of this transition between normal
and enhanced fpk s moves in toward ν = 1 as n is increased. For ν < 1, a ν-region
of enhanced fpk is visible for all n. Similar to the transition in fpk for ν > 1, the
transition in fpk seen for ν < 1 also moves in toward ν = 1 as n is increased.
Fig. 6.10 is S/fpk vs ν for all measured n for sample 54-1. The black lines show the
theoretical prediction for full participation of the quasicarriers at density n∗ = nν ∗ /ν
54
Figure 6.6: a) Spectra from sample 54-1 showing the real part of the diagonal conductivity (Re[σxx ]) vs frequency taken at many ν. ν of blue traces
is marked on the right, at a temperature of 30 mK, n as-cooled, and normalized to ν = 1 with each spectrum proportionally offset vertically from
the last and the filling factor step between each spectra is ν = 0.005. b)
The peak frequency of the pinning mode resonance (fpk ) vs. ν. c) S/fpk vs
ν. The black lines are the theoretical prediction for full participation in the
resonance.
as in Eqs. 4.2. As shown in Fig. 6.10, there does not appear to be any change in
participation in the resonance that corresponds to the feature we see in fpk vs ν.
55
Figure 6.7: Waterfalls of spectra from sample 54-1 show the real part of the
diagonal conductivity (Re[σxx ]) vs frequency taken at many ν. ν of blue
traces marked on the right, normalized using a ν = 1 spectrum, n marked
at the top of each waterfall, and each is spectrum is proportionally offset
vertically. a) and b) The step in ν between spectra is 0.01. c) The step in
ν between spectra is 0.005. There was no data taken above ν = 1.05 for
n = 2.17 × 1011 cm−2 .
6.3.2
Sample 54-3
Sample 54-3 is sample 54-1 with the addition of a front gate. In principal the
front and back gates let us vary n while maintaining a symmetric charge distribution
in the well. However, on applying voltage to the front gate SdH oscillations were no
longer well developed. In practice, we kept the charge distribution nearly symmetric
56
Figure 6.8: Waterfalls of spectra from sample 54-1 showing the real part of
the diagonal conductivity (Re[σxx ]) vs frequency, taken at many ν. ν of blue
traces marked on the right, 30 mK, normalized using a ν = 1 spectrum, n
marked at the top of each waterfall, and each spectrum is proportionally
offset vertically. a) and b) The step in ν between spectra is 0.01. c) The
step in ν between spectra is 0.005.
as took the data on sample 54-3. Study of the gate voltage dependence of the density
of 54-3 indicates the gate voltage-density ratios are about the same for the back gate
and front gate. The data presented in this section was taken using equal front gate
and back gate voltages. Simulations indicate that the sample as-cooled (both gate
voltages zero) is near its balanced state, δn/n < 0.1.
The data with the sample 54-3 as cooled (n = 2.43 × 1011 cm−2 and zero gate
57
Figure 6.9: fpk in GHz vs ν as n is increased, from Figs. 6.7 and 6.8.
voltages) is essentially the same as sample 54-1. Fig. 6.11a is a waterfall showing
(Re[σxx ]) vs B. As for sample 54-1 there is a resonance peak for ν on either side of
one. The resonance frequency decreases uniformly as ν moves away from ν = 1 as is
typical for an IQHEWC resonance [2].
In Fig. 6.11c, we plot a waterfall of spectra from ν = 1 to ν = 0.8 for n =
2.43 × 1011 cm−2 , and symmetric charge distribution in the well. The resonance in
58
Figure 6.10: Sample 54-1, S/fpk vs. LL filling factor for all measured n.
(See section 4.3.)
Fig. 6.11c initially appears similar to the one in Fig. 6.11b, but the frequency of
the resonance peak then increases near ν = 0.87. In Fig. 6.11d we plot a waterfall
of spectra from higher ν (1 to 1.2) to illustrate that as n is increased, the resonance
peak observed at higher ν does not appear to change.
Fig. 6.12 is composed of six image plots of Re[σxx ] spectra on the ν-f plane, with
the color representing Re[σxx ] as n is increased and Δ is minimized. As n increases
59
Figure 6.11: a-c) Re[σxx ] vs f at many ν and T = 30 mK. ν of black traces
marked on the right. a) Spectra from ν = 0.8 to 1.2 with a ν step between
each spectrum of ν = 0.0143. b) Spectra from ν = 0.8 to 1.0 with a step of
ν = 0.0111 between spectrum. c) Spectra from ν = 1.0 to 1.2 with a step of
ν = 0.0091 between each spectrum.
we can see a region of enhanced resonance frequency develop for ν < 1. This region
moves in from low ν towards ν = 1 as n increases. For this sample there is no similar
region of enhanced f for ν > 1.
Fig. 6.13 shows fpk vs ν for ν = 0.8 to 1.0 and n from 1.7 to 3.0 × 1011 cm−2 . As
n is increased an enhancement in fpk appears first at a ν = 0.87 then the ν of the
transition to enhanced fpk moving in towards ν = 1 as n is increased.
60
Figure 6.12: Image plots of f vs ν with the color representing Re[σxx ] for a
variety of n. n listed on in image plot in units of 1011 cm−2 .
6.4
54 nm WQW – summary
The data from the 54 nm WQW samples have a number of common features. The
54 nm WQW samples (54-1, 54-2, 54-3) have spectra that exhibit a pinning mode
61
Figure 6.13: fpk vs ν, n is listed along the right of the plot in units of 1011
cm−2 . All fp k taken from spectra taken while maintaining symmetric charge
distribution in the well.
resonance. At sufficient n all these 54 nm WQW samples exhibit an n dependent
transition to a region of increased frequency in fpk vs ν for ν < 1. For samples in
which n was changed only with back gate bias, there is also a transition to a region
of enhanced f that appears for ν > 1. Defining a filling factor (νc ) of the transition
to enhanced fpk we summarize the evolution of the transition with n in Fig. 6.14, the
figure shows that νc increases toward with n. Defining the transition filling factor (νc )
62
as that of the peak in fpk vs ν. For all of the 54 nm samples transitions to regions of
enhanced fpk move in toward ν = 1 as n is increased.
Figure 6.14: Location of the transition to a higher frequency range in ν vs
n for fpk vs ν for samples 54-1 and 54-3 for ν < 1.
6.5
42 nm WQW – pinning mode resonances
around ν = 1
This section presents data for sample 42-1, a 42 nm WQW with both a front and
a back gate, taken in a top-loading dilution refrigerator at a temperature of 50 mK.
The data were taken in the milikelvin facility at the National High Magnetic Field
Laboratory, in superconducting magnet number one (SCM1).
For this sample as for sample 54-3 we could measure clean SdH oscillations. Again
when we were trying to keep the charge distribution symmetric, we apply gate voltages
so that the change in overall density due to each gate is the same.
63
Figure 6.15: Sample 42-1, 50 mK, n = 3.04 × 1011 cm−2 , and no bias on
either gate. a) Many spectra taken at a different ν normalized to ν = 1. ν is
marked on the left of the graph for black spectra, spectra are proportionally
vertically offset, and the step between spectra is ν = 0.00625. b) fpk vs. ν.
c) S/fpk vs. ν.
