Download Microwave Spectroscopy of Two-Dimensional Electrons in Tilted Magnetic Field

Microwave spectroscopy of
two-dimensional electrons in tilted
magnetic field
Han Zhu
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of Physics
Adviser: Daniel C. Tsui
March 2010
c Copyright by Han Zhu, 2010.
All Rights Reserved
Abstract
A two-dimensional electron system in a perpendicular magnetic field exhibits a large
family of phases with crystalline or partial crystalline order. These include the Wigner
solid (WS) at small filling factor ν of the lowest Landau level (LL), WS formed of
dilute quasiparticles near integer ν, and the bubble and the stripe phases in the third
or higher LL’s.
These phases are pinned by sample disorder, and exhibit collective oscillation
modes, or pinning modes, which give rise to resonances in microwave conductivity
spectra, as found in previous studies. The resonances serve not only as evidence for
the particular phases, but also as tools for probing charge ordering, dynamics, and
disorder effects.
This thesis presents our microwave spectroscopic study of these phases in the
presence of an in-plane magnetic field, Bip , which can affect the electron orbitals and
spins.
We find that applying Bip expands the range of ν in which WS occurs in the lowest
LL. In the second LL, near integer ν, applying Bip similarly expands the range of ν
for the quasiparticle WS, accompanied by a significant increase in the pinning energy. The stripe phase shows a resonance only if measured with a microwave electric
field polarized perpendicular to the stripes. The resonance polarization can switch
in Bip , indicating reorientation of the stripes; the strength of the disorder pinning
depends on Bip , and plays an important role in orienting the stripes. The bubble
phase shows isotropic resonance at zero Bip ; applying Bip isotropically shifts up the
resonance frequency, and induces anisotropy in the resonance intensity. Because the
experimental systems have finite layer thickness, we ascribe these effects to Bip altering the electron orbital wave functions, which changes electron-electron interaction
and electron-disorder interaction.
Applying Bip also increases the Zeeman gap, and affects the spin degree of freedom.
iii
We find the pinning-mode resonance frequency of the solid near ν = 1 depends on the
Zeeman gap, in a manner consistent with expectation for a solid formed of skyrmions,
textures involving multiple flipped spins.
At the end of the thesis we also report pinning-mode evidence for WS of e/3
quasiparticles near ν = 1/3.
iv
Acknowledgements
During my graduate studies I have had the fortune to benefit from two great advisors,
Prof. Daniel C. Tsui at Princeton, and Dr. Lloyd W. Engel at National High Magnetic
Field Laboratory, and also from many brilliant students and postdocs who also offered
valuable training and help.
I would like to express my deepest gratitude to Prof. Tsui, for his guidance,
both in physics and in life. I came to Tsui group without much sense of what an
experimentalist is like. During my first experimental project on microwave cyclotron
resonance, Prof. Tsui would sit down with me for hours, and the training he gave
me ranged from explaining a lock-in amplifier to putting error bars. Over the years,
his guidance has always been sparkling with wisdom. The education I received from
Prof. Tsui is truly unforgettable and will continue to influence me profoundly in the
rest of my life.
I have also had the privilege to learn from Dr. Engel when I worked at the Magnet
lab. We are both passionate about the physics of Wigner crystals, microwaves, and
probably doing experiments in general. I thank Dr. Engel not only for the physics
and skills he taught me, but also for the inspirational style of education which made
research so enjoyable.
I have also benefited immensely from the training and help I received from many
students and postdocs in both Tsui group and Engel group. Wanli Li, Keji Lai, Ravi
Pillarisetty, and Gabor Csathy kindly helped me through studying dc transport 101
and cryogenics 101. G. Sambandamurthy trained me on operating dilution refrigerator, and gave many extremely useful lessons on survival skills as experimentalists.
Zhigang Jiang has always been a great resource of help on both quantum Hall physics
and experimental techniques. Yong Chen offered me many of his insights on Wigner
crystals and made available to me all his previous data. Zhihai Wang wrote many of
the Labview procedures (later revised by Pei-hsun Jiang) without which taking the
v
30,000 odd traces I took would be a much bigger pain. I am especially grateful to
Pei-hsun Jiang and Brenden Magill who accompanied me at the Magnet Lab, and
Tzu-Ming Lu who joined Tsui group the same time as I did, for always being ready
to help in my experiments.
There are many more people I am grateful to. I thank Loren Pfeiffer, Kenneth
West, Kirk Baldwin, Michael Manfra at the Bell Labs for providing the state-of-theart GaAs samples which fueled this research. I thank Prof. Ong and his crew, Lu
Li, Minhyea Lee, Joseph Checkelsky, for being so generous with their time and lab
resources during my work on cyclotron resonance in Jadwin Hall in 2005. I thank the
Helium supply team, the user support teams, the machine shop, and the electronics
shop at the Magnet Lab for their expert experimental assistance which have been
essential for all these works. And I thank Prof. Huse for being the Reader of this
thesis.
I thank all my friends, at Princeton and in Florida. Without them life would have
been unimaginably poorer, considering that I am away from my own country and
that I did most of the experiments in places without a window.
Finally, I would like to express my most sincere gratitude to my parents, for
everything, and to my wife, Qinzhe, for being by my side all the time and sharing her
life and dreams with me.
vi
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
1 Wigner solids and beyond
1
1.1
Wigner solids in zero magnetic field . . . . . . . . . . . . . . . . . . .
1
1.2
In magnetic field: 2D Wigner solids and beyond . . . . . . . . . . . .
3
1.2.1
Wigner solids in quantum Hall systems . . . . . . . . . . . . .
3
1.2.2
The charge-density-wave-like phases in higher LL’s . . . . . .
5
1.3
Microwave/rf spectroscopy of electron solids . . . . . . . . . . . . . .
7
1.4
The tilted field measurements . . . . . . . . . . . . . . . . . . . . . .
10
1.5
Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2 Samples and experimental methods
14
2.1
Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.2
Micrwave measurements . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.2.1
Coplanar wave guide . . . . . . . . . . . . . . . . . . . . . . .
14
2.2.2
The external circuit and the rotator . . . . . . . . . . . . . . .
18
2.2.3
Sample temperature . . . . . . . . . . . . . . . . . . . . . . .
21
3 Microwave/rf resonances of electron solid phases
22
3.1
Disorder pinning modes: a brief account . . . . . . . . . . . . . . . .
22
3.2
Microwave/rf pinning mode resonances of the WS . . . . . . . . . . .
26
vii
4 Tilt-induced Wigner solid in the lowest two Landau levels
4.1
4.2
4.3
4.4
30
Tilt-induced Wigner solid in the lowest Landau level . . . . . . . . .
31
4.1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.1.2
Effect of tilting on a system with relatively high density . . . .
33
4.1.3
Effect of tilting on a system with relatively low density . . . .
38
4.1.4
A phase diagram in tilted field . . . . . . . . . . . . . . . . . .
40
Tilt-induced Wigner solid in the second Landau level . . . . . . . . .
43
4.2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
4.2.2
Tilt-induced resonance at ν = 2 + 1/5 and 3 − 1/5 . . . . . . .
45
4.2.3
Tilt-strengthened disorder pinning . . . . . . . . . . . . . . . .
51
Interplay between the Wigner solid and the fractional quantum Hall
liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
Summary and discussions . . . . . . . . . . . . . . . . . . . . . . . .
57
5 Pinning modes of a Skyrme crystal near ν = 1
60
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
5.2
Pinning mode resonances near ν = 1 in perpendicular field . . . . . .
62
5.3
Pinning mode resonances near ν = 1 in tilted field . . . . . . . . . . .
64
5.4
Summary and discussions . . . . . . . . . . . . . . . . . . . . . . . .
70
6 Tilt-induced anisotropies in the third Landau level
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2
Pinning-mode resonances of the stripe and the bubble phases at zero
72
74
tilting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
6.3
The effect of tilting on the stripe anisotropy . . . . . . . . . . . . . .
83
6.4
Tilt induced anisotropy in the bubble phase . . . . . . . . . . . . . .
92
6.5
Summary and dicussions . . . . . . . . . . . . . . . . . . . . . . . . .
96
viii
7 Pinning mode resonances within the ν = 1/3 fractional quantum Hall
effect
98
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
7.2
Pinning mode resonance found near ν = 1/3 . . . . . . . . . . . . . . 100
7.3
Interpretation: a pinned solid of e/3 charge carriers with Coulomb
interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.4
Summary and discussions . . . . . . . . . . . . . . . . . . . . . . . . 107
8 Summaries and perspectives
108
A Higher order resonance peaks of the Wigner solids
114
B Supplementary data in the lowest Landau level in tilted field
118
C Coplanar wave-guides and microstrips
122
C.1 Coplanar wave guides . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
C.2 Microstrips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
D Supplementary information about the samples, cryostats and magnets
126
D.1 Cooling-down procedures . . . . . . . . . . . . . . . . . . . . . . . . . 126
D.2 “C120” (14T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
D.3
“SCM1” (18T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
D.4
“PDF” [Resistive (33T) and Hybrid (45T) cells] . . . . . . . . . . . . 129
E List of publications resulting from this thesis
ix
131
Chapter 1
Wigner solids and beyond
1.1
Wigner solids in zero magnetic field
The uniform system of electrons is one of the basic models of condensed matter
physics. For a group of electrons, if the Coulomb repulsion dominates the kinetic
energy, the favorable state shall be one in which the electrons can optimally allign
themselves so that their mutual repulsion can be minimum. It is not difficult to
imagine such a state as one in which electrons remain equally separated over space,
which means they have a crystal order.
The kinetic energy from thermal agitations can be suppressed simply by cooling.
At temperature T = 0, the Pauli exclusion principle renders the kinetic energy of the
electrons a function of their density. In 1934, Eugene Wigner (158) gave a simple
argument and showed that a crystalline phase, subsequently named after him, is
the ground state for a low density electron system in a neutralizing background.
In a T = 0, d-dimensional system of electrons with density n and inter-electron
√
separation r ∼ 1/ d n, the Coulomb repulsion scales as 1/r, and the kinetic energy
is characterized by the Fermi energy EF ∼ 1/kF2 , where kF ∼ 1/r is the Fermi wave
number. Clearly, the Coulomb energy dominates for sufficiently low n. A crystalline
1
order helps the electrons minimize repulsion by keeping them apart all the time.
More detailed calculations of the energetics reveal that a triangular lattice is optimal
for the two-dimensional (2D) case, and a body-centered-cubic (bcc) lattice for the
three-dimensional (3D) [see review in (49)].
It is difficult to make laboratory examples of 3D Wigner solid (WS) of electrons
(150), which requires n on the order of 1018 /cm3 (14) and an ultra-clean environment
to protect the positional order. There has been, though, an interesting demonstration of the 3D WS using not electrons, but charged polystyrene spheres in aqueous
suspension, which was indeed found to form a bcc lattice (164). A low-n 2D electron
system (2DES) is easier to obtain. A well-known experimentally realizable system
was a 2DES of density ∼ 108 /cm2 spread on the surface of liquid helium (72), and a
crystal was observed which had the predicted triangular lattice structure (10). Such
a crystal was classical because the thermal energy is much larger than the Fermi
energy. A quantum 2D WS (in zero magnetic field) in solid-state materials (3) has
been long sought after (21). A commonly used parameter to describe the system is
√
rs , the ratio of inter-electron separation r = 1/ πn to the Bohr radius aB , which can
be shown to be a measure of the Coulomb energy in units of the Fermi energy. A
Monte-Carlo study showed that a WS is favorable for rs > 37 (39, 152), corresponding
to GaAs-based 2DES density n on the level of ∼ 108 /cm2 [or 2D hole system (2DHS)
density p on the level of ∼ 109 /cm2 (for the larger band mass of the holes)]. In
ultra-clean semiconductor heterostructures, experiments have recently been pushed
to such densities [for example, (33)].
2
1.2
In magnetic field: 2D Wigner solids and beyond
It was realized in the 1970’s (59, 103) that a strong magnetic field can greatly facilitate
the Wigner crystallization in a 2DES which otherwise has too high density to show a
WS in zero field (39, 152). As a strong B is applied perpendicular to the plane, the in√
plane length scale of the electron orbits decreases as ∼ 1/ B. When eventually that
becomes much smaller than the inter-electron separation, the electrons are like point
charges and can optimally allign themselves to minimize repulsion. With advancing
technologies in magnets, low temperature, and semiconductor heterstructures, experimental evidence of such B-induced WS phases started to accumulate following the
discovery of the quantum Hall effects.
1.2.1
Wigner solids in quantum Hall systems
Here I recollect a few aspects about the quantum Hall effects (79, 128, 137) in relation
to the WS. In a perpendicular B field, the 2DES has its density of states discretized
into a ladder of LL’s. Each LL has the same degeneracy proportional to B, and the
gap between neighboring LL’s is ~ωc , where ωc = eB/m∗ is the cyclotron frequency,
and m∗ is the band mass of the electrons. Because of a nonzero Landé g factor, each
LL1 is further spin-plit. For a 2DES with density n, the filling factor ν of the spin-split
LL’s is nh/2eB. Such a system is well known to exhibit the integer (IQHE) (156)
and fractional (FQHE) (155) quantum Hall effects in dc transport: Near integer and
some rational fractional values of ν, the Hall resistance Rxy measured with sweeping
B exhibits a squence of plateaus with the quantized values of (1/ν)h/e2 , accompanied
by vanishing longitudinal resistance Rxx .
It was recognized early on (59, 103) that in the lowest LL, the FQHE sequence
1
Throughout this thesis, when we refer to the lowest, second or higher LL, they stand for the
orbital LL’s, unless explicitly mentioned.
3
ultimately gives way to a WS state for B sufficiently strong, or ν sufficiently small.
Much theoretical work has been done toward estimating exactly where the WS phase
replaces FQHE liquids as the ground state. The transition has been estimated to be
around ν ∼ 1/6 (84). But exact prediction is challenging (24, 66, 82, 84, 87, 141, 166,
174) because at low ν the energy curves of the two types of states can come so close
to each other that a small error in the estimation can vastly affect the prediction.
There have also been propositions (15, 16, 79, 113, 170) that correlations responsible
for the FQHE’s can still be relevant for the solid state.
In experiments, unlike the remarkable quantum Hall effects, the case for a WS
was built gradually [see reviews in (23, 49, 92, 139)]. At low ν, the FQHE sequence
is found to be terminated by an insulating phase (161), with the last FQHE state at
ν = 1/5 in a GaAs-based 2DES (80, 161), and ν = 1/3 in a 2DHS (33, 135, 136).
The insulator at low ν is consistent with the expectation for a WS phase. In the
presence of disorder, no truly long-range positional order can survive in the 2D world
(77), and a 2D WS will deform over space and get pinned by the disorder (see more
discussions on pinning in Chap. 3). At T = 0, the pinned solid pieces cannot slide
and thus are dc insulators. But the insulating behavior alone is not enough evidence
to identify a WS. When a strong sample disorder outweighs interaction, the electrons
will be individually trapped in the disorder potential, also making an insulator. The
case of a many-body phase having characteristics of crystal at the low-ν termination
of the FQHE sequence has been built by a combination of many experiments (13, 64,
67, 69, 80, 96, 118, 135, 136, 161, 163) that measured the dc transport, I-V curve,
narrow-band noise spectrum, radio-frequency (rf), and photoluminesence properties,
and also by the broad-band microwave/rf spectroscopy, which is the main tool used
in the experiments reported in this thesis.
It is also possible to find WS on the edges of the IQHE and FQHE states. In the
case of the IQHE’s, when ν is near integer values, there is a dilute system of electrons
4
or holes in the partially filled top LL. These electrons or holes must be localized (76)
and not contributing to the dc transport, in order for the Hall resistance to show
the quantized plateau. These electrons or holes can be described by a partial filling
factor ν ∗ = |ν − nearest integer|. If ν ∗ is sufficiently small, the electrons or holes have
their wave function size much smaller than their separation. Such a system satisfies
the condition for a WS (105), similarly as the WS found at low ν of the lowest LL.
But, since this subsystem is just a small portion of the 2DES, the positional order
can be more easily destroyed by disorder. It is only recently, with the production of
ultra-high mobility samples, that spectroscopic evidence of a solid phase near integer
ν was found [(18, 90), and also see Figs. 4.2, 6.1 and 7.1 for examples of such resonant
spectra].
The FQHE states are strongly-correlated liquids with fractionally charged excitations, as first captured by an accurate trial wave function proposed by Laughlin
(85). It is well accepted that much of the physics is contained in a model which
considered a new type of particles, the composite fermions (CF’s) (79). A CF is a
combined state of an electron with an even number of flux quanta. The FQHE’s of
electrons at fractional fillings of the LL’s can be understood as the IQHE’s of CF’s
at interger fillings of the CF-LL’s (78, 79). By analogy to the solid phase near the
electron IQHE states, a solid phase is also expected near the electron FQHE states,
or the CF IQHE states. Such a solid phase is expected to be formed by fractionally
charged quasiparticles (74). In Chap. 7, I shall present the spectroscopic evidence
for such a solid phase revealed in microwave measurements.
1.2.2
The charge-density-wave-like phases in higher LL’s
For the third or higher LL’s, Hartree-Fock (56, 83, 111), variational (55), and numerical (73) studies suggested that the physics is different from the FQHE-dominated
lowest LL (for review, see (42, 52, 53)). The main reason is the different short-range
5
interactions, since the electron wave function has a Gaussian shape in the lowest LL,
and a ring-like shape in high LL’s. If only a small portion ν ∗ of the top LL is filled by
a dilute system of electrons or holes, the short-range interaction is not important. In
these situations, we have found evidence for a WS phase (18, 90). As ν moves away
from integer in the third or higher LL’s, the rich sequence of FQHE states that flourished in the lowest LL are absent. Instead, two other types of states were discovered
in dc transport (40, 42, 102): states with highly anisotropic longitudinal magnetoresistance near ν = 9/2, 11/2, 13/2..., and, on the flanges of the anisotropic states,
“reentrant integer quantum Hall” (RIQH) states which are similar in dc transport to
the adjacent IQHE states. The anisotropic and the RIQH states have been identified
as the stripe and bubble phases which were predicted earlier as charge density waves
(CDW’s) for the third or higher LL’s (55, 56, 83, 111). The bubble phase has a predicted structure of triangular lattice like a WS (73), with each lattice site occupied by
a cluster of M electrons (or holes). In the presence of disorder, similar to the WS, the
lattice of bubbles will be pinned and not contributing to dc transport, which explains
the reentrant IQHE. Microwave/rf spectroscopic evidence for the pinning modes of
the bubble lattice was found by (91). The predicted stripe state was first understood
as a uni-directional CDW (56, 83, 111) and later has been discussed as an electron
liquid crystal ((57)). A highly anisotropic rf resonance was observed by our group in
the stripe phase and has been interpreted as its pinning mode (134).
In the second LL, the electron wave function is intermediate between being Gaussian and ring-like. Not surprisingly, a delicate competition between FQHE states and
CDW states has been found (43, 123, 165). The FQHE states at partial LL fillings
ν ∗ = 1/5, 2/5, and 1/3, and those at even-denominator fillings ν = 5/2 and 7/2, alternate with RIQH states which are thought to be pinned bubble phases. These states
are particularly close to each other in energy (65). As shall be discussed in Chapt.
5, we found that applying an in-plane field turns the ν = 2 + 1/5, 4/5 FQHE states
6
into WS phases which are stabilized by a strengthened electron-disorder interaction.
1.3
Microwave/rf spectroscopy of electron solids
It can be difficult to study a disorder-pinned solid using dc transport alone because
of its small dc conductance at low temperatures. In the presence of disorder, even
arbitrarily weak, a 2D solid cannot sustain long-range positional order, or an infinite
correlation length (77). The lattice breaks into finite pieces pinned by the disorder
potential. If disorder is strong and dominates over the electron interaction, it destroys
the crystal order on all scales. The situation we are more interested in is when the
disorder is weak and permits local crystal order. In such a weak-pinning picture,
although the pinned solid pieces cannot slide as a whole in a weak dc electric field
(therefore insulating), they perform collective oscillation modes, or pinning modes,
within the pinning potential. The broad-band microwave/rf 2 spectroscopy has proved
to be a particularly useful tool as it can couple to these modes. High mobility samples
(low disorder level) have been found to show striking resonances in their microwave/rf
absorption spectra, which have been interpreted as the pinning modes and have served
as signatures of the solid phases, including those occuring at low ν and near IQHE
and FQHE states, and the CDW-like phases in high LL’s.
A 2D WS in magnetic field has a magnetophonon mode and a magnetoplasmon
mode (10). The magnetoplasmon mode is gapped at the cyclotron frequency and is
well above the range of frequency of our spectroscopic measurements. The magnetophonon mode is gapless in the absence of disoder. Once disorder is turned on, the
correlation length of the WS is reduced to a finite scale. That limitation enforces a
cut-off at small wave vector and opens a gap in the dispersion relation of the magnetophonon mode (58, 116). When one measures the microwave absorption of the
2
Conventionally, rf stands for frequencies lower than 300 MHz and microwave is used for higher
frequencies. In this thesis sometimes I use the term “microwave” in the broader sense, to include
frequencies lower than 300 MHz.
7
system in the long wave length limit by applying a spatially near-uniform microwave
field, one is generating excitations over this gap.
The main technical challenge to observe a pinning mode is that it requires a
broad-band measurement of a small signal at milli-Kelvin temperatures. Some of
the earliest attempts (5) to probe the magneto-phonon excitations of the WS suffered
from instrumental problems and did not convincingly reveal the WS resonances (4, 64,
148, 149, 162, 163). Later experiments employing surface acoustic waves (118, 159),
lower frequency (low rf) measurements (97–99), or microwave strip-lines (with more
careful data extraction) (64, 75) found features that can be attributed to a pinned
WS.
The spectroscopic measurement technique which produced the results reported in
this thesis was first developed in the 1990’s (44–46, 92, 93) and has been continuously
optimized till today. It enables a measurement of the frequency dependence of the
longitudinal conductivity of the 2DES over a frequency range from ∼ 15 MHz to
∼ 25 GHz. Details of this technique are described in Chap. 2. The coupling to the
2DES is provided by a thin-metal-film coplanar wave guide (CPW) deposited onto
the sample surface. A vector network analyzer sends and receives microwave signal
and yields the full complex transmission tensor. The quantity of the most interest
to us is the loss in the transmitted power. The loss due to absorption by the 2DES
is singled out by comparing the transmitted power at the 2DES state of interest, to
the power when the 2DES is not expected to absorb any signal (e.g., depleted, or at
a prominent IQHE or FQHE state). Chap. 2 and Appendix C show the standard
transmission-line analysis used to compute the diagonal conductivity of the 2DES
from its absorption of the microwave.
As an example, Fig. 1.1 shows the frequency f dependence of the real diagonal
conductivity, Re [σxx (f )], obtained in a 2DHS (sample “H30”) with density p =
3.15×1010 /cm2 and a low-temperature mobility µ = 0.8×106 V/cm2 s. The spectrum
8
14
Re[σxx(f)] (µS)
12
10
8
6
ν=0.14
4
2
0
ν=0.33
400
800
f (MHz)
1200
Figure 1.1: The real diagonal conductivity spectra Re [σxx (f )] for ν = 0.14 and
ν = 1/3, measured in sample “H30”. The ν = 0.14 trace is offset by 2 µS for clarity.
The measurement was done in “C120” (see Chap. 2 and Appendix E for information),
in perpendicular field, and at temperature of 40 mK.
9
obtained at ν = 0.14, where the 2DHS is well into the low-ν insulating phase, is put in
comparison with the spectrum at ν = 1/3, an FQHE state. Both spectra are obtained,
as described above, by normalizing the transmitted microwave power measured in the
presence of the 2DHS layer against a reference measured after the 2DHS is depleted by
a positive back-gate voltage. In contrast to the essentially flat spectrum at ν = 1/3,
the ν = 0.14 state shows a striking resonance, which is interpreted as the pinning
mode of a WS phase (see discussions of the pinning modes in Chap. 3).
In more recent measurements of samples of high mobility in relatively wide quantum wells (“QW65” and “QW50”, see Tab. 2.1 for sample parameters) (19), the
low-ν phase with resonant microwave/rf spectra was found to show two different resonances: A relatively high frequency “A” resonance occuring above ν ∼ 1/8, which
crosses over to a relatively low freuquency “B” resonance at lower ν. The two distinct
types of resonances suggest two different solid phases, but their respective nature is
unknown (see preliminary data and discussions on the effect of tilting in Appendix
B).
1.4
The tilted field measurements
If the sample can be tilted at an arbitrary angle in the magnetic field, there will be
two degrees of freedom in the perpendicular component of the field, B⊥ , the in-plane
component, Bip , and the total field, Btot . The tilted field technique has proved to be
useful in quantum Hall physics, because it offers a convenient way to manipulate both
the orbital wave function and the spin of the 2DES. If the 2DES layer is infinitely
thin, the orbital degree of freedom is frozen, the effect of applying Bip is only to
increase the Zeeman gap. If the 2DES has a finite thickness, application of Bip also
couples the in-plane and the perpendicular dynamics of the electrons.
In this thesis I report a line of microwave/rf spectroscopic measurements done in
10
a tilted magnetic field, studying the WS phases (including those at low ν and those
near ν = 1, 2, and 3), and the “stripe” and the “bubble” phases. The spin and orbital
effects from the application of Bip play different roles in these situations, sometimes
producing quite dramatic results, as shall be presented in the following chapters.
1.5
Outline of the Thesis
Chapter 2 describes the samples and the experimental techniques, particularly the
“microwve sample rotator” which enables much of the works reported in this thesis.
Chapter 3 recaps our current understanding of the weak pinning of 2D electron
solid phases, and discusses the factors that determine the properties of the resonance,
such as the peak frequency and integrated intensity.
