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Angle -dependent high magnetic field microwave spectroscopy of low dimensional conductors and superconductors

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ANGLE-DEPENDENT HIGH MAGNETIC FIELD MICROWAVE
SPECTROSCOPY OF LOW DIMENSIONAL CONDUCTORS AND
SUPERCONDUCTORS
By
SUSUMU TAKAHASHI
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2005
UMI Number: 3204496
UMI Microform 3204496
Copyright 2006 by ProQuest Information and Learning Company.
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
ProQuest Information and Learning Company
300 North Zeeb Road
P.O. Box 1346
Ann Arbor, MI 48106-1346
Copyright 2005
by
Susumu Takahashi
To my parents, Koh and Teruko Takahashi,
and my family, Ryoko, Kai and Riku
ACKNOWLEDGMENTS
Since the fall of 2001, I have spent a great amount of time working on my
research with many people at the University of Florida (UF) and other places.
Without their assistance, encouragement and guidance, I could not have completed
this dissertation.
First, I would like to thank my advisor, Professor Stephen O. Hill. I have
received numerous benefits by his patient guidance and continuous support over
three and a half years. Steve’s enthusiastic discussion has always encouraged me
to tackle difficult but creative research projects. I was also supported by Steve
to perform many experiments at the National High Magnetic Field Laboratory
(NHMFL), Tallahassee, FL, and to attend conferences in various places. These
experiences are my priceless treasure.
I also would like to express my thanks to other faculty at the University of
Florida. I thank my supervisory committee, Prof. David Tanner, Prof. Peter J.
Hirschfeld, Prof. Mark W. Meisel, and Prof. Daniel R. Talham, for a number of
useful discussion and valuable comments. I also thank Prof. Amlan Biswas for
providing a PCCO sample. I must also thank the technical staff at the University
of Florida. In particular, I thank the machine shop for making a rotating cavity
and giving great suggestions for designing the apparatus.
Many experiments in this thesis were carried out at the NHMFL. I would like
to thank the scientists and staff for supporting our experiments. In particular, I
would like to thank Prof. James Brooks, Dr. Louis Claude Brunel and Dr. Hans
Van Tol for kindly lending equipments for our experiments at the NHMFL.
iv
I also had the great fortune to have discussions with and receive suggestions
from many conference attendees and visitors. I am especially grateful to Dr.
Phillipe Goy of ABmm, Prof. Toshihito Osada of the University of Tokyo, Prof.
Woun Kang of Ewha Womans University, Prof. Victor M. Yakovenko of the
University of Maryland and Prof. Andrei G. Lebed of the University of Arizona.
I also thank other members of the Hill group and people at UF, Dr. Rachel
Edwards, Dr. John Lee, Dr. Konstantin Petukhov, Jon Lawrence, Norm Anderson,
Tony Wilson, Amalia Betancur-Rodiguez, Saiti Datta, Sung-Su Kim, Dan Benjamin, Emmitt Thompson, Costel Rotundu, Tara Dhakal, Naveen Margankunte,
Hidenori Tashiro, Yoshihiro Irokawa and Stephen Flocks, for kind assistance in
my experiments, and help with my writing and friendship. In particular, I would
like to acknowledge Dr. Alexey Kovalev for useful discussion and assistance with
experiments in the initial stage of my research.
Finally, I would like to thank both my and my wife Ryoko’s families for their
constant support, encouragement and love. I would also like to thank Ryoko
and my sons, Kai and Riku, for feeding me, giving me happiness and loving me.
Without them, I would not have even been able to survive Gainesville wildlife.
v
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
CHAPTER
1
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
1.2
1.3
.
.
.
.
.
.
.
.
1
2
2
4
4
6
11
12
PERIODIC ORBIT RESONANCE . . . . . . . . . . . . . . . . . . . . .
14
2.1
2.2
2.3
Experimental Techniques to Study Fermi Surfaces . . . . . . . . .
Cyclotron Resonance . . . . . . . . . . . . . . . . . . . . . . . . .
Cyclotron Resonance Involving an Open Fermi Surface: Periodic
Orbit Resonance . . . . . . . . . . . . . . . . . . . . . . . . . .
Periodic Orbit Resonance for a Quasi-two-dimensional Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
POR and Angle-dependent Magnetoresistance Oscillations . . . .
Quantum Effects in the Conductivity . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
15
EXPERIMENTAL SETUP . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.1
3.2
3.3
3.4
3.5
42
45
53
65
1.4
1.5
1.6
2
2.4
2.5
2.6
2.7
3
Overview of Low Dimensional Systems . . . . . . . . . . . . .
Fermi Surfaces of Low Dimensional Conductors . . . . . . . .
Quasi-one-dimensional and Quasi-two-dimensional Materials .
1.3.1 The Quasi-one-dimensional Conductor (TMTSF)2 ClO4
1.3.2 The Quasi-two-dimensional Conductor κ-(ET)2 X . . . .
Instability in Low Dimensional Conductors . . . . . . . . . . .
Superconductivity in Low Dimensional Materials . . . . . . . .
Impurity Effect on the Superconductivity . . . . . . . . . . . .
Overview of Microwave Magneto-optics .
Experimental Setup . . . . . . . . . . . .
Rotating Cavity . . . . . . . . . . . . . .
Model of the Resonant Cavity . . . . . .
Positioning Low Dimensional Conductors
the Cylindrical Cavity . . . . . . . . .
vi
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
and Superconductors
. . . . . . . . . . . .
.
.
.
.
.
.
.
.
1
. .
. .
. .
. .
in
. .
22
27
30
37
41
69
3.6
3.7
3.8
4
PERIODIC ORBIT RESONANCES IN QUASI-ONE-DIMENSIONAL
CONDUCTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
4.2
4.3
4.4
5
.
71
72
72
77
77
79
80
81
82
82
. 89
. 93
. 102
Overview of the Superconductivity in (TMTSF)2 ClO4 . . . . . . .
DC (ω ≈ 0) Transport Measurements for Different Cooling Rates .
Study of the Periodic Orbit Resonance at Different Cooling Rates
Analysis of the Scattering Rate Γ . . . . . . . . . . . . . . . . . .
Relation Between Tc and the Pair Breaking Strength α . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
103
106
109
110
116
117
PERIODIC-ORBIT RESONANCE IN QUASI-TWO-DIMENSIONAL
CONDUCTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.1
6.2
6.3
6.4
6.5
6.6
7
The Quasi-one-dimensional Conductor, (TMTSF)2 ClO4 . . . . .
Semiclassical Description of the Periodic Orbit Resonance and
the Lebed Effect . . . . . . . . . . . . . . . . . . . . . . . . .