6.5.1
42 nm WQW – symmetric charge distribution
In this section, we present data taken while keeping the charge distribution approximately symmetric, similar to the data for sample 54-3. Fig. 6.15a shows spectra
taken at many ν, n as cooled, and both gates at 0 V. There is a resonance peak on
64
either side of ν = 1, its peak frequency decreases as ν moves away from ν = 1, and
it vanishes as ν approaches ν = 4/5 and ν = 6/5. If we plot fpk vs ν, as shown in
Fig. 6.15b, there is a slight bump at ν = 0.84. When we plot S/fpk vs. ν, as shown
in Fig. 6.15c, we observe that S/fpk falls near the line for full participation, and that
it does not have a feature at ν = 0.84 where we observe a bump in fpk vs ν.
The spectra in Fig. 6.16b and 6.16c show a small range of weakly enhanced resonance frequency around ν = 0.84. This small range of increased frequency becomes
more visible as n is increased. Fig. 6.17 shows fpk vs ν, for n = lower spectra were
insufficiently well developed for ν greater than about ν = 1.12 for us to discern a
resonance. There are two features in fpk vs. ν; a small bump occurs at ν ∼ 0.84 for
n ≥ 3.26 × 1011 cm−2 , and a small bump occurs for ν ∼ 1.15 for n ≥ 3.49 × 1011 cm−2 .
6.5.2
42 nm WQW – asymmetric charge distribution
We can explore the effect of Δ on the pinning mode resonance if we hold n constant while making the charge distribution asymmetric, by using combinations of gate
voltages such that δn = 0. Fig. 6.18 shows three sets of spectra taken at many ν,
normalized to ν = 1, taken at 50 mK, with n held constant near 3.25 × 1011 cm−2 .
Fig. 6.18a and Fig. 6.18c both show spectra taken at increased Δ compared to Fig.
6.18b to explore the effect of making the charge distribution assymetric.
To get a broad overview of the effect of changing the symmetry of the well on the
pinning mode for a variety of n, we can use a matrix of image plots as shown in Fig.
6.19. All of the image plots in Fig. 6.19 of f vs ν (color representing Re[σxx ]) are for
n where we observe a kink in fpk vs ν. Plots where the well is kept symmetric on the
left, plots of asymmetric charge distributions to the right of the thick black line, and
plots of the same n are separated by the dashed vertical lines. Examining Fig. 6.19,
we can see that changing the symmetry of the charge distribution in the well does
not seem to affect the fpk of the pinning mode resonance. Increasing Δ appears to
65
Figure 6.16: Sample 42-1, waterfalls of spectra taken at many ν. ν of black
traces are shown on the left, offset vertically proportional to ν, taken at 50
mK, normalized to ν = 1, at similar Δ, with n at the top of each waterfall,
and a step of ν = 0.00625 between spectra.
66
Figure 6.17: Sample 42-1 fpk vs. ν, traces are offset vertically by 0.5 GHz
per trace with n = 3.58 not offset, and n is in units of 1011 cm−2 .
increase the width and decrease the total conductivity of the pinning mode resonance.
On the left in Fig. 6.19, plots of S/fpk vs ν for various n show that the bump in fpk
is associated with a change in participation in the pinning mode resonance, but that
there does not seem to be any affect on the bump from making the charge distribution
in the well asymmetric other than to decrease the overall participation at the highest
n.
67
Figure 6.18: Sample 42-1, waterfalls of spectra taken at 50 mK, normalized
to a ν = 1 spectrum. ν of black traces is shown on the left, offset vertically
proportional to ν, a step of ν = 0.00625 between spectra, and n 3.25 ×
1011 cm−2 . a) Asymmetric charge distribution, with gates set to give δn =
0.22 × 1011 cm−2 . b) Symmetric charge distribution, gates balanced. c)
Asymmetric charge distribution, gates set to give δn = −0.22 × 1011 cm−2 .
6.5.3
42 nm WQW – summary
To summarize the 42 nm WQW data, we have observed a pinning mode on either
side of ν = 1 that changes only slightly as n is changed. As in the 54 nm WQW
68
Figure 6.19: Image plots of f vs. ν with the intensity of the plot representing
Re[σxx ] from sample 42-1. n and δ in units of 1011 cm−2 as marked on each
plot; plots on the left have symmetric charge distribution in the well (δn
0); plots right of the thick vertical line have asymmetric charge distribution
with postive δn corresponding to more charge added by the front gate.
69
Figure 6.20: Sample 42-1, with fpk vs. ν on the left, and S/fpk vs. ν
on the right. Symmetric states are black; non-symmetric with the voltage
biased more on the back gate in blue, non-symmetric with the voltage biased
towards the front gate in red; with the densities on the graph in units of
1011 cm−2 .
spectra, we identify the pinning mode resonance (minus the additional feature seen
in fpk vs ν for n ≥ 3.26 × 1011 cm−2 ) as a pinned IQHEWC. The bump we observed
in fpk vs ν appears on both sides of ν = 1, at least at the largest n. Unlike sample
54-1 or 54-3 there is little sensitivity of the ν of the bump to changing density. It is
70
unclear if the small bump in fpk vs ν seen in sample 42-1 is related to the large kink
fpk vs ν seen in the 54 nm samples.
6.6
Summary and analysis
The 42 nm WQW (42-1) and the 54 nm WQW (54-1, 54-2, 54-3) have spectra
that exhibit a pinning mode resonance, and these pinning mode resonances imply
the existence of electron solids in our samples. This pinning mode resonance has
features that evolve as n is changed. The transition we observe in the 54 nm WQWs
spectra from a region in ν of lower fpk to a region of higher fpk , we interpret as a
transition between an IQHEWC and an unknown electron solid. The location in ν of
this transition filling factor moves in toward ν = 1 as n is increased.
As shown in Fig 6.14 for ν < 1, samples 54-1 and 54-3 have νc vs n in agreement.
For ν > 1 sample 54-1 shows a transition for n ≥ 2.83 × 1011 cm−2 . Sample 54-3 does
not show the same transition for ν > 1. The difference between the samples is that
for sample 54-3, n is changed with front and back gate voltages set to at least roughly
preserve a symmetric charge distribution in the well. From the thesis of Shabani
[103], we know that an asymmetry in the charge distribution can stabilize insulating
states. Thus the charge distribution asymmetry in sample 54-1 is likely responsible
for making the unknown electron solid state preferable for ν > 1 as n is increased.
We do not understand why charge asymmetry would stabilize the unknown electron
solid for lower n for ν > 1, but does not change νc vs n for ν < 1.
We can draw a parallel to the RIQHE that Liu et al. [7] observed in a piece of 42
nm WQW from the same wafer as our sample. The RIQHE he observed appeared first
for ν < 1, then moved in toward ν = 1 as n was increased, similarly to the behavior of
the transition in fpk versus ν (for ν < 1) in the 54 nm samples we measured. As they
increased n further, they observed an RIQHE appear for ν > 1 and move in toward
ν = 1, again similar to the behavior of the transition in fpk vs ν that we report in our
71
samples. Liu et al. proposed that the RIQHE in the 42 nm WQW they measured
was from a transition between the ν = 1 IQHEWC and an electron solid comprised
of CFs [7]. Our results build on and expand those of Liu et al. While they are only
able to infer the existence of multiple electron solid phases, we observe pinning mode
resonance that has ν-regions of enhanced fpk that we identify as a new electron solid
phase.
We will now focus on the electron solid with enhanced fpk in 54 nm WQW, and
discuss the possibilities for its composition. The enhancement of fpk is either from
a decrease in the effective electron-electron interaction between quasi-carriers in the
electron solid, an increase in the strength of the effective disorder on the quasi-carriers
composing the electron solid, or a combination of both.