Chapter 4 presents the effect of applying Bip by tilting the sample on the competition between the WS phase and the FQHE states. In the lowest LL, tilting is found
to favor the WS phase. Tilting has a more significant effect for larger electron density,
or smaller lattice constant, which suggests this effect is due to finite thickness of the
2DES layer. In the second LL, tilting is also found to induce a WS at ν = 2 + 1/5
and 3 − 1/5, which are FQHE states at zero tilt. But unlike the case in the lowest
LL, here the data suggest the effect of tilting is from strengthened disorder pinning,
as indicated by the increase of resonance peak frequency, which lowers the energy of
the WS phase.
Chapter 5 presents the effect of Bip on the solid phase near ν = 1 and that just
below ν = 2, both belonging to the lowest orbital LL. Application of intermediately strong Bip , increasing the Zeeman gap, is found to have negligible effect on the
resonances just below ν = 2, but to increase the pinning frequencies near ν = 1,
particularly for smaller quasiparticle/hole densities. The data demonstrate that the
charge dynamics near ν = 1, characteristic of a crystal order, are affected by the spin
11
degree of freedom, in a manner consistent with the expectation for a Skyrme crystal.
Chapter 6 presents the effect of Bip on the stripe and bubble phases. The stripe
phase is found to show an rf pinning-mode resonance only for the rf electric field polarized perpendicular to the stripe orientation. Applying Bip can switch the polarization
along which the pinning mode resonance is observed, indicating the reorientation of
the stripe phase. The resonance frequency, which is a measure of the strength of
the disorder pinning, increases with Bip . The magnitude of this increase indicates
that disorder interaction is playing an important role in orienting the stripes. The
bubble phase shows isotropic resonance in zero Bip . Applying Bip causes the bubble
resonance peak frequency to isotropically increase, but induces a strong anisotropy
in the resonance amplitude. This chapter shows Bip (of the magnitude used in these
experiments) has fundamentally different effects on the stripe and the bubble phases.
Chapter 7 presents the observation, in perpendicular field, of microwave resonances for ν within a range of ∼ ±0.015 around ν = 1/3. The resonances are similar
to those observed near integer ν, on the edges of the IQHE’s. The resonances near
ν = 1/3 are interpreted as the pinning modes exhibited by the sought-after solid phase
formed by the dilute e/3-charged carriers with long-range Coulomb interaction.
Appendix A reports a continuing study of the higher order resonance peaks of
the WS phases, which reside on the higher-frequency side to the main resonance.
The data show that the peak frequencies of the higher order resonances can have a
qualitatively different dependence on ν from that of the main resonance.
Appendix B describes an ongoing research on the effect of tilting on the WS phase
that already exists in perpendicular field. The WS at zero tilting, as found by previous
studies (19), exhibits two distinct types of resonances, a higher f resonance “A” for
relatively high ν, which crosses over into a lower f resonance “B” for relatively low
ν. The current data seem to suggest that a Bip of intermediate magnitude may cause
an “A”-type resonance to cross over into a “B”-type.
12
Other appendices contain additional information on the CPW, microstrips, cooldown procedures, and notes on the different cryogenic setups where these measurements took place.
All the data reported in this thesis including the analysis done for them are stored
in
https://files.magnet.fsu.edu/cms/hzhu
They can be accessed via the intranet of the National High Magnetic Field Laboratory, or upon request [to Lloyd Engel ([email protected]) or to the author at
his life-long email address [email protected]].
13
Chapter 2
Samples and experimental methods
2.1
Samples
The measurements reported in this thesis used 2DES’s and 2DHS’s in modulation
doped AlGaAs/GaAs/AlGaAs quantum wells of high quality, grown with molecular
beam epitaxy (MBE) [for review, see (41, 125, 127)] by L. N. Pfeiffer, K. W. West,
K. Baldwin at Bell Labs (now in the new lab housed in the Princeton Engineering
Quadrangle). Table 2.1 contains a list of the wafer numbers of the samples used, the
important parameters, and the part of data in this thesis they produced.
2.2
2.2.1
Micrwave measurements
Coplanar wave guide
The purpose of the measurement is to obtain the ac diagonal conductivity of the 2DES
at rf to microwave frequencies. We carry out such measurements using coplanar wave
guides (CPW’s) which are deposited on the sample surface and coupling capacitively
to the 2DES. The CPW’s, invented by C. P. Wen (157) four decades ago, have
found wide application in millimeter wave integrated circuits because they can be
14
wafer
name
QW15 (1009-01.1)
QW65 (830-99.1)
QW30 (120-05.1)
QW50 (720-99.1)
H30 (1-1299.1)
structure
nominal n (p)
(cm−2 )
15 nm QW 5 × 1010
(n-type)
65 nm QW 5 × 1010
(n-type)
30 nm QW 27 × 1010
(n-type)
50 nm QW 11 × 1010
(n-type)
30 nm QW 10 × 1010
(p-type)
nominal µ
(cm2 /Vs)
1 × 106
Data
8 × 106
Chap. 3 & 4 &
Appen. B
Chap. 4 & 5 & 6
29 × 106
Chap. 3
10 × 106
Chap. 5 & 7
0.8 × 106
Chap. 1 & Appen. A
Table 2.1: Summary of samples studied. Each sample has a “nickname” and a Pfeiffer’s wafter number. The densities (n for 2DES or p for 2DHS) and low-temperature
mobilities µ are as-cooled values. Also listed for each sample are the parts of data
presented in this thesis that each of them produced.
conveniently printed on a board (131). In our samples, because the 2DES’s lie on a
plane typically d ∼ 0.1 − 0.5 µm below the sample surface, the CPW measurement
is an ideal technique for measuring the microwave conductivities (46, 93).
A schematic of a CPW on a sample is shown in Fig. 2.1 (a-b). The CPW consists
of a driven center line separated from two grounded side planes. The slot width w
ranges from 20 to 80 µm. The width of the center line s is kept to be 1.5 times the
slot width so that in the absence of the 2DES, the impedance is matched to 50 Ω,
same as the coaxial cables (co-ax for short) and other external transmission lines
used to transmit the microwave signal to and from the sample. The CPW’s on the
samples cut from wafer “QW30” have a straight center line of w = 78 µm, and of
length l between 3 to 6 mm, for the purpose of controlling the polarization of the
microwave electric field (the need for that will be explained in Chap. 6). The other
samples all have a meandering-type center line with length l up to 28 mm, for a larger
signal/noise ratio.
The microwave signal is sent from an Agilent network analyzer. Before it reaches
the sample, the signal goes through a number of different coaxial cables, attenuators,
15
(a)
(b)
w s
(c)
Lodx
Codx
Ydx
Figure 2.1: (a) A schematic of the measurement setup, with a meandering-type CPW
on the sample. (b) A cross-section view of the CPW. (c) A model of the measuring
circuit, where L0 is the inductance per unit length of the CPW, C0 the capacitance
per unit length, and Y the admittance per unit length due to the 2DES.
16
microstrips, and circuit-board CPW’s (a low-pass filter is being planned too), which
I shall describe with more details later. In the remaining of this section, assuming we
have measured the absorption of the microwave signal by the 2DES, I shall focus on
how to convert that to the diagonal conductivity of the 2DES.
In the absence of the 2DES, the CPW has inductance L0 and capacitance C0
per unit length, and has a geometry designed to match its impedance to 50 Ω for
suppressing reflection at the points where the CPW joins the external transmission
lines. The CPW is made of gold film at least 300 nm thick to have negligible sheet
resistance. In the presence of the 2DES, we make the following assumptions about
the measurement:
1. The 2DES has a small conductivity Re(σxx ) and appears as only a weak load
on the CPW. This guarantees that the impedance Z0 of the sample CPW is
still close to 50 Ω, with low reflections at the joints, and limits the power-toconductivity conversion in the linear regime.
2. There is no leakage down to the 2DES.
3. The measurement is done at sufficiently high frequency ω, so that 2σxx /w ≪
ωC0 ).
The CPW can be modeled as a circuit shown in Fig. 2.1(c). The 2DES appears
as a load with admittance per unit length Y ≈ 2σxx /w. When the above assumptions
are satisfied, Appendix D.1 shows that
Re [σxx (f )] = (w/2lZ0 ) ln(P/P0 ),
(2.1)
where P (or P0 ) is the loss in the transmitted signal with (or without) the 2DES,
assuming that one can turn on and off the 2DES absorption without affecting the
external transmission lines.
17
In a real measurement, the loss of the power (initial output minus what returns
to the analyzer) mostly occurs at the various attenuators and the co-ax cables (5 to
10 m long, with nonnegligible resistance), and some parts of the transmission lines
are even deliberately made lossy for a more efficient cooling. The absorption by the
2DES is usually less than 10% of the total loss, P . A reference, P0 , is substracted
from P to single out the 2DES absorption. In some measurements, we can take a
reference after depleting the 2DES by applying a sufficiently large negative voltage
at the back gate (or a large positive voltage to deplete a 2DHS). We have found that
the prominent IQHE and FQHE states have essentially flat spectra with respect to
the depleted state (see the example shown in Fig. 1.1), which makes it possible to
use those states as alternative reference.
2.2.2
The external circuit and the rotator
In the earlier version of the spectroscopic experiments (46, 92), the microwave signal
was generated at room temperature, sent one-way into the cryostat to the sample
and detected in the cold space. Later developments combining high quality (low loss,
low heat load) co-ax and careful cold-sinking have enabled round-trip transmission
of microwave signal. The signal is sent from the output port of the network analyzer
at room temperature, and eventually returns to the input port. The external circuit
to be described in this section includes all the components between the analyzer and
the CPW on the sample.
The experiments were done in different environments, including a “C120” bottomloading dilution refridgerator (dilfridge for short) equipped with a 14 T magnet, an
“SCM1” top-loading dilfridge with an 18 T magnet, and a “PDF” top-loading dilfridge
with a long narrow mixing chamber to fit the small bores of the resistive (33/35 T)
and the hybrid (45) magnets (more details in Appendix D).
In the following, I use the probe for the “PDF” dilfridge as an example to explain
18
Figure 2.2: Picture of a sample mounted on a block. The sample is 3 × 5 mm in size
and is equipped with a CPW of l = 15.5 mm and slot w = 60 µm. The CPW on the
sample is indium soldered to the circuit-board CPW on the block. The circuit-board
CPW then joins (mechanically via pressed pins) the semirigid co-ax at the side of the
block (not shown). Also seen are (gold) dc wires connected to the (indium) ohmic
contacts on the sample edges.
the path for the microwave signal. From the output port of the network analyzer, the
signal at the power level of about −30 dBm first goes through a low-loss flexible co-ax
of length ∼ 3.5 m before it reaches the top of the dilfridge probe. A 40 dB attenuator
there reduces the power level of the signal. Then the signal goes down into the probe
which is inserted into the dilfridge. The signal travels through a low-loss semirigid
co-ax cable. This cable is joined by a lossy cable before it reaches the position of
the 1K pot, for the dual purposes of cooling the cable pin and not heating up the
pot. At the end of the probe, the lossy cable is replaced by a low-loss cable that
goes to the sample block. After passing through the sample, the signal returns to the
analyzer input port via an almost identical route, except that an amplifier instead of
the attenuator sits on top of the probe.
What remains to be described is how to connect the co-ax to the CPW on the sam-
19
（ａ）　“ＰＤＦ”　ｒｏｔａｔｏｒ
（ｂ）　“ＳＣＭ１”　ｒｏｔａｔｏｒ
Figure 2.3: Pictures of the rotators for (a) the “PDF” dilfridge and (b) the “SCM1”
dilfridge. The “PDF” rotator body is made of high strength plastic (Parmax), and
the “SCM1” rotator body is made of brass.
ple. Two types of measurements can be done, that with the sample plane fixed to be
perpendicular to the magnetic field, or with the sample rotatable. For the perpendicular field measurement, as shown in Fig. 2.2, the sample is mounted on a coin-silver
block, on which a circuit-board CPW is embedded and is connected to the sample
by indium soldering. The pin of the external co-ax is clamped down to the circuitboard CPW and makes a well-matched connection. For the tilted field measurement1 ,
the sample is mounted on a half-circular shaped block, on which a microstrip (see
Appendix D.2) is fixed, and is connected to the sample via a microstrip-to-CPW
joint. After the microstrip leaves the block carrying the microwave signal, it is free to
twist for about 2 inches. Then it meets the co-ax via a standard coax-to-microstrip
launcher. The sample block is fixed to a rotator. The rotation is controlled by a
simple system consisting of a BeCu spring with one end fixed to the rack and the
other end tied to a Cavalier string, which goes around the pulley and then is tied to
a stainless-steel
2
rod spanning the length of the probe. An actuator is mounted on
1
The rotator for the “PDF” dilfridge was a recent development, and the tilted-field data reported
in this thesis mainly came from the “C120” and “SCM1” dilfridges.
2
Or G10 plus stainless steel, to replicate the structure of the dilfridge or probe for having the
same thermal expansion.
20
top of the probe and can pull or push. A picture showing just the sample part can
be found in Fig. 2.3 (a).
The rotators used in “C120” and “SCM1” [Fig. 2.3 (b)] dilfridges have different
designs, thanks to the ampler space allowed by those dilfridges. The sample is still
mounted on the coin-silver cube used for perpendicular field measurements, as shown
in Fig. 2.2. If switching to a tilted field measurement, the whole block is mounted
on a rotator controlled by a similar spring/rod/actuator system described above [Fig.
2.3(b)]. The block and the co-ax are joined by a flexible microstrip that permits
rotation of the block.
The flexible microstrips, the most important part in the microwave rotators, are
home made and designed to have an impedance of 50 Ω, matched to all the other microwave components. The joints between microstrips and the sample CPW’s, circuitboard CPW’s, and the co-ax cables have been optimized to minimize reflection.
2.2.3
Sample temperature
In the bottom-loading “C120” dilfridge, the microwave measuring setup is thermally
anchored to the mixing chamber, whereas in the top-loading “SCM1” or “PDF” dilfridge, the sample is immersed in the helium liquid. Due to the heating effect of
microwave and the heat load carried by the co-ax cables, the actual sample temperature can be higher than the mixing chamber temperature, by a larger margin
in a bottom-loading dilfridge. To minimize the heating effect of the microwave, we
do measurements in the low-power limit, when further lowering the power does not
affect the signal except only increasing noise. Usually the power on the sample level
is ∼ 1 pW. The actual sample temperature is estimated by monitoring the measurement result while tuning the mixing chamber temperature. We estimate the base
sample temperature to be ∼ 40 mK in the “C120” dilfridge, ∼ 55 mK in the “SCM1”
dilfridge, and ∼ 60 to 70 mK in the “PDF” dilfridge.
21
Chapter 3
Microwave/rf resonances of
electron solid phases
The microwave/rf resonances, such as the example presented in Fig. 1.1, and many
more to be presented in the rest of this thesis for the various WS or CDW-like phases,
are interpreted as the disorder pinning modes. This chapter gives a brief theoretical
account of this striking phenomena.
3.1
Disorder pinning modes: a brief account
We consider a 2D WS consisting of electrons which in magnetic field B has a wave
function size lB . The sample has a disorder potential, V (r), with strength V0 and
correlation length ξ0 . A weak disorder cannot wipe out the crystal order completely,
but over a long distance it has an accumulating effect and breaks the crystal into
pinned pieces (77). The pinned solid pieces perform collective oscillation modes, or
pinning modes, around the pinning disorder potential (22, 51, 54, 58, 116). The
pinning modes can be viewed as the disorder-gapped magneto-phonon excitations of
the WS.
At B = 0, a 2D WS consisting of point charges (10) has a longitudinal mode
22
ωL (q) (basically the plasmon mode, which exists also in phases other than solid), and
a transverse mode ωT (q) (the result of a non-zero shear modulus, a defining trait of
a solid). Both modes are gapless in a clean system:
ωL (q) = (ne2 /2m∗ ǫ0 ǫr )1/2 q 1/2 ,
(3.1)
ωT (q) = (µT /nm∗ )1/2 q,
(3.2)
and
where ǫr = 13 for GaAs, µT = 0.245(e2 n3/2 /4πǫ0 ǫr ) is the shear modulus. In a
perpendicular magnetic field B, the two modes are mixed due to the Lorentz force
into two hybridized branches (116),
2
ω±
1
1
= (ωc2 + ωL2 + ωT2 ) ±
2
2
q
(ωc2 + ωL2 + ωT2 )2 − 4ωT2 ωL2 ,
(3.3)
where ωc is the cyclotron resonance frequency. In typical magnetic fields of our
measurements, ωc ≫ ωL and ωT , and the result is a magneto-plasmon mode which is
gapped at ωc , and a gapless magneto-phonon mode,
ω− (q) = ωL (q)ωT (q)/ωc ,
(3.4)
which has a q 3/2 dispersion relation in the long wavelength limit. In a typical measurement, ωc is at or above the order of 100 GHz, and our microwave measurements
in the GHz or sub-GHz range only excite the ω− mode.
The presence of disorder limits the correlation length of the WS and results in a
cut-off at low q, opening a gap in the ω− mode. Earlier treatments (58, 116) characterized the pinning disorder as a harmonic potential with characteristic frequency
23
ω0 . The ω− mode becomes
ω− (q) =
q
(ω02 + ωL2 (q))(ω02 + ωT2 (q))/ωc ,
(3.5)
with a q = 0 gap of ω02 /ωc .
The pinning-mode resonance with peak frequency fpk shown in the microwave
measurements has been interpreted as essentially measuring this gap 1 . In order to
predict the pinning-mode frequency fpk = (1/2π)ω02 /ωc , one has to know ω0 , the
phenomenological angular frequency characterizing the disorder potential. If ω0 is
independent of B, fpk ∝ 1/ωc ∝ 1/B, which does not always hold as found by the
pinning mode measurements previously done by our group for the lowest-LL WS (17).
fpk can even increase with B in some samples. This is because of the oversimplification
by assuming ω0 as an always B-independent constant.
The interaction between electron and disorder is subject to the fact that, if B
is not infinitely strong, the electron wave function has a finite in-plane length scale
p
lB ∼ ~e/B (the magnetic length) (22, 51, 54). If lB is much smaller than the
intrinsic correlation length of the sample disorder, ξ0 , the electrons can still be treated
as point charges; otherwise the disorder responsible for pinning the electrons is an
effective one and is determined by a convolution of the electron density profile and
the disorder profile.
Chitra et al. (22) treated the disorder pinning when considering an electron
wave function with the size of the magnetic length lB , and a disorder potential V (r)
with a Gaussian correlation function hV (r)V (r′ )i = ∆D(r − r′ ), where D(r − r′ ) ∼
(1/ξ02 ) exp[−(r − r′ )2 /ξ02 ]. We can try to construct the form of the peak frequency of
the pinning-mode resonance, fpk , by a dimensional analysis 2 . First we consider the
situation where lB ≪ ξ0 . Because fpk ∼ ω02 /ωc , what needs to be known is ω02 . The
1
In some cases, as shown in (19), the measurements are not in the long wavelength limit and one
has to consider the dispersion.
2
Suggested by P. Littlewood during a discussion.
24
relevant quantities are electron mass m∗ , shear modulus µT , and the disorder parameters ∆ and ξ0 . The quantity that strengthens the pinning is ∆, and the quantities
that weaken the pinning are m∗ , µT (stiffer solid is less sensitive to disorder), and
ξ0 (larger ξ0 means a larger Larkin domain), and they can be used to construct the
following term, ∆/m∗ µT ξ06 , which has the dimension of [T ]−2 . From this,
fpk ∼ ω02 /ωc ∼
∆
∼ 1/B.
µT ξ06 eB
(3.6)
If lB ≫ ξ0 , due to the averaging effect of the electron wave function, one has to replace
ξ0 with lB , and Eq. (3.6) becomes
fpk ∼
∆e2 B 2
∆
=
∼ B2.
6
µ T lB
eB
µT ~3
(3.7)
Eqs. (3.6) and (3.7) agree with (22). We see that the B-dependence of fpk can
be anywhere between 1/B to B 2 . In our experiments, the B-dependence of fpk does
not have a universal form from sample to sample. But an important conclusion we
can draw, which Eqs. (3.6) and (3.7) agree on, is that fpk ∼ ∆/µT , which means
the pinning mode frequency is higher for stronger disorder, and lower for stronger
interaction (stiffer solid). This may be the most important feature of the weak-pinning
theory and shall be referred to many times throughout this thesis. Because µT has
been determined for a classical system to be ∼ n3/2 (10), that predicts fpk ∼ n−3/2 .
This relation has not been universally verified in experiments, mainly because it is
difficult to vary n (e.g., by applying a back-gate voltage) while keeping the other
factors constant.
Another important result, which shall be called on many times in the following
chapters, is a sum rule derived by Fukuyama and Lee (58) with regard to the integrated intensity S obtained by integrating the Re [σxx (f )] spectrum. The sum rule
25
has a simple form 3 ,
S/fpk = neπ/2B.
(3.8)
This result is from a treatment of an isotropic, classical CDW with cosine variance
of the charge density in a strong magnetic field so that ωc ≫ ωL ωT . The real WS
encountered in our experiments is expected to assume a different form than this
model, but in the previous experiments of our group, we still found a good agreement
with this sum rule in the low-ν WS phase in high quality samples, where a resonance
is well developed (17). As shall be presented in the following chapters, a less perfect
but still strong consistency with this sum rule is found for the more subtle solid phases
near the IQHE ((18, 90), Chaps. 4 and 7) and the FQHE states (Chap. 7). But this
sum rule cannot be directly applied if the phase considered is anisotropic and shows
direction-dependent resonance (Chap. 6).
3.2
Microwave/rf pinning mode resonances of the
WS
In our microwave/rf spectroscopy experiments, the resonances observed in the low-ν
insulating phase (44–46, 92, 93, 169) have been interpreted as the disorder pinning
modes (22, 51, 54), and have served as evidence for a collectively pinned solid phase.
To illustrate features of the pinning mode resonances of the WS phase, we use as
an expample one of such resonances exhibited by sample “QW65” (see Tab. 2.1 for
the sample parameters). Fig. 3.1 (a) shows the resonance at ν = 0.066 in sample
“QW65”. A strong indication of the collective nature of the resonance is that the fpk
value of the resonance corresponds to an energy of merely 4 mK (times the Boltzmann
constant kB ), and yet this resonance was observed at T = 75 mK, and survives at T up
3
The original result obtained by Fukuyama and Lee (58) had a trivial error by a factor of 2
(22, 54).
26
(a)
(b)
12
10
Re[ σxx (f)] ( µ S)
15
65 nm QW
10
-2
n=5.3x10 cm
B=33 T
ν=0.066
8
15 nm QW
10
-2
n=4.5x10 cm
B=30 T
ν=0.063
10
6
5
4
2
0
0
0
40
80
0
120
2
4
6
8
10
12
f (GHz)
f (MHz )
Figure 3.1: The real diagonal conductivity spectra Re [σxx (f )] for (a) sample “QW65”
and (b) sample “QW15”. The measurements where done in the “PDF” dilfridge and
at T = 75 mK.
to 250 mK (20). If the resonance was from the localization-delocalization transition
of electrons individually trapped in disorder, one would expect to see the resonance
flatten out at temperatures comparable to its peak frequency, which is clearly not the
case.
After establishing the collective nature of the pinning, we can try to estimate
the correlation length of the solid. The essence of the phenomenon of pinning of
WS is the electron-disorder interaction. The length scale important for the pinning
behavior is the Larkin length, Ll , at which the deviations of electrons from the perfect
lattice reach the correlation length of the disorder. This is different from the usual
definition of a domain size, La , the length scale at which the deviations exceed the
lattice constant a.
Ll can be estimated from fpk , as described below, although the estimation is
necessarily model dependent. Ll is a static quantity, and fpk is a dynamic quantity,
p
but from fpk we can obtain ω0 as 2πfpk ωc , which characterizes the strength of
pinning. Because in a WS with long-range Coulomb interaction, the shear modulus
27
µT is much smaller than the compression modulus (10), the deformation of the lattice
caused by the disorder is mainly from shearing. Treating a Larkin domain containing
L2l n electrons as a mass-spring system immediately gives ω02 ∼ µT /L2l nm∗ , or
Ll ∼
r
µT
.
neBfpk
(3.9)
We can also consider ω0 to be the cut-off frequency of the transverse phonon mode,
which means ω0 = ct 2π
, where ct is the speed of the transverse phonon, given by
Ll
p
µT /nm∗ . That gives
s
2πµT
Ll =
.
(3.10)
neBfpk
Estimated from the resonance shown in Fig. 3.1(a), we obtain Ll to be 21 times the
lattice constant, which means a Larkin domain contains about 103 electrons.
What type of disorder pins the WS in experiments has been a long-standing and
important question. This is also highly relevant in many of our discussions of the tilted
field measurements, as to be presented in the following chapters. Theoretical studies
(51, 132) suggest that remote charged impurities, which could strongly affect the
electron mobility at B = 0 and FQHE physics, are generally not effective in pinning
the (weakly-pinned) WS. Fertig (51) proposed some dilute, short-range (comparable
to lB ) interface “pits” as a more possible source of pinning disorder.
While our experiments cannot rule out other sources of disorder, they do support
that disorder from the interface roughness is important for the pinning. To illustrate
the importance of interface, we compare sample “QW65”, a 65 nm quantum well,
and sample “QW15”, which was grown in the same way by MBE, with the only
difference being a much smaller width of the quantum well (15 nm). Sample “QW15”
has the same kind of doping (a source of scattering) as “QW65”, but due to its
tighter vertical confinement, “QW15” has a low-temperature mobility of µ = 1 ×
106 cm2 /Vs, much lower than the µ = 8 × 106 cm2 /Vs of “QW65”. Fig. 3.1(b) shows
28
the resonance at ν = 0.063 from sample “QW15”, in comparison with the resonance
from sample “QW65” in (a). The pinning mode frequency shows a striking increase
by two orders of magnitude, from 85 MHz in “QW65” to 7 GHz in “QW15”. Because
the charged impurities are the same for the two samples from the simlar MBE growth,
the difference in the pinning modes is attributed to the different vertical confinements.