Observation of the Periodic Orbit Resonances in (TMTSF)2 ClO4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
70
NON-MAGNETIC IMPURITY EFFECTS ON THE SUPERCONDUCTIVITY IN (TMTSF)2 ClO4 . . . . . . . . . . . . . . . . . . . . . . . . 103
5.1
5.2
5.3
5.4
5.5
5.6
6
3.5.1 In-plane Measurements. . . . . . . . . . . . . . . . . . . . .
3.5.2 Interlayer Measurements. . . . . . . . . . . . . . . . . . . .
3.5.3 Configuration for Interlayer Measurements Using the Magnetic Component of the Microwaves . . . . . . . . . . . .
Microwave Response of Low dimensional Conductors and Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1 Skin Depth Regime . . . . . . . . . . . . . . . . . . . . . .
3.6.2 Metallic Depolarization Regime . . . . . . . . . . . . . . .
Measurement of the Change of the Complex Impedance Ze . . . .
3.7.1 Frequency-lock Method . . . . . . . . . . . . . . . . . . . .
3.7.2 Phase-lock Method . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Quasi-two-dimensional Conductors κ-(ET)2 X . . . . . . . .
Periodic-orbit Resonance in κ-(ET)2 X . . . . . . . . . . . . . . .
Experiments for κ-(ET)2 Cu(NCS)2 . . . . . . . . . . . . . . . .
Experiments for κ-(ET)2 I3 . . . . . . . . . . . . . . . . . . . . .
Angle-resolved Mapping of Fermi Velocity: A Proposed Experiment for Nodal Q2D Superconductors . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
118
121
125
129
. 132
. 139
SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
vii
APPENDIX
SEMICLASSICAL CALCULATION OF THE ELECTRICAL
CONDUCTIVITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
A.1
A.2
A.3
A Simple Quasi-one-dimensional Model . . . . . . . . . . . . . . . 144
A General Quasi-one-dimensional Model . . . . . . . . . . . . . . 147
A Simple Quasi-two-dimensional Model . . . . . . . . . . . . . . . 149
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
viii
Table
LIST OF TABLES
page
3–1 Available magnet systems at UF and the NHMFL. . . . . . . . . . . .
48
3–2 Probes used for the cavity perturbation technique. . . . . . . . . . . .
51
3–3 Resonance parameters for several different cavity modes. . . . . . . . .
61
4–1 Lattice parameters and the AMRO notations for the n-th nearest neighbors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6–1 Unit cell parameters for κ-(ET)2 X. . . . . . . . . . . . . . . . . . . . . 119
ix
Figure
LIST OF FIGURES
page
1–1 Illustration of the FS by varying the bandwidths tb and tc . . . . . . .
3
1–2 Illustration of the crystal structure of (TMTSF)2 ClO4 and the TMTSF
molecule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1–3 Illustration of the crystal structure of κ-(ET)2 Cu(NCS)2 and the ET
molecule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1–4 Illustration of the Peierls instability in a 1D system. . . . . . . . . . .
8
1–5 Illustration of the confinement effect on the trajectories of electrons
in a magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1–6 T-H phase diagram for (TMTSF)2 ClO4 . . . . . . . . . . . . . . . . . .
10
2–1 The Fermi surface for the two-dimensional electron system. . . . . . .
16
2–2 Real part of σxx as a function of frequency for various values of ωτ . .
19
2–3 The Fermi surface for the two-dimensional electron system with an
arbitrary direction of the magnetic field. . . . . . . . . . . . . . . .
21
2–4 Oscillatory group velocity vz for the Q1D POR. . . . . . . . . . . . .
23
2–5 POR in α-(ET)2 KHg(SCN)4 . . . . . . . . . . . . . . . . . . . . . . .
26
2–6 Oscillatory group velocity vz for Q2D POR. . . . . . . . . . . . . . .
28
2–7 AMRO for Q2D conductors. . . . . . . . . . . . . . . . . . . . . . . .
31
2–8 AMRO for Q1D conductors. . . . . . . . . . . . . . . . . . . . . . . .
33
2–9 The Lebed effect in both the dc and ac conductivity.
. . . . . . . . .
34
2–10 Yamaji oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2–11 Numerical calculation of the conductivity for a Q2D FS. . . . . . . .
36
2–12 Landau tube with a magnetic field. . . . . . . . . . . . . . . . . . . .
38
2–13 DOS of a Q2D conductor in a magnetic field in units of ~ωc . . . . . .
39
3–1 Frequency range for the Q2D POR in magnetic fields accessible at
the NHMFL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
x
3–2 Overview of the experimental setup. . . . . . . . . . . . . . . . . . . .
46
3–3 A schematic diagram of the 3 He probe. . . . . . . . . . . . . . . . . .
50
3–4 Vertical temperature distribution in the cryostat for the QD PPMS
7 T and Oxford Instruments 17 T magnets. . . . . . . . . . . . . .
52
3–5 A schematic diagram of the rotating cavity system. . . . . . . . . . .
55
3–6 Photographs of the rotating cavity system. . . . . . . . . . . . . . . .
57
3–7 Schematic diagrams showing various different sample mounting configurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3–8 Angle-dependence of the cavity resonant properties. . . . . . . . . . .
63
3–9 A simple description of the resonant cavity. . . . . . . . . . . . . . . .
67
3–10 Schematic diagram illustrating the various possibilities for exciting
in-plane and interlayer currents in a Q2D plate-like sample. . . . .
70
3–11 Positioning of the sample in the TE011 mode. . . . . . . . . . . . . .
73
3–12 Typical changes in the amplitude and phase of the microwaves transmitted through the cavity. . . . . . . . . . . . . . . . . . . . . . . .
79
4–1 Electronic properties of the (TMTSF)2 X. . . . . . . . . . . . . . . . .
83
4–2 Illustration of crystal axes for (TMTSF)2 ClO4 . . . . . . . . . . . . . .
86
4–3 The dc AMRO experiment in (TMTSF)2 ClO4 . . . . . . . . . . . . . .
88
4–4 The oblique real-space crystal lattice. . . . . . . . . . . . . . . . . . .
90
4–5 Resonance conditions and the POR by sweeping the angle and magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
4–6 Overview of the orientations in the experiments. . . . . . . . . . . . .
96
4–7 Microwave absorption as a function of the magnetic field. . . . . . . .
97
4–8 Angle dependence of the quantity ν/Bres . . . . . . . . . . . . . . . . .
99
4–9 Angle sweep and field sweep measurements for (TMTSF)2 ClO4 . . . . 100
4–10 Summary of the p/q = 0, 1 and 1 POR data for sample C.