Fig. 6.21 shows spectra (Re[σxx ] vs f ) taken at n = 2.41 × 1011 cm−2 , various
temperatures, and two different ν from sample 54-1. If we compare Fig. 6.21a (taken
from a region of ν where the IQHEWC is) and Fig. 6.21b (taken from a region of ν
where the unknown electron solid is), we can see that the pinning mode resonance
in the region where the unknown electron solid is the dominant phase melts earlier.
While the enhanced-fpk region at ν = 0.85 shows the resonance disappear at lower
temperature then the unenhanced-fpk resonance at ν = 0.95 we can not ascribe it to
a different solid phase at ν = 0.85. This is because the melting point of a Wigner
solid depends upon the filling factor (ν ∗ ), in this case as described in Ref. melting.
A decrease in quasi-carrier density of the IQHEWC would reduce the strength of
the electron-electron interaction and increase fpk [12]. If there is a change in quasicarrier density, we would expect a corresponding dip in S/fpk vs. ν since Fukuyama’s
sum rule gives us a measure of the density of chargecarriers are participating. From
Fig. 6.10, we can observe that S/fpk vs ν indicates participation at the full quasi
carrier density n∗ .
The unknown electron solid is probably not a bubble or stripe phase. Bubbles and
72
Figure 6.21: Sample 54-1, (Re[σxx ] vs f ) taken at n = 2.41 × 1011 cm−2 ,
and various temperatures as marked on graph. a) taken from a region of ν
near ν = 1 where fpk is not enhanced. b) from a region of ν where fpk is
enhanced.
stripes are not predicted to be stable in the lowest LL, and have never been observed
in the lowest LL [12, 79, 80].
Another possibility is an electron solid composed of skyrmions [83, 84]. These
electron solid have been seen around ν = 1 [82]. The unknown electron solid in our
samples is unlikely to be an electron solid composed of skyrmions, as skyrmions will
shrink when the Zeeman energy increases and our samples have a g-factor 4 to 5 times
higher than the 30 nm QW in which the skyrmion electron solid was observed.
It is conceivable that under the experimental conditions, a two-component bilayer
solid could occur. Narasimhan and Ho [16] have described bilayer solid phases in73
cluding the effects of interlayer tunneling with phase diagrams in the space of the
interlayer separation d/a, and the tunneling Δ/EC , where a is a typical intercarrier spacing (a2 = 2/n∗ ), and EC is the Coulomb energy of point charges at that
separation. For large tunneling, one-component solids (no interlayer staggering) are
predicted, since charge transfer has an energy cost related to the tunneling (subband)
energy. In our experiments taking νc = 0.9, and n = 2.8 × 1011 cm−2 , a = 80 nm and
d/a = 0.4, giving EC = 16 K. From simulations Δ ∼ 25 K at this n, so Δ/EC ∼ 1.5.
For d/a ∼ 0.4, Narasimhan and Ho [16] calculate a transition to two-component rectangular lattice occurs at Δ/EC ∼ 0.6, so we expect the transition to be well within
the one component regime. It is not clear how Δ increased by unbalance will affect
the bilayer solid phase diagram.
In a low density (n = 1.0 × 1011 cm−2 ) 50 nm WQW 2DES system, a transition
between a phase (referred to as the A-phase) with a pinning mode that shows dependence on wavevector and the terminal WC (referred to as the B-phase) has been
observed [3]. This A-phase was observed below ν = 2/9, was reentrant around the 1/5
FQHE state, and then crossed over into the B-phase which dominates for ν ≤ 0.12.
One of the possibilities for the A-phase is a CF WC or a charge density wave composed of CF. The ν range of this A-phase (specifically 0.2 < ν < 0.12) appears similar
to the ν ∗ range in which we see our unknown electron solid. A transition between an
IQHEWC and an electron solid composed of CFs has been proposed as the source of a
n dependent RIQHE observed near the ν = 1 plateau [7]. Some calculations indicate
that various types of CF Wigner crystal are the applicable ground states for the low
ν (ν 1/5) solid phase [107–109].
To see if the new electron solid we observed in our WQW samples was related to
this A-phase, we prepared sample 54-2 with an 80 μm wide slot CPW on its surface.
fpk vs ν in Fig. 6.22 shows no change between sample 54-1 (30 μm) and sample
54-2 (80 μm). Hence, the new electron solid in our WQW does not show the same
74
Figure 6.22: fpk vs. ν of the pinning mode for samples 54-1 (circles) and
54-2 (squares). Both from spectra taken at n = 2.41 × 1011 cm−2
wavevector dependence as this A-phase of Ref. [3]. Even though we do not observe
any wavevector dependence in our unknown electron solid, an electron solid composed
of CF still a possibility.
75
CHAPTER 7
SIGNIFICANCE AND FUTURE WORK
This chapter places the results presented in this dissertation into broader context
and discusses some possibilities for future work.
7.1
7.1.1
EMPs in magnetic field gradients and
MEMPs
Significance
In systems that show the quantum Hall effects, edges are of crucial, fundamental importance. The picture of the edge of these systems has developed over the
long history of the study of 2DES in high field, starting from the original Halperin
picture [110] of the edge, and continuing with the use of the Landauer-Büttiker formalism [111] to describe non-local transport effects. The nature of the edge states
and of the reconstruction of the edges has been illuminated by calculations [53, 112],
and by experiments on tunneling between edge states [113–117], and on tunneling
between edge states and bulk material [118]. In particular, experiments addressing
fundamental physics of the fractional quantum Hall effect, including fractional charge
[119, 120], and more recently the nature of non-Abelian states [121, 122] have relied
on edge-state transport and tunneling.
EMPs, while capable exhibiting features due to edge reconstruction [44], can be
mostly explained by simple hydrodynamic models [21, 24, 29, 48], which combine
76
the electrostatics near the edge with the Hall conductivity. On one level the large
effort we expended in studying magnetic edges, was an attempt to access 2DES on
a small length scale without a density step, and so to study edge states with altered
balance between electrostatics, exchange and the spatially varying compressibility of
the 2DES.
Less remotely, EMPs in magnetic field gradients and MEMPs are novel plasma
excitations, which are clearly related to the way the 2DES is confined, as shown in
our work on the EMP in gradient field. They are different from normal EMPs and
also from the well-known [50] (magneto)plasmon excitations. Coupling to spin at
magnetic edges with only an in-plane external field has been reported [54]. Particularly in low magnetic fields (with the sort of magnetic film we used), it is possible to
envision magnetic quantum dots [123], in which confinement is exclusively magnetic,
or augmented by the gradient field of a ferromagnet. MEMPs or EMPs in magnetic
field gradient are closely related to the lower hybrid modes that such magnetic dots
would exhibit, and microwave measurements could provide a contactless means of
coupling to such dots.
7.1.2
Future work, EMPs in magnetic field gradients and
MEMP search
In light of the work we present in this dissertation, the approach to better understanding of EMPs in magnetic field gradients is reasonably clear.
First, we would like to improve our understanding of the ferromagnetic permalloy
film, either by measuring the magnetization of the film directly (using a superconducting quantum interference detector or a magnetic force microscope) or by measuring it
indirectly through the DC magnetoresistance of the 2DES under the magnet. A better understanding of the magnetization of the permalloy film will allow us to evaluate
if the ∂Bz /∂x term in Eq. 3.9 is correct.
77
Second, we would like to tilt the sample to provide an in-plane field to allow us to
change the inhomogeneity from a dip to a spike based on the direction of the in-plane
field as well as increase the overall magnitude of the magnetic inhomogeneity from
the permalloy film. This will allow us to change the sign and value of the ∂Bz /∂x
term and see if the velocity for the EMP we measure changes comparably.