29
Chapter 4
Tilt-induced Wigner solid in the
lowest two Landau levels
This chapter presents the effect of tilting on the competition between the WS phase
and the FQHE states. In the lowest LL, we find the ν range that exhibits pinningmode resonance of WS expands with tilting. Within the same quantum well, the
effect of tilting is stronger when the 2DES has larger electron density, or smaller
lattice constant. This fact suggests the effect of tilting is due to finite thickness of
the 2DES layer.
We also examine the effect of tilting in the second LL, just above ν = 2 or just
below ν = 3, where the electrons or holes in the top (partially filled) LL form a WS,
as evidenced by the pinning mode resonances found in previous experiments (18).
Tilting results in the expansion of the range of ν that exhibits the pinning-mode
resonance of WS, similar to the case in the lowest LL. Meanwhile, we find tilting
also causes the pinning mode frequency to increase, an indication of strengthening of
disorder pinning, which stabilizes the pinned WS.
We defer to the next chapter a report of the study of the solid phase near ν = 1.
Our experiment in tilted field supports the prediction that the solid is formed of
30
skyrmions.
4.1
Tilt-induced Wigner solid in the lowest Landau level
4.1.1
Introduction
In the lowest LL, a 2DES exhibits two types of ground states, the FQHE liquid
states and the WS state (128, 137). The FQHE states are incompressible, strongly
correlated quantum liquids having zero shear modulus and showing dissipationless
transport, whereas the WS is a compressible solid with a shear mode and is an
insulator due to pinning.
In perpendicular magnetic field, dc transport experiments done with GaAs-based
2DES’s find the last state in the FQHE sequence to be at ν = 1/5, before giving way
to a low-ν insulating phase. A re-entrant insulating phase is also found for a narrow
range of ν above 1/5. The insulating phases have been understood as disorder-pinned
WS, as supported by a combination of many experiments (13, 64, 67, 69, 80, 96,
118, 161, 163) [see review in (49)], and also by the pinning-mode resonances in the
microwave conductivity spectra (see data and discussions in Chaps. 1 and 3).
A 2DHS was found to show an insulating phase for ν below 1/3 and a re-entrant
insulating phase for a range of ν between 1/3 and 2/5 (33, 135, 136). In the ν < 1/3
insulating phase, a resonance was found in the microwave spectrum (93), similar to
the ones seen in electron systems. These experiments establish the WS as the ground
state below ν = 1/3. Our microwave experiments in 2DHS samples are yet to see any
resonance in the re-entrant insulating phase.
Dc transport measurements in tilted fields have been done with both 2DES’s and
2DHS’s, and showed that tilting expands the filling factor range for the insulating
31
phase. W. Pan et al. (120) studied in tilted field a 2DHS of density p = 1.6×1010 /cm2
hosted in a quantum well of width W = 30 nm. At zero tilting, ν = 1/3 is a welldeveloped FQHE state, and below that the system is insulating. As the sample normal
axis is tilted at θ = 80◦ from the field direction, the ν = 1/3 state becomes insulating,
and shows a drastic increase in its low-temperature longitudinal resistivity: at the
temperature of 30 mK, the ρxx value at ν = 1/3 rises from ∼ 0.4 kΩ/ at θ = 0◦ to
∼ 180 kΩ/ at θ = 80◦ . But ν = 1/3 is still a local minimum in the Rxx vs. ν trace.
More recently, Piot et al. (126) studied in tilted field a 2DES of density n =
1.52 × 1011 /cm2 in a quantum well of width W = 40 nm. At zero tilting, the ν = 1/5
state clearly shows the FQHE, and the system is insulating both for ν < 1/5 and for
a narrow range of ν above 1/5. Upon tilting, higher ν values, which show FQHE’s
in perpendicular field, become insulating. At θ = 68◦ of tilting, the boundary in
ν separating the FQHE-dominated phase and the insulating phase moves up from
ν ∼ 1/5 to ν ∼ 2/3.
Both W. Pan et al. and Piot et al. interpreted the tilt-induced insulating phases
as tilt-induced WS. This effect was predicted by Y. Yu and S. Yang (172) earlier as
due to finite thickness W of the quantum well that confines the electrons. Similar
p
to the magnetic length lB⊥ = ~/eB⊥ associated with the perpendicular field B⊥ ,
p
one can define a magnetic length lBip = ~/eBip associated with the in-plane field
Bip . When lBip becomes comparable to, or even smaller than W/2, the electrons
acquire freedom to adjust their position along the normal axis of the quantum well.
Considering a parabolic well, Yu and Yang (172) performed Monte Carlo simulations
for a Laughlin-liquid state in a tilted field, and found the liquid-state energy is lowered
as θ increases. Meanwhile, they used a variational wave function for the WS state in
the tilted field, and found the solid-state energy also decreases with tilting, by a larger
amount than the liquid state. This clearly predicts that tilting favors the WS state.
The authors also demonstrated that tilting has a larger effect for a wider quantum
32
well.
In this experiment, we find the ν range that exhibits pinning-mode resonance of
WS expands with tilting, which serves as spectroscopic evidence that tilting induces
WS. The emergence of a pinning-mode resonance in the conductivity spectrum clearly
and conveniently marks the onset of the WS phase, which we find increases in ν with
tilting. We systematically study the effect of tilting at various electron densities n,
within the same quantum well of width W . If we measure length in units of the
√
lattice constant, a ∼ 1/ n, then a larger n, or a smaller a, means a larger W/a, an
effectively wider quantum well. In this experiment, we find tilting to have a stronger
effect for larger n with W fixed. Comparing our results with the data from Piot et
al. (126) which used a sample with both n and W different from ours, we find the
effect of tilting is controlled by W/a.
4.1.2
Effect of tilting on a system with relatively high density
The data came from sample “QW65” (see Tab. 2.1 for its parameters). The measurements were done in several “SCM1” magnet times, but the sample state showed
only slight variations from cool-down to cool-down, not affecting the main features
presented here. The data presented in this subsection was taken at the as-cooled
density n of 5.3 × 1010 cm2 .
I start by presenting the spectra at θ = 0◦ , and focusing on the range of ν near
1/5 where the FQHE liquid states are in particularly close competition with the WS
phase. Fig. 4.1 (a) shows what we like to call a “carpet” plot of conductivity spectra
taken at many different ν between ν = 0.223 and ν = 0.190, changing in steps of 0.001
1
. For clarity, each spectrum is vertically offset from the preceding one by 1 µS. The
spectrum at ν = 0.190 exhibits a well developed resonance. As ν increases towards
1/5, this resonance weakens and shifts to lower frequencies, and eventually flattens
1
As in all other “carpet” plots shown in this thesis, the electron density n is fixed, and ν is tuned
by sweeping B.
33
(a)
(b)
ν=
100
1/5
0.196
fpk (MHz)
0.192
2/9
50
Re[σxx(f)] (µS)
1/5
0
0.18
0.19
0.20
0.204
0.208
0.21
0.22
0.23
ν
(c)
3.5
3.0
0.212
S/fpk (µS)
5 µS
0.216
2.5
1/5
2.0
2/9
1.5
1.0
50
100
150
200
250
0.220
0.5
2/9
0.0
0.18
300
0.19
0.20
0.21
0.22
0.23
ν
f (MHz)
Figure 4.1: (a) The real conductivity spectra at many ν values between ν = 0.223
and ν = 0.190 (by tuning B), changing in steps of 0.001. The spectra were measured
at θ = 0◦ and T = 50 mK. The dashed curve marks the position of the resonance
peaks. (b) fpk values of the resonances shown in (a), plotted as a function of ν. (c)
S/fpk values of the resonances in (a), plotted vs. ν.
34
out at ν = 1/5, which is an FQHE liquid (with no shear mode). The state at ν = 2/9
also shows a flat spectrum, consistent with it also being an FQHE liquid. There is a
range of ν between 1/5 and 2/9 showing resonant spectra. As ν approaches 1/5 or
2/9, the resonance amplitudes and peak frequencies both decrease.
Fig. 4.1 (b) plots fpk vs. ν for the resonances presented in (a), and it clearly
shows the decrease in fpk in the vicinity of ν = 1/5 or 2/9. Panel (c) plots the ratio
of the integrated intensity S over fpk extracted from the resonances in (a). The sum
rule obtained by Fukuyama and Lee (58), as described in Chap. 3, predicts S/fpk
to be proportional to the charge density participating in the pinning divided by B.
As found by earlier microwave studies (17), at low ν, deep into the solid phase, the
agreement between the sum rule and the experimentally obtained S/fpk values is
usually good. But for higher ν, in the solid phase but close to the FQHE-dominated
region, the experimentally obtained S/fpk has been found to be smaller than what
the sum rule predicts (17). For this sample, the Fukuyama-Lee prediction of S/fpk is
13 µS at ν = 0.2 (S/fpk ∝ ν), but the values shown in Fig. 4.1 (c) are just about 3 µS,
and decreases to zero as ν approaches 1/5 or 2/9. This suggests that in the range of
ν covered in Fig. 4.1, either only a small portion of the total charge forms pinned
solids, or the pinning in this range of ν is very different from that of the lower-ν WS.
Fig. 4.2 shows the effect of applying an in-plane magnetic field, Bip , by tilting the
sample so that its normal axis is at an angle θ to the field direction. θ is calculated
from observing the fields of prominent integer quantum Hall states. Spectra contained
in the same panel were taken at a fixed θ. Thus Bip = B⊥ tan θ is proportional to B⊥
(inversely to ν).
For comparison, Fig. 4.2(a) reproduces from Fig. 4.1 the spectra measured at
θ = 0◦ for ν between 2/9 and 0.19.
Fig. 4.2(b) presents the spectra at θ = 36◦ , and a few differences from the θ = 0◦
spectra are noticeable. First, at θ = 36◦ the resonance remains visible up to higher
35
θ=0
(b)
o
ν=
0.190
Re[σxx(f)] (µS)
0.210
0.210
0.214
0.214
0.218
2/9
50 100 150 200
2/9
5 µS
0.218
0.210
0.214
0.218
2/9
0.226
0.230
0.234
0.238
0.242
0.246
0.206
0.206
0.226
0.230
50 100 150 200
f (MHz)
ν=
0.196
1/5
0.206
1/5
1/5
5 µS
ν=
0.190
o
θ=44
0.196
0.196
f (MHz)
(c)
o
θ=36
10 µS
(a)
50 100 150 200
f (MHz)
Figure 4.2: (a)-(c) show the real conductivity spectra Re[σxx (f )] at various ν as
labeled on the right, measured at tilting angles (a) θ = 0◦ , (b) θ = 36◦ (Bip , proportional to 1/ν, is 7.70 T at ν = 1/5), and (d) θ = 44◦ (Bip = 10.24 T at ν = 1/5). The
measurement temperature was 55 mK. For clarity, each succeeding trace is vertically
offset by 1 µS.
36
14
T=55 mK
o
θ=63.5
Re[σxx] (µS)
12
10
8
6
4
2
0
100
200
300
400
ν
0.39
0.40
0.41
0.42
0.43
0.44
0.45
500
Bip (T)
10.90
10.63
10.37
10.12
9.89
9.66
9.45
f (MHz)
Figure 4.3: The real conductivity spectra Re[σxx (f )] of the 2DES at the ν values
marked in the graph. The 2DES has density n = 5.3×1010 cm2 , and the measurements
were done at θ = 63.5◦ . For clarity each spectrum has been vertically offset by 1 µS
from the adjacent one.
ν, with the boundary separating the resonant and non-resonant spectra shifted from
ν = 2/9 to 0.23. A resonance is partially developed at ν = 2/9, having smaller
amplitude and lower frequency than the resonances immediately above or below it in
ν. Second, the spectrum at ν = 1/5 remains flat, but closely above or below it in ν
the resonances become better developed than at θ = 0◦ , increasing in both amplitude
and frequency. Third, deeper into the solid phase, the ν = 0.190 resonance is also
sensitive to the tilt, and it shifts to lower frequency and has a different lineshape.
With the sample further tilted to θ = 44◦ , as shown in Fig. 4.2(c), the boundary in
ν for resonant spectrum moves up to 0.242. As ν decreases from 0.242, the resonance
gradually grows in amplitude and increases in fpk . Once ν gets below ∼ 0.226, fpk
stops increasing, but the resonance keeps getting sharper. At this tilting angle, the
fractional fillings ν = 2/9 and 1/5 can no longer be distinguished by examining the
spectra.
37
Tilting the sample to even higher angles further shifts up the boundary in ν
for resonant spectra. Fig. 4.3 shows the spectra measured at θ = 63.5◦ , for ν values
decreasing from 0.45 toward 0.39, changing in steps of 0.01. At ν = 0.39, the spectrum
is clearly resonant, with fpk at 280 MHz.
2
As ν increases, the resonance gradually
weakens, and it flattens out at ν ∼ 0.42, which we take to be the boundary in ν for
the WS phase.
4.1.3
Effect of tilting on a system with relatively low density
A voltage of −289 V applied to the back gate reduces n from the as-cooled value of
5.3 × 1010 cm2 to 3.0 × 1010 cm2 . Fig. 4.4 (a) shows the spectra taken at θ = 0◦
for eight ν values, decreasing from 0.22 to 0.15 in equal steps of 0.01. The spectrum
becomes resonant as ν gets below 0.2, indicating the onset of a WS phase. With
the sample tilted at θ = 73.6◦ , Fig. 4.4 (b) shows spectra taken at a set of ν values
between 0.30 and 0.26, again changing in steps of 0.01. The spectrum at 0.26 is
clearly resonant, but that at 0.30 only shows a very broad and weak resonance-like
structure.
To have an overview of the effect of tilting on the n = 3.0 × 1010 cm−2 state,
Fig. 4.5 contains traces of Re[σxx ] measured at fixed frequency of 200 MHz, plotted
against perpendicular field B⊥ . The traces, with vertical offset for clarity, are taken
at θ = 0◦ , 45.7◦ , 65.0◦ , and 73.6◦ , from bottom to top. The ν = 1/3 state, which is
marked on the graph at B⊥ = 3.6 T, remains a well developed minimum of Re[σxx ]
even at the highest tilting angle achieved, 73.6◦ , indicating that the ν = 1/3 state
remains an FQHE liquid state. As tilting increases, Re[σxx ] on the larger-B⊥ side
of ν = 1/3 increases relative to the Re[σxx ] value at ν = 1/3, consistent with the
expansion of the WS phase.
2
The small wigglings are due to the microwave reflections occuring at the joints in the microwave
circuit.
38
10
10
n=3.0×10 /cm
o
θ=0 , T=55 mK
Re[σxx] (µS)
8
6
ν=
0.15
0.17
0.19
0.20
4
2
200
10
Re[σxx] (µS)
8
2
400
600
800
0.21
0.22
1000
2
10
n=3.0×10 /cm
o
θ=73.6
T=55 mK
6
ν Bip (T)
0.26 15.68
0.27 15.10
4
0.28 14.56
2
0
0.29 14.06
0.30 13.59
200
400
600
800
1000
f (MHz)
Figure 4.4: (a) The real conductivity spectra Re[σxx (f )] of the 2DES at various ν
values as marked in the graph. The 2DES has density n = 3.0 × 1010 cm2 , and the
measurements were done at θ = 0◦ . (b) The Re[σxx (f )] spectra taken for the ν values
marked in the graph, with the sample tilted to 74◦ . For clarity each spectrum has
been vertically offset by 1 µS from the adjacent one.
39
ν=1
ν=1/3
o
10
n=3.0×10 /cm
f=200 MHz
T~65 mK
Re[σxx] (µS)
73.6
2
o
65.0
o
40.7
2 µS
o
0
0
1
2
3
4
5
6
7
Perpendicular field (T)
Figure 4.5: The real conductivity spectra Re[σxx ] at 200 MHz, vs. the perpendicular
field B⊥ . The traces are taken at different tilting angles as marked in the graph,
and they are plotted with offset for clarity. Also marked on the graph are the positions corresponding to ν = 1 and ν = 1/3. Due to the eddy current heating, the
measurement temperature was slightly higher.
4.1.4
A phase diagram in tilted field
The previous subsections presented the effects of tilting when the 2DES has density
n = 5.3×1010 cm2 and 3.0×1010 cm2 . By changing the back-gate voltage, this sample
has also been studied for n = 4.3 × 1010 cm2 and 3.6 × 1010 cm2 . For each density,
the sample is tilted at a series of angles, from θ = 0◦ , to the highest angle allowed
by the maximum total magnetic field, 18 T. At each tilting angle, a set of Re[σxx (f )]
spectra are measured, such as the ones shown in Figs. 4.3 and 4.4. The value of ν at
which the pinning-mode resonance emerges is taken to be the boundary, νc , for the
WS phase. Given all these data, we are able to produce the phase diagram shown in
Fig. 4.6.
In Fig. 4.6, the horizontal axis is νc , and the vertical axes are 1/ cos θ and θ.
It contains four solid curves, each corresponding to a different 2DES density n, as
40
o
74
3.5
n=3.0
3.0
n=3.6
1/cos(θ)
o
n=4.3
70
Piot et al.
2.5
o
2DES density
10
2
n (10 /cm )=5.3
2.0
65
θ
o
60
o
55
o
50
1.5
o
40
1.0
0.2
0.3
0.4
0.5
0.6
0
o
νc
Figure 4.6: For different n (as marked on the graph), the position of νc with varying
1/ cos(θ) (left vertical axis) or θ (right vertical axis). The dashed curve is reproduced
from Piot et al. (126).
marked on the graph. Each data point on the curves marks the estimated νc . For
ν below νc (to its left), the Re[σxx ] spectrum is resonant and suggests a WS phase.
For any fixed n, νc increases with 1/ cos θ (or θ), meaning the range of ν for the WS
phase expands as θ increases.
A prominent feature we notice in the phase diagram, Fig. 4.6, is that the dependence of νc on θ weakens as n decreases. If we measure lengths in units of the lattice
√
constant a, which, assuming a triangular lattice, is given by (2/ 3n)1/2 , then the result of decreasing n is a smaller W/a, or an effectively narrower quantum well. Thus
Fig. 4.6 shows tilting has a weaker effect on νc for smaller W/a. We compare the
phase diagram obtained in this experiment with that obtained by Piot et al. (126),
which is plotted as the dashed curve in Fig. 4.6. Piot et al. studied a 2DES with
n = 1.52 × 1011 /cm2 in a W = 40 nm quantum well. Even though its n and W are
both different from sample “QW65” used in our experiment, its W/a is just 3% larger
41
than the n = 5.3 × 1010 cm2 state of our sample. As we see in Fig. 4.6, the phase
boundaries of these two states are very close. Given the (small) difference in W/a,
we would predict the phase boundary in the sample used by Piot et al. to lie slightly
lower than the phase boundary in our sample for n = 5.3 × 1010 cm2 , but as shown
in Fig. 4.6, their positions are reversed.3
The phase diagram shown in Fig. 4.6 also hints that, as W/a decreases, the system
may have contrasting behavior at a large tilting angle. For the n = 5.3 × 1010 cm2
state, we do not see any saturation of the tilting effect on νc up to θ = 63.5◦ . In the
study by Piot et al. of the system with similar W/a at tilting up to 71◦ , the shift of
νc with θ showed no sign of saturation either. For the n = 3.0 × 1010 cm2 state, at
θ = 74◦ , the large in-plane field Bip corresponds to a magnetic length lBip of merely
6 nm at ν = 1/3, much smaller than W/2 (32.5 nm). As shown in both Fig. 4.6 and
the B scans in Fig. 4.5, there appears to be a saturation of the expansion of the WS
phase at large tilting angles. Certainly, we have to use caution to make assertions like
this, because this experiment is limited by the maximum magnetic field (18 T). Given
the present data, the dinstinction shown in Fig. 4.6 is already striking, considering
the relatively small reduction of W/a from the n = 5.3 × 1010 cm2 state (W/a = 1.39)
to the n = 3.0 × 1010 cm2 state (W/a = 1.07).
3
It shall be noted that the two experiments adopt different ways of determining the solid-liquid
phase boundary. In this experiment, we take the filling factor where an appreciable resonance shows
up as the boundary, and Piot et al. in their dc transport experiment (126) define the boundary as
the largest ν value for Rxx to exceed h/2e2 = 12.91 kΩ.
42
4.2
Tilt-induced Wigner solid in the second Landau level
4.2.1
Introduction
The FQHE and WS phases also compete to be the ground state of the system for ν
near integers. When ν is just away from an integer, the top partially filled LL hosts a
dilute system of electrons (for ν above the integer) or holes (for ν below the integer),
with a partial filling factor ν ∗ = |ν − nearest integer| ≪ 1. These electrons or holes
must be localized (76), in order for the IQHE to manifest in transport. If these
electrons or holes are sufficiently dilute (ν ∗ sufficiently small), and provided that the
sample disorder is sufficiently weak, they tend to form a WS, driven by dominating
Coulomb interaction (56, 83, 105, 111). Evidence for the crystallization comes from
the microwave pinning mode resonances found near, but not precisely at ν = 1, 2, 3
or 4 in samples of high mobility (low disorder) (18, 90) (see Figs. 4.8 for examples of
such resonances).
As ν further deviates from the integer, the electrons or holes in the partially filled
LL may reach a large enough density to favor FQHE ground state. In samples with
high mobility, FQHE is found for ν = 1±1/5, ν = 2+1/5, 3±1/5, 4−1/5. Beyond the
second LL, it is generally believed that FQHE’s do not occur (56, 83, 111), although
a weak ν = 4 + 1/5 FQHE was hinted by a local minimum in the magnetoresistance
which show up at that filling factor only for a temperature window of 80 ∼ 120 mK
(61). Instead the third or higher LL’s are dominated by CDW-like phases, such as
the stripe and bubble phases, which are the topics of Chap. 6.
The “melting” of the solid phase near integer ν into the ν ∗ = 1/5 FQHE liquids
can be affected by disorder. In the presence of disorder, the solid deforms itself
to adjust to the disorder landscape, and gains energy, which stabilizes the solid. In
experiments, the electron-disorder interaction can be tuned by controlling the amount
43
and profile of disorder with sample engineering, or alternatively, as we shall see in
this chapter, by manipulating the electron wave function with application of Bip .
Disorder can have a drastic effect when the energies of the competing phases are
close to each other. A particularly close competition among near-degenerate phases
is found in the second LL (43, 165), where 2D electron liquids, solids, and CDW’s,
all join to play. Dc transport find FQHE’s at partial fillings of the top LL ν ∗ = 1/5,
2/5, and 1/3, and at even-denominator LL fillings ν = 5/2 and 7/2 (114). These
FQHE states alternate with re-entrant IQHE states which are interpreted as pinned
“bubble” phases. Close to integer ν, the ground state is a pinned solid, as evidenced
by the observed microwave pinning mode resonances. This solid phase competes with
a series of FQHE liquids: while the ν ∗ = 1/9 and 1/7 FQHE liquids do not develope,
the FQHE shows up in dc transport at ν ∗ = 1/5.
A recent dc transport experiment (34) showed that tilting the sample significantly
alters the ground state in the second LL. At zero tilting, the IQHE state at ν = 2 is
marked by the vanishing magnetoresistance Rxx and quantized Hall resistance Rxy =
(1/2)h/e2 . ν = 2 + 1/5 is an FQHE state at zero tilting, showing vanishing Rxx , and a
fractionally quantized plateau of Rxy at [1/(2 + 1/5)]h/e2 . Tilting is found to destroy
the FQHE states at ν = 2 + 1/5, and merge its Rxy value into the IQHE plateau of
(1/2)h/e2 .
The remaining of this section presents the microwave spectroscopy in the second
LL in tilted field. Tilting is found to have two effects. First, it expands the filling
factor range close to integer filling that shows the pinning mode resonances, and
induces a resonance not seen in the originally (zero-tilt) flat spectra at ν = 2 + 1/5
and 3 − 1/5. The emerging resonance is a clear signature of a tilt-induced solid
phase. Second, the pinning resonances that already exist at zero tilt shift to higher
frequencies with tilting. Because the pinning mode frequency measures the strength
of pinning, the results suggest that the disorder-carrier interaction is tilt-dependent,
44
and may be driving the liquid-solid phase transition.
4.2.2
Tilt-induced resonance at ν = 2 + 1/5 and 3 − 1/5
We studied sample “QW30”, a 2DES in a 30-nm AlGaAs/GaAs/AlGaAs quantum
well, with density 2.7×1011 /cm2 and low-temperature mobility 29×106 cm2 /Vs. The
sample is of similar quality to the ones previously studied in dc transport (34, 43, 165),
and has a straight-line CPW, with slot width w = 30 µm and length 5 mm.
We use the same terminology throughout this thesis about tilting: the magnetic
field with total magnitude Btot has an in-plane component Bip and a perpendicular
component B⊥ . By comparing B⊥ and the total field Btot at prominent integer
quantum Hall states, we obtain the rotation angle θ from cos θ = B⊥ /Btot . The inplane field Bip = Btot × sin θ, and, in our setup, is perpendicular to the propagation
direction of the CPW. All the conductivity spectra were measured at T ∼ 40 mK.
Fig. 4.7 shows Re[σxx ] between ν = 2 and ν = 4, measured at the fixed frequency
f = 400 MHz, with the sample tilted to θ = 0◦ , 16◦ , and 31◦ . The data are taken at a
slightly higher sample temperature of about 60 mK. The traces are vertically offset for
clarity, and the positions of the integer and some fractional filling factors are marked.
The traces all show well developed conductivity minima at the integer fillings, ν = 2,
3, and 4. At θ = 0◦ (top trace), the conductivity exhibits peaks between ν being
integer and integer ±1/5. When the sample is tilted to θ = 16◦ (middle trace) and
31◦ (bottom trace), the conductivity peaks shift away from integer fillings. This shift
is more obvious for ν between 2 and 3 than for ν between 3 and 4. At θ = 31◦ ,
the fractional fillings ν = 7/3 and 8/3 show better-resolved conductivity minima,
consistent with a recent dc transport measurement (37) which found tilting increases
the energy gaps at these FQHE states.