. . . . . . 101
5–1 Illustration of the ClO4 anion and the crystal structure of (TMTSF)2 ClO4
below TAO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5–2 dc transport measurements at different cooling rates. . . . . . . . . . 107
xi
5–3 Summary of the dc transport measurements. . . . . . . . . . . . . . . 108
5–4 Microwave absorption as a function of magnetic field at different rates. 110
5–5 Least-square fit to microwave absorption. . . . . . . . . . . . . . . . . 112
5–6 Cooling rate and temperature rate dependence of the scattering rate.
113
5–7 Comparison between the resistance Rzz and the scattering rate Γ as
a function of the cooling rate. . . . . . . . . . . . . . . . . . . . . . 115
5–8 Tc vs. the scattering rate Γ. . . . . . . . . . . . . . . . . . . . . . . . 116
6–1 Phase diagram for the organic conductors κ-(ET)2 X. . . . . . . . . . 119
6–2 Fermi surface (FS) and trajectories of an electron under a magnetic
field for κ-(ET)2 Cu(NCS)2 . . . . . . . . . . . . . . . . . . . . . . . 120
6–3 Warping on a Q2D FS. . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6–4 Numerical calculation of the ac conductivity for a Q2D FS. . . . . . . 124
6–5 Overview of the orientations in the experiments on κ-(ET)2 X. . . . . 125
6–6 Experimental data for κ-(ET)2 Cu(NCS)2 . . . . . . . . . . . . . . . . . 128
6–7 Angle-dependence of the POR and the SdH oscillations for two kinds
of rotations in κ-(ET)2 Cu(NCS)2 . . . . . . . . . . . . . . . . . . . . 129
6–8 Experimental data for κ-(ET)2 I3 . . . . . . . . . . . . . . . . . . . . . 130
6–9 Angle-dependence of the POR in κ-(ET)2 I3 . . . . . . . . . . . . . . . 131
6–10 Self-crossing orbits and open trajectories. . . . . . . . . . . . . . . . . 134
6–11 Angle-resolved mapping of vF . . . . . . . . . . . . . . . . . . . . . . . 137
A–1 Representation of rotation of a magnetic field relative to a Q1D FS. . 145
A–2 Representation of rotation of the magnetic field for a Q2D FS. . . . . 150
xii
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ANGLE-DEPENDENT HIGH MAGNETIC FIELD MICROWAVE
SPECTROSCOPY OF LOW DIMENSIONAL CONDUCTORS AND
SUPERCONDUCTORS
By
Susumu Takahashi
December 2005
Chair: Stephen O. Hill
Major Department: Physics
This dissertation presents studies of angle-dependent high-field microwave
spectroscopy of low dimensional conductors and superconductors. Over the past
20 years, low dimensional conductors and superconductors have been investigated
extensively because of their unusual superconducting, electronic and magnetic
ground states. In order to understand these phenomena, it is important to study
the topology of the Fermi surface (FS). We employ a novel type of cyclotron
resonance to study the FS, the so-called periodic orbit resonance (POR).
In Chapter 2, we explain the details of the POR effect using a semiclassical
description. An important aspect of this POR effect is that it is applicable not only
to a quasi-two-dimensional (Q2D) FS, but also to a quasi-one-dimensional (Q1D)
FS.
In Chapter 3, our experimental techniques are presented. We outline a rotating
cylindrical cavity, which enables angle-dependent cavity perturbation measurements
in ultra-high-field magnets, and two-axis rotation capabilities in standard high-field
superconducting split-pair magnets.
xiii
In Chapters 4 and 5, the results of studies of the Q1D conductor (TMTSF)2 ClO4
are shown. Using the POR, we determined the Fermi velocity vF and revealed new
information concerning the nature of the so-called Lebed effect in Chapter 4. In
Chapter 5, we studied the non-magnetic impurity effect and its influence on the
possible spin-triplet superconductivity in (TMTSF)2 ClO4 .
In Chapter 6, measurements of the POR are performed in the Q2D conductors
κ-(ET)2 X [X=Cu(NCS)2 and I3 ]. In X=I3 , POR involving the magnetic breakdown
effect was observed for the first time.
xiv
CHAPTER 1
INTRODUCTION
1.1
Overview of Low Dimensional Systems
Electronic band structures give us an idea of the conducting properties of
materials. For the case of an insulator, the allowed energy bands are either filled
or empty and the band gap between the filled and empty bands is large enough
to prevent thermal excitation of electrons to the empty band: thus no current
flows in response to an external field. For the case of a metal, one or more bands
are partially filled. In this case, electrons can move easily in response to an external field [1]. For the case of certain kinds of metals, the band structure is highly
anisotropic; i.e., electrons in such metals can move along one direction much
more easily than the other directions. These materials are called low dimensional
systems. In particular, these systems are often classified as either quasi-twodimensional (Q2D) or quasi-one-dimensional (Q1D). Recently, the study of these
systems has been attractive because many interesting phenomena have been discovered in them, including unconventional superconductivity, metal-insulator transitions, antiferromagnetism, spin-density-waves (SDW) and charge-density-waves
(CDW). In order to understand these low dimensional systems, it is important to
study their detailed electronic properties, as well as their superconducting properties. Although the full band structure gives overall information on the electronic
property, it is often enough to investigate the topology of the Fermi surface (FS) of
the systems because states near the FS dominate the low temperature conducting
properties.
1
2
1.2
Fermi Surfaces of Low Dimensional Conductors
For the sake of getting a picture of the FS of a low dimensional system, we
start by considering an isotropic band structure, i.e., the ratio of transfer energies
in a tight binding model tx : ty : tz = 1 : 1 : 1. In general, the shape of the FS
can be complicated, even when the conducting properties are three-dimensional
and free electron-like. However, we here consider a perfectly spherical FS as the
simplest case, as shown in the upper left picture of Fig. 1–1. Starting with this
FS, we change the anisotropy. When the bandwidth along the z-direction becomes
smaller, e.g., tx : ty : tz = 1 : 1 : 1/2, the FS sphere is stretched along the zdirection. The smaller the z-axis band width, the more the z-direction of the FS is
stretched, as shown in the upper middle figure. Eventually the z-component of the
FS connects with the FS in the next Brillouin zone, and then the FS becomes open
along the z-direction, namely a cylinder-like shape with a small corrugation of the
cylinder, as shown in the upper right figure. In this case, the transfer energy ratio
will be highly anisotropic, e.g., tx : ty : tz = 1 : 1 : 1/100. Such an anisotropic FS
is said to be Q2D. Examples of materials with Q2D FSs are κ-(ET)2 Cu(NCS)2 and
κ-(ET)2 I3 (See Chap. 6.). Next, we change the band width along the y-direction.