Third, we would like to vary the gate bias to move where the density gradient is
located in relation to the magnetic field inhomogeneity. Varying the location of the
density gradient is another way to probe the dependence of vp on Bz and ∂Bz /∂x
and explore the accuracy of our model of an EMP propagating in a magnetic field
gradient.
With improved understanding of the EMP in magnetic field gradient, it may be
that the parameters (dispersion, damping, chirality etc.) of a MEMP will be predicted
[124]. In that case a more focused search for MEMPs can be undertaken.
7.2
7.2.1
A new electron solid in a WQW
Significance
Electron solid phases stabilized by the electron-electron interaction, as discussed
at length in Chapter 4, exist in many forms in systems that show the quantum Hall
effect. The shift to enhanced fpk that we have seen substantiates the suggestion in
ref. [7], that the WQW samples exhibit a previously unknown solid phase. Besides
demonstrating the existence of the pinning mode, and its sensitivity to the transition
between solids, our work essentially extends ref. [7] to n for which it shows the νrange of the IQHE centered at ν = 1 is significantly extended. Hence we studied a
transition between solids within the IQHE around ν = 1. The most striking feature
of the new solid is the dependence of the transition filling νc on n, and the close of
approach of νc to ν = 1 is increased.
78
It is likely that the region nearest ν = 1, without enhanced fpk , is a solid of
quasicarriers similar to those found in a narrower (30 nm) QW of similar n [2] and
in a 50 nm QW of lower n [82]. This follows since narrower QWs require larger n
[7, 125] to show the RIQHE features in dc that were reported in ref [7], and at low n
the fpk vs ν curves do not show an enhanced region.
The conclusion of Chapter 6 compared the various known electron solids with the
observed new solid phase, but there is as yet no clear identification of the nature of the
new solid. It is clear that the combination of a wide well and sufficient n is required
to see the new phase, though even in the WQWs we studied the ν = 1 IQHE gap is
due to spin, and not to subband energy. The softening of the short-range effective
interaction is likely of importance, and Jastrow correlations as in a CF WC [107–109],
may play a role, possibly in combination with order from spin or some modulation or
correlation involving the vertical wave function but more subtle than a two component
crystal. The key fact that we have elucidated here is that the transition can approach
ν = 1 so closely, and so correspond to small quasicarrier density n∗ .
7.2.2
Future work in WQW 2DES
Future work at higher fields (∼ 30 T) will allow us to see if there are also multiple
solid phases in the WQW 2DES at the low ν termination of the FQH series. If
so, we can compare these phases with the behavior of the new electron solid we have
observed around ν = 1, and also with the “A and B” phases [3] discussed in Section 4.5.
Our collaborators on the WQW project already plan high field DC magnetotransport
measurements of similar WQW 2DES, and we hope to use their results as a starting
point for higher field measurements.
For future work in exploring this new electron solid at B < 15 T there are three
approaches.
The first is to measure WQW 2DES with similar mobility and n, but with different
well widths. Larger well widths may have other effects than reducing the density
79
required to see the new phase [7, 125] We could search for a relationship between the
width of the well and νc vs n. The magnitude of the enhancment in fpk shift may be
dependent on the well width either through the altered sheer modulus of the solid or
through altered disorder. In particular disorder due to interface roughness [78] well
width is likely to be sensitive to well width [126] .
The second (related) approach is to vary the charge asymmetry in the well systematically to see if νc , as a function of n, changes. We have already seen that increasing
the charge asymmetry causes our new electron solid to appear for ν > 1, and we
would like systematically study the effect of δn on νc .
Third, we would like to apply an in-plane magnetic field. The addition of an
in-plane field has been observed to stabilize electron solids at higher ν [127]. While
interpretation of data for in-plane field is complex due to mixing of subband and
Landau wavefunctions, it may be possible to sort these effects out, and if the addition
of an in-plane field alters the dependence of νc on n, it could give us valuable clues
about the composition of the new electron solid.
80
BIBLIOGRAPHY
[1] P. D. Ye, L. W. Engel, D. C. Tsui, R. M. Lewis, L. N. Pfeiffer, and K. West.
Correlation lengths of the wigner-crystal order in a two-dimensional electron
system at high magnetic fields. Phys. Rev. Lett., 89:176802, Oct 2002. (document), 1.1
[2] Yong Chen, R. M. Lewis, L. W. Engel, D. C. Tsui, P. D. Ye, L. N. Pfeiffer,
and K. W. West. Microwave resonance of the 2d wigner crystal around integer
landau fillings. Phys. Rev. Lett., 91:016801, Jul 2003. (document), 1.3, 1.3, 4.2,
4.4, 4.3, 6.3.2, 7.2.1
[3] Yong P. Chen, R. M. Lewis, L. W. Engel, D. C. Tsui, P. D. Ye, Z. H. Wang,
L. N. Pfeiffer, and K. W. West. Evidence for two different solid phases of twodimensional electrons in high magnetic fields. Phys. Rev. Lett., 93:206805, Nov
2004. (document), 4.2, 4.5, 6.6, 7.2.2
[4] K. B. Cooper, M. P. Lilly, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West. Insulating phases of two-dimensional electrons in high landau levels: Observation
of sharp thresholds to conduction. Phys. Rev. B, 60:R11285–R11288, Oct 1999.
(document), 4.6, 4.6
[5] S. A. Mikhailov and Oleg Kirichek. Edge Excitations of Low-Dimensional
Charged Systems. Nova Science Publishers, New York, 2000. (document), 1.2,
2.1, 2.2, 2.3, 2.3, 1, 2, 3, 2.4, 3.8.1
[6] Yoseph Imry and Shang-keng Ma. Random-field instability of the ordered state
of continuous symmetry. Phys. Rev. Lett., 35:1399–1401, Nov 1975. (document),
4.3
[7] Yang Liu, C. G. Pappas, M. Shayegan, L. N. Pfeiffer, K. W. West, and K. W.
Baldwin. Observation of reentrant integer quantum hall states in the lowest
landau level. Phys. Rev. Lett., 109:036801, Jul 2012. (document), 1.3, 4.6, 5.1,
5.5, 6, 6.6, 6.6, 7.2.1, 7.2.2
81
[8] T. Vančura, T. Ihn, S. Broderick, K. Ensslin, W. Wegscheider, and M. Bichler.
Electron transport in a two-dimensional electron gas with magnetic barriers.
Phys. Rev. B, 62:5074–5078, Aug 2000. 1.2, 2.5, 2.5
[9] E. Wigner. On the interaction of electrons in metals. Phys. Rev., 46:1002–1011,
Dec 1934. 1.3, 4.2
[10] M. Shayegan. Properties of the Electron Solid, pages 71–108. Wiley-VCH Verlag
GmbH, 1997. 1.3
[11] B. Tanatar and D. M. Ceperley. Ground state of the two-dimensional electron
gas. Phys. Rev. B, 39:5005–5016, Mar 1989. 1.3, 4.2
[12] C.-C. Li, L. W. Engel, D. Shahar, D. C. Tsui, and M. Shayegan. Microwave conductivity resonance of two-dimensional hole system. Phys. Rev. Lett., 79:1353–
1356, Aug 1997. 1.3, 4.2, 4.5, 6.6
[13] Hidetoshi Fukuyama and Patrick A. Lee. Pinning and conductivity of twodimensional charge-density waves in magnetic fields. Phys. Rev. B, 18:6245–
6252, Dec 1978. 1.3, 4.3
[14] D. C. Tsui, H. L. Stormer, and A. C. Gossard. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett., 48:1559–1562, May 1982.