Fig. 4.8 contains two “carpet” plots, each showing twenty-six spectra of the real
conductivity, Re[σxx (f )], for the range of ν between 2.00 and 2.25, measured with
45
14 8
19 11 10 16
ν=4 5 3 3 5 3 5 3
7
3
11
5
2
5.0
5.5
Re[σxx] (µS)
f=400 MHz, T~60 mK
o
0
16
o
o
31
2 µS
3.0
3.5
4.0
4.5
B (T)
Figure 4.7: The real part of diagonal conductivity Re[σxx ] measured at 400 MHz,
plotted as a function of perpendicular field B⊥ , with the sample tilted to θ = 0◦ (top
trace), 16◦ (middle trace, Bip at ν = 2 is 1.55 T) and 31◦ (bottom trace, Bip at ν = 2
is 3.24 T). Traces are vertically offset for clarity.
46
ν=
2.25
(b)
(a)
o
o
θ=31
θ=0
Re[σxx(f)] (µS)
2.20
2.15
2.10
2.05
20 µS
2.00
1
2
f (GHz)
3
4
1
2
f (GHz)
3
4
Figure 4.8: The spectra of conductivity for various ν between 2.00 (bottom) and 2.25
(top), changing in steps of 0.01, with the traces vertically offset for clarity. Data in
(a) are taken at θ = 0◦ and (b) at θ = 31◦ (giving Bip = 3.24 T at ν = 2 and 2.88 T
at ν = 2.25). The measurement temperature was 40 mK.
the sample (a) perpendicular in field, or (b) tilted to 31◦ . In panel (a), at θ = 0◦ ,
the spectrum is resonant for ν between 2.05 and 2.18. For ν between 2 and 2.05, it
is possible that a resonance exists but is too weak to be identified. The resonance
flattens out beyond ν = 2.18, and the ν = 2.20 spectrum is essentially featureless.
The resonance has been interpreted (18, 90) as the pinning mode of the WS formed
by the electrons in the partially filled second LL, and the range of ν with resonant
spectrum clearly marks this WS phase. The overall decrease in fpk as ν increases is
consistent with the picture of weak pinning, because a larger charge density means
stronger interaction, which relatively weakens the role of disorder. Panel (b) shows
the spectra for the same ν range at θ = 31◦ , and the difference from panel (a) is
obvious. The resonance is visible also at ν values higher than 2.20 which show flat
spectra in (a), and tilting significantly increases the resonance peak frequencies.
Fig. 4.9 highlights the spectra at four filling factors between ν = 2 and 3. Panel
47
ν=2.15 ν=2+1/5 ν=2.85 ν=3-1/5
Bip(T) at
ο
θ=16
1.44 T
1.41 T
1.09 T
1.11 T
θ=31
3.01 T
2.95 T
2.28 T
2.32 T
ο
o
31
o
16
Re[σxx(f)] (µS)
0
o
8
(a)
ν=
2.15
(b)
(c)
ν=
2.85
(d)
6
ν=
2+1/5
4
2
0
6
4
ν=
3-1/5
2
0
1
2
3
f (GHz)
1
2
3
Figure 4.9: The spectra of conductivity at (a) ν = 2.15, (b) ν = 2 + 1/5, (c) ν = 2.85,
and (d) ν = 3 − 1/5, with the sample tilted to θ = 0◦ (solid curve), 16◦ (short
dash) and 31◦ (long dash). The table on top lists Bip value for each ν and θ. The
measurement temperature was 40 mK.
48
(a) presents the conductivity spectra at ν = 2.15, measured at θ = 0◦ , 16◦ and 31◦ .
At θ = 0◦ , the spectrum shows a resonance at 0.48 GHz. This resonance is interpreted
as the pinning mode of the WS formed by the electrons in the top LL (18, 90). As
the sample is tilted, we see a doubling of fpk at θ = 16◦ , and a fourfold increase at
θ = 31◦ , and the resonance also gets appreciably broader. Fig. 4.9 (b) shows the case
for ν = 2 + 1/5. At θ = 0◦ , its conductivity spectrum is essentially flat, in agreement
with the state being an FQHE liquid (34, 43, 165). This spectrum becomes resonant
once the sample is tilted. The amplitude and peak frequency of the induced resonance
both increase with tilting.
Fig. 4.9 (c) shows the case for ν = 2.85. At θ = 0◦ , the spectrum shows a
resonance with a 0.53 GHz peak, resembling the one at ν = 2.15. Similarly, this
resonance at ν = 2.85 has been interpreted (18, 90) as the pinning mode of the WS
formed by the top-LL holes. As the sample is tilted, the resonance shifts to higher
frequency, but by a less remarkable amount than the ν = 2.15 resonance for the
same θ. Fig. 4.9 (d) shows the effect of tilting on the ν = 3 − 1/5 spectrum. Like
ν = 2 + 1/5, ν = 3 − 1/5 is also an FQHE state and shows a flat spectrum at θ = 0◦ .
Tilting the sample also gradually induces a resonance at ν = 3 − 1/5, similar to the
effect it has on the ν = 2 + 1/5 state, but at ν = 3 − 1/5 a larger tilting angle is
needed to have the same appreciable effect.
Fig. 4.10 gives an overview of the properties of the resonances at different θ for
ν between 2 and 2.25, where tilting has the most appreciable effect. At θ = 0◦ , the
resonance is visible for ν ∗ = ν − 2 between 0.05 and 0.18, and has been interpreted
(18, 90) as the pinning mode of a top-LL WS phase. Data for ν ∗ below 0.05 are absent
because the resonance is too weak, or not visible. For ν ∗ above 0.18, the resonance
flattens out, signaling the melting of the solid phase to the ν ∗ = 1/5 FQHE liquid.
As the sample is tilted, the resonance remains visible out to higher ν ∗ . In Fig. 4.10
(a), we see that fpk decreases as ν ∗ increases, consistent with the picture of a weakly
49
fpk (GHz)
4
(a)
o
0
3
o
16
o
2
31
1
2.5
(b)
∆f (GHz)
2.0
1.5
1.0
0.5
S/fpk (µS)
15
10
(c)
F-L
prediction
5
0
0.0
0.1
∗
0.2
ν
Figure 4.10: (a) The resonance peak frequency fpk , (b) the full width at half-maximum
∆f , and (c) the ratio of the integrated intensity S over fpk , are plotted all as functions
of the partial filling factor ν ∗ , at sample tiltings θ = 0◦ (circles), 16◦ (squares) and
31◦ (triangles). The Fukuyama-Lee prediction (58) is shown as the solid line in (c).
See Fig. 5.3 for the Bip values.
50
pinned solid, as increasing electron-electron interaction relatively weakens the role of
disorder (see discussions in Chap. 3). When the sample is tilted to 16◦ and 31◦ , in
Fig. 4.10(a) we see significant upshifting of the peak frequencies. As Fig. 4.10(b)
shows, the resonance linewidth ∆f also increases with tilting. The increase of both
fpk and ∆f clearly shows that the solid is pinned more strongly by disorder.
The effect of tilting as expanding the range of ν for the solid phase is clear in Fig.
4.10(c), which shows the ratio of the integrated intensity S of the resonance to the
peak frequency fpk . The Fukuyama-Lee sum rule (58) predicts S/fpk is proportional
to the density of the pinned charge divided by the magnetic field. Because the charge
being considered is in the partially filled top LL, the sum rule (58) takes the form
of S/fpk = (e2 π/2h)ν ∗ . Fig. 4.10 (c) compares this prediction to the S/fpk values
extracted from the resonances. At θ = 0◦ , S/fpk linearly increases with ν ∗ for 0.05 <
ν ∗ < 0.12, consistent with the picture of a WS of growing charge density. As ν ∗ gets
above 0.12, S/fpk decreases, and vanishes at ν ∗ = 0.18. This suggests that the solid
melts for ν ∗ between 0.12 and 0.18. As the sample is tilted to 16◦ , and further to
31◦ , S/fpk remains approximately linear in ν ∗ up to larger values of ν ∗ , suggesting
that the solid is winning over the ν ∗ = 1/5 FQHE liquid state as the ground state.
(At θ = 16◦ and 31◦ , data for small ν ∗ are not shown, because the full lineshape of
the resonance exceeds the frequency range of our measurement and does not allow an
accurate estimation of S.)
4.2.3
Tilt-strengthened disorder pinning
The effect that tilting induces WS phase in the second LL appears similar to the one
reported in Sec. 4.1 in the lowest LL. But in the second LL, the data suggest the
emergence of WS with tilting is driven by strengthened disorder pinning which lowers
the energy of the WS.
In the lowest LL, the phase diagram (Fig. 4.6) shown in Sec. 4.1.4 demonstrates
51
that the effect of tilting is tied to the finite layer thickness of the 2DES, and shows
the controlling parameter is the width W of the quantum well measured in units of
√
the lattice constant a = (2/ 3n)1/2 , where n is the 2DES density. As shown in Fig.
4.6, the effect of tilting significantly weakens when W/a is reduced from 1.39 to 1.07.
In the present sample, W = 30 nm, and for the system of dilute electrons or holes in
the top LL near integer ν, W/a is 0.32 (for ν = 2.10) and 0.44 (for ν = 2.20). Both
are much smaller than the values seen in Sec. 4.1 (due to the narrower quantum well
and also due to the much smaller density n∗ in the subsystem in the second LL). This
suggests that the finite-thickness effect is much weaker for the present system.
Another illuminating experimental fact which we shall dwell on in the next chapter
is that, within the same sample, “QW30”, we have also studied in tilted field the WS
phase just below ν = 2, which belongs to the lowest orbital LL. Tilting at 30◦ is not
found to have any appreciable effect on that solid phase, and neither does it affect
the resonance frequency (despite a larger n∗ , a smaller a, and a larger W/a for the
same ν ∗ ). This is in contrast to the observation for the second LL.
We see in Figs. 4.8-4.10 that, besides expanding the range of ν with resonant
spectrum, the other effect of tilting is the increase in both fpk and ∆f . Qualitatively,
it indicates the strengthening of the disorder pinning, which stabilizes the solid phase.
As shown in Fig. 4.9, the fpk at ν = 2.15 is more sensitive to tilting than the fpk at
ν = 2.85, and the induced resonance at ν = 2 + 1/5 is also more sensitive to tilting
than the one at ν = 3 − 1/5. By contrast, as we shall see in the next chapter, for
ν just below 2 (belonging to the lowest LL), tilting does not affect fpk , neither does
it expand the solid phase. This consistency leads us to propose that the observed
expansion in ν ∗ of the WS phase in the second LL is driven by the strengthened
disorder pinning, which stabilizes the pinned solid.
If one can quantitatively obtain the energy Ep gained by the solid phase from
disorder pinning, one can use it to give a rough estimate of the energy difference (in
52
the clean limit) between the solid phase and the ν ∗ = 1/5 FQHE liquid. Because
the observed fpk is a dynamic quantity, and Ep is a static quantity, the estimation
has to be indirect. Following the discussions in Chap. 3, the pinning potential is
characterized by a phenomenological frequency ω0 , which is related to the pinning
mode frequency ωpk = 2πfpk and the cyclotron frequency ωc = eB/m∗ as ω02 = ωpk ωc .
The average pinning energy per electron is estimated by m∗ ω02 ξ 2 , where ξ is the
correlation length of the pinning potential. The disorder landscape of the sample
has an intrinsic correlation length ξ0 . If ξ0 is much larger than the magnetic length
p
lB = ~/eB, the electrons can be treated as point charges, and ξ = ξ0 . In this case
we cannot calculate Ep without knowing ξ0 . If lB ≫ ξ0 , due to an averaging effect of
the electron wave function, ξ = lB , and we have Ep = ~ωpk .
The lB values for typical filling factors studied here are around 11 ∼ 14 nm.
Although we have no direct measurement of ξ0 of this sample, the typical lateral size
of roughness found in the GaAs-on-AlAs interface in MBE grown 2DES samples (133)
is below 10 nm. Thus, ~ωpk is likely a good estimate of Ep . If ξ0 turns out to be
larger, ~ωpk is an underestimate.
As shown in Fig. 4.10, the upshift of fpk increases with tilt, but is insensitive
to ν ∗ . The emergence of the WS phase is a gradual process and we do not know
where the pinning energy exactly compensates for the clean-system energy difference
between the FQHE and the WS. But given Fig. 4.9 it is safe to say this energy is on
the order of ~∆ωpk ∼ 50 mK. Accurate calculations of the energies of the FQHE state
and the WS are considered to be difficult (55, 65). In the literature, the clean-system
energy difference at ν = 2 + 1/5 varies widely from ∼ 200 mK (55) to ∼ 2 K (65).
53
4.3
Interplay between the Wigner solid and the
fractional quantum Hall liquids
Before concluding, in this section I discuss features in the pinning-mode resonance
when a WS is apparently affected by the FQHE, mainly in light of the data presented
in the preceding sections, and also with some supplementary data.
Of particular interest are the transitions between FQHE liquids and the WS (166).
At low temperatures, such transitions can be triggered by tuning ν and θ. In the
lowest LL and at zero tilt, the WS gives way to FQHE liquids at ν = 1/5 and 2/9.
As shown in Fig. 4.1, upon approaching ν = 1/5 and 2/9, the resonance weakens
and shows both decreasing fpk and S/fpk . As shown in Fig. 4.2, tilting induces a
resonance in the originally flat spectra at ν = 1/5 and 2/9, both being FQHE states
at zero tilt. Tilting at 44◦ induces a resonance at both ν values, but at 36◦ , the
spectrum is resonant at ν = 2/9 but not yet at ν = 1/5. Thus we see a smaller tilting
angle is needed to induce a resonance at ν = 2/9 than at ν = 1/5. At θ = 36◦ , the
partially developed ν = 2/9 resonance has both lower fpk and amplitude than the
resonances immediately above or below it in ν.
As Sec. 4.2 shows, the low-temperature transitions between FQHE liquids and the
WS also occur in the second LL, and we find a similar behavior of the pinning modes.
Fig. 4.10 (c) shows that, starting from ν ∗ = 1/5 and θ = 0◦ , the system goes from an
FQHE liquid phase into a WS phase either as ν ∗ decreases or as θ increases. This is
similar to what we see at ν = 1/5 in the lowest LL. Here the transition also occurs
gradually, as shown by the smooth change of S/fpk with ν ∗ or θ (see discussions in
Sec. 4.2).
Another variable is the sample temperature T . In the WS phase, a precursor of
an FQHE liquid can be induced by raising T , as demonstrated by W. Pan et al. (122)
for small ν (1/7, 1/9, etc.). Here I show a set of data that provides a perspective on
54
o
θ=44
4
(a) ν=0.195
Bip=10.50T
3
(d)
60
fpk (MHz)
2
1
0
55
50
Re[σxx (f)] (µS)
T (mK)
(b) ν=0.200
Bip=10.24 T
4
3
T (mK)
45
55
55
2
65
(e)
1
75
5.5
0
85
5.0
S/fpk (µS)
4
(c) ν=0.208
Bip=9.85 T
3
2
1
65
75
85
4.5
4.0
3.5
3.0
0
50
100
150
200
250
0.200
ν
f (MHz)
0.208
Figure 4.11: The real conductivity spectra Re[σxx (f )] at filling factors (a) ν = 0.195,
(b) ν = 0.200, and (c) ν = 0.208, measured with the sample tilted to θ = 44◦
(Bip = 10.24 T at ν = 1/5), and at different temperatures T : 55 mK (solid), 65
mK (dot), 75 mK (dash), and 85 mK (dash-dot). (d) and (e) plot the fpk and S/fpk
values of the resonances for ν between 0.195 to 0.208, measured at θ = 44◦ , and at
55 mK (solid circles), 65 mK (open circles), 75 mK (solid squares), and 85 mK (open
squares).
55
this phenomenon from pinning mode resonances. In Fig. 4.2, we see that the ν = 1/5
state, which has flat spectrum at θ = 0◦ , shows a resonance at θ = 44◦ that looks
no different from those at immediately higher or lower ν. Figs. 4.11 (a-c) compare
the temperature dependence of the resonance at ν = 0.200 and those at ν = 0.195
and 0.208. At the base temperature of T = 55 mK, the three resonances have similar
amplitudes, peak frequencies, and lineshapes. As T increases, the ν = 0.200 resonance
shows the most appreciable decrease in amplitude and fpk . At 85 mK, the ν = 0.200
state is readily differentiable from the other two. This distinction is also clearly
shown in panels (d) and (e) of Fig. 4.11, which plot the fpk and S/fpk values of the
resonances for ν between 0.195 to 0.208, at varying temperatures. The ν = 0.200
state shows the steepest temperature dependence. At 85 mK, both fpk and S/fpk
have a minimum located at exactly ν = 0.200.
To summarize, as a WS melts into an FQHE liquid, the pinning-mode resonance
gradually weakens and shifts to lower frequency, before eventually disappearing. In
such high quality samples, density inhomogeneity is small and is unlikely to account
for the range of ν showing gradual change in S/fpk , neither can it explain the lowering
of fpk . This can be a trivial effect due to the disorder profile which is nonuniform by
definition. Another possibility concerns an intrinsic feature of the liquid-solid transition in a 2DES with Coulomb interaction. It was recently proposed, based on rather
general arguments, that such transitions should occur via a series of intermediate
phases (146). These intermediate phases are suggested to be some electronic “microemulsion”, which consists of short-length-scale mixtures of solid and liquid. For ν in
a WS phase but close to FQHE states, the weaker, lower-fpk resonance may be the
pinning mode exhibited by such an intermediate phase. Although a detailed theory
for the collective excitations in such a “micro-emulsion” phase is not yet available,
one may expect that its liquid fraction should grow upon approaching FQHE; and
the liquid fraction can damp the WS resonance and reduce fpk (109, 116). Both
56
predictions are consistent with our observations of the behaviors of S/fpk and fpk .
4.4
Summary and discussions
This chapter reports spectroscopic evidence that tilting favors the WS phases both
in the lowest LL and also near integer ν in the second LL.
In the lowest LL, at zero tilting, the conductivity spectrum is resonant for ν below
1/5 and also for a narrow range of ν between 1/5 and 2/9, and the resonance serves
as signature of a WS phase. The ν = 1/5 and 2/9 states show flat spectra, consistent
with them being FQHE liquids with zero shear modulus. Tilting the sample causes
the range of ν with resonant spectrum to expand to include higher ν values which
show flat spectra at zero tilting. The induced resonance is interpreted as the pinning
mode of a tilt-induced WS phase.
In the lowest LL, the expansion of the filling factor range for the WS phase is
not always accompanied by an increase of fpk . In particular, as Fig. 4.2 shows, fpk
of the ν = 0.19 resonance even decreases by 40% with 44◦ of tilting. fpk indicates
the strength of pinning, which serves to stabilize the pinned solid. The behavior of
fpk with tilting cannot explain why tilting favors the WS phase. Rather the phase
diagram shown in Fig. 4.6 suggests that the expansion in ν of the WS phase is tied
to finite thickness of the 2DES layer. We have studied, within the same quantum
well of width W , a series of states with varying densities n. If we measure length
√
in units of the lattice constant a = (2/ 3n)1/2 , this enables us to vary the relative
quantum well width W/a. We found tilting has a weaker effect for a smaller W/a, or
an effectivly narrower well. In particular, at W/a = 1.07, the data seem to suggest
that the expansion of the WS phase with tilting saturates at high tilting angles. In
this case, tilting up to 73.6◦ still does not induce a WS phase at ν = 1/3, although
at this tilting angle Bip corresponds to a magnetic length much smaller than W/2.
57
Limited by the total field available, we are not able to verify if the effect of tilting
also eventually saturates for larger W/a.
Appendix B presents an inspiring set of data about the effect of tilting on the WS
below ν = 0.2, and some preliminary discussions regarding the ν-dependent nature
of the WS phases.
In the second LL, at zero tilting, the spectrum is clearly resonant for ν ∗ below
∼ 0.18, indicative of a WS as the ground state. At ν = 2 + 1/5 and 3 − 1/5, tilting
induces a resonance not seen at zero tilt, which is interpreted as the pinning mode
of an emerging WS phase. Comparison with the tilted-field measurements in the
lowest LL suggests that the observed effect of tilting in the second LL is not due
to the finite layer thickness. On the other hand, tilting is found to strengthen the
disorder pinning of the WS, which stabilizes the pinned solid and is possibly driving
the system from the FQHE liquid into the WS. Going a step further, if this transition
is solely disorder-driven, we can use the pinning energy of the emerging WS as a
rough estimate of the energy difference between the two phases in a clean system.
We estimate that the emerging WS states at ν = 2 + 1/5 and ν = 3 − 1/5 have a
pinning energy on the order of 50 mK.
The tilt-dependence of the electron-disorder interaction in the second LL could
be understood by considering the electron wave function in a quantum well. If the
disorder responsible for pinning is due to interface roughness, as conjectured by Fertig
(51), tilting could increase pinning by increasing wave function amplitude at the
quantum well interfaces. In the second LL, with the magnetic field perpendicular
to the sample, the x-y-direction (in-plane) wave function has a node at the center,
whereas the z direction wave function is peaked at the center. Tilting the field
effectively mixes the x-y direction and z direction wave functions, pushing the wave
function toward the quantum well interfaces, and thus could increase the interface
roughness pinning. Such an effect can be inferred from the wave functions in tilted
58
field obtained in (147) for a parabolic quantum well.
59
Chapter 5
Pinning modes of a Skyrme crystal
near ν = 1
Microwave pinning-mode resonances found near integer ν have been taken as signatures of the WS phase formed of the dilute quasiparticles or quasiholes in the partially
filled top LL (18, 90). This chapter examines the effect of Bip on the pinning mode
resonances near ν = 1 and just below ν = 2, both belonging to the lowest orbital LL.
We find application of Bip , increasing the Zeeman energy, has negligible effect on the
resonances just below ν = 2, but increases the pinning-mode resonance frequencies
near ν = 1, particularly for smaller quasiparticle/hole densities. The charge dynamics near ν = 1, characteristic of a crystal order, are affected by spin, in a manner
consistent with the expectation for a Skyrme crystal.
5.1
Introduction
Spin textures, structures of spin rotating coherently in space, are of importance to
a number of different classes of materials (35). An example of such a spin texture
with wide application in descriptions of magnetic order is the skyrmion, a topological
defect first used to describe baryons (143). Skyrmions can form arrays which have
60
been considered in a variety of magnetic systems, and which have been seen in bulk
material (112) in magnetic field, in neutron scattering.
Skyrmions are particularly important in 2DES’s with an additional spin or pseudospin degree of freedom (50, 63, 86, 145). In typical 2DES’s, it turns out that the
external magnetic field couples rather weakly to the spin degrees of freedom and low
energy spin fluctuations are not completely frozen out by the (small) Zeeman splitting
1
.
At exactly ν = 1, the system has a fully spin polarized ground state because
this makes the spatial wave function antisymmetric which minimizes the Coulomb
repulsion. The excitations at ν = 1 were predicted to be pairs of skyrmions and
antiskyrmions, consisting of multiple rotating spins. The spin texture of a skyrmion
is smoothly distorting over space and locally the spins are kept parallel, which is
favored by the exchange effect at a cost in the Zeeman energy. A skyrmion has
unit charge, bundled along with the spin texture which spreads out in space. The
skyrmion charge is also predicted (50) to have a spread-out distribution different from,
for example, the LL orbitals that would characterize an isolated electron in 2DES.
Skyrmions were identified in 2DES as excitations near ν = 1 in experiments
(2, 7, 110) that measured the electron spin polarization vs. ν, and the presence of
skyrmions was also shown to affect the measured energy gaps in transport (107, 138).
The wide applicability of skyrmions in 2DES is attested to also by their presence in
layer-index pseudospin of bilayers (11, 28), valley pseudospin (142) in AlAs, or in
predictions for graphene involving both valley pseudospin and spin (167).
When the skyrmions are sufficiently dilute, they are expected to crystallize, driven
by Coulomb repulsion, similar to the case of dilute electrons. But for skyrmions, the
predicted state has tightly interwoven spin and charge crystal order. Such a state has
1
For a quick comparison, for a GaAs-based√2DES with n = 1 × 1010 /cm2 , the cyclotron gap ~ωc
at B = 1 T is 19.7 K, the Coulomb energy e2 πn/4πǫ is 23 K, and the Zeeman energy gµB B at 1
T is merely 0.3 K.
61
been of great interest in 2DES (12, 32, 70, 115, 124, 129) and has been considered
theoretically for graphene (30). NMR (38, 62, 144, 153) and heat capacity (8, 9, 108)
experiments in the Skyrme crystal range near ν = 1 in 2DES have focused on the
strong coupling of nuclear spins to electron spins, likely via a soft spin wave which
was very recently unveiled in Raman scattering (60). The soft spin mode has been
interpreted as arising from XY-spin orientational order (32).
The main experimental result described in this chapter is that the charge dynamics near ν = 1 show characteristics of a Skyrme crystal; we find these dynamics to be
affected by the Zeeman energy and by the charge density in a manner consistent with
calculations (1, 12, 32) done for Skyrme crystals. The microwave conductivity spectroscopy itself has no direct spin sensitivity. Our ability to observe this demonstrates
as well that the charge distribution of a skyrmion in a crystal differs from that of an
ordinary Landau quasiparticle.