While the band width is reduced, the y-direction of the FS tube is stretched step
by step, as seen in the lower left figure. Eventually, the FS becomes a couple of
plane-like sheets with small corrugations, as shown in the lower right panel of
Fig. 1–1. These corrugations are related to the small transfer energies along the y
and z-directions, e.g., tx : ty : tz = 1 : 1/100 : 1/100, representing a Q1D FS. This is
the case for (TMTSF)2 ClO4 described in Chap. 4.
1.3
Quasi-one-dimensional and Quasi-two-dimensional Materials
Here we introduce examples of low dimensional materials which have the FSs
described in the previous section.
3
3D FS
Q2D FS
z
y
Q1D FS
x
Figure 1–1. Illustration of the FS by varying the bandwidths tb and tc . Starting
from a spherical FS (ta :tb :tc = 1:1:1), we change the FS anisotropy. As
the band width along the z-direction becomes smaller, eventually the
z-component of the FS connects with the FS in the next Brillouin zone,
and then the FS becomes open along the z-direction, i.e., a cylinderlike shape with a small corrugation of the cylinder, as shown in the
upper right figure. Such an anisotropic FS is a so-called Q2D FS.
Next, we change the band width along the y-direction. As the band
width is reduced, eventually the FS becomes a pair of plane-like sheets
with small corrugations, as shown in the lower right figure. These corrugations are related to the small transfer energies along the y and
z-directions, e.g., tx : ty : tz = 1 : 1/100 : 1/100. This represents a Q1D
FS.
4
1.3.1
The Quasi-one-dimensional Conductor (TMTSF)2 ClO4
The organic metal (TMTSF)2 ClO4 belongs to the family of quasi-onedimensional (Q1D) Bechgaard salts [2], having the common formula (TMTSF)2 X.
TMTSF is an abbreviation for tetramethyl-tetraselenafulvalene, and the anion X
is AsF6 , ClO4 , PF6 , ReO4 , etc. The (TMTSF)2 X compounds are the so-called 2:1
charge transfer salts which transfer one electron from two TMTSF molecules to one
X anion. The crystal structures of the (TMTSF)2 X series are similar. Fig. 1–2(a)
shows the crystal structure of (TMTSF)2 ClO4 . As shown to the left of Fig. 1–2(a),
the planar TMTSF molecules are stacked along the a-direction. The conducting
properties come from the overlap of π-orbitals on the TMTSF molecules, as shown
in Fig. 1–2(b). Since the π-orbitals are oriented perpendicular to the TMTSF
molecule, the molecules couple strongly along the a-direction via the overlap of
partially occupied π-orbitals so that this direction becomes the most conducting
direction. On the other hand, the overlap along the b and c-directions is weak. In
particular, the overlap is extremely weak along the c-direction because the coupling
between TMTSF molecules is hindered by the insulating ClO4 anion sheets, as
shown in the right panel in Fig. 1–2(a). It turns out that the conducting properties are extremely one-dimensional. In the case of (TMTSF)2 ClO4 , the transfer
integrals are approximately ta : tb : tc = 250 meV : 20 meV : 1 meV [2]. This
common crystal structure is the origin of the Q1D conducting properties for all of
the (TMTSF)2 X compounds. Thus, (TMTSF)2 X, including X=ClO4 , PF6 etc., are
good examples for studying Q1D FSs.
1.3.2
The Quasi-two-dimensional Conductor κ-(ET)2 X
The organic superconductors κ-(ET)2 X (X = Cu(NCS)2 , I3 etc.) belong to the
ET family of charge-transfer salts (CTS), where ET represents bisethylenedithiotetrathiafulvalene [(CH2 )2 ]2 C6 S8 (alternatively denoted BEDT-TTF). In contrast
to the planar TMTSF molecule (shown in Fig. 1–2), the ET molecule shown in
5
a) Crystal structure of (TMTSF)2ClO4
a
c
c
ac-plane
ClO4 anion
b
bc-plane
TMTSF
Molecule
b) TMTSF Molecule
π-orbit
Figure 1–2. Illustration of the crystal structure of (TMTSF)2 ClO4 and the
TMTSF molecule. (a) The crystal structure of (TMTSF)2 ClO4 viewed
along the b-axis and the a-axis. The a-direction is the most conducting direction due to the strongest coupling between the π-orbitals on
neighboring TMTSF molecules. (b) A schematic of the planar TMTSF
molecule. The π-orbitals extend from the Se atoms perpendicular to
the plane of the molecule.
6
(b)
(a)
S
C
Cu(NCS)2 anion
c
ET molecule
b ET molecule
Figure 1–3. Illustration of the crystal structure of κ-(ET)2 Cu(NCS)2 and the ET
molecule. (a) Structure of the ET molecule. In contrast to the TMTSF
molecule, the ET molecule is not planar. (b) The crystal structure of
κ-(ET)2 Cu(NCS)2 viewed along the a∗ -axis.
Fig. 1–3(a) is not exactly planar. This non-planar nature of the ET molecule
complicates the morphology of the ET crystal structures compared with other 2:1
(TMTSF)2 X salts. For instance, there are four kinds of crystal morphologies in
the case of (ET)2 I3 , i.e., the α-, β-, θ- and κ-phases, where α, β, θ and κ denote
different crystal structures. Fig. 1–3(b) shows the crystal structure viewed along
the a∗ -axis for the κ-phase of (ET)2 Cu(NCS)2 . For the case of κ-(ET)2 Cu(NCS)2 ,
the ET molecules dimerize in a face to face arrangement. The dimer pairs then
pack orthogonally so that the overlap of the π-orbitals of the ET molecules is fairly
isotropic in the bc-plane. On the other hand, the overlap along the a-direction is
much smaller because it is prevented by the insulating Cu(NCS)2 anion layers.
It turns out that the conducting properties in κ-(ET)2 Cu(NCS)2 are highly twodimensional, or quasi-two-dimensional (Q2D), i.e., σk /σ⊥ ∼ 1000 or tk /t⊥ ∼
30 [3].
1.4
Instability in Low Dimensional Conductors
According to recent studies of low dimensional systems, the topology of the
high temperature FS contributes to the nature of the ground states significantly.