1.3, 4.1
[15] H. C. Manoharan, Y. W. Suen, M. B. Santos, and M. Shayegan. Evidence for
a bilayer quantum wigner solid. Phys. Rev. Lett., 77:1813–1816, Aug 1996. 1.3,
5.1
[16] Subha Narasimhan and Tin-Lun Ho. Wigner-crystal phases in bilayer quantum
hall systems. Phys. Rev. B, 52:12291–12306, Oct 1995. 6.6
[17] M. Shayegan. Case for the Magnetic-Field-Induced Two-Dimensional Wigner
Crystal, pages 343–384. Wiley-VCH Verlag GmbH, 2007. 1.3, 4.2
[18] J. K. Jain. Composite-fermion approach for the fractional quantum hall effect.
Phys. Rev. Lett., 63:199–202, Jul 1989. 1.3, 4.1
[19] J. K. Jain. Incompressible quantum hall states. Phys. Rev. B, 40:8079–8082,
Oct 1989.
82
[20] J. K. Jain. Theory of the fractional quantum hall effect. Phys. Rev. B, 41:7653–
7665, Apr 1990. 1.3, 4.1
[21] I. L. Aleiner and L. I. Glazman. Novel edge excitations of two-dimensional
electron liquid in a magnetic field. Phys. Rev. Lett., 72:2935–2938, May 1994.
2.1, 2.3, 7.1.1
[22] O. G. Balev and Nelson Studart. Temperature effects on edge magnetoplasmons
in the quantum hall regime. Phys. Rev. B, 61:2703–2710, Jan 2000.
[23] O. G. Balev, Nelson Studart, and P. Vasilopoulos. Edge magnetoplasmons in
periodically modulated structures. Phys. Rev. B, 62:15834–15841, Dec 2000.
2.3
[24] O. G. Balev and P. Vasilopoulos. Edge magnetoplasmons for very low temperatures and sharp density profiles. Phys. Rev. B, 56:13252–13262, Nov 1997.
7.1.1
[25] Alexander L. Fetter. Edge magnetoplasmons in a bounded two-dimensional
electron fluid. Phys. Rev. B, 32:7676–7684, Dec 1985. 2.3
[26] Alexander L. Fetter. Edge magnetoplasmons in a two-dimensional electron fluid
confined to a half-plane. Phys. Rev. B, 33:3717–3723, Mar 1986.
[27] Alexander L. Fetter. Magnetoplasmons in a two-dimensional electron fluid: Disk
geometry. Phys. Rev. B, 33:5221–5227, Apr 1986. 2.3
[28] S A Mikhailov and V A Volkov. Inter-edge magnetoplasmons in inhomogeneous two-dimensional electron systems. Journal of Physics: Condensed Matter,
4(31):6523, 1992. 2.3
[29] Mikhailov S. A. Volkov V. A. Theory of edge magnetoplasmons in a twodimensional electron gas. JETP Letters, 42:556–560, Dec 1985. 2.3, 7.1.1
[30] M. Wassermeier, J. Oshinowo, J. P. Kotthaus, A. H. MacDonald, C. T. Foxon,
and J. J. Harris. Edge magnetoplasmons in the fractional-quantum-hall-effect
regime. Phys. Rev. B, 41:10287–10290, May 1990. 2.1, 2.3
[31] S. J. Allen, H. L. Stormer, and J. C. M. Hwang. Dimensional resonance of the
two-dimensional electron gas in selectively doped gaas/algaas heterostructures.
Phys. Rev. B, 28:4875–4877, Oct 1983. 2.1, 2.3
83
[32] N. Q. Balaban, U. Meirav, Hadas Shtrikman, and V. Umansky. Observation
of the logarithmic dispersion of high-frequency edge excitations. Phys. Rev. B,
55:R13397–R13400, May 1997. 2.3, 5
[33] K. Bollweg, T. Kurth, D. Heitmann, V. Gudmundsson, E. Vasiliadou, P. Grambow, and K. Eberl. Detection of compressible and incompressible states in quantum dots and antidots by far-infrared spectroscopy. Phys. Rev. Lett., 76:2774–
2777, Apr 1996. 2.3
[34] T. Demel, D. Heitmann, P. Grambow, and K. Ploog. Nonlocal dynamic response
and level crossings in quantum-dot structures. Phys. Rev. Lett., 64:788–791, Feb
1990. 2.3
[35] T. Demel, D. Heitmann, P. Grambow, and K. Ploog. One-dimensional plasmons
in algaas/gaas quantum wires. Phys. Rev. Lett., 66:2657–2660, May 1991. 2.3
[36] G. Ernst, N. B. Zhitenev, R. J. Haug, and K. von Klitzing. Dynamic excitations
of fractional quantum hall edge channels. Phys. Rev. Lett., 79:3748–3751, Nov
1997. 2.3
[37] D. C. Glattli, E. Y. Andrei, G. Deville, J. Poitrenaud, and F. I. B. Williams.
Dynamical hall effect in a two-dimensional classical plasma. Phys. Rev. Lett.,
54:1710–1713, Apr 1985. 2.3
[38] I. M. Grodnensky, D. Heitmann, K. von Klitzing, and A. Yu. Kamaev. Dynamical response of a two-dimensional electron system in a strong magnetic field.
Phys. Rev. B, 44:1946–1949, Jul 1991. 2.3
[39] H. Kamata, T. Ota, K. Muraki, and T. Fujisawa. Voltage-controlled group
velocity of edge magnetoplasmon in the quantum hall regime. Phys. Rev. B,
81:085329, Feb 2010.
[40] D. B. Mast, A. J. Dahm, and A. L. Fetter. Observation of bulk and edge
magnetoplasmons in a two-dimensional electron fluid. Phys. Rev. Lett., 54:1706–
1709, Apr 1985. 2.3
[41] V. M. Muravev, A. A. Fortunatov, I. V. Kukushkin, J. H. Smet, W. Dietsche,
and K. von Klitzing. Tunable plasmonic crystals for edge magnetoplasmons of
a two-dimensional electron system. Phys. Rev. Lett., 101:216801, Nov 2008.
84
[42] A V Polisskii, V I Talyanskii, N B Zhitenev, J Cole, C G Smith, M Pepper, D A
Ritchie, J E F Frost, and G A C Jones. Low-frequency edge magnetoplasmons
in the quantum hall regime under conditions of reduced coulomb interaction.
Journal of Physics: Condensed Matter, 4(15):3955, 1992. 2.3, 2.3, 3.8.1, 3.8.1
[43] P. K. H. Sommerfeld, P. P. Steijaert, P. J. M. Peters, and R. W. van der
Heijden. Magnetoplasmons at boundaries between two-dimensional electron
systems. Phys. Rev. Lett., 74:2559–2562, Mar 1995. 2.4
[44] G. Sukhodub, F. Hohls, and R. J. Haug. Observation of an interedge magnetoplasmon mode in a degenerate two-dimensional electron gas. Phys. Rev. Lett.,
93:196801, Nov 2004. 2.3, 2.4, 7.1.1
[45] V. I. Talyanskii, A. V. Polisski, D. D. Arnone, M. Pepper, C. G. Smith, D. A.
Ritchie, J. E. Frost, and G. A. C. Jones. Spectroscopy of a two-dimensional
electron gas in the quantum-hall-effect regime by use of low-frequency edge
magnetoplasmons. Phys. Rev. B, 46:12427–12432, Nov 1992. 2.3, 4
[46] V. I. Talyanskii, M. Y. Simmons, J. E. F. Frost, M. Pepper, D. A. Ritchie, A. C.