5.2
Pinning mode resonances near ν = 1 in perpendicular field
The microwave spectroscopy has revealed the pinning modes of pinned solid phases
within the IQHE’s near ν = 1, 2, 3 and 4 in low-disorder 2DES’s (18, 90) (also see
Chap. 4.2). Fig. 5.1 shows a set of conductivity spectra obtained in sample “QW30”
for ν (a) around 1 and (b) just below 2. The resonances in the spectra, which
flatten out for temperatures above 150 mK, are understood as the pinning modes
of the solid of quasiparticles/holes (18). For brevity, we shall refer to the range of
ν just below 2 showing resonance as ν = 2− , and the resonant ν ranges just above
(below) 1 as ν = 1+ (1− ). In both panels, fpk decreases with increasing partial filling
factor ν ∗ = |ν−nearest integer|. This is consistent with the picture of a weakly pinned
crystal, because higher ν ∗ means denser quasiparticles/holes and stronger interaction,
62
(a)
ν=
1.16
(b)
ν=
2.00
1.08
1.96
1.04
1.92
1.00
1.88
0.96
20 µS
Re[σxx] (µS)
1.12
1.84
1.80
20 µS
0.92
1
0.88
2
3
3
4
f (GHz)
0.84
1
2
4
f (GHz)
Figure 5.1: From sample “QW30”, (a) at θ = 0o , the real conductivity spectra at
various ν, increasing from 0.82 to 1.18 in steps of 0.02. (b) Also taken at θ = 0o ,
the real conductivity spectra at ν from 1.78 to 2.00, increasing in steps of 0.02. The
measurement temperature was 40 mK.
63
and thus a weaker role of disorder (see dicussions in Chap. 3).
The ν = 2− solid is well accepted to be a solid of single Landau quasiholes, and
the solid near ν = 1 is predicted to be one of skyrmions (12). Comparing Fig. 5.1
(a) and (b), we notice that, close to integer ν, the ν = 2− resonances have a steeper
ν ∗ -dependence of fpk than the ν = 1+ and 1− resonances. But besides that, Fig. 5.1
(a) and (b) show major similarities. Just in perpendicular field, there is no other
evidence that can easily differentiate the two solid phases.
5.3
Pinning mode resonances near ν = 1 in tilted
field
The tilted-field experiment described in this section addresses the nature of the solid
around ν = 1, and shows that the pinning mode near ν = 1 is affected by skyrmion
formation. These effects become clear through systematic study of the dependence of
the pinning mode on Bip , which at fixed ν reduces the skyrmion spin (50, 63, 86, 145).
Bip of the magnitude used in this experiment has essentially no effect on the solid
just below ν = 2, expected to be a solid of single Landau quasiholes. In contrast, Bip
increases the pinning frequency of the solid near ν = 1. In addition, the skyrmion
density (the same as the charge density) in the solid increases for ν farther from
exactly 1, and we find Bip increases the pinning mode frequency only for small enough
skyrmion density, or wide enough skyrmion separation. This shows that the spin
size of skyrmions is affected by their mutual proximity (32), an illustration of the
intertwined nature of charge and spin in this state.
We used 2DES’s from two samples: sample “QW30”, a 30 nm quantum well,
with the electron density n = 2.7 × 1011 / cm2 , and a low-temperature mobility µ =
27 × 106 cm2 /Vs, and sample “QW50”, a 50 nm quantum well, with a lower n =
1.1 × 1011 / cm2 and µ = 15 × 106 cm2 /Vs. “QW30” has a straight-line CPW with
64
ν=1.05 31o
o
θ=0
ν=1.91
31
o
2.5 (b)
f (GHz)
<K0>
~1.5
2.0
o
1.5
1.0
θ=0
0.5
0
1
o
2
3
3
4
1.3
(c)
o
<K0>
~2.5
2
fpk (31 )/fpk (0 )
Re[
s
1
fpk (GHz)
o
0
2 µS
xx(f)]
(µS)
(a)
4
filling factor
+
ν=1
-
1.2
ν=1
1.1
ν=2
-
1.0
5
1.0
10
2.0
3.0
4.0
2
n* (10 /cm )
Figure 5.2: (a) For sample “QW30”, the real conductivity spectra at ν = 1.05 and
ν = 1.91, measured at θ = 0o (solid) and 31o (dash). (b) For sample “QW30” at
θ = 0o , the resonance peak frequencies fpk as a function of quasiparticle density
n∗ = n(ν ∗ /ν) for the ν = 2− (open squares), ν = 1+ (solid circles), and ν = 1− (open
circles) solids. The skyrmion sizes, hK0 i, estimated by (32), are marked in the graph.
(c) The relative change in fpk from θ = 0o to 31◦ . At θ = 31◦ , Bip is 6.49 T at ν = 1.
The measurement temperature was 40 mK.
slot width 78 µm and length 5 mm, and “QW50” has a meander-line CPW with
slot width 30 µm and length 28 mm. The sample sits on the rotator, and as in
all our tilted field experiments, the tilting angle θ is calculated from comparing the
total fields Btot and perpendicular fields B⊥ of prominent quantum Hall states. The
measurements were done in the “C120” dilfridge and inside a 14 T magnet, and the
sample temperature was 40 mK.
For sample “QW30”, Fig. 5.1 in the prevous section shows the spectra in per-
65
pendicular field for ν = 2− , 1+ , and 1− . Still for sample “QW30”, we repeat the
measurement in a tilted magnetic field. We fix B⊥ and hence ν, and increase Btot
by a factor of 1/ cos θ. The Zeeman energy also increases, since it is proportional to
Btot . Fig. 5.2 (a) shows the spectra at ν = 1.05 and 1.91, each taken at θ = 0o
and 31o . At these two ν, the quasi-particles/holes are within the same orbital LL,
and have equal density n∗ ∼ 1.3 × 1010 /cm2 , calculated as (ν ∗ /ν)n. At θ = 0o , the
ν = 1.05 resonance peak is at about half the frequency of the ν = 1.91 resonance.
Tilting to 31o has negligible effect on the ν = 1.91 resonance, but shifts the ν = 1.05
resonance to higher frequency. The ν = 0.95 resonance, not shown here, behaves
almost identically to the ν = 1.05 resonance.
Fig. 5.2(b) uses the θ = 0o data in Fig. 5.1, and plots fpk vs quisiparticle/hole
density n∗ , for all three solid phases at ν = 1+ , 1− and 2− . All the three curves show
fpk decreases as n∗ increases, as we have seen in Fig. 5.1. The ν = 1+ and ν = 1−
curves agree well, indicating good symmetry between quasiparticles and quasiholes.
The ν = 1+ and 1− curves also overlap with the ν = 2− curve for high n∗ . But for
low n∗ (< 2.5 × 1010 /cm2 ), the ν = 1+ and 1− resonances have lower fpk than the
ν = 2− resonance for the same n∗ .
Fig. 5.2(c) plots the ratio, fpk (31o )/fpk (0◦ ), of the θ = 31o and 0◦ fpk values.
fpk increases with tilting only for the ν = 1+ and 1− resonances with low n∗ (n∗ <
2.3 ± 0.3 × 1010 /cm2 , with the error arising mainly from uncertainty in ν of data at
different angles). The two curves of fpk vs n∗ , for ν = 1+ and 1− , agree with each
other, indicating a still good particle-hole symmetry. Tilting has negligible effect on
the ν = 2− resonances, or the ν = 1+ and 1− resonances with high n∗ .
Sample “QW50” has lower n than “QW30”, and as discussed below, is predicted
to be capable of supporting larger skyrmions. Near integer ν, “QW50” also displays
resonances in the real conductivity spectra. Fig. 5.3 shows the resonances at three
typical ν: (a) 0.96, (b) 1.04, and (c) 1.90, each taken at θ = 0o and θ = 63o . Tilting
66
1.0
(a)
ν=0.96
Re[σxx(f)] (µS)
0.5
θ=0
0.0
1.5 (b)
o
θ=63
ν=1.04
o
1.0
0.5
0.0
2.0 (c)
1.5
1.0
0.5
0.0
ν=1.90
1
2
f (GHz)
3
4
Figure 5.3: The conductivity spectra of sample B, taken at θ = 0◦ (solid curve) and
θ = 63◦ (dashed curve), at filling factors (a) ν = 0.96, (b) ν = 1.04, and (c) ν = 1.90.
At θ = 63◦ , Bip = 9.0 T at ν = 0.96, 8.3 T at ν = 1.04, and 4.5 T at ν = 2. The
measurement temperature was 40 mK.
67
−
o~
θ=0 g=0.012
1.8
fpk (GHz)
1.4
1.2
ν=1
o~
θ=50 g=0.019
(a)
4.5
<K0>
~3.5
1.6
+
ν=2
(b)
3.0
−
o~
θ=63 g=0.026
2.0
(c)
2.5
2.0
2.5
1.0
ν=1
1.5
0.8
0.6
0.0
1.0
0.0
1.0
10
0.0 0.4 0.8 1.2 1.6
2
n* (10 /cm )
Figure 5.4: From sample B, fpk as a function of n∗ for the ν = 2− (open squares),
ν = 1+ (solid circles), and ν = 1− (open circles) resonances, at (a) θ = 0o , (a)
θ = 50o , and (c) θ = 63o . The skyrmion sizes, hK0 i, estimated by (32), are marked
in the graph.
shifts the ν = 0.96 and ν = 1.04 resonances to higher frequencies, in clear contrast to
the negligible effect it has on the ν = 1.90 resonance.
For sample “QW50”, at θ = 0◦ , 50◦ and 63◦ respectively, Figs. 5.4 (a-c) show fpk
vs n∗ for ν = 1+ , 1− and 2− . Here we also see that fpk decreases as n∗ increases,
consistent with a weak-pinning picture and the ν = 1+ and 1− resonances show good
agreement with each other consistent with particle hole symmetry. The ν = 2−
resonances show no change as the sample is tilted. The ν = 1+,− curves lie below the
ν = 2− curve more for lower n∗ . As θ increases, the ν = 1+,− curves move upward
toward the 2− curve, and for 63o , the curves are close together except at the smallest
n∗
By analogy to the WS pinning modes found at very low ν (<∼ 1/5) (see previous
chapters), the observed resonances at ν just away from integers have been taken as
signatures of solid phases formed of quasiparticles/holes (18, 90). Here we show that
the dependence on Bip indicates a clear difference between the crystals of ν = 1+ and
68
1− from that of ν = 2− , and provides evidence for a Skyrme solid near ν = 1.
The effect of Bip near ν = 1 cannot be explained without invoking the spin. This
is clear from the negligible effect of Bip for the solids other than those with low n∗ near
ν = 1. The ν = 2− solids have been further studied at θ = 51o for sample “QW30”
and at θ = 67o for “QW50”, and still show no change, but the same Bip already has
obvious effect near ν = 1. ν = 2− , 1+ and 1− all belong to the lowest orbital LL.
If the effect of Bip were due to the orbital wave function and finite thickness of the
quantum well, it would be unlikely that the same effect would be completely absent
for the ν = 2− solid and the ν = 1+ and 1− crystals with high n∗ .
The effect of Bip near ν = 1 finds a natural explanation in the predicted solid of
skyrmions (12, 32, 70, 115, 124, 129). The skyrmion size, hK0 i, denotes the number
of flipped spins (relative to the maximally spin-polarized state with the same charge)
(32, 50). Two key parameters control hK0 i. One is the Zeeman/Coulomb energy ratio
(50), g̃ = gµB Btot / (e2 /ǫl0 ), where |g| ≈ 0.44 is the g factor in GaAs, µB the Bohr
p
magneton, ǫ ≈ 13 the dielectric constant of GaAs, and l0 = ~/eB⊥ the magnetic
length. Larger g̃ makes flipping spins more costly, and favors smaller hK0 i. The
second parameter is n∗ : increasing n∗ by tuning ν away from 1 brings skyrmions
closer and limits hK0 i. Côté et al. (32) calculated hK0 i, predicting the reduction of
hK0 i on increasing g̃ or n∗ .
For the samples used in this experiment, “QW30” has g̃ ∼ 0.019 at ν = 1 in
perpendicular field, and “QW50” with a lower n has a smaller g̃ ∼ 0.012, because
√
g̃ ∼ n. Upon tilting, because g̃ ∝ Btot , g̃ increases by a factor of 1/ cos θ. Based on
(32), the calculated hK0 i for different g̃ and n∗ are marked in Fig. 5.2(b) and 5.4.
Consistently for the two samples, increasing g̃ beyond that needed to bring hK0 i
(obtained from n∗ and θ according to ref. (32)) down to around 2.0, produces no
change on the ν = 1+ and 1− resonances. We interpret this as due to the skyrmions
becoming small enough that their resonance cannot be distinguished from that of
69
ordinary Landau quasiparticles. This is particularly clear in sample “QW50”, in
which the fpk vs n∗ curves for ν = 1+ and 1− at large θ or large n∗ match the curves
for 2− . The curves come together for hK0 i below about 2, even though this condition
occurs at different n∗ when g̃ changes. In sample “QW30”, for n∗ = 2.3 × 1010 cm−2 ,
the largest n∗ for which we observe tilt-induced change in fpk that we ascribe to
skyrmion formation, we estimate hK0 i ≈ 2.6 at 0o , and hK0 i ≈ 2.1 at 31o .
At fixed n∗ , decreasing Bip produces larger predicted hK0 i, and we find a lower
fpk . This suggests a solid formed by larger skyrmions is more weakly pinned. This
effect can be due to two possible mechanisms which may both be operational: one
from skyrmion-disorder interaction and the other from skyrmion-skyrmion interaction. Larger skyrmions average disorder over a larger area, resulting in an effectively
weaker disorder responsible for the pinning, and hence a lower fpk . fpk is also sensitive
to the shear modulus, which is determined by the interaction between the skyrmions.
When the skyrmions grow bigger and begin to overlap, the inter-skyrmion repulsion
is expected to get stronger and to increase the shear modulus of the solid (1). And
within the weak-pinning picture (Chap. 3), a stiffer solid has lower fpk , because it is
more difficult for the solid to deform to adjust to the disorder landscape.
5.4
Summary and discussions
In this chapter, we have studied the effects of Bip on the pinning mode resonances
exhibited by the solid phase formed at ν = 2− , expected to be a solid of single Landau
quasiholes, and those at ν = 1+ and ν = 1− , predicted to be solids of skyrmions. Bip
of the magnitude used in this experiment has no discernible effect for ν = 2− . By
contrast, the pinning modes near ν = 1, which are clear evidence of a solid phase,
appear to be consistently determined by the predicted skyrmion sizes, indicating the
solid around that filling is formed of skyrmions at least for small n∗ . As tilting causes
70
the predicted skyrmions to shrink, we observe upshift of the pinning frequency, which
saturates as the skyrmions approach single spin flips. The dependence of the pinning
on the spin is a consequence of the coupled charge and spin degrees of freedom for
crystallized skyrmions.
As ν departs from one, a phase transition is predicted, from a triangular-lattice ferromagnet of spin helicity to a square-lattice antiferromagnet (12, 32, 70, 115, 124, 129),
in favor of lower exchange energy at a cost in the Madelung energy (10). Calculations
in Ref. (32) predict this to happen roughly at hK0 i ∼ 2.5 in Fig. 5.2(b), ∼ 5 in Fig.
5.4(a), ∼ 2.5 in Fig. 5.4(b), and ∼ 1.5 in Fig. 5.4(c). This prediction is made for
a clean system. In our observation, the fpk vs. n∗ curves show no jumps. We note
that for such crystals with Coulomb interaction, the shear modulus is small relative
to the bulk modulus (10). Near the phase boundary, the energy difference between
the two types of lattices is small. Thus a disorder potential can relatively easily make
a square lattice oblique, smearing the difference between a square and a triangular
lattice. It may require a much cleaner system to observe a sharp phase boundary
between square and triangular lattices.
71
Chapter 6
Tilt-induced anisotropies in the
third Landau level
When three or more orbital LL’s are occupied, the top-LL carriers are predicted to
form collective phases with characteristics of CDW’s, including the stripe and the
bubble phases [(55, 56, 73, 83, 111, 130), also see reviews in (42, 52, 53)]. Such
predictions have received strong support from dc transport measurements in ultrahigh-mobility 2DES’s. Near ν = 9/2, 11/2, 13/2,..., Lilly et al. (102) and Du et al.
(40) found strong anisotropy in dc transport at low temperatures. These anisotropic
states have been understood as the stripe phase in which charge density has striped
modulatoins over space. The dc transport experiments in high LL’s also reveal remarkable phenomena away from half-odd integer fillings. In ranges of ν near 1/4 or
3/4 filling of the top LL, both Lilly et al. (102) and Du et al. (40) reported that
the resistivity isotropically falls to zero, while the Hall resistance is quantized at the
value of the adjacent IQHE plateaus. These re-entrant IQHE states have been taken
as evidence for a crystal lattice formed of electron bubbles. Sec. 6.1 contains a review of the previous dc transport experiments, followed by discussions in light of the
developing theoretical picture of the stripe and the bubble phases.
72
Secs. 6.2-6.4 present our spectroscopic studies of the stripe and the bubble phases.
Both phases exhibit striking microwave/rf resonances, which are understood as the
pinning modes of the phases as they are pinned by disorder. Like the pinning-mode
resonances of the WS, these resonances serve as spectroscopic signatures of the particular phases, and the properties of the resonances release important information
about the electrons participating in the pinning modes.
We start with presenting in Sec. 6.2 the microwave/rf spectra in perpendicular
field when the Fermi level is in the third or higher LL. In the stripe phase, we observe
a resonance in the spectrum measured along the “hard” axis, in which larger dc
resistivity occurs, while in the orthogonal, “easy”-axis, no resonance is discernible
(134). The resonance in the “hard” axis is understood as the pinning mode of the
stripe phase. In the bubble phase, the spectrum shows an esentially isotropic pinningmode resonance (91).
Sec. 6.3 presents the effect of Bip on the anisotropic pinning-mode resonances of
the stripe phase. The polarization along which the resonance is observed switches
as Bip is applied, indicating clearly the reorientation of the stripes. The resonance
frequency, which is a measure of the pinning interaction between the 2DES and disorder, increases with Bip , and the magnitude of this increase is larger when Bip is
parallel to the stripe orientation. This observation indicates that disorder interaction
is playing an important role in determining the stripe orientation. We also find S/fpk
of the stripe-phase resonance grows with Bip , suggesting increased participation of
the electrons in the pinning mode.
Sec. 6.4 presents the effect of Bip on the pinning-mode resonances of the bubble
phase. The resonances are essentially isotropic in zero Bip . As Bip is applied, the
resonance peak frequency and linewidth both increase with Bip while remaining essentially isotropic. Bip induces a strong anisotropy in the resonance amplitude and
intensity: the resonance is stronger when the microwave electric field EM is polarized
73
perpendicular to Bip , and weaker when EM and Bip are parallel.
6.1
Introduction
Evidence in dc transport for the CDW-like phases in the third or higher LL appeared
only after the introduction of high quality GaAs-based 2DES samples with mobilities
exceeding 107 cm2 /Vs, and only has been seen at temperatures <∼ 0.1 K (40, 102).
Near ν = 9/2, 11/2, 13/2,..., the system shows strongly anisotropic dc transport, with
the longitudinal resistance much higher for current flowing in the “hard” direction
than in the perpendicular “easy” direction. As ν deviates from half-odd integer
values in the third or higher LL, the anisotropic state is replaced by a re-entrant
IQHE state with vanishing longitudinal resistance and quantized Hall resistance, like
the nearest IQHE’s, but cleanly separated from them. Moving even closer to integer
ν, the longitudinal resistances briefly become non-zero again before vanishing into
the IQHE states.
In the anisotropic states, most experiments (26, 40, 101, 102, 119, 160) using
GaAs-based samples with various electron densities n = 1.5 ∼ 3 × 1011 /cm2 found
that the “easy” direction is along the [110] axis of the GaAs host lattice, and the
orthogonal “hard” direction is along [11̄0]. Using a heterostructure insulated gate
field effect transistor (HIGFET) with tunable n, J. Zhu et al. (173) found that the
“easy” direction switches from being along [110] to being along [11̄0] as n is raised
above ∼ 2.9 × 1011 /cm2 .
The “hard”/“easy” axes can switch as Bip is applied, as found by a series of dc
transport experiments (25–27, 101, 119, 121, 151, 173). When Bip is applied along
the zero-Bip “easy” axis, the “hard”/“easy” axes are found to trade places, leaving
the new “easy” axis perpendicular to Bip . The effect of Bip applied along the zeroBip “hard” axis appears to be more sample-dependent. The longitudinal resistances
74
along the zero-Bip “hard” and “easy” axes approach each other (27, 101, 119) and in
some cases they cross over (101). In the HIGFET used by J. Zhu et al. (173), for
n <∼ 3 × 1011 /cm2 , the zero-Bip “easy” axis is along [110] and Bip is not found to
change that direction; but for n >∼ 3 × 1011 /cm2 , the zero-Bip “easy” axis is along
[11̄0], and applying Bip either along [11̄0] or [110] causes the “easy” and “hard” axes
to interchange.
The CDW picture originally proposed by Koulakov et al. (55, 56, 83) and by
Moessner and Chalker (111) with Hartree-Fock (HF) theory provides a framework for
understanding the experimental observation described above. Unlike the lowest LL
which is dominated by FQHE’s, CDW-like states were found to be favorable when
three or more orbital LL’s are occupied. An important difference between the lowest
and higher LL’s is that the electron wave function in high LL’s have nodes, which
reduce short-range repulsion and allow the system to phase separate. As the longrange part of the Coulomb interaction is not affected by the existence of the nodes,
a finite length scale is established for the phase separation. This length comes out to
be on the order of the magnetic length, lB .
When three or more LL’s are occupied, the HF studies (55, 56, 83, 111) predicted
the ground states for the top-LL carriers as the following. For small partial filling
factors ν ∗ = |ν − nearest integer|, only the long-range interaction is important and a
Wigner crystal is expected. As ν ∗ increases, the ground state is a triangular lattice
of multi-electron bubbles. Like Wigner crystal of electrons, this crystal of bubbles is
pinned by residual disorder as well, and does not contribute to dc transport, which
accounts for the observed re-entrant IQHE. As ν ∗ approaches 1/2, the HF theory
predicted the ground state to be a unidirectional CDW. For the case of ν = 9/2,
the system forms alternating stripes of ν = 4 and ν = 5, with the period found
to be comparable to lB . Dc resistance is presumably lower along the stripes and
higher across them. If the stripes are somehow consistently oriented over the whole
75
sample area, anisotropic transport is expected. The observed striking anisotropy in
longitudinal resistance near ν = 9/2, 11/2,..., has been taken as strong evidence for
the formation of charge density stripes oriented along the lower-resistance “easy”
direction, and orthogonal to the higher-resistance “hard” direction.
A large theoretical literature has developed in the aftermath of the experimental
discovery of the anisotropic transport. Finite-size exact diagonalization calculations
by Rezayi et al. (130) supports the early HF results that stripe formation is favorable
at half-odd integer fillings in the third or higher LL. In the original HF prediction
(55, 56, 83, 111), along the stripes the charge density is uniform. A series of works
(47, 94, 95, 104, 171) pointed out that the stripes are unstable to charge density
modulations along their orientation, and proposed that (at least locally) the ground
state is a highly anisotropic lattice, referred to as the “stripe crystal”, which also can
give rise to anisotropic transport. The crystal order, however, is predicted to melt
in the presence of quantum fluctuations (171) and thermal fluctuations (104) at the
usual experimental temperatures. Tsuda et al. (154) also found that the crystal order
is unstable against injection of electrical currents typically used in the dc transport
measurements.
Fradkin and Kivelson (57) have pointed out interesting analogies with liquid crystals. In the original picture predicted by the HF studies (55, 56, 83, 111), translational
symmetry is broken across the stripes, analogous to a smectic state. Quantum fluctuations can restore the broken translational symmetry, resulting in a nematic state
with just orientational symmetry broken. The earlier HF studies (55, 56, 83, 111)
predicted a melting temperature of stripes of a few Kelvins, much higher than the
∼ 100 mK at which the transport anisotropy is found to diappear. A possible picture
is that at higher temperatures stripe pieces still exist but do not yet have a macroscopic orientation, and ∼ 100 mK is the temperature below which the stripe pieces
start to show orientational order.
76
The observation of strong transport anisotropy suggests that there is a (presumably small) symmetry breaking field native to the sample which orients the
stripes. The mechanism is still not understood, despite a great deal of attention
(25, 26, 160, 173). When Bip is applied, it acts as a symmetry breaking field and can
interchange the “hard” and “easy” axes in the sample, as demonstrated by studies
of dc transport mentioned above (25–27, 101, 119, 121, 151, 173). HF calculations
(81, 147) aimed to explain the Bip effect have adopted the unidirectional CDW picture,
and obtained a per-electron energy difference between stripe orientations perpendicular and parallel to Bip . This anisotropy energy, denoted EA , is found from the effect
of Bip on the wave function in the direction (z) perpendicular to the 2D plane in a
2DES with finite thickness. EA turns out to favor stripes perpendicular to Bip . This
is in accord with the dc transport studies (25–27, 101, 119, 151, 173), in which for
most of the samples surveyed, switching the stripe direction at least occurs for Bip
applied along the zero-Bip stripes. But that theoretical framework does not address
experimental results for Bip applied perpendicular to the zero-Bip stripes, which show
that dc resistances along the zero-Bip “hard” and “easy” axes can approach each other
(27, 101, 119, 173) and even cross over in some cases (101, 173), leaving the axis of
lower dc resistance parallel to Bip .
As we shall see in this chapter, the stripe and bubble phases both show striking
microwave/rf resonances. These resonances serve as clear signatures of the particular
phases and also bring out rich information about them.
For the stripe phase, Sec. 6.2 presents its microwave/rf spectra measured in
perpendicular field, and Sec. 6.3 examines the effect of Bip . In perpendicular field,
for ranges of ν near 9/2, 11/2, ..., we observe a resonance only in the Re[σxx (f )]
spectrum, where x̂ denotes the “hard” axis of higher dc resistance, while no resonance
is visible in the Re[σyy (f )] spectrum measured in the “easy” (ŷ) axis. We interpret
the resonance in the “hard” axis as the pinning mode of the stripe phase.
77
We find application of Bip along either the zero-Bip “hard” or “easy” axis to interchange the resonant and non-resonant polarization, clearly indicating the switching of
the stripe orientation. From the resonance we extract two additional pieces of information, the peak frequency fpk and the integrated intensity S. We find applying Bip
increases fpk , which is a measure of the disorder pinning energy, and the magnitude
of the increase depends on the direction of Bip . Taking stripes to be oriented along
the non-resonant direction and perpendicular to the resonant direction, we find fpk
is higher when the stripe is apparently parallel to Bip , and lower when the stripe is
perpendicular to Bip . Because pinning energy stablizes the pinned stripe phase, this
observation suggests disorder pinning favors stripes parallel to Bip , which is opposite
to the anisotropy energy EA obtained by the HF calculation for a disorder-free system. We suggest the switching of stripes in Bip is driven by the interplay of (at least)
these two effects.