7
Examples include the Peierls instabilities, charge-density-wave (CDW) states, spinPeierls instabilities, antiferromagnetic (AF) states and spin-density-wave (SDW)
states. In particular, nesting of the FS is important. For example, in the case of
1D systems, the FS consists of a couple of flat sheets, as shown in Fig. 1–4(a). The
instability in a 1D system is related to this shape of the FS, since any points on
one FS can be mapped into the other FS by a single wave vector Qx = 2kF , the
so-called nesting vector. The correlation of electrons on the FS becomes divergently
strong at Qx = 2kF . This is the so-called Kohn anomaly. As a result, the electronphonon interaction becomes divergently strong at the nesting vector Qx = 2kF
with decreasing temperature. Therefore, the phonon mode at Qx = 2kF becomes
soft, as shown in Fig. 1–4(b). This soft phonon frequency goes to zero at low
temperature, resulting in a static lattice distortion with Qx = 2kF , the so-called
Peierls distortion. This Peierls transition can be described by a mean field theory,
and the transition temperature Tp is given by a BCS-type gap equation [2]. This
distortion also affects the electronic state. An example for a half-filled 1D system
is shown in Fig. 1–4(c). At high temperatures (T > Tp ), there is one electron per
site, and the electron can readily move to other sites, so that the system is metallic.
However, at low temperatures (T < Tp ), the system undergoes the Peierls distortion
with Qx = 2kF = π/2a becoming dimerized. This changes the half-filling to full
filling. As a result, electrons cannot move to other sites, and the system becomes
an insulator. This metal-insulator transition is called the Peierls transition, which
can also happen at any band filling. This transition leads to a modulation of the
charge density, known as a CDW. Furthermore, the nesting also affects other
interactions, e.g., the electron-electron interaction etc. As a result, a similar
transition can happen involving the spin degrees of freedom. This type of transition
is known as the SDW, or AF transition. These transitions are often observed in
Q1D and Q2D organic conductors, e.g., the (TMTSF)2 X and ET-salts [2].
(a)
kz
(b)
1D FS
ky
kx
Dispersion relation of Phonon
8
Q = 2kF
(c)
High T
itinerant
metallic state
no hopping
insulating state
a
Low T (< Tp)
2a
n = half-filling
Figure 1–4. Illustration of the Peierls instability in a 1D system. (a) The FS is
represented by a pair of open FS sheets. Any points on one FS can
be mapped into the other FS by a single wavevector Qx = 2kF , the
so-called nesting vector. (b) Kohn anomalies in 1, 2 and 3 dimensional
system. The phonon dispersion is plotted as a function of wavenumber.
The anomaly is seen at Q = 2kF . In the case of 1D, the phonon dispersion becomes zero at Q = 2kF . This causes a static lattice distortion,
the so-called Peierls distortion (Reprinted figure with permission from
Kagoshima [4]. Copyright 1981 by the Institute of Pure and Applied
Physics (Japan).). (c) Because of the nesting, the electron-phonon
interaction becomes divergently strong, so that it causes a lattice distortion, the so-called Peierls distortion. For the case of half-filling,
the system dimerizes below the transition temperature Tp due to the
Peierls distortion. This affects the electronic structure. The distortion results in a fully occupied band, which does not allow electrons
to move to other sites. As a result, the system becomes an insulator.
Thus, the Peierls transition leads to a metal-insulator transition.
9
B
W = 4tbb/ħωc
∝ 1/Β
W
b
a
B = Low
W = Large
B = High
W = Small
Figure 1–5. Illustration of the confinement effect on the trajectories of electrons
in a magnetic field. Upon increasing the magnetic field, the transverse
width of the trajectories of electrons decreases. This results in an increase in the one-dimensional properties, i.e., the nesting on the FS
becomes stronger.
In some cases, the nesting property is strongly increased via a magnetic field.
In fact, a reentrance of the SDW phase under a high magnetic field is observed in
the Q1D organic conductor (TMTSF)2 X. This phase is the so-called field-induced
SDW (FISDW) state. For (TMTSF)2 X, the transfer energy along the b-direction,
tb , is significant, tb /ta ∼ 0.1, so that the motion of electrons is somewhat twodimensional. Application of a magnetic field leads to confinement of the 2D
motion. Fig. 1–5 shows trajectories of electrons in a real space. By applying a
stronger magnetic field, the width of the trajectories becomes increasingly smaller.
Eventually, the field confines the width of the trajectory to within one unit cell in
the b-direction. Then the motion of the electrons is effectively one-dimensional.
In the case of (TMTSF)2 X, this dimensional crossover effect is related to the
confinement in the ab-plane, so that the FISDW phase has a minimum critical field
when the field is perpendicular to the ab-plane. In the case of (TMTSF)2 ClO4 [5],
the critical field of the FISDW phase TSDW is 7 T at T = 2 K, as shown in Fig. 1–
6. More details of the FISDW state are explained by the standard model [6, 7, 8].
10
FISDW
Figure 1–6. T-H phase diagram for (TMTSF)2 ClO4 . The FISDW phase is observed in high magnetic field, e.g., the critical field BSDW ∼ 7 tesla at
T = 2 K. The FISDW effect is caused by the field dependent nesting
(confinement) effect on the ab-plane. The another phase transition was
also proposed at higher magnetic fields using normal and Hall resistance and magnetization measurements, e.g., the critical field of the
second phase ∼ 27 tesla at T = 2 K (Reprinted figure with permission
from McKernan et al. [5]. Copyright 1995 by the American Physical
Society.).
11
1.5
Superconductivity in Low Dimensional Materials
As mentioned, exotic superconductivity has been observed in many low dimensional materials, e.g., the HTSC, κ-(ET)2 X, Sr2 RuO4 and (TMTSF)2 X. These
exotica may be represented by anisotropic superconducting energy gaps, spin-triplet
Cooper pairs, non electron-phonon interaction mediated Cooper pairing mechanisms, coexistence of superconductivity and magnetism. For example, (TMTSF)2 X
has recently been considered to be a spin-triplet superconductor since it shows no
77
Se Knight shift through the superconducting transition temperature [9], exceeding
the Pauli paramagnetic limit in the upper critical field [10], and non-magnetic
impurity effects [11]. κ-(ET)2 Cu(NCS)2 is considered to be a d-wave superconductor because it shows a four-fold symmetry in the magneto-thermal conductivity
tensor [12].
In the case of low dimensional organic superconductors, which are highly
relevant materials to this thesis, the topology of the FS (i.e., nesting) may be
important for generating such anisotropic superconducting energy gaps. Recently,
many theoretical studies have highlighted the importance of such nesting properties
for theories of unconventional superconductors, e.g., low dimensional organic
superconductors [13, 14].