Churchill, and G. A. C. Jones. Experimental investigation of the damping of
low-frequency edge magnetoplasmons in gaas-alx ga1−x as heterostructures. Phys.
Rev. B, 50:1582–1587, Jul 1994. 2.3
[47] A. M. C. Valkering, P. K. H. Sommerfeld, and R. W. van der Heijden. Effect
of the classical electron coulomb crystal on interedge magnetoplasmons. Phys.
Rev. B, 58:4138–4142, Aug 1998. 2.4
[48] N. B. Zhitenev, R. J. Haug, K. v. Klitzing, and K. Eberl. Experimental determination of the dispersion of edge magnetoplasmons confined in edge channels.
Phys. Rev. B, 49:7809–7812, Mar 1994. 7.1.1
[49] U. Zülicke, Robert Bluhm, V. Alan Kosteleckýand, and A. H. MacDonald.
Edge-magnetoplasmon wave-packet revivals in the quantum-hall effect. Phys.
Rev. B, 55:9800–9816, Apr 1997. 2.1
[50] Tsuneya Ando, Alan B. Fowler, and Frank Stern. Electronic properties of twodimensional systems. Rev. Mod. Phys., 54:437–672, Apr 1982. 2.2, 2.2, 7.1.1
[51] Mikhailov S. A. Volkov V. A. Edge magnetoplasmons: low-frequency weakly
damped excitations in two-dimensional electron systems. Sov. Phys.-JETP,
67:1639–1653, 1988. 2.3, 3.8.1
85
[52] N. Q. Balaban, U. Meirav, and I. Bar-Joseph. Absence of scaling in the integer
quantum hall effect. Phys. Rev. Lett., 81:4967–4970, Nov 1998. 2.3
[53] D. B. Chklovskii, B. I. Shklovskii, and L. I. Glazman. Electrostatics of edge
channels. Phys. Rev. B, 46:4026–4034, Aug 1992. 2.3, 3.8.1, 7.1.1
[54] A. Nogaret, S. J. Bending, and M. Henini. Resistance resonance effects through
magnetic edge states. Phys. Rev. Lett., 84:2231–2234, Mar 2000. 2.5, 2.5, 7.1.1
[55] J. H. Smet, K. von Klitzing, D. Weiss, and W. Wegscheider. dc transport of
composite fermions in weak periodic potentials. Phys. Rev. Lett., 80:4538–4541,
May 1998. 2.5
[56] J Reijniers and F M Peeters. Snake orbits and related magnetic edge states.
Journal of Physics: Condensed Matter, 12(47):9771, 2000. 2.5
[57] J. Reijniers and F. M. Peeters. Resistance effects due to magnetic guiding orbits.
Phys. Rev. B, 63:165317, Apr 2001. 2.5
[58] O. G. Balev and I. A. Larkin. Balev and larkin are interested in doing a through
theoretical analysis of an emp in an inhomogeneous magnetic field. private
communication, 2012. 3.8, 3.8.1
[59] O. G. Balev and I. A. Larkin. Private communication, 2012. 3.8.1
[60] Ivan A. Larkin and John H. Davies. Edge of the two-dimensional electron gas
in a gated heterostructure. Phys. Rev. B, 52:R5535–R5538, Aug 1995. 3.8.1
[61] M Volmer and J. Neamtu. simulation of magnetization curves in magnetic thin
films using the stoner-wolfarth model. Romanian Reports in Physics, 56:367–
372, 2004. 3.8.1, 3.8.1
[62] A. Matulis, F. M. Peeters, and P. Vasilopoulos. Wave-vector-dependent tunneling through magnetic barriers. Phys. Rev. Lett., 72:1518–1521, Mar 1994.
3.8.1
[63] L I Glazman and I A Larkin. Lateral position control of an electron channel in
a split-gate device. Semiconductor Science and Technology, 6(1):32, 1991. 3.8.1
[64] T. Ando, Y. Matsumoto, and Y. Uemura. Theory of Hall Effect in a TwoDimensional Electron System. Journal of the Physical Society of Japan, 39:279,
August 1975. 4.1
86
[65] K. v. Klitzing, G. Dorda, and M. Pepper. New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance.
Phys. Rev. Lett., 45:494–497, Aug 1980. 4.1
[66] R. E. Prange and S. M. Girvin. The Quantum Hall Effect. Springer- Verlag,
1990. 4.1
[67] R. B. Laughlin. Anomalous quantum hall effect: An incompressible quantum
fluid with fractionally charged excitations. Phys. Rev. Lett., 50:1395–1398, May
1983. 4.1
[68] H. A. Fertig. Properties of the Electron Solid, pages 71–108. Wiley-VCH Verlag
GmbH, 2007. 4.2
[69] D. M. Ceperley and B. J. Alder. Ground state of the electron gas by a stochastic
method. Phys. Rev. Lett., 45:566–569, Aug 1980. 4.2
[70] R. Chitra and T. Giamarchi. Zero field wigner crystal. Eur. Phys. J. B,
44(4):455–467, 2005. 4.2
[71] Pui K. Lam and S. M. Girvin. Liquid-solid transition and the fractional
quantum-hall effect. Phys. Rev. B, 30:473–475, Jul 1984. 4.2
[72] Y. E. Lozovik and V. I. Yudson. Crystallization of a two-dimensional electron
gas in a magnetic field. Soviet Journal of Experimental and Theoretical Physics
Letters, 22:11, July 1975. 4.2
[73] D. Levesque, J. J. Weis, and A. H. MacDonald. Crystallization of the incompressible quantum-fluid state of a two-dimensional electron gas in a strong
magnetic field. Phys. Rev. B, 30:1056–1058, Jul 1984. 4.2
[74] R. M. Lewis, Yong P. Chen, L. W. Engel, D. C. Tsui, L. N. Pfeiffer, and K. W.
West. Microwave resonance of the reentrant insulating quantum hall phases in
the first excited landau level. Phys. Rev. B, 71:081301, Feb 2005. 4.6, 4.6
[75] T. Sajoto, Y. P. Li, L. W. Engel, D. C. Tsui, and M. Shayegan. Hall resistance
of the reentrant insulating phase around the 1/5 fractional quantum hall liquid.
Phys. Rev. Lett., 70:2321–2324, Apr 1993.
[76] Kun Yang, F. D. M. Haldane, and E. H. Rezayi. Wigner crystals in the lowest
landau level at low-filling factors. Phys. Rev. B, 64:081301, Aug 2001. 4.2
87
[77] Lynn Bonsall and A. A. Maradudin. Some static and dynamical properties of
a two-dimensional wigner crystal. Phys. Rev. B, 15:1959–1973, Feb 1977. 4.3
[78] H. A. Fertig. Electromagnetic response of a pinned wigner crystal. Phys. Rev.
B, 59:2120–2141, Jan 1999. 4.3, 7.2.2
[79] R. M. Lewis, Yong Chen, L. W. Engel, D. C. Tsui, P. D. Ye, L. N. Pfeiffer, and
K. W. West. Evidence of a first-order phase transition between wigner-crystal
and bubble phases of 2d electrons in higher landau levels. Phys. Rev. Lett.,
93:176808, Oct 2004. 4.4, 4.5, 4.6, 4.6, 6.6
[80] R. M. Lewis, P. D. Ye, L. W. Engel, D. C. Tsui, L. N. Pfeiffer, and K. W. West.
Microwave resonance of the bubble phases in 1/4 and 3/4 filled high landau
levels. Phys. Rev. Lett., 89:136804, Sep 2002. 4.5, 6.6
[81] S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. Rezayi. Skyrmions and
the crossover from the integer to fractional quantum hall effect at small zeeman
energies. Phys. Rev. B, 47:16419–16426, Jun 1993. 4.5
[82] Han Zhu, G. Sambandamurthy, Yong P. Chen, P. Jiang, L. W. Engel, D. C.