The other piece of information from the resonances is S. While fpk increases with
Bip , we find S/fpk also grows with Bip . Although how to quantitatively interpret
S/fpk for such an highly anisotropic state is still an open question, the observed
increase of S/fpk with Bip is suggestive of more electrons participating in the pinning
mode.
For the bubble phase, Sec. 6.2 shows its microwave/rf spectra in perpendicular
field and Sec. 6.4 presents the effect of Bip . As previously reported (91), the bubble
phase shows an esentially isotropic resonance which is interpreted as its pinning mode.
As we show in Sec. 6.4, the effect of Bip on the bubble phase is qualitatively different from the effect Bip has on the stripe phase. As Bip is applied, the bubble-phase
resonance peak frequency and linewidth both increase with Bip and in the meanwhile
remain essentially isotropic. Bip induces a strong anisotropy in the resonance amplitude and intensity, with the resonance being stronger when the microwave electric
field EM has polarization perpendicular to Bip , and weaker when EM and Bip are
78
(b)
ν=
4.8
ν=
4.8
4.7
4.7
4.6
9/2
50µS
4.4
200
400
600
Re[σyy(f)] (µS)
Re[σxx(f)] (µS)
(a)
4.6
9/2
4.4
4.3
4.3
4.2
4.2
200
800 1000
400
600
800 1000
f (MHz)
f (MHz)
Figure 6.1: Real diagonal conductivity spectra for many filling factors ν between 4.10
and 4.90; each succeeding trace offset upward by 5 µS: (a) “Hard”-direction conductivity Re(σxx ) measured in sample 1; (b) “easy”-direction conductivity Re(σyy ) from
sample 2. Filling factors ν are marked to the right of each panel. The measurements
in this chapter were all done in the “C120” dilfridge, at the temperature of 40 mK.
parallel.
6.2
Pinning-mode resonances of the stripe and the
bubble phases at zero tilting
We studied sample “QW30”, a 2DES in a 30 nm quantum well, with density 2.7 ×
1011 /cm2 , and low-temperature mobility 29×106 cm2 /Vs. The CPW transmission line
launches a microwave/rf electric field EM polarized perpendicular to its propagation
direction. In order to measure conductivities along orthogonal axes of the host GaAs
79
crystal, we have cleaved two samples from the same wafer at adjacent positions, and
patterned CPWs along perpendicular directions. We denote by x̂ the GaAs crystal
axis [11̄0], which for this sample is the dc “hard” direction at ν = 9/2, perpendicular
to the stripes, and denote by ŷ the crystal axis [110], the dc “easy” direction parallel
to the stripes. Sample 1 (2) has EM along x̂ (ŷ), and is used to measure conductivity
σxx (σyy ).
Fig. 6.1 shows the spectra of (a) Re[σxx (f )] and (b) Re[σyy (f )], at many filling
factors between 4.10 and 4.90, increasing in steps of 0.01. Each succeeding trace is
offset vertically by 5 µS for clarity. For 4.2 ≤ ν ≤ 4.4 and also 4.6 ≤ ν ≤ 4.8,
resonances are present in both Re[σxx (f )] and Re[σyy (f )], and have been interpreted
as the pinning modes of the bubble phases (91). As ν increases from 4.2 to 4.3, or
decreases from 4.8 to 4.7, the peak frequency changes from above 500 MHz to about
180 MHz. The resonances in Re[σyy (f )] are less sharp than those in Re[σxx (f )],
likely due to small density inhomogeneity in sample 2 and the strong filling factor
dependence of the resonance. As ν = 4.3 → 4.36, or ν = 4.7 → 4.64, the peak
frequency only shifts slightly with ν. For 4.36 < ν < 4.4, or 4.6 < ν < 4.64, as
shown in panel (a), the resonance in Re[σxx (f )] weakens and broadens, and the peak
frequency decreases from 180 MHz to 100 MHz as ν moves toward 9/2. As shown in
panel (b), the resonance in Re[σyy (f )] diminishes as ν approaches 4.4 from below or
4.6 from above. The behavior for ν = 4.3 → 4.4, or ν = 4.7 → 4.6 may be due to
the formation of one or more of intermediate phases, or even an M = 3 bubble phase
(29, 31, 48), as the stripe range is approached.
Where the bubble phase resonances are well-developed, both Re[σxx (f )] and Re[σyy (f )]
exhibit one or two smaller, higher frequency peaks. One possibility is that these
peaks are produced by the spatial harmonics of the in-plane microwave/rf electric
field lauched by the CPW (see more discussions of such peaks for the WS phases in
Appendix A). An alternative interpretation as due to “optical” modes of the bub-
80
600
σxx
σyy
500
fpk (MHz)
400
300
200
100
0
4.2
4.4
4.6
4.8
ν
Figure 6.2: The peak frequencies of the resonances in the Re[σxx ] spectra (solid
squares) and Re[σyy ] spectra (open squares), between ν = 4.2 and 4.8.
bles, associated with excitations of degrees of freedom internal to the bubbles (29), is
unlikely since the optical modes are predicted to lie at much higher frequency (& 50
GHz).
Between ν = 4.4 and 4.6, Figs. 6.1 (a) and (b) are in stark contrast. A sharp
resonance is present in the spectra of Re[σxx (f )] shown in panel (a), peaked around 90
MHz, and weakly dependent on ν, whereas no resonance is discernible in Re[σyy (f )]
shown in panel (b). This range is in reasonable agreement with theoretical estimates
(29, 47, 65, 140) for the occurrence of the stripe phase. The estimates of the range of
the stripe phase are all symmetric about ν = 9/2, with the lower boundaries varying
from ν = 4.35 (65) to 4.42 (29).
Fig. 6.2 summarizes the ν dependence of the resonance peak frequency fpk . We
estimate the absolute error in ν in the figure as ±0.02. Except for ν in the stripe
range between 4.4 and 4.6, there is excellent quantitative agreement of fpk between
the “hard” and “easy” orientations, as expected for the bubble phases.
In a similar fashion as Fig. 6.1, Fig. 6.3 shows conductivity spectra taken at
81
(a)
ν=
6.8
(b)
ν=
6.8
6.7
50µS
0
200
400
600
Re[σyy(f)] (µS)
Re[σxx(f)] (µS)
6.6
6.7
6.6
13/2
13/2
6.4
6.4
6.3
6.3
6.2
6.2
800 1000
0
f (MHz)
200
400
600
800 1000
f (MHz)
Figure 6.3: Real diagonal conductivity spectra for many filling factors ν between 6.10
and 6.90; each succeeding trace offset upward by 4 µS: (a) “Hard”-direction conductivity Re(σxx ) measured in sample 1; (b) “easy”-direction conductivity Re(σyy ) from
sample 2. Filling factors ν are marked to the right of each panel. The measurement
temperature T was 40 mK.
82
various filling factors between 6.10 and 6.90, increasing in steps of 0.01. The range of
6.45 < ν < 6.55, centering on ν = 13/2, is marked by an anisotropy similar to that
around 9/2 as shown in Fig. 6.1, with a resonance present only in the Re[σxx (f )]
spectra, but not in Re[σyy (f )]. This is interpreted as from the stripe phase around
ν = 13/2, similar to the one occuring for ν around 9/2, but in a narrower filling factor
range. For ν < 6.4 and ν > 6.6, resonances are nearly isotropic from x̂ to ŷ, and are
regarded as from the bubble phase.
The theoretically proposed bubble lattice can be viewed as a WS of clusters of
M > 1 electron guiding centers (55, 56, 65, 73, 83, 111), and its pinning mode is
qualitatively similar to that of the WS phase (M = 1) (31, 89, 91). The stripe phase
is a very different object because it shows (partial) solid order in only one direction.
There have been two calculations of the pinning modes of the stripe phase in the
presence of disorder (94, 95, 117). Though both predict pinning modes for rf electric
field parallel to the stripes, in seeming contrast with our not observing an “easy”direction resonance, the theories indicate that “easy”-direction pinning mode can be
comparatively weak under our experimental conditions. In (94, 95), pinning modes
are found only in a state that has pinning both along the stripes and perpendicular to
them. The predicted pinning has about the same frequency parallel and perpendicular
to the stripes, though the mode parallel to the stripes is weaker by a factor of about
4. The work of (117) obtains only the conductivity along the stripes, and predicts the
“easy”-direction pinning mode frequency and amplitude both vanish at small wave
vector q.
6.3
The effect of tilting on the stripe anisotropy
The tilted-field experiments reported in this section used the same samples as in the
previous section, and here σxx and σyy have the same meanings. The sample sits on
83
Bip || y (a) σ
xx
(b) y
E
Re(σxx),Re(σyy) (µS) at ν=9/2
15
σyy
10
x
E
5
Bip=0.51 T
Bip=0 T
0
(d)
(c)
15
10
Bip=2.32 T
Bip=0.73 T
5
2.71 T
0
0.83 T
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
f (GHz)
Figure 6.4: Spectra of the real conductivities Re(σxx ) (solid lines) and Re(σyy ) (dashed
lines), for Bip along ŷ and increasing in magnitude from (a) to (d). σxx is measured
in sample 1, and σyy in sample 2. The inset shows a schematic of the microwave
transmission line and the nominal stripe orientation at zero Bip . The measurement
temperature was 40 mK.
a rotator and is cooled to 40 mK. Like other tilted field measurements, the rotation
angle θ is calculated from the magnetic fields of prominent quantum Hall states. At
perpendicular field B⊥ , Bip = B⊥ tan θ. Bip can be directed to be along either x̂ or
ŷ (in separate cool-down’s), so we present data from a total of four combinations of
EM and Bip directions.
Fig. 6.4 shows spectra of the real diagonal conductivities, Re(σxx ) and Re(σyy ),
at ν = 9/2, with Bip applied along ŷ, parallel to the zero-Bip stripe orientation.
For reference, Fig. 6.4(a) shows spectra taken at Bip = 0, measured from the same
samples and cool-downs used for Bip > 0 in the rest of Fig. 6.4; a 90 MHz resonance
is present in the spectrum of Re(σxx ), for which EM is polarized in the “hard” axis,
84
Bip || x
10
Re(σxx),Re(σyy) (µS) at ν=9/2
8
10
(b) Bip=0.29 T
8
6
σxx
4
σyy
2
12
(a) Bip=0 T
y
6
x
4
2
0
0
(c) Bip=0.55 T
15
8
(d) Bip=2.57 T
6
10
4
5
2
0
0
0.2
0.4
0.6
0.8
2.40 T
0.2
0.4
0.6
0.8
f (GHz)
Figure 6.5: Spectra of the real conductivities Re(σxx ) (solid lines) and Re(σyy ) (dashed
lines), for increasing Bip along x̂ from (a) to (d). The inset shows the nominal stripe
orientation at zero Bip .
85
but there is no resonance in Re(σyy ), for which EM is polarized in the “easy” axis.
Application of Bip ≈ 0.51 T, as shown in Fig. 6.4(b), does not switch the axis on
which the resonance is observed; the resonance remains visible only in Re(σxx ), with
the peak conductivity, σpk , and peak frequency, fpk , increased. In Fig. 6.4(c), the
Re(σxx ) resonance at Bip = 0.73 T is also well-developed with fpk increased further,
but σpk reduced. Also in Fig. 6.4(c), at Bip = 0.83 T, Re(σyy ) shows a weak, lowfrequency peak. Fig. 6.4(d) shows that at the larger Bip ’s of 2.32 and 2.71 T, the
resonance in Re(σxx ) is absent, while there is a broad resonance in Re(σyy ), indicating
that switching of the polarization of the resonance has occurred.
Fig. 6.5 shows conductivity spectra for Bip applied along x̂, perpendicular to the
zero-Bip stripe direction. Fig. 6.5(a) shows the Bip = 0 spectra for reference, taken
in the same cool-downs used to obtain the Bip > 0 data in that figure. The samples
used to obtain the data in Fig. 6.5 were the same as those used in Fig. 6.4, but had to
be remounted and cooled again to obtain the data in Fig. 6.5 (to switch direction on
the rotator). As expected for different cool-downs of the same samples, the spectra
in Figs. 6.4(a) and 6.5(a) are in good agreement, with a resonance only in Re(σxx ).
Figs. 6.5(b-d) again show that Bip switches the polarization axis of the resonance.
The switching appears to be taking place at Bip = 0.29 T, for which spectra in panel
(b) show well-developed resonances both in Re(σxx ) and in Re(σyy ). The switching
is complete by Bip = 0.55 T: as seen in Fig. 6.5(c), Re(σyy ) shows a resonance at
peak frequency 170 MHz, while Re(σxx ) shows none. Fig. 6.5(d) presents data for
the much larger Bip around 2.5 T; resonances are again present in both directions.
Fig. 6.6 presents plots of the resonance peak conductivity σpk , peak frequency
fpk , and the ratio of the integrated intensity S over fpk , all as functions of Bip .
First of all, the evolutions of the σpk and S/fpk values with Bip , as shown in panels
(a-d), clearly show the switching of the resonance polarization. For Bip applied either
along x̂ or along ŷ, there are distinct ranges of Bip in which a resonance is present
86
σyy
15
σxx
(b)
Bip || y
10
10
5
0
(d)
20
15
σyy
10
σxx
Bip || x
5
0
0.0
5
0
S/fpk (µS)
σpk (µS)
(c)
15
S/fpk (µS)
σpk (µS)
(a) 20
0.5
1.0
1.5 2.0
Bip (T)
15
10
5
0
0.0
2.5
0.5
1.0 1.5 2.0
Bip (T)
2.5
(e) 500
fpk (MHz)
400
(f)
300
Bip (T) ∆ (mK)
200
0.3 ||x 2.4
0.85 ||y 6
2.5 ||x 12
100
0
0.0
0.5
1.0
1.5
2.0
2.5
Bip (T)
Figure 6.6: The top panels are from the resonances in Re(σxx ) (open triangles) and
Re(σyy ) (open circles) with Bip applied along ŷ. (See Fig. 6.4 for the spectra.) (a)
plots the resonance peak conductivity σpk and (b) plots the ratio of the integrated
intensity S over peak frequency fpk , both as functions of Bip . The middle panels are
from the resonances in Re(σxx ) (solid triangles) and Re(σyy ) (solid circles) with Bip
applied along ŷ. (See Fig. 6.5 for the spectra.) (c) plots σpk and (d) plots S/fpk ,
both as functions of Bip . In the bottom panels, (e) plots fpk vs. Bip , with the same
symbols as in (a-d). (f) shows ∆, the per-carrier pinning energy difference between
stripes parallel and perpendicular to Bip ; magnitudes and directions of Bip at which
∆ is assessed appear in the left column.
87
exclusively in Re(σxx ) or Re(σyy ). This switching of the resonance polarization is
most naturally interpreted as a reorientation of the stripes, by analogy with the Bip induced switching of the “hard” and “easy” axes observed in dc transport studies
(25–27, 101, 119, 121, 151, 173).
Second, as shown in panel (e), fpk depends on both the direction and magnitude
of Bip . While all the curves in panel (e) show fpk increasing with Bip , the rate of
this increase is faster in the curves for σxx with Bip along ŷ and for σyy with Bip
along x̂. Hence when Bip is significant, these curves, for which EM polarization is
perpendicular to Bip , have larger fpk than the curves with parallel EM and Bip .
In the context of weak pinning theories (22, 51, 54, 58, 116), hfpk is a measure
of the pinning energy—the average energy per carrier due to the pinning disorder
potential (see discussions in Chap. 4.2.3). This holds for the electron wave function
size larger than the intrinsic correlation length of the disorder, which we have argued
in Chap. 4.2.3 is likely to be true for the present experimental system, otherwise
hfpk is an underestimate. To affect fpk , Bip must modify the effect of disorder,
by modifying the electron wave function within the confining quantum well. If the
pinning disorder is from interface roughness, as proposed by Fertig (51) for the WS,
Bip could increase pinning by increasing wave function amplitude at the quantum well
interfaces. Such an effect can be inferred from the wave functions in a quantum well
in the higher LL’s, in the presence of Bip , as presented in (147) (also see discussions
in Chap. 4.4 on similar effects in the second LL).
The change in fpk with application of Bip is large enough to be comparable to the
calculated EA (26, 81, 147), implying that a disorder interaction, specifically pinning
energy, plays an important role in determining the orientation of the stripe state. For
the Bip -ranges where resonance switches polarization, pinning energy anisotropy ∆
due to Bip is directly obtained as the difference of fpk measured with EM polarized
perpendicular and parallel to Bip ; taking the resonances to be occurring when the
88
measuring field EM is polarized perpendicular to the stripes,
∆ = h[fpk (stripe||Bip ) − fpk (stripe ⊥ Bip )] = h[fpk (EM ⊥ Bip ) − fpk (EM ||Bip )].(6.1)
The ∆ and Bip values are presented in panel (f) of Fig. 6.6. For comparison, EA ,
calculated (26) for the same carrier density and quantum well thickness as in the
present sample, is about 6 mK per carrier at Bip = 0.8 T. The pinning energy tends
to stabilize the orientation of the stripes parallel to Bip , and so is competing against
EA , which favors the stripes perpendicular to Bip . The balance of (at least) these two
factors, EA and ∆, may explain the complex switching behavior we observed. Since
EA obtained from the HF calculations depends on the finite thickness of the 2DES
layer, and ∆ depends on the disorder pinning, they can both vary from sample to
sample. This may explain some of the sample-dependent behavior of the stripe states
that has been noted in dc transport experiments (see review in Sec. 6.1).
Our third observation is that, while fpk increases with Bip , S/fpk also grows, as
shown in Fig. 6.6 (b) and (d). In the Fukuyama-Lee sum rule (58) for a weakly pinned
isotropic CDW, S/fpk is proportional to the pinned charge density. At ν = 9/2, had
the electrons on the top half-filled LL formed an isotropic CDW, the FukuyamaLee sum rule predicts S/fpk = 30 µS, which is much higher than the ∼ 7µS value
observed for the “hard”-axis resonance at Bip = 0. The sum rule certainly cannot be
applied directly to the highly anisotropic stripe phase in this case, and a quantitative
interpretation of the observed S/fpk value is not yet available. But qualitatively,
increasing S/fpk with Bip is suggestive of more charge participating in the pinning
mode.
We devote the rest of this section to a discussion of the Bip -ranges in which the
switching of the resonance polarization is taking place. In these ranges, resonances
in Re(σxx ) and Re(σyy ) can both be present for an applied Bip magnitude and direc-
89
Bip
(a)
E
ο
θ=0
(b)
ο
θ=7
(c)
20 µS
ο
θ=13 ν=
4.62
4.59
Re[σxx(f)]
4.56
4.53
9/2
4.47
4.44
4.41
4.38
100 200 300
100 200 300
100 200 300
E
(d)
ο
θ=0
(e)
ο
θ=7
20 µS
(f)
ο
θ=13 ν=
4.62
4.59
4.56
Re[σyy(f)]
4.53
9/2
4.47
4.44
4.41
4.38
100 200 300
100 200 300
200 400 600
f (MHz)
Figure 6.7: For in-plane field Bip along x̂, each panel contains a set of spectra [Re(σxx )
in panels (a)-(c) and Re(σyy ) in panels (d)-(f)] at various ν between 4.38 and 4.62,
increasing in steps of 0.01. Each succeeding spectrum is vertically offset by 4 µS for
clarity. Spectra in (a) are measured at θ = 0◦ , (b) at θ = 7◦ with Bip = 0.29 T at
ν = 9/2, and (c) at θ = 13◦ with Bip = 0.55 T at ν = 9/2 (Bip at a fixed θ is inversely
proportional to ν). Panels (d-f) are also measured at θ = 0◦ , 7◦ and 13◦ , respectively.
Spectra in each panel are measured in the order of increasing ν, or decreasing B⊥ .
Sample temperature is 40 mK.
90
tion. The spectra in Fig. 6.5(b) are the most striking example of two well-developed
resonances with different fpk and linewidths present for the two polarizations, under
the same conditions.
Fig. 6.7 presents “carpet” plots of spectra at the same tilting angles of Fig. 6.5(ac), with Bip applied along x̂. Each panel contains spectra at various ν between 4.38
and 4.62, measured in the order of increasing ν, or decreasing B⊥ . Panels (a-c) show
Re(σxx ), and each panel is measured at a fixed θ: (a) measured at θ = 0◦ , (b) at
θ = 7◦ with Bip = 0.29 T at ν = 9/2, and (c) at θ = 13◦ with Bip = 0.55 T at
ν = 9/2. Panels (d-f) show Re(σyy ), also at θ = 0◦ , 7◦ and 13◦ , respectively. From (a)
to (c) and from (d) to (f), the resonance switches from being present exclusively in
Re(σxx ) to being present exclusively in Re(σyy ). In panels (b) and (e), the switching
is taking place, and resonances are present in both Re(σxx ) and Re(σyy ). As we see in
(b), the spectra are not symmetric in ν about 9/2, being weaker for ν below 9/2 and
stronger for ν above 9/2. We find in panel (e) an opposite, though less appreciable,
asymmetry about ν = 9/2. The asymmetry about ν = 9/2 is probably because the
traces within a carpet plot are measured at the same θ, and Bip for each trace is
inversely proportional to ν, being slighly larger for ν just below 9/2 than for ν just
above 9/2.
There are different possible ways for a stripe to change orientation. If Bip drives
a stripe to rotate continuously, one would then expect to find some transitional value
of Bip where the “hard” and “easy” axes of high and low dc resistances are rotated
by 45◦ relative to their Bip = 0 orientation, but that was against findings in previous
dc transport measurements (102).
As we see in Figs. 6.7(b) and (e), where switching occurs, the coexisting resonances
in Re(σxx ) and Re(σyy ) have different peak frequencies and linewidths, suggesting
there are two distinct pinning modes. Taking the presence of a resonance as an
indicator of a region of stripes perpendicular to the resonance polarization, the two
91
resonances can be interpreted as arising from two coexisting portions of stripes with
perpendicular orientations, having different pinning modes. Such coexistence would
be consistent with energy minima at the orthogonal [110] and [11̄0] stripe orientations;
such minima in stripe state energy vs. orientation were proposed in (25), which
reported metastable states with the “easy” axis along either [110] or [11̄0] in dc
transport.
6.4
Tilt induced anisotropy in the bubble phase
In ranges of ν on either side of the stripe phase, the bubble phase shows nearly
isotropic pinning-mode resonance in perpendicular field (91), as presented in Sec.
6.2. This section shows that Bip induces anisotropy in the bubble phase pinning
modes.
Fig. 6.8 shows the conductivity spectra at ν = 4.29, which is in the bubble phase.
The other filling factors in the bubble phases, including those near ν = 4.7, 5.3, and
5.7, behave similarly under tilting. In different cool-downs, both the applied in-plane
magnetic field Bip and the polarization of the microwave electric field EM can be along
either x̂ ([11̄0]) or ŷ ([110]), making a total of four combinations of directions. Each
panel in Fig. 6.8 corresponds to one of the four direction combinations of Bip and
EM , and shows the real diagonal conductivity spectra at a series of Bip increasing
from 0 to about 2.8 T (at ν = 4.29, Bip = 2.52 tan θ T). The spectra all exhibit
well-developed pinning-mode resonance of the bubble phase.
As Bip is applied, a common feature we find in all the cases shown in Fig. 6.8 is
the remarkable increase in fpk , from about 200 MHz at Bip = 0 T to about 1 GHz at
Bip ∼ 2.8 T, accompanied by the increase of resonance linewidth ∆fpk as well. We
can divide the four panels in Fig. 6.8 into two groups: for (a) and (c), Bip and EM are
perpendicular to each other, and for (b) and (d), Bip and EM are parallel. At roughly
92
y
50
x
Bip (T)
0
0.50
1.08
1.70
2.86
(a)
40
30
20
10
Bip
EM
0
50
Bip (T)
0
0.58
1.00
1.65
2.55
(b)
40
30
20
Re[σxx,yy (f)]
10
Bip
EM
0
50
40
Bip (T)
0
0.57
1.12
2.77
(c)
30
20
EM
Bip
10
0
50
Bip (T)
0
0.46
0.86
1.59
2.51
40
(d)
30
20
10
EM Bip
0
0
200
400
600
800
1000 1200 1400
f (MHz)
Figure 6.8: The real diagonal conductivity spectra, Re[σxx,yy (f )], at the fixed filling
factor ν = 4.29, for different values of Bip increasing from 0 T. Each panel corresponds
one of the four different directions of the microwave electric field EM and the in-plane
magnetic field Bip , as specified on the right. The measurement temperature was 40
mK.
93
Resonance in σxx
S/fpk(µS)
σpk (µS)
∆f (GHz)
fpk (GHz)
1.0
Resonance in σyy
1.0
Bip || y
Bip || x
0.8
0.6
Bip || x
0.8
Bip || y
0.6
0.4
0.4
0.2
0.2
0.4
0.3
0.2
0.1
0.6
0.4
0.2
35
30
25
20
15
40
30
20
20
14
12
10
8
6
16
12
8
0.0
1.0
2.0
3.0
0.0
Bip (T)
1.0
2.0
3.0
Bip (T)
Figure 6.9: From the resonances at ν = 4.29 shown in Fig. 6.8, from top to bottom
the panels show fpk , ∆f , and σpk , all as functions of Bip . In the panels on the lefthand side, EM is along x̂, and Bip is parallel (solid circles) or perpendicular (solid
squares) to EM ; in the panels on the right-hand side, EM is along ŷ, and Bip is parallel
(hollow circles) or perpendicular (hollow squares) to EM .
94
the same Bip , the resonance amplitude σpk is larger with Bip and EM perpendicular
to each other than with Bip and EM parallel.