Similar to the anisotropic conducting properties of low dimensional superconductors, their superconducting properties are often anisotropic. For the case
of the quasi-two-dimensional (or layered) superconductors, the superconducting
anisotropy parameter γ is represented by the ratio between the interlayer and
in-plane penetration depth or coherence length, i.e., γ = λ⊥ /λk or γ 0 = ξk /ξ⊥ respectively. In the extreme cases, γ can be several hundred: e.g., Bi2 Sr2 CaCu2 O8+y ,
γ ∼ 50 − 200 [15]; and κ-(ET)2 Cu(NCS)2 , γ ∼ 100 − 200 [16]. This high degree
of anisotropy is related to a weak Josephson coupling between the superconducting
layers. The characteristic interlayer Josephson plasma frequency is often in the
12
range of microwave frequencies. This can be seen as a Josephson plasma resonance
(JPR). We have studied the JPR phenomena for the Q2D organic superconductor κ-(ET)2 Cu(NCS)2 . The results were reported in Kovalev et al. [17] and
Benjamin et al. [18].
1.6
Impurity Effect on the Superconductivity
For unconventional superconductors, the non-magnetic impurity effect
is one of the main evidences for unconventional superconductivity. We have
therefore performed an experiment to study the non-magnetic impurity effect in
(TMTSF)2 ClO4 . This result is described in Chap. 5.
The original work on the impurity effect was developed for magnetic impurities in conventional superconductors in the 1950’s and 1960’s. In 1959, Anderson
extended the Bardeen-Cooper-Schreiffer (BCS) theory [19] for very dirty superconductors, where elastic scattering from non-magnetic impurities is large compared
with the superconducting energy gap [20]. In this theory, Anderson showed that
the Cooper pair with the momentum (k ↑, −k ↓) is immune to non-magnetic scattering. Independently, the impurity effect was discussed by Abrikosov and Gor’kov
in the 1960’s (AG theory) [21]. The AG theory predicts a strong suppression and
disappearance of the transition temperature by magnetic impurities, and also predicts changes of various thermodynamic properties, e.g., the existence of a gapless
regime in the superconducting state.
Impurity scattering destroys the coupling between Cooper pairs. This is
more generally explained in terms of the effect of broken time-reversal symmetry.
For instance, in the case of BCS-type pairs (isotropic gaps with the momentum
(k ↑, −k ↓), the scattering by non-magnetic impurities, which acts on the electric charges of the Cooper pairs identically, does not destroy the Cooper pairs
because the scattering has little effect on the antisymmetry of the pair (Anderson’s
theorem). On the other hand, scattering by magnetic impurities can act on the
13
spin and ultimately flip one of the spins. As a result, the scattering changes the
pair to a parallel spin alignment (the time-reversal symmetry is broken), and the
Cooper pair is destroyed. Furthermore, extensive studies show that the broken
time-reversal symmetry is not only caused by magnetic impurities, but also in thin
films in parallel magnetic fields, by exchange fields, and proximity effects, etc. [22].
According to the AG theory, the suppression of the superconducting critical
temperature Tc is simply related to the scattering time τK of the destructive
perturbation. The function to express the relation is the so-called universal
function given by the following equation.
ln(
Tc
1
1
α
) = ψ( ) − ψ( +
),
Tc0
2
2 2πkB Tc
(1–1)
where Tc0 = Tc (0) is the superconducting critical temperature without impurities,
ψ(z) = Γ0 (z)/Γ(z) is the digamma function, and α is the pair-breaking strength,
2α =
~
,
τK
(1–2)
where τK is the scattering time for depairing.
Impurity effects were observed not only in the BCS-type superconductors, but
also in many exotic superconductors. In particular, in the case of the unconventional superconductors, effects were observed even due to non-magnetic impurities.
That is why this non-magnetic impurity effect is considered to be an evidence of
unconventional superconductivity. For example, the spin-triplet superconductor
Sr2 RuO4 shows non-magnetic impurity effects [23].
In this chapter, we explained the motivation of the study of low dimensional
materials briefly. In the next chapter, we introduce a periodic orbit resonance effect
to probe the FS. Chapter 3 shows our experimental setup. Chapter 4 and 5 explain
the study of the FS and superconductivity for the Q1D material (TMTSF)2 ClO4 .
Chapter 6 show the study of the FS for the Q2D materials κ-(ET)2 X.
CHAPTER 2
PERIODIC ORBIT RESONANCE
2.1
Experimental Techniques to Study Fermi Surfaces
Since the Fermi surface (FS) can explain many aspects of conductors, superconductors and itinerant magnetic systems, many techniques have been
developed to study FS topology. Examples include: angle-resolved photoemission spectroscopy (ARPES) [24], the de-Haas-van Alphen effect (dHvA) [25], the
Shubnikov-de Haas effect (SdH) [25], angle-dependent magnetoresistance oscillations (AMRO) [26, 27, 28, 29, 30], cyclotron resonance (CR) [31] and periodic orbit
resonance (POR) [32, 33]. ARPES is now a leading technique to investigate FSs
and has contributed significantly to the investigation of the FS in high temperature
superconductors. Using ARPES, one can measure not only the FS, but also the
band structure. However, since the skin depth of the photon in ARPES is much
shorter than a unit cell, it suffers from the fact that it is a surface probe with a
depth resolution often not more than a unit cell. ARPES is also restricted to measuring in-plane band structures of quasi-two-dimensional (Q2D) systems because
of the resolution of interlayer scattering processes. The SdH and dHvA effects
probe the effective mass, m∗ , and the quasiparticle lifetime, τ . However these
quantities are averaged over the FS. The SdH and dHvA effects are also limited to
three-dimensional (3D) and Q2D systems because the technique needs closed orbits
in order to give rise to Landau quantization. Since the SdH and dHvA phenomena
come from quantized Fermi levels under a high magnetic field, as explained in
Sec. 2.6, one can observe these effects only at low temperature, i.e., kB T < ~ωc ,
where ωc is the cyclotron frequency. AMRO can provide additional topographic
14
15
information in reciprocal space such as the Fermi wave vectors kF . AMRO is also
applicable in the case of quasi-one-dimensional (Q1D) systems.
CR can be used to determine the averaged m* and τ over the FS. Since CR
requires closed orbits, the technique is limited to 3D and Q2D systems. Our new
experimental technique, the so-called POR, is closely related to CR. The difference
between the POR and CR is that the POR can also come from open orbits. Details
are explained in Sec. 2.4. The POR can therefore be observed in Q1D systems.
In the case of Q1D systems, the POR probes the average Fermi velocity vF and
the scattering time τ , as shown in Chap. 4. In the case of Q2D systems, the POR
probes both the average m∗ and the k-dependent Fermi velocity vF (k), as well as
τ (k), as shown in Chap. 6.