Tsui, L. N. Pfeiffer, and K. W. West. Pinning-mode resonance of a skyrme
crystal near landau-level filling factor ν = 1. Phys. Rev. Lett., 104:226801, Jun
2010. 4.5, 6.6, 7.2.1
[83] L. Brey, H. A. Fertig, R. Côté, and A. H. MacDonald. Skyrme crystal in a
two-dimensional electron gas. Phys. Rev. Lett., 75:2562–2565, Sep 1995. 6.6
[84] R. Côté, A. H. MacDonald, Luis Brey, H. A. Fertig, S. M. Girvin, and H. T. C.
Stoof. Collective excitations, nmr, and phase transitions in skyrme crystals.
Phys. Rev. Lett., 78:4825–4828, Jun 1997. 4.5, 6.6
[85] J. S. Xia, W. Pan, C. L. Vicente, E. D. Adams, N. S. Sullivan, H. L. Stormer,
D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, and K. W. West. Electron correlation
in the second landau level: A competition between many nearly degenerate
quantum phases. Phys. Rev. Lett., 93:176809, Oct 2004. 4.6, 4.6
[86] R.R. Du, D.C. Tsui, H.L. Stormer, L.N. Pfeiffer, K.W. Baldwin, and K.W.
West. Strongly anisotropic transport in higher two-dimensional landau levels.
Solid State Communications, 109(6):389 – 394, 1999. 4.6
88
[87] M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West.
Evidence for an anisotropic state of two-dimensional electrons in high landau
levels. Phys. Rev. Lett., 82:394–397, Jan 1999. 4.6
[88] J. P. Eisenstein, K. B. Cooper, L. N. Pfeiffer, and K. W. West. Insulating and
fractional quantum hall states in the first excited landau level. Phys. Rev. Lett.,
88:076801, Jan 2002. 4.6
[89] A. L. Graninger, D. Kamburov, M. Shayegan, L. N. Pfeiffer, K. W. West, K. W.
Baldwin, and R. Winkler. Reentrant ν = 1 quantum hall state in a twodimensional hole system. Phys. Rev. Lett., 107:176810, Oct 2011. 5.1
[90] Yang Liu, D. Kamburov, M. Shayegan, L. N. Pfeiffer, K. W. West, and K. W.
Baldwin. Anomalous robustness of the ν = 5/2 fractional quantum hall state
near a sharp phase boundary. Phys. Rev. Lett., 107:176805, Oct 2011. 5.1
[91] Yang Liu, J. Shabani, D. Kamburov, M. Shayegan, L. N. Pfeiffer, K. W. West,
and K. W. Baldwin. Evolution of the 7/2 fractional quantum hall state in
two-subband systems. Phys. Rev. Lett., 107:266802, Dec 2011. 5.1
[92] Yang Liu, J. Shabani, and M. Shayegan. Stability of the q/3 fractional quantum
hall states. Phys. Rev. B, 84:195303, Nov 2011. 5.1
[93] J. Shabani, T. Gokmen, Y. T. Chiu, and M. Shayegan. Evidence for developing fractional quantum hall states at even denominator 1/2 and 1/4 fillings in
asymmetric wide quantum wells. Phys. Rev. Lett., 103:256802, Dec 2009. 5.1
[94] J. Shabani, Y. Liu, and M. Shayegan. Fractional quantum hall effect at high
fillings in a two-subband electron system. Phys. Rev. Lett., 105:246805, Dec
2010. 5.1
[95] M Shayegan, H C Manoharan, Y W Suen, T S Lay, and M B Santos. Correlated
bilayer electron states. Semiconductor Science and Technology, 11(11S):1539,
1996. 5.1
[96] Y. W. Suen, H. C. Manoharan, X. Ying, M. B. Santos, and M. Shayegan. Origin
of the ν = 1/2 fractional quantum hall state in wide single quantum wells. Phys.
Rev. Lett., 72:3405–3408, May 1994. 5.1
89
[97] Y. W. Suen, J. Jo, M. B. Santos, L. W. Engel, S. W. Hwang, and M. Shayegan.
Missing integral quantum hall effect in a wide single quantum well. Phys. Rev.
B, 44:5947–5950, Sep 1991. 5.1
[98] T. S. Lay, Y. W. Suen, H. C. Manoharan, X. Ying, M. B. Santos, and
M. Shayegan. Anomalous temperature dependence of the correlated ν = 1
quantum hall effect in bilayer electron systems. Phys. Rev. B, 50:17725–17728,
Dec 1994. 5.1
[99] K. Moon, H. Mori, Kun Yang, S. M. Girvin, A. H. MacDonald, L. Zheng,
D. Yoshioka, and Shou-Cheng Zhang. Spontaneous interlayer coherence in
double-layer quantum hall systems: Charged vortices and kosterlitz-thouless
phase transitions. Phys. Rev. B, 51:5138–5170, Feb 1995. 5.1
[100] Wanli Li, D. R. Luhman, D. C. Tsui, L. N. Pfeiffer, and K. W. West. Observation of reentrant phases induced by short-range disorder in the lowest landau
level of alx ga1−x As/al0.32 ga0.68 As heterostructures. Phys. Rev. Lett., 105:076803,
Aug 2010. 5.1
[101] I. Yang, W. Kang, S. T. Hannahs, L. N. Pfeiffer, and K. W. West. Vertical
confinement and evolution of reentrant insulating transition in the fractional
quantum hall regime. Phys. Rev. B, 68:121302, Sep 2003. 5.1
[102] H. C. Manoharan. New particles and phases in reduced-dimensional systems.
PhD thesis, Princeton University, 1998. 5.2
[103] J. Shabani. Fractional Quantum Hall Effect in Wide Quantum Wells. PhD
thesis, Princeton University, 2011. 5.2, 6.6
[104] C. P. Wen. Coplanar waveguide: A surface transmission line suitable for nonreciprocal gyromagnetic device applications. IEEE Trans. Microwave Theory
and Tech, 17:1087, 1969. 6.2
[105] L. W. Engel, D. Shahar, Ç. Kurdak, and D. C. Tsui. Microwave frequency
dependence of integer quantum hall effect: Evidence for finite-frequency scaling.
Phys. Rev. Lett., 71:2638–2641, Oct 1993. 6.2
[106] Zhihai Wang, Yong P. Chen, Han Zhu, L. W. Engel, D. C. Tsui, E. Tutuc, and
M. Shayegan. Unequal layer densities in bilayer wigner crystal at high magnetic
fields. Phys. Rev. B, 85:195408, May 2012. 6.2
90
[107] Chia-Chen Chang, Gun Sang Jeon, and Jainendra K. Jain. Microscopic verification of topological electron-vortex binding in the lowest landau-level crystal
state. Phys. Rev. Lett., 94:016809, Jan 2005. 6.6, 7.2.1
[108] R. Narevich, Ganpathy Murthy, and H. A. Fertig. Hamiltonian theory of the
composite-fermion wigner crystal. Phys. Rev. B, 64:245326, Dec 2001.
[109] Hangmo Yi and H. A. Fertig. Laughlin-jastrow-correlated wigner crystal in a
strong magnetic field. Phys. Rev. B, 58:4019–4027, Aug 1998. 6.6, 7.2.1
[110] B. I. Halperin. Quantized hall conductance, current-carrying edge states, and
the existence of extended states in a two-dimensional disordered potential. Phys.