For ν = 4.29, Fig. 6.9 shows the resonance peak frequency fpk , half-height width
∆f , peak conductivity σpk , and the ratio of the integrated intensity S over fpk , all
plotted vs. Bip . These data are from the spectra shown in Fig. 6.8 (a-d), for all the
four direction combinations of Bip and EM .
In all cases, as panels (a) and (b) show, both fpk and ∆f increase with Bip , and
both appear to be highly independent of the directions of Bip and EM relative to
the crystal axis or relative to each other. As discussed in Chaps. 4.2.3 and 6.3,
fpk is a measure of the pinning energy, which is jointly determined by the carriercarrier interaction and the carrier-disorder interaction. Increasing fpk indicates a
larger pinning energy, probably because the in-plane field pushes the 2D electrons
closer to the GaAs/Alx Ga1−x As interface, as discussed in Chaps. 4.4 and 6.3.
In contrast to the essentially isotropic increase of fpk and ∆f with Bip , panel (c)
shows that σpk becomes anisotropic in Bip : σpk is larger for Bip and EM perpendicular
to each other and smaller for Bip and EM parallel. The anisotropy is also clear in
S/fpk of the resonances, as shown in panel (d). Starting from Bip = 0, the resonance
is isotropic with S/fpk ≈ 8.5 µS, which is about half of the value predicted by the
Fukuyama-Lee sum rule (58) assuming that all the electrons on the top LL form a
pinned CDW. For Bip and EM perpendicular to each other, S/fpk increases with Bip
for the range of Bip used in this experiment. For Bip and EM parallel, S/fpk first
decreases as Bip increases from 0 to about 1.5 T, and then starts to increase with Bip
but still is smaller than the S/fpk values for Bip and EM perpendicular.
95
6.5
Summary and dicussions
In this chapter, we have studied in the presence of Bip the stripe and the bubble
phases, both of which are identified by striking microwave/rf pinning-mode resonances.
At zero Bip , the stripe phase shows a resonant rf spectrum if measured with EM
polarized perpendicular to the nominal stripe orientation, and no resonance is visible
if measured with EM parallel to the stripe. Applying Bip causes the resonant EM
polarization to switch, indicating clearly the switching of the stripe orientation. The
presence of resonances in both polarizations around the Bip of the apparent switching
of the stripe orientation suggests there are likely coexisting regions of perpendicularly
oriented stripes at the transition.
It appears to be a common feature in the second, third, or higher LL’s that the
disorder pinning energy, which is measured by fpk , increases with Bip . For the stripe
phase, our measurements show that the increase of pinning energy depends on the
relative orientation of the stripe and Bip , and favors stripes parallel to Bip . On the
other hand, the previous HF calculations (81, 147) for a disorder-free system show
that Bip causes an anisotropy in the per-electron energy EA , which favors stripes perpendicular to Bip . The competition of (at least) these two effects drives the switching
behavior of the stripes in Bip .
Besides reorienting the stripes, we find Bip also increases S/fpk of the stripephase pinning-mode resonance, which is suggestive of more charge participating in
the pinning mode. But the established weak-pinning picture for Wigner crystal cannot
be directly applied to the anisotropic stripe phase, and a quantitative interpretation
of S/fpk is lacking.
We also showed that Bip of the magnitudes used in this experiment has fundamentally different effects on the stripe and the bubble phases. The bubble phase at zero
Bip shows an essentially isotropic pinning-mode resonance (91). As Bip is applied,
96
the resonance fpk ’s measured along different EM polarizations increase with Bip while
remaining isotropic, but σpk and S/fpk of the resonance depends on the polarization
direction of the measuring field EM . The resonance is stronger for EM perpendicular
to Bip , and weaker for Bip and EM parallel.
97
Chapter 7
Pinning mode resonances within
the ν = 1/3 fractional quantum Hall
effect
This chapter describes resonances observed in the microwave spectra of the real diagonal conductivities when a 2DES is within a range of ∼ ±0.015 from filling factor
ν = 1/3 of the lowest LL. The resonances are similar to those observed near integer ν.
They are interpreted as the collective pinning modes exhibited by a disorder-pinned
solid phase near ν = 1/3, formed by a dilute system of e/3-charged carriers with
long-range Coulomb interaction.
7.1
Introduction
The IQHE’s and FQHE’s exhibited by a 2DES subject to a perpendicular magnetic
field both result from the opening up of energy gaps and charge localization. For the
IQHE, the energy gap is the LL gap. Electron-electron interaction, while less important for the LL gap, can play a crucial role in the mechanism for charge localization.
When ν is just away from an integer, there is a dilute subsystem of electrons or holes
98
in the top LL, which must be localized (76), in order for the IQHE to manifest as
quantized transport. If these electrons or holes are sufficiently dilute, and if the disorder is sufficiently weak, they tend to form a solid, driven by dominating Coulomb
repulsion. Evidence for the solid phase came from the pinning-mode resonances near
integer ν, first observed in (18, 90) (also see data near ν = 1, 2, and 3 presented in
Chap. 4.2 and 5).
In the composite fermion (CF) framework (79), the ν = 1/3 FQHE of electrons
can be viewed as the IQHE of CF’s at νCF = 1. In this case, a CF is the bound
states of one electron and two quantized vortices, and νCF is the filling factor of the
CF-LL’s, which is related to ν as ν = νCF /(2νCF + 1). As ν deviates from 1/3, or as
νCF deviates from 1, a dilute subsystem of CF quasiparticles or quasiholes reside on
the top (partially filled) CF-LL, and they show fractional charges of e/3. Evidence
for the e/3 fractional charge has been found in measurements of resonant tunnelling
(68), shot noise (36), and spatial scanning of chemical potential (106) at ν = 1/3.
These “extra” e/3 charges must be localized to account for the quantized transport
of the FQHE (106). As in the case for the IQHE of electrons, interaction can play a
crucial role in the charge localization responsible for the FQHE of electrons at ν = 1/3,
or the IQHE of CF’s at νCF = 1. In the long-range limit, the interaction is expected
to be the Coulomb repulsion between e/3 charges, and this is what is important
in the dilute limit. Thus, near ν = 1/3 or νCF = 1, the system is in a situation
similar to that near integer ν: there exists a dilute subsystem of e/3 charges with
dominating Coulomb interaction, and this subsystem satisfies the condition for the
Wigner crystallization, as pointed out by Halperin long ago (74). In the presence of
disorder, such a solid will be pinned and thus collectively localized, which is consistent
with the quantized transport found near ν = 1/3.
The main result of this experiment is the observation of resonances in the spectra
of real diagonal conductivity on the edges of the ν = 1/3 FQHE state (at zero
99
ν=
1.16
Re[σxx(f)] (µS)
1.12
1.08
1.04
5 µS
1.00
1
2
f (GHz)
3
4
Figure 7.1: The real diagonal conductivity spectra Re [σxx (f )] for various ν values
increasing from 1 (bottom trace) to 1.16 (top trace), at equal steps of 0.02. Ajacent
traces are offset by 2 µS from each other for clarity. Given 63o of tilting, the skyrmion
spin is calculated in (32) to be ∼ 1.5 at ν = 1.13 and ∼ 2.0 at ν = 1.06. The
measurement was done in the “C120” dilfridge, at T = 40 mK.
tilting). Resonances are observed for ν within a range of ∼ ±0.015 about ν = 1/3,
corresponding to νCF ∼ ±0.15 about νCF = 1. The resonances are similar to those
observed for ν within ∼ ±0.16 around ν = 1, and are interpreted as pinning modes
of the solid phase of the e/3 quasiparticles or quasiholes.
7.2
Pinning mode resonance found near ν = 1/3
This experiment used sample “QW50”, a 50 nm wide AlGaAs/GaAs/AlGaAs quantum well, with electron density n = 1.1 × 1011 / cm2 , and a low-temperature mobility
µ = 15 × 106 cm2 /Vs.
For comparison, Fig. 7.1 shows the conductivity spectra near ν = 1. As discussed
in Chap. 5, the excitations at ν = 1 are expected to be skyrmions. To limit the
skyrmion formation by increasing the Zeeman gap, we carried out the measurements
100
with the sample tilted to θ = 63◦ in the magnetic field. Fig. 7.1 shows the spectra of
Re [σxx (f )], spanning the frequency range of 0.1 GHz - 4 GHz, for various ν between
ν = 1.02 and 1.16, changing in equal increments of 0.02. (The spectra are essentially
symmetric about ν = 1, which is used as reference of the microwave absorption.)
Resonances are clearly visible in the spectra. Such resonances were first reported by
Y. Chen et al. (18) from measurements in perpendicular magnetic fields, and have
been studied in tilted fields as described in Chap. 5. They have been interpreted as
evidence for disorder-pinned solid phase of the quasiparticles. As |ν − 1| gets above
0.16, the resonance weakens, signalling the melting of the solid phase.
Fig. 7.2 presents the microwave Re [σxx (f )] spectra near ν = 1/3. Panel (a)
shows spectra for various ν decreasing from 0.331 to 0.319, and (b) shows spectra
for ν increasing from 0.335 to 0.353, both in equal steps of 0.002. The spectra are
measured in perpendicular magnetic fields, and are normalized to the spectrum at
ν = 1/3 which serves as reference of microwave absorption. In both panels, as ν starts
to deviate from 1/3, a resonance emerges in the spectrum, first with a relatively high
peak frequency. As |ν − 1/3| increases, the resonance increases in amplitude and
shifts to lower frequency. Beyond |ν − 1/3| = 0.015, the conductivity spectrum loses
the resonant shape and becomes a monotonically decreasing function with frequency.
7.3
Interpretation: a pinned solid of e/3 charge
carriers with Coulomb interaction
The evolution of the resonances close to ν = 1/3 bears a strong similarity to that
observed near ν = 1. As we shall discuss in this section, the resonances near ν = 1/3
can be most naturally understood as pinning modes of solid phase formed of the e/3
quasiparticles or quasiholes of the ν = 1/3 FQHE.
Figs. 7.3(a) and (b) plot the ν dependencies of the peak frequencies, fpk , of the
101
(a)
(b)
ν=
ν=
Re [σxx] (µS)
0.313
0.2
µS
0
1
2
3
f (GHz)
4
0.353
0.317
0.349
0.321
0.345
0.325
0.341
0.329
0.337
1/3
1/3
5
0
1
2
3
f (GHz)
4
5
Figure 7.2: The real diagonal conductivity spectra Re [σxx (f )] for ν near 1/3. (a) ν
decreases from 1/3 (bottom trace) to 0.331, and further to 0.313 (top trace) in equal
steps of 0.002. (b) ν increases from 1/3 (bottom trace) to 0.335, and further to 0.353
(top trace) in equal steps of 0.02. Ajacent traces are offset by 0.1 µS from each other
for clarity. The measurement was done in the “SCM1” dilfridge, at T = 50 ∼ 55 mK.
102
(10
11
0.10 0.00
-2
(10
cm )
-2
cm )
0.10 0.00 0.10
0.10
(a) 1.4
(b )
1.2
f pk (GHz)
f pk (GHz)
11
1.0
0.8
1.5
1.0
0.5
0.6
-0.05
0.05
-0.015
0.15
ν-1
ν-1/3
( d) 1.2
6
S/f pk (µ S)
S/f pk (µ S)
( c)
0.015
4
2
0
-0.05
0.05
0.15
1.0
Fukuyama
0.8
-Lee
0.6
0.4
0.2
0.0
-0.015
0.015
ν-1/3
ν-1
Figure 7.3: The resonance peak frequencies fpk versus filling factor near (a) ν = 1 and
(b) ν = 1/3. In both panels, the quasiparticle number density n∗ is marked on the
upper horizontal axis. Integrated intensity over peak frequency S/fpk as a function
of filling near (c) ν = 1 and (d) ν = 1/3, in comparison with the Fukuyama-Lee
prediction (dashed lines).
103
resonances near ν = 1 and ν = 1/3, respectively. fpk decreases as |ν − 1| or |ν − 1/3|
increases, as is clear in the spectra shown in Figs. 7.1 and 7.2. This trend is consistent
with the picture of weak disorder pinning (see discussions in Chap. 3). As ν deviates
from 1 or 1/3, the quasiparticles or the quasiholes accumulate, and the interaction
strength increases as the square of charge density, which relatively weakens the role
of disorder and causes the pinning frequency to decrease. We also note that fpk falls
into the same range of values (around 1 GHz) for ν either near 1 or near 1/3, a fact
we shall discuss further below.
Panels (c) and (d) of Fig. 7.3 plot the ν dependence of the ratio of the integrated
intensity S over fpk . S is calculated by integrating the spectrum against frequency,
and then subtracting the contribution from an apparent frequency-independent background. In the treatment by Fukuyama and Lee (F-L) of weak pinning [(58), Chap.
3.2], S/fpk is found to be proportional to the charge density participating in the pinning, divided by the magnetic field. Near ν = 1, the charge density of the electrons
or holes in the top LL is (|ν − 1| /ν)ne, which we use to calculate the F-L prediction
of S/fpk (plotted as the dashed line in Fig. 7.3(c)). The S/fpk values extracted from
the measured spectra are in good agreement with the F-L prediction, up to ν = 1.1
or down to ν = 0.9, suggesting a majority of the electrons or holes in the top LL are
participating in the pinning.
Panel (d) shows the S/fpk values of the resonances near ν = 1/3. Because the
conductivity spectra are no longer resonant beyond |ν − 1/3| = 0.015, we only calculate S/fpk for |ν − 1/3| < 0.015. For comparison, the F-L prediction of S/fpk is
calculated given the charge density (|ν − 1/3| /ν)ne. For ν close to 1/3, the measured value of S/fpk follows the same trend as the F-L prediction, and its magnitude
is slightly larger than one half of the F-L prediction. For ν approaching 1/3 ± 0.015,
because the resonance verges on disappearing to the low-frequency side and assumes
a heavily damped shape, the S/fpk obtained from the measured spectrum is probably
104
an overestimate.
Therefore, within the weak pinning framework, the ν-dependencies of fpk and
S/fpk shown in Figs. 7.3(b) and (d) suggest that the amount of charge participating
in the pinning mode grows with |ν − 1/3| (when it is small).
The range of ν near 1/3 that shows the resonance is an important piece of information. Near the ν = 1 IQHE, the solid phase is found to be favorable for |ν − 1|
smaller than ∼ 0.16, beyond which the microwave resonances weaken [see Fig. 7.3(c)
and (18, 90)]. Near ν = 1/3, the resonances are observed for a much narrower range,
|ν − 1/3| < 0.015. In the CF framework, this corresponds to a range of |νCF −1| < 0.15
around a fully filled νCF = 1 CF-LL.
We also note that in the lowest LL a WS phase is favorable for ν <∼ 0.2, which is
not very different from the |ν − 1| or |νCF − 1| ranges within which resonant spectra
provide evidence for solid phases. For a 2D system of charge carriers with Coulomb
interaction, a solid phase is favorable when a strong magnetic field results in a small
ratio of l/r, where l is the in-plane length scale of the charge-carrier wave function,
and r is the charge-carrier separation (59, 84, 103). For a 2DES in the lowest LL,
l/r is directly related to ν, and for the subsystem near ν = 1 or νCF = 1, l/r of the
electron or CF quasiparticles is directly related to |ν − 1| or |νCF − 1|.
Lastly, we give a semi-quantitative analysis of the resonance fpk ’s within the weak
pinning framework. We notice in Fig. 7.3 that the fpk values for both the resonances
near ν = 1 and those near ν = 1/3 fall into the same range. In both cases, fpk
strongly depends on ν, or the number density of quasiparticles n∗ [both parameter
marked in Fig. 7.3(a) and (b)]. But if we compare a state near ν = 1 and a state
near ν = 1/3 with the same n∗ , we can see that their fpk values roughly agree.
In the following we show that the agreement can be understood within the weakpinning picture, only on the condition that we take the solid near ν = 1/3 to be formed
by e/3-charged carriers with long-range Colomb interaction. As discussed in Chap. 3,
105
the weak-pinning theory gives fpk as the disorder-induced gap in the otherwise gapless
magneto-phonon mode (10, 22, 51, 54, 58, 116). fpk has been calculated (22) for a
disorder potential energy V (r) with a Gaussian correlation function hV (r)V (r′ )i =
∆D(r − r′ ), where D(r − r′ ) ∼ (1/ξ02 ) exp[−(r − r′ )2 /ξ02 ]. (As shown in Chap. 3, the
form of fpk can be constructed also by a simple dimensional analysis). We denote the
charge of the carrier by e∗ (e∗ = e around ν = 1 and e/3 near ν = 1/3). If ξ0 ≫ l,
the in-plane size of the charge-carrier wave function, we have
fpk ∼
∆
;
µT ξ06 e∗ B
(7.1)
if l ≫ ξ0 , we have to replace ξ0 with l due to the averaging effect of the carrier wave
function, and
fpk ∼
∆
,
µT l 6 e ∗ B
(7.2)
Just for the purpose of comparison, we do not have to know which case applies,
because an e carrier at ν = 1 and an e/3 carrier at ν = 1/3 have essentially the
same l, and ξ0 is an intrinsic parameter of the sample. µT is the shear modulus,
which, if we use the result for a point-charge system as approximation (correct for a
dilute system), is proportional to (e∗ )2 (n∗ )3/2 . The sample disorder does not change
by itself, but it is reasonable to assume the force between disorder and charge carrier
is electronic, so that V (r) ∝ e∗ and ∆ ∝ (e∗ )2 . Finally, e∗ B remains the same when
ν changes from 1 to 1/3 (and e∗ changes from e to e/3). Putting all these together,
we see that the weak-pinning picture predicts the same fpk for the solid phases near
ν = 1 and that near ν = 1/3, for the same n∗ of the carriers. This is consistent with
our observation in Figs. 7.3 (a,b).
106
7.4
Summary and discussions
To summarize, for ν within a range of ±0.015 about 1/3, we found that the real
diagonal conductivity spectra of a 2DES exhibit microwave resonances. By analogy
to the similar resonances previously found near integer ν, we interpret the resonances
near ν = 1/3 as the pinning modes of a solid phase formed of e/3 quasiparticles or
quasiholes.
Several aspects of the resonance reveal important information about the pinned
solid. First, S/fpk of the resonances is reasonably consistent with the FukuyamaLee sum rule calculated for a WS of charge density growing with |ν − 1/3|. Second,
the resonances are found within ±0.015 near the 1/3 filling of the lowest electron
LL, which corresponds to ±0.15 about complete filling of the lowest CF-LL. This
filling factor range of the CF-LL is similar to the filling factor range of solid phases
near integer fillings of electron LL’s. Third, within the same sample, we find the
resonance near ν = 1/3 has roughly the same fpk as the resonance near ν = 1 for the
same quasiparticle number density n∗ . This agreement in fpk can be quantitatively
understood within the weak-pinning picture if considering the solid near ν = 1/3 as
formed of e/3 charges with long-range Coulomb interaction.
107
Chapter 8
Summaries and perspectives
This thesis presents microwave spectroscopic studies of various phases with crystalline
order or partial crystalline order exhibited by quantum Hall systems, in the presence
of an in-plane magnetic field. Most of the phases studied can be categorized as
isotropic solid of charge carriers, including the WS of electrons at low LL filling, solid
of quasiparticles or quasiholes near integer ν (or, as shown in Chap. 7, near ν = 1/3),
and solid of multi-electron bubbles in the third or higher LL’s. We have also studied
the unique, highly anisotropic stripe phases near half-odd integer fillings in the third
or higher LL’s.
The main experimental tool we use is microwave spectroscopy. The above-mentioned
phases are all marked by striking resonances in the microwave spectra of real diagonal
conductivities. The resonances have been understood as the pinning modes exhibited
by these phases of 2D electrons pinned by sample disorder. They not only serve as
spectroscopic evidence for the particular phases, but also contain rich information
about charge ordering, dynamics, and interaction with disorder.
Using this technique, we observe (Chap. 7) that, within ±0.015 about ν = 1/3,
the microwave spectra exhibit pinning-mode resonances similar to those observed near
integer ν. These resonances are strong evidence for the long-sought-after solid phase
108
formed by dilute e/3-charged quasiparticles or quasiholes.
In major part of the research reported in this thesis, we have added to the parameter space one more dimension, the in-plane magnetic field Bip . To do so we
have designed a series of sample rotators that enable tilting the sample within a
magnet while maintaining low-reflection, broad-band microwave transmission. When
the sample normal axis is tilted at an angle θ to the field direction, the magnetic
field Btot has a perpendicular component B⊥ = Btot cos θ and an in-plane component
Bip = Btot sin θ.
As we have discussed in this thesis, Bip can affect in a number of ways the carriers in a solid phase. Because the carrier systems used in the experiments are not
strictly 2D, Bip affects the orbital wave function of the carriers, and hence affects the
interaction between carriers and the interaction between carrier and pinning disorder.
Because sample tilting enables us to separately vary B⊥ and Btot , it also offers a way
to independently control ν and the Zeeman energy.
The following summarizes our studies of the effects of Bip on the WS phases (both
the solid at low ν of the lowest LL and those near integer ν) and the stripe and bubble
phases.
1. We find application of Bip favors the WS phases (Chap. 4). In the lowest LL, in
perpendicular field, the WS phase terminates the FQHE series at low ν. Tilting
is found to expand the range of ν for the WS phase, which in our experiments is
conveniently identified by the range of ν showing the pinning-mode resonance.
The effect of Bip is more significant for larger electron density, or smaller lattice
constant, which reveals the role of finite thickness of the electron layer.
We also examine the effect of tilting in the second LL, just above ν = 2 or just
below ν = 3, where the dilute electrons or holes in the top (partially filled) LL
form a WS. Tilting similarly results in the expansion of the range of ν that
exhibits the pinning-mode resonance of WS. But a major difference from the
109
case of the lowest LL is that, in the second LL, we find the resonance frequency
fpk significantly increases with Bip . The increase of fpk indicates strengthened
disorder pinning, which stablizes the pinned WS.
2. In tilted field, we study in Chap. 5 the pinning-mode resonance of the solid
phase that forms in the lowest LL near ν = 1. When the quasiparticle density
is low, tilting the sample shifts the resonance to higher frequencies, in marked
contrast to the case when the quasiparticle density is high or the case of the
resonance just below ν = 2, where tilting has negligible effects. These results
support that the solid phase near ν = 1 is formed of skyrmions, which are
textures involving multiple flipped spins and having charge distribution more
spread out than a single Landau orbital. The observed shift of the resonance
fpk with tilting is explained in terms of the skyrmion size, which changes with
the Zeeman energy and the skyrmion density. The apparent dependence of the
pinning modes on the Zeeman energy reveal the coupled charge and spin degrees
of freedom in the solid of skyrmions.
3. When the Fermi level is in the third or higher LL’s, the stripe and bubble
phases replace FQHE liquids as the ground state. We find (Chap. 6) that
the stripe phase shows an rf pinning-mode resonance only for the rf electric
field EM polarized perpendicular to the stripe orientation, while no resonance
is discernible for EM polarized parallel to the stripes.
At Bip = 0, the orientational order of the stripes is ascribed to a (small)
symmetry-breaking field intrinsic to the sample, the origin of which remains
unclear. Application of Bip overcomes this intrinsic symmetry-breaking field
and can reorient the stripes. Reorientation of the stripes is indicated both
by previous measurements that showed interchanging directions of higher and
lower dc resistivities, and by our rf measurements that showed interchanging
110
directions of resonant and non-resonant EM polarizations.
Our rf measurements suggest that disorder plays an important role in orienting
the stripes in the presence of Bip . We find fpk of the stripe-phase resonance
increases with Bip , by a larger amount for Bip applied parallel to the apparent
stripe orientation. Since fpk measures the disorder pinning energy, this finding
indicates that disorder interaction favors stripes parallel to Bip . On the other
hand, previous Hartree-Fock calculations (81, 147) find that, in a disorder-free
system, stripes have lower energy if oriented perpendicular to Bip . Our result
and refs. (81, 147) jointly suggest that the stripe orientation in Bip is governed
by the interplay of (at least) these two mechanisms.
Besides fpk , we read from the pinning-mode resonance another important piece
of information, the integrated intensity S. While applying Bip shifts up fpk , a
sign of strengthened pinning, we find S/fpk also increases. Although a quantitative understanding of S/fpk is lacking in the case of stripe phase, the increase
of S/fpk is suggestive of more electrons participating in the pinning mode.
The bubble phase shows isotropic resonance at zero Bip . Applying Bip causes
the bubble-phase resonance fpk to isotropically increase, and induces anisotropy
in the resonance amplitude and intensity: the resonance is stronger for Bip and
EM perpendicular than for Bip and EM parallel. This shows that the bubble
phase becomes anisotropic, but having a distinct type of anisotropy from the
stripe phase.
Microwave spectroscopy, as we have seen, is a particularly powerful tool to study
the WS, bubble and stripe phases, and to reveal information not accessible in dc
transport. There are ample reasons to believe that this tool will continue to produce
new knowledge, and here I give my version of a very limited list of some questions
which the microwave sample rotator may help answering.
111
• The pinning mode resonances observed in the insulating phase at low ν of the
lowest LL have provided evidence for the WS phase. A good understanding of
this WS is of great importance since it serves as the prototype for various other
solid phases, for example, those occuring near integer ν. In wide quantum well
samples of high quality, Y. Chen et al. (19) found that there are actually two
distinct pinning modes, suggesting two different solid phases. The two phases,
labeled “A” and “B”, occur for different ranges of ν: “A” phase dominates
for 2/9 > ν > 0.18 (but disappearing around 1/5), “B” phase dominates for
ν < 0.12, and in the transitional range (0.18 > ν > 0.12), the two phases
coexist. “A” phase appears to have a particularly large correlation length (the
pinning-mode resonance sensitive to the change of CPW slot width from 30 µm
to 60 µm) (19).