2.2
Cyclotron Resonance
Cyclotron resonance (CR) is known as one of the most useful tools to probe
effective masses experimentally. It was first used in the 1950’s to study metallic
elements, e.g., Cu [34, 35, 36], Al [37, 38, 39], Sn [34, 40]. The CR technique is still
being used extensively in some fields today, e.g., the two-dimensional electron gas
(2DEG) system in a GaAs/(Ga,As)Al heterojunction [41]. In order to describe the
CR phenomena, one can start by considering semiclassical electron dynamics with
a simple model: a two-dimensional (2D) electron gas, i.e., the band index n is a
constant of the motion and the FS is a complete cylinder, as shown in Fig. 2–1, and
an energy dispersion represented by
E(k) =
~2 2
(k + ky2 ),
2m∗ x
(2–1)
where m∗ is the effective mass. We now consider a dc magnetic field applied along
the z-axis perpendicular to the conducting plane, i.e., B = (0, 0, B), as shown in
Fig. 2–1. Electrons in the system experience a Lorentz force due to the magnetic
field, and start to change their momentum. The motion of the electrons is given by
16
z
B
x
y
Trajectory
Fermi
velocity
Fermi
surface
Figure 2–1. The Fermi surface for the two-dimensional electron system. The magnetic field is applied along the z-axis. Because of the Lorentz force,
electrons move on the Fermi surface. The trajectory of the motion is
oscillatory. The Fermi velocity, perpendicular to the Fermi surface, is
represented by the arrows.
17
the equation of motion for the Lorentz force,
~k̇ = −e(v × B),
(2–2)
and the definition of the group velocity is
vg =
1
∇k [E(k)].
~
(2–3)
In the present case, using the energy dispersion in Eq. 2–1, the explicit expressions
corresponding to the above equations are,
eB
k˙x = − ∗ ky (t),
m
(2–4a)
eB
k˙y = ∗ kx (t),
m
(2–4b)
k˙z = 0,
(2–4c)
and
~
kx (t).
m∗
~
vy (t) = ∗ ky (t).
m
vx (t) =
vz = 0.
(2–5a)
(2–5b)
(2–5c)
Thus, solving the differential equations 2–4 and 2–5, one can see that the resultant
motion is oscillatory, i.e., kx (t) = k cos(ωc t+φ(k)) and ky (t) = k sin(ωc t+φ(k)); the
oscillatory velocities, vx = v cos(ωc t + φ(k)) and vy = v sin(ωc t + φ(k)); and kz = 0
and vz = 0, where ωc = eB/m∗ is the characteristic frequency of the oscillatory
motion, the so-called cyclotron frequency. The cyclotron frequency, νc = ωc /2π ∼
28 (GHz/T) ×
me
B,
m∗
for many materials usually goes into the microwave frequency
range for typical magnetic fields (B < 10 T), since their effective masses are often
in the range of m∗ ∼ 0.1me − 10me where me is the electron mass. There is no
kinetic energy change, because the motion is caused by the magnetic field. This
18
oscillatory motion is illustrated in Fig. 2–1. The shape of the constant energy
surfaces is cylindrical because of the 2D nature of the electronic property in the 2D
electron gas. At E = EF , this surface corresponds to the FS. The group velocity,
represented by arrows in Fig. 2–1, is always perpendicular to the FS. As a result,
the group velocity rotates a full 360◦ as the electron rotates around the surface, as
shown in Fig. 2–1.
Cyclotron motion affects transport properties. We now calculate the electrical
conductivity resulting from this motion, using a Boltzmann equation within the
relaxation time approximation [31, 33],
2e2
σpq (ω, B) =
V
Z
∂fk
dk [−
]vp (k, 0)
∂Ek
Z
3
0
1
dtvq (k, t) exp( − iω)t,
τ
−∞
(2–6a)
or an alternative expression is,
2e2
σpq (ω, B) =
V
Z
∂f (E)
dE[−
]N (E)
∂E
where
Z
v p (ω, k) ≡
Z
dk 2 vp (k, 0)v q (ω, k),
(2–6b)
0
1
dtvp (k, t) exp( − iω)t,
τ
−∞
(2–6c)
where p and q are indices of a cartesian coordinate system, i.e., x, y or z. In
Eq. 2–6c, exp(−iωt) is a oscillatory term to give arise the resonance, and exp(1/τ )
is a damping term to dephase the oscillatory motion. In the low temperature
∂fk
limit,− ∂E
'
k
1
δ(k
~v
(E)
− kF ) or − ∂f∂E
' δ(E − EF ), so that the motion of electrons
affecting the conductivity is restricted to the FS. The conductivity along the x-axis
is then given by the following expression, (See Appendix A.1 for details of the
calculation.)
2e2
σxx (ω, B) =
V
=
Z
Z
2
d SF vx (k, 0)
0
1
dtvx (k, t) exp( − iω)t,
τ
−∞
1
2e2 vo2
[
V i (ω + ωc ) +
1
τ
+
1
].
i(ω − ωc ) + τ1
(2–7a)
(2–7b)
19
ωτ=3
Re σxx(ω,B)
ωτ=2
ωτ=1
ωτ=0.8
ωτ=0.5
0
1
2
3
4
5
ωc/ω
Figure 2–2. Real part of σxx as a function of frequency for various values of ωτ .
The peak in the conductivity represents CR. The resonance condition
is ω = ωc . For ωτ >1, the CR is well-pronounced. However, the CR
becomes too weak to observed for ωτ <1. Thus ωτ has to be greater
than 1 in order to observe CR.
Thus, the real part of σxx is given by
Re σxx (ω, B) =
1
σdc
1
+
],
[
ωc 2
2
2 1 + (1 + ω ) (ωτ )
1 + (1 − ωωc )2 (ωτ )2
(2–8)
where σdc is the dc (ω = 0) conductivity and τ is the relaxation time of the
electrons on the FS. In the present case, σxx = σyy and σzz = 0. The real
part of the conductivity, Re σxx , is plotted in Fig. 2–2 as a function of ω/ωc .
There is a clear Lorentzian-like peak at the resonance condition, ω/ωc = 1 (or
ω = ωc ). This peak in Re σxx is caused by the cyclotron motion. This is cyclotron
resonance. Since ωc = eB/m∗ , in the present case, ωc goes to zero when B → 0.
The conductivity at B = 0 is therefore given by
Re σxx (ω, B = 0) =
σdc
.
1 + (ωτ )2
(2–9)
Eq. 2–9 is of course the ac Drude conductivity which has a peak at ω = 0 [31].