Rev. B, 25:2185–2190, Feb 1982. 7.1.1
[111] M. Büttiker. Absence of backscattering in the quantum hall effect in multiprobe
conductors. Phys. Rev. B, 38:9375–9389, Nov 1988. 7.1.1
[112] C. de C. Chamon and X. G. Wen. Sharp and smooth boundaries of quantum
hall liquids. Phys. Rev. B, 49:8227–8241, Mar 1994. 7.1.1
[113] C. L. Kane and Matthew P. A. Fisher. Transport in a one-channel luttinger
liquid. Phys. Rev. Lett., 68:1220–1223, Feb 1992. 7.1.1
[114] M. Grayson, D. C. Tsui, L. N. Pfeiffer, K. W. West, and A. M. Chang. Continuum of chiral luttinger liquids at the fractional quantum hall edge. Phys. Rev.
Lett., 80:1062–1065, Feb 1998.
[115] O. M. Auslaender, A. Yacoby, R. de Picciotto, K. W. Baldwin, L. N. Pfeiffer,
and K. W. West. Experimental evidence for resonant tunneling in a luttinger
liquid. Phys. Rev. Lett., 84:1764–1767, Feb 2000.
[116] I. J. Maasilta and V. J. Goldman. Line shape of resonant tunneling between
fractional quantum hall edges. Phys. Rev. B, 55:4081–4084, Feb 1997.
[117] F.P. Milliken, C.P. Umbach, and R.A. Webb. Indications of a luttinger liquid
in the fractional quantum hall regime. Solid State Communications, 97(4):309
– 313, 1996. 7.1.1
[118] A. M. Chang, L. N. Pfeiffer, and K. W. West. Observation of chiral luttinger
behavior in electron tunneling into fractional quantum hall edges. Phys. Rev.
Lett., 77:2538–2541, Sep 1996. 7.1.1
91
[119] J. A. Simmons, H. P. Wei, L. W. Engel, D. C. Tsui, and M. Shayegan. Resistance fluctuations in narrow algaas/gaas heterostructures: Direct evidence
of fractional charge in the fractional quantum hall effect. Phys. Rev. Lett.,
63:1731–1734, Oct 1989. 7.1.1
[120] V. J. Goldman and B. Su. Resonant tunneling in the quantum hall regime:
Measurement of fractional charge. Science, 267(5200):1010–1012, 1995. 7.1.1
[121] A. Bid, N. Ofek, H. Inoue, M. Heblium, C. L. Kane, V. Umansky, and D. Mahalu. Observation of neutral modes in the fractional quantum hall regime.
Nature, 466:585, July 2010. 7.1.1
[122] R. L. Willett, L. N. Pfeiffer, and K. W. West. Alternation and interchange
of e/4 and e/2 period interference oscillations consistent with filling factor 5/2
non-abelian quasiparticles. Phys. Rev. B, 82:205301, Nov 2010. 7.1.1
[123] D. Uzur, A. Nogaret, H. E. Beere, D. A. Ritchie, C. H. Marrows, and B. J.
Hickey. Probing the annular electronic shell structure of a magnetic corral.
Phys. Rev. B, 69:241301, Jun 2004. 7.1.1
[124] O. G. Balev and I. A. Larkin. “magnetic gradient” edge magnetoplasmons in
non-uniform magnetic field. unpublished, 2012. 7.1.2
[125] Yang Liu and M. Shayegan. private communication, 2012. 7.2.1, 7.2.2
[126] H. Sakaki, T. Noda, K. Hirakawa, M. Tanaka, and T. Matsusue. Interface
roughness scattering in gaas/alas quantum wells. Applied Physics Letters,
51(23):1934–1936, 1987. 7.2.2
[127] W. Pan, G. A. Csáthy, D. C. Tsui, L. N. Pfeiffer, and K. W. West. Transition
from a fractional quantum hall liquid to an electron solid at landau level filling
ν = 13 in tilted magnetic fields. Phys. Rev. B, 71:035302, Jan 2005. 7.2.2
92
BIOGRAPHICAL SKETCH
Brenden A. Magill
Education
(Expected) Ph.D. Physics, Florida State University, 05/2013.
M.S. Physics, Florida State University, 05/2005.
B.S. Physics, San Diego State University, 05/2003.
Employment
Florida State University and NHMFL(09/2003-Present) Graduate Research
Assistant.
Charge density excitations bound to a linear magnetic field inhomogeneity.
Microwave spectroscopy of epitaxial graphene.
Edge magentoplasmons in a magnetic field gradient.
Microwave spectroscopy of “wide” quantum well 2DES.
Florida State University University (08/2003-06/2006) Teaching Assistant
Performed teaching duties and laboratory assistance in undergraduate physics
labs.
San Diego State University (01/2001-05/2003) Undergraduate Research Assistant.
Undergraduate Research Assistant.
93
Current Research
I am currently involved in a variety of research that can be separated into
three broad categories. These categories are: pump probe measurements of narrow
gap semiconductors, using surface plasmon resonance to functionalize surfaces for
nano-assembly, and setting up a cyrostat and probe to do optically detected magnetic
resonance measurements.
References
Dr. Lloyd W. Engel: Scholar/Scientist, National High Magnetic Field Laboratory, 1800 E. Paul Dirac Dr., Tallahassee, Fl, 32310, engel@magnet.fsu.edu,
850-644-6980
Dr. Nick Bonesteel: Professor of Physics, Florida State University, 315 Keen
Building, Tallahassee, Fl, 32310, bonestee@magnet.fsu.edu, 850-644-7805
Dr. Stephen McGill: Assistant Scholar/Scientist, National High Magnetic
Field Laboratory, 1800 E. Paul Dirac Dr., Tallahassee, Fl, 32310, mcgill@magnet.fsu.edu,
850-644-5890
Presentations/Posters
Moon, B.H.; Magill, B.A.; Engel, L.W.; Tsui, D.C.; Pfeiffer, L.N. and West K.W.,
Pinning mode of 2D electron system with short-range alloy disorder, American
Physical Society March Meeting, Dallas, TX, March 21-25 (2011)
Magill, B.A.; Polyanskii, A.A.; Engel, L.W.; Lilly, M.P.; Simmons, J.A.; Reno,
J.L., Microwave absorsoption of a 2D electron system in a spatially varying magnetic field, American Physical Society March Meeting, Portland, OR, March 17-19
(2010)
Magill, B.A.; Engel, L.W.; Lilly, M.P.; Simmons, J.A.; Reno, J.L., Microwave
modes of a two dimensional electron system in a spatially varying magnetic field,
American Physical Society March Meeting, Pittsburg, PA, March 16-20 (2009)
94
Magill, B.A.; Engel, L.W.; Lilly, M.P.; Simmons, J.A.; Reno, J.L., Microwave
modes of a two dimensional electron systems near macroscopic ferromagnets, American Physical Society March Meeting, New Orleans, LA, March 10-14 (2008)
Magill, B.A.; Engel, L.W.; Lilly, M.P.; Simmons, J.A.; Reno, J.L., Microwave
modes of a two dimensional electron system in a spatially varying magnetic field,
American Physical Society March Meeting, Denver, CO, March 5-9 (2007)
Urbano, R.R.; Pires, M.A.; Bittar, E.M.; Rettori, C.; Pagliuso, P.G.; Magill, B.A.;
Oseroff, S., ESR of Gd3+ in the Intermediate Valence Y bAl3 and its Reference
Compound LuAl3 , American Physical Society March Meeting, Los Angelas, CA,
March 21-25 (2005)
95
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