An appealing picture is that the “A” phase, occuring at relatively high ν, may
be a solid phase with substantial FQHE-liquid-like correlations, which is absent
in the “B” phase. Theoretical works (15, 16, 79, 113, 170) found the correlated
solid to have lower energy than the uncorrelated one for roughly the range of ν
in which “A” phase is observed. Further work will be needed to more directly
demonstrate the possible role of correlations in the “A” phase.
Appendix B contains a preliminary, yet encouraging set of data that show a
significant effect of Bip on the “A” phase. The data seem to suggest that the
“A” phase transforms into the “B” phase as Bip is applied. (Note that in Chap.
4 we also found that applying Bip makes FQHE-liquid correlation less favorable.)
Potentially this may be a useful piece of information for understanding the “A”
and “B” phases.
• In the second LL, dc transport measurements (43, 165) found re-entrant IQHE
states near ν = 2.30, 2.42, 2.58, and 2.70. Near these filling factors, microwave
112
spectroscopic studies (88) also observed weak resonances. These measurements
are consistent with the formation of a bubble phase crystal centered around
these fillings. Csathy et al. (34) found that tilting destroys the re-entrant IQHE
states, suggesting the bubble crystal melts. The cause for the melting is not
clear, and it should be rewarding to examine this phenomenon with microwave
spectroscopy.
• We have studied in Chap. 6 the behaviors of the stripe and bubble phases in
tilted fields. The applied Bip in the experiment is smaller than 3 T, and we
find applying Bip can switch the stripe orientation and induce anisotropy in the
bubble phase. The polarization for which the stripe phase shows a resonance
can be different from the polarization for which the bubble phase resonance is
stronger. The physics behind the different anisotropies induced by Bip for the
stripe and bubble phases is not understood. Pushing the experiments to larger
Bip may provide more clues.
113
Appendix A
Higher order resonance peaks of
the Wigner solids
It has been noted that in samples of high mobility, the sharp pinning-mode resonances
of WS are sometimes accompanied by weaker second and even third peaks at higher
frequencies f . Such higher f resonances were first reported in (19), and their origin
is not clear. One possibility is that such higher f peaks are simple harmonic series of
the main peaks. But this interpretation cannot account for the following facts found
in previous measurements (19).
(1) The higher f peaks do not appear to fit simple (integer multiples in frequency)
harmonic series of the main peaks.
(2) The CPW’s deposited on the sample has a finite slot width w. This results in
finite wave vectors q = 2π/w and its multiples in the microwave electric fields. But,
as shown by (19), the higher f peaks measured with CPW slot width w do not fit
the main peaks measured for the same sample with CPW slot width w/2.
(3) The higher f peaks can have different dispersion relation, fpk (q), from that of
the main peak (19).
This appendix presents results from recent measurements which, in addition to
114
10
-2
p=10.3×10 cm
Re[σxx(f)] (µS)
5 µS
B=45T,ν=0.092
0.10
0.12
0.14
0.16
0.18
0.20
200
400
600
800
1000
f (MHz)
Figure A.1: The real diagonal conductivity spectra Re [σxx (f )] for various ν decreasing from 0.20 (bottom trace) to 0.092 (top trace), with the ν values marked by the
side of each trace. Ajacent traces are offset by 2 µS from each other for clarity. The
measurement temperature was 80 ∼ 100 mK.
the above observations, show that the higher f peaks can have qualitatively different
ν dependence from the main peaks.
We used sample “H30”, which is a p-type 30-nm-wide quantum well. The 2DHS
has an as-cooled density of p = 9.2 × 1010 /cm2 , and we have applied a back-gate
voltage Vbg to tune it between 5.9×1010 /cm2 (at Vbg = +375 V) and 12.6×1010 /cm2
(at Vbg = −375 V). The measurements were done in the “PDF” dilfridge and inside
the 45 T magnet. Due to a less stable condition during the measurements, the sample
temperature varied from 80 mK to 120 mK, but this fluctuation does not significantly
alter the fpk values.
115
Fig. A.1 shows a set of resonances for the state with 10.3 × 1010 /cm2 , for various
filling factors within the WS phase. Besides the main peak with lower peak frequency
(1)
(2)
fpk , each trace also shows a second peak at frequency fpk and even a weaker third
(3)
peak at fpk . There are no apparent simple relations among the fpk values of the three
(1)
peaks. As ν decreases from 0.20 to 0.092, fpk increases from 175 MHz to 300 MHz,
(2)
(3)
whereas fpk and fpk show a much weaker dependence on ν.
We have carried out similar measurements for other densities of the 2D holes,
and the fpk values are summarized in Fig. A.2. For lower density states, the third
(1)
peaks are not discernible. Fig. A.2 clearly shows that fpk ’s increase strongly as ν
(2)
(3)
decreases, but fpk ’s and fpk ’s have a much weaker dependence on ν, and even can
weakly decrease with decreasing ν.
Besides the three facts listed at the beginning of this appendix, the observation
(1)
(2,3)
that fpk ’s and fpk ’s can have qualitatively ν dependence adds to the mystery of the
origin of these higher-f weak peaks.
116
Squares: 1st Circles: 2nd Triangles: 3rd
600
p
10
(×10 /cm2)
fpk (1), (2), (3) (MHz)
500
5.9
7.0
400
9.2
300
10.3
12.6
200
0.05
0.10
0.15
0.20
ν
Figure A.2: The fpk values are plotted as a function of ν, for different 2DHS densities,
(1)
p, which are marked to the right. Among the symbols, fpk are represented by squares,
(2)
(3)
fpk by circles, and fpk by triangles.
117
Appendix B
Supplementary data in the lowest
Landau level in tilted field
In a previous study(19), samples “QW65” and “QW50” were found to exhibit two
apparently distinct types of resonances in the high B insulating phase. They are
interpreted as coming from two different pinned electron solid phases (labeled as “A”
and “B”). “A” resonance is observable for ν < 2/9 (disappearing around the ν = 1/5
FQHE) and then crosses over to the different “B” resonance which dominates at
sufficiently low ν. “A” resonances have higher fpk ’s than “B” resonances, and at the
cross-over region, the spectrum can show double peaks. It is still unknown what types
of solid phases are yielding the different resonances.
In Chap. 4 we showed that tilting expands the WS phase to higher ν values which
are dominated by FQHE liquids at zero Bip . Bip also affects the filling factors which
already show a resonance at zero Bip . As shown in Fig. 4.2, a 44◦ tilting also causes
the peak frequency of the ν = 0.190 resonance (type “A”) to decrease by 40%.
This appendix provides an inspiring set of data that address the effect of tilting
for ν already in the WS phaseaat zero tilting. The data were from sample “QW65”,
and the measurements were done in the “SCM1” dilfridge and inside an 18 T magnet.
118
θ=0
o
T=60 mK
15
Re[σxx] (µS)
ν=
0.12
0.14
10
0.16
0.18
5
0.20
0.22
0.23
0.25
0
100
200
300
400
500
f (MHz)
Figure B.1: The real diagonal conductivity spectra Re [σxx (f )] for various ν decreasing from 0.25 (bottom trace) to 0.12 (top trace), with the respective ν values marked
by the side of each trace. The spectra were measured at 0◦ tilting and at a temperature of 60 mK. Ajacent traces are offset by 1 µS from each other for clarity.
During this cool-down, the sample had a less than perfect state, and ν = 1/5 did not
show a good FQHE state. Thus although the data are intriguing, they only provide
a hint on the physics, which awaits further study.
Fig. B.1 shows the spectra measured in perpendicular field, for various ν in the
WS phase. The resonant range of ν includes 1/5, and the low-ν resonances are less
sharp then they have been seen before (19), suggesting the sample state is not perfect.
As ν decreases from 1/5, the resonance peak shifts down, and during this process the
structure of the spectrum seems to suggest a cross-over from a higher f peak to a
119
5
ν=0.19
o
48.6
o
45.2
o
38
o
31.8
o
0
Re[σxx(f)] (µS)
4
3
2
ν=0.14
8
o
31.8
o
0
6
4
2
1
0
0
0
100
200
300
400
500
0
100
200
300
400
500
f (MHz)
Figure B.2: The real diagonal conductivity spectra Re [σxx (f )] for (a) ν = 0.19
and (b) ν = 0.14, at various tilting angles as marked in the graph. For ν = 0.19,
B⊥ = 11.2 T, and Bip = 6.9 T at 31.8◦ , 8.8 T at 38◦ , 11.3 T at 45.2◦ , 12.7 T at
48.6◦ . For ν = 0.14, B⊥ = 15.1 T, and Bip = 9.4 T at 31.8◦ . The measurement was
at T = 60 mK.
lower f peak. This structure reminds us of the “A” and “B” resonance peaks seen in
a good state of the sample (19).
The sample is then tilted in the magnetic field. Fig. B.2 (a) shows the spectra for
ν = 0.19. As the sample is tilted, the resonance peak with relatively high f and low
amplitude σpk at θ = 0◦ evolves into a low-f , high-σpk resonance at θ = 48.6◦ . During
the evolution, at θ = 38◦ , the resonance is suggestive of a double peak structure and
a cross-over from a high f peak to a low f one. (b) shows the spectra for ν = 0.14,
due to the limitation of total field (18 T maximum), we could only tilt the sample
to about 32◦ . Upon tilting, while the resonance grows in σpk , its fpk does not show
significant change.
The spectra shown in Fig. B.2 seems to suggest that, for relatively high ν, tilting
causes a cross-over from a higher-f peak to a lower-f peak. For relatively low ν, the
resonance already shows the low f peak and its frequency does not shift significantly
with tilting. Although during this cool-down the sample did not have a good state,
120
the observations seem to hint that tilting may cause a cross-over from resonance “A”
to resonance “B”, for relatively high ν values. The respective nature of each of these
two resonances is still unknown, and this encouraging data set shows that tilting may
be a useful tool to study them.
To explain the observed effect of tilting needs an exact understanding of the
WS, and also more experiments. Here I remark on a few theoretical considerations
pertaining to the WS phases and their possible behavior in tilted field.
(1) While a WS of bare electrons is favorable at low ν, the WS at relatively high
ν (closer to 1/5) has been proposed to possess the type of correlation responsible for
the FQHE (15, 16, 79, 113, 170). The possible effects of such correlations on the shear
modulus, the pinning modes, and how they can be affected by Bip are not clear. It
is possible that, as tilting moves such filling factors as ν = 0.19 deeper into the WS
phase, an uncorrelated solid become favorable over a correlated solid.
(2) As proposed by S.-J. Yang (168), applying Bip breaks the 2D rotational symmetry, and the [110] axis of the triangular lattice tends to align with the Bip direction,
resulting in larger crystal domains, which are more difficult to pin.
(3) Another possible way for Bip to affect pinning is by modifying the orbital
wave function of the electrons. If the pinning disorder is from the roughness at the
quantum well interface, as conjectured by Fertig (51), Bip may affect the electrondisorder interaction by changing the electron density profile across the quantum well.
121
Appendix C
Coplanar wave-guides and
microstrips
Both Coplanar wave-guides and microstrips are standard planary components of microwave circuits. Schematics of their geometries are shown in Fig. C.1
C.1
Coplanar wave guides
Here I summarize a few basic facts about Coplanar Wave-guide (CPW) as a transmission line. More complete treatments can be found in (131, 157) and the general
transmission line theory is covered in (100).
Consider a CPW (schematically shown in Fig. C.1(a) for a local cross section) with
center-conductor width s, slot width w, and fabricated on a substrate of thickness h
and relative dielectric constant ǫr . The surrounding is assumed to be vacuum or air
(ǫr = 1, which is also a good approximation for helium). In our usual applications,
s, w, h ≪ λ (the microwave wavelength). Also the CPW metal films have very small
resistance and the side planes can be approximately treated as infinitely extended.
In this case, the CPW, when not loaded by 2DES, is a loss-less transmission line
with two parameters L0 and C0 , where L0 is the inductance per unit length, C0 is
122
(a) CPW
w
s
w
εr
(b) Microstrip
h
s
t
h
εr
Figure C.1: Schematics of (a) a CPW and (b) a microstrip. The orange-colored parts
are metal layers and the grey parts are dielectric material with dielectric constant ǫr .
the capacitance per unit length (between the center conductor to ground). For the
standard transmission line theory [see Chap. 3, Sec. 1 of (100)], we have vph =
p
√
1/ L0 C0 as the phase velocity and Z0 = L0 /C0 the characteristic impedance. L0
and C0 are only determined by the CPW geometry and ǫr . An excellent approximation
can be obtained with conformal mapping (157). In practice, it is often convenient to
use an online-available CPW designing calculator (for example,
http://www1.sphere.ne.jp/i-lab/ilab/tool/cpw e.htm).
We used two types of CPW’s in our measurements. The block that hosts the
sample has an embeded circuit-board CPW (see Fig. 2.2) that serves to bridge the
sample and the outer circuit. The substrate (thickness h = 25 mils) of the circuit
board has ǫr = 10, and the CPW typically has w = 22 mils and s = 16 mils. Two
CPW’s meet at the edge of the block (visible in Fig. 2.2) and the joint is made by silver
epoxy. The current geometry of the circuit-board CPW allows low reflection, low loss
microwave transmission up to about 20 GHz. To push the microwave measurements
to higher frequencies, one may need to reduce the dimensions of the circuit-board
CPW (perhaps fabricated with lithography). Also one may need to innovate new
ways to turn/join the CPW’s.
123
The other type is the CPW deposited on the sample surface. As discussed in
Chap. 2.2, in the absence of a 2DES, we want the CPW to be impedance matched to
50 Ω. The 2DES with its conductivity Re[σxx ] works as a shunt and absorbs a small
part of the microwave signal. The CPW’s deposited on the samples usually are of
one of three kinds, “M30” (w = 30 µm, s = 45 µm, on samples from wafers “QW65”,
“QW50”, and “H30”), “M60” (w = 60 µm, s = 90 µm, on sample “QW65”), “S80”
(w = 78 µm, s = 117 µm, on samples “QW65 and “QW30”).
Throughout this thesis, we have calculated the real part of the diagonal conductivity of the 2DES from its absorption of the microwave power. Eq. (2.1) for
that conversion is obtained from the following analysis. The CPW can be modeled
as a circuit shown in Fig. 2.1(c). The 2DES appears as a load with admittance
per unit length Y ≈ 2σxx /w. Given this model, from the standard transmission
line theory ((100), Chap. 3, Sec. 1), one finds the propagation constant to be
p
γ ≈ jωL0 (Y + jωC0 ). For a weakly conductive 2DES, Y is much smaller than
p
p
ωC, and Re(γ) ≈ L0 /C0 σxx /w ≈ Z0 σxx /w, where Z0 = L0 /C0 = 50 Ω is the
characteristic impedance the CPW is designed to have. Along the CPW length l,
the transmission parameter, S21, which is the voltage ratio of the input and output
signal, is given by exp(−γl). The power attenuation is,
2l
P
= |S21|2 = exp[−2Re(γ)l] ≈ exp(−Z0 Re(σxx )),
P0
w
(C.1)
where P is the output power and P0 is the input power. From Eq. (C.1), one can
conveniently calculate Re(σxx ) as Eq. (2.1). In typical situations, the condition for
Y to be much smaller than ωC is satisfied for f = ω/2π greater than 20 MHz.
124
C.2
Microstrips
Microstrips, like CPW’s, are also standard planary transmission lines for microwaves.
It was invented in 1952 by Grieg and Engelmann (71). Another basic reference is (6).
A microstrip transmission line consists of a conductive strip of width s and thickness
t and a wider ground plane, separated by a dielectric layer (“substrate”) of thickness
h, as shown in Fig. C.1(b).
The electromagnetic field exists partly in the substrate, and partly in the vacuum
above it. In general, the dielectric constant ǫr of the substrate will be greater than
that of the air, so that the wave is travelling in an inhomogeneous medium. This
behavior is commonly described by stating the effective dielectric constant, ǫeff , of the
microstrip, which is the dielectric constant of an equivalent homogeneous medium
(6). For s ≪ h,
ǫeff =
ǫr + 1
ǫr − 1
+ 0.04
,
2
2
(C.2)
and for s ≫ h,
ǫeff = ǫr .
(C.3)
The impedance of a microstrip slightly varies with the frequency of the signal, but
this is not important for our work. We find it convenient to use an online-available
calculator of microstrip impedance,
http://emclab.mst.edu/pcbtlc2/microstrip.html
To fabricate the microstrips, we used a DuPont-made laminate (Pyralux AP 8565)
with copper layers on both front and back. For impedance match at 50 Ω, s is
optimally 14 mil for the parameters of the laminate. Then the fabrication is done
using the standard process for making PC-boards.
125
Appendix D
Supplementary information about
the samples, cryostats and magnets
D.1
Cooling-down procedures
Before a cool-down starts, the sample sits in dark for a period of time ranging from
half an hour (“SCM1”) to two hours (“C120”). For sample “QW30”, extended darkwaiting time is necessary for achieving a homogeneous state.
A general belief is that the 2DES should be cooled down as slowly and smoothly as
the experiment allows, for the system to homogenize. My usual procedure is to cool
down the samples from 300K to 100K at 2 K/min and then from there at 1.6 K/min.
Another procedure used by other collegues was cooling from 300 K to 77 K in at least
one hour ( <∼ 4 K/min); then from 77 K to 4 K in another hour or more ( <∼ 1
K/min). The cooling from 4 K to 1.6 K used to take hours in “C120” dilfridge due to
the time spent on pumping out the exchange gas from the inner vaccuum chamber.
A newly installed charcoal sorb greatly reduces that time. Further cooling to base
temperature is determined by dilfridge operation and usually takes a few hours.
During the cool-down, we ground all dc contacts of the sample and also attenuate
126
the co-ax ports (effectively grounding the pin). It is essential to check that the grounds
are the same.
All the samples studied in this thesis were cooled in dark and LED illumination
is generally found to make the 2DES state inhomogeneous, likely due to the CPW
metal films present. For those samples for which illumination is essential for inducing
a good state, a technique under development is to shine light to the back of the sample
using an infrared LED. For doing this a hole need be drilled in the block to allow the
infrared light to go through.
In the following sections we note some issues specific for the magnet cells/cryostats
used at the National High Magnetic Field Laboratory (the Magnet lab). As mentioned
in Chap. 2, at the magnet lab we mainly used three dilfridges, the “C120”, “SCM1”,
and “PDF”. There is also a He3 system available and can be useful for sample characterization. This chapter contained new developments during the past four years
and serves as a supplementary material to Appendix E.1 in (17).
D.2
“C120” (14T)
Room C120 at the Magnet Lab houses our bottom-loading dilfridge, and a superconducting magnet, both products of Oxford Instruments. The mixing chamber (MC)
of the dilfridge can reach a base T of 20mK. However, the sample/block (sitting
in vacuum and thermally connected to the MC) can only be cooled to 35mK. The
main heat load comes from the co-ax cables and noise. Over the years, such heat
load has been reduced using, for example, superconducting (Nb) co-ax cables and
connectors to block the heat conduction in various places in the dilfridge. Among
recent developments, there is a redesigned cold-finger that enables interchangeable
perpendicular-field and rotator measurements. It has a silver bar running from the
MC to the sample block and a two-stage filter board for the dc contact wires, for bet-
127
ter cooling. An additional stage of 6 − 10 dB attenuators are now used at a position
below the MC, working at mK temperatures.
The C120 magnet, with a replacement scheduled to arrive in 2010, is designed to
have a maximum field of 14T (16T if pumping the λ plate). The maximum B sweeping
rate that will not heat the sample appreciably is 0.07 T/min. Faster sweeping rates
will heat up the sample/block due to eddy current. Should faster sweeping become
essential, a sufficient waiting time is needed to guarantee thermal equilibrium between
the sample and the MC (this time has to be determined on-site, becasue dilfridge
running conditions can vary).
The temperature from 300 K to 1 K is read by a Silicon diode. Below 1 K, the
temperature is currently read by a ruthenium oxide thermometer on the MC, which is
far from the magnetic field region. We have confirmed that the sample/block (using
both an on-block thermometer and measured features from the 2DES) temperature
can follow the MC temperature down to 35 mK. The temperature can be raised using
an externally controlled MC heater. A temperature stabilization time of 20 min is
found to be generally sufficient. In addition, a powerful on-block heater enables rapid
thermal cycling of the sample without pulling out the dilfridge.
D.3
“SCM1” (18T)
“SCM1” (superconducting magnet 1) uses a top-loading dilfridge, and a superconducting magnet of maximum 18 T (20 T with λ plate), both from Oxford. The
sample/block is immersed in the 3He-4He mixture and can (almost instantly) be in
thermal equilibrium with the MC down to 45 − 50 mK, as we have checked. We have
typically been sweeping at 0.2 − 0.4 T/min without noticing significant heating. For
accurate B-trace measurements the sweep rate is more limited by the possible hysteresis of the magnet than the heating concern. In “SCM1”, the microwave response
128
of the semirigid co-ax cables (in the probe) has a non-negligible dependence on the
helium level (particularly when the level is near 80%) in the dewar (this is in contrast to C120, where the fridge co-ax is in vacuum and the microwave response has
negligible helium level dependence). For accurate spectroscopy measurements, such
a dependence needs to be taken into account and can be corrected by, for example,
recording reference spectra at different helium levels.
The ruthenium oxide thermometer in MC (which is subjected to B) has a significant, almost linear positive magnetoresistance for B >∼ 3 T. A reasonably good
approximation we have used to correct for the magnetoresistance is given as the following:
R(B = 0) = R(B)/(1 + 0.11 × B/18),
(D.1)
in which B is in Tesla and R(B) is the resistance reading at B > 3 T, and R(B = 0)
is the estimated resistance reading at 0T and can be converted to a temperature
using the B=0 R-T calibration. The temperature can be raised with an externally
controlled MC heater. A temperature stabilization time of 10 − 15 minutes is found
to be generally sufficient for T <∼ 400 mK. Above ∼ 400 mK, keeping T stable over
a long term is more difficult. However, for the relatively short time to measure each
spectrum (typically 2-3 minutes), this usually does not pose a significant problem.
D.4
“PDF” [Resistive (33T) and Hybrid (45T)
cells]
The resistive cells have Florida-Bitter resistive magnets with maximum B = 33 T.
The hybrid cell has at present the worlds highest DC B-field magnet, using a11.5
T superconductin magnet outsert and a 33.5 T resistive magnet insert to give a
combined maximum field of 45T.
129
Milli-Kelvin experiments in one of these cells are done in a top loading portable dilution fridge (“PDF”) inserted in the magnet bore. The “PDF” has an unusually long
MC tailored to suit the magnets. The sample temperature is read from a thermometer placed right beside the sample. For a more efficient cooling, it is of paramount
importance to ensure that He3-He4 mixture can flow freely between the sample space
and about one meter above into the MC where the He3-He4 phase separation lies.
Extra care is needed when using the rotator because of the extra clogging. With our
microwave probe, the sample can typically be cooled to 55 − 70 mK. Although the
resistive magnet (and the resistive insert of the hybrid magnet) can be sweeped much
faster than a superconducting magnet, we have normally kept the sweep rate below
1 T/min to prevent eddy-current heating. The magneto-resistance of the ruthenium
oxide thermometers (on the probe) can be corrected for in similar ways as in Eq. D.1.
Because of the small bore size of the magnets, the MC has a smaller diameter
than the “C120” and “SCM1” dilfridge, and the usual sample block (Fig. 2.2 can
barely fit in for a perpendicular field measurement. For a rotator run, the sample has
to be remounted on a differently designed, smaller block.
In the resistive or hybrid cells, the fringe B field can be significant. For safety
reasons, the user is prevented from getting near the probe/cryostat when the resistive
magnet is energized. The network analyzer and all other instruments sit outside of
the 100 Gauss border, and is connected to the top of the probe by a pair of long co-ax
cables.
130
Appendix E
List of publications resulting from
this thesis
1. Han Zhu, P. Jiang, G. Sambandamurthy, L. W. Engel, D. C. Tsui, L. N.
Pfeiffer, K. W. West, Microwave spectroscopy of Wigner crystal induced by
in-plane magnetic field, manuscript in preparation.
2. Han Zhu, Yong Chen, P. Jiang, L. W. Engel, D. C. Tsui, L. N. Pfeiffer, K.
W. West, Spectroscopic evidence for a solid phase of e/3-charged carriers near
Landau level filling factor 1/3, manuscript in preparation.
3. Han Zhu, Yong Chen, G. Sambandamurthy, L. W. Engel, D. C. Tsui, L. N.
Pfeiffer, K. W. West, Pinning mode resonance of a Skyrme crystal near Landau
level filling factor ν=1, arXiv:0907.5218.
4. Han Zhu, G. Sambandamurthy, L. W. Engel, D. C. Tsui, L. N. Pfeiffer, K. W.
West, Pinning mode resonances of 2D electron stripe phases: Effect of in-plane
magnetic field, Physical Review Letters, vol. 102, 136804 (2009).
5. Han Zhu, G. Sambandamurthy, L. W. Engel, D. C. Tsui, L. N. Pfeiffer, K. W.
West, Pinning mode resonances of 2D electron stripe phases at high Landau
131
levels, Physica B, vol. 404, 367 (2009).
6. G. Sambandamurthy, Han Zhu, and Y. P. Chen, L. W. Engel, D. C. Tsui, L.
N. Pfeiffer, and K. W. West, Pinning modes of the stripe phases of 2d electron
systems in higher Landau levels, International Journal of Modern Physics B,
vol. 23, 2628 (2009).
7. G. Sambandamurthy, R. M. Lewis, Han Zhu, and Y. P. Chen, L. W. Engel,
D. C. Tsui, L. N. Pfeiffer, and K. W. West, Observation of Pinning Mode of
Stripe Phases of 2D Systems in High Landau Levels, Physical Review Letters,
vol. 100, p. 256801 (2008).
8. Han Zhu, K. Lai, D. C. Tsui, S. P. Bayrakci, N. P. Ong, M. Manfrac, L.
Pfeiffer, K. West, Density and well width dependences of the effective mass of
two-dimensional holes in (100) GaAs quantum wells measured using cyclotron
resonance at microwave frequencies, Solid State Communications, vol. 141, 510
(2007).
132
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