20
Thus, the peak corresponding to CR is the same as that of the ac Drude conductivity. The width and height of the CR peak depend on the parameter, ωτ , as shown
in Fig. 2–2. The peak is well-pronounced if ωτ >1. However the peak becomes
smeared out if ωτ <1. As with any resonance phenomena, the ωτ condition is related to phase memory, or dephasing. The ωτ < 1, eletrons dephase faster than the
time taken to complete an orbit. Thus, the oscillatory motion is highly damped and
the exponential dominates Eq. 2–6c. On the other hand, if ωτ > 1, electrons may
execute many orbits before dephasing, and the oscillatory term dominates Eq. 2–6c,
resulting in the resonance. Thus, the key to observing CR is an oscillatory velocity
and ωτ > 1.
We can now consider the case of a tilted magnetic field, e.g., B = (B sin θ, 0, B cos θ).
(See Fig. 2–3.) The equation of motion and the group velocity are now given by
eB cos θ
ky (t),
k˙x = −
m∗
eB cos θ
k˙y =
kx (t),
m∗
eB sin θ
k˙z =
ky (t),
m∗
(2–10a)
(2–10b)
(2–10c)
and
~
kx (t),
m∗
~
vy (t) = ∗ ky (t),
m
vx (t) =
vz = 0.
(2–11a)
(2–11b)
(2–11c)
Similarly, the resultant motion therefore follows an oscillatory trajectory, kx (t) =
k cos(ω2D t + φ(k)), ky (t) = k sin(ω2D t + φ(k)), and kz (t) = k tan θ cos(ω2D t + φ(k));
the oscillatory velocities, vx = v cos(ω2D t + φ(k)) and vy = v sin(ω2D t + φ(k)); and
vz = 0. The cyclotron frequency is given by
ω2D =
eB cos θ
,
m∗
(2–12)
21
z
B
z
θ
x
y
Trajectory
Fermi
velocity
Fermi
surface
Figure 2–3. The Fermi surface for the two-dimensional electron system. The magnetic field is applied along an arbitrary direction relative to the z-axis,
denoted by the angle θ. The plane of the trajectory also becomes
tilted. Correspondingly, the Fermi velocity remains perpendicular to
the FS.
22
which now depends on the angle θ. Again σxx = σyy , similar to Eq. (2–7b), and
σzz = 0 with the resonance condition ω = ±ω2D , or,
ω=
2.3
eB
| cos θ|.
m∗
(2–13)
Cyclotron Resonance Involving an Open Fermi Surface: Periodic
Orbit Resonance
In Sec. 2.2, we considered a closed trajectory on a 2D FS to illustrate the CR
effect. However, the resonance phenomenon does not require closed orbit motion,
because the essential ingredient of the resonance is an oscillatory group velocity.
Therefore, the resonance can even be seen in a system with only open trajectories.
This resonance effect is the so-called periodic orbit resonance (POR) [33], or also
the Fermi-surface traversal resonances (FTR) by Ardavan et al. [32]. The POR
was originally predicted by Osada et al. [42] for a quasi-one-dimensional (Q1D)
system. After that, many theoretical works have been performed using different
models, e.g., Hill [33], Blundell et al. [43], Moses and McKenzie [44, 45]. We now
consider the POR for the case of a quasi-one-dimensional (Q1D) FS which consists
of completely open FS sheets. The highly anisotropic energy dispersion can be
written in the form
E(k) = ~vF (|kx | − kF ) − 2ty cos(ky b) − 2tz cos(kz c),
(2–14)
where EF À ty and tz , vF is the Fermi velocity, and b and c are lattice constants
along the y and z-directions respectively. ty and tz are the transfer energy associated with the lattice vectors Ry and Rz . This energy dispersion describe a Q1D
FS. The 1st term of the energy dispersion is responsible for a flat shape of the
FS, and the 2nd and 3rd term are responsible for the warping along the lattice
vectors. The physical meaning of each transfer energy ty and tz is therefore the
Fourier component of the warping. As an example, the energy anisotropy in the
23
Fermi velocity
(a)
Corrugation
axis
z
θ
Q1D Fermi surface
(b)
B(θ)
no oscillatory velocity
B(0)
Figure 2–4. Oscillatory group velocity vz for the Q1D POR. (a) The trajectory
of electrons on the Q1D FS in the case of a tilted magnetic field. The
corrugation is produced by the transfer integral along the z-direction.
The z component of the group velocity becomes oscillatory. This oscillatory group velocity causes the resonance in the conductivity σzz . The
periodicity of the POR is angle-dependent. (b) Trajectory of electrons
on the Q1D FS in the case of a magnetic field along the z-axis. vz is
not oscillatory in this case, so that no POR is seen.
Q1D organic conductor, (TMTSF)2 ClO4 is tx : ty : tz = 250 : 20 : 1 meV [2].
Fig. 2–4 shows such a Q1D FS consisting of a planar sheet which is corrugated due
to the small transfer energy along the least conducting direction. A magnetic field
is now applied parallel to the corrugated plane, as shown in Fig. 2–4(a). A charged
particle will move according to the Lorentz Force. Because the FS is corrugated,
and the direction of the Fermi velocity (indicated by arrows in Fig. 2–4.) is always
perpendicular to the FS, the motion results in an oscillatory Fermi velocity. This
is the origin of the Q1D POR. The period of the motion can be varied by changing
magnetic field strength, or the angle between the magnetic field and the direction
24
of the corrugation. For instance, if the magnetic field is applied along the direction
of the corrugation, there is no POR, because there is no oscillatory velocity, as
shown in Fig. 2–4(b).
We now consider the resonance condition for the POR in the case of a Q1D FS
with a magnetic field applied parallel to the FS sheet. The corrugation direction
is assumed along the z-axis. The applied magnetic field is expressed by B =
(0, B sin θ, B cos θ) where θ is a angle between the magnetic field and the z-axis.
Recalling the equation of motion, Eq. (2–2),
2tb beB cos(θ)
2tc ceB sin(θ)
k˙x = −
sin[ky (t)b] +
sin[kz (t)c],
2
~
~2
vF eB cos(θ)
k˙y = sgn(kx )
,
~
vF eB sin(θ)
,
k˙z = −sgn(kx )
~
(2–15a)
(2–15b)
(2–15c)
and
vx = sgn(kx )vF ,
2ty b
sin[ky (t)b],
~
2tz c
vz (t) =
sin[kz (t)c].
~
vy (t) =
(2–16a)
(2–16b)
(2–16c)
Solving the above equations, the z-component of the trajectory and velocity is
given by
kz = −sgn(kx )[
vF eB
sin(θ)]t + kz (0),
~
(2–17)
and
vz (t) =
2tz c
vF eBc
sin[kz (0)c − sgn(kx )
sin(θ)t] ≡ 2tz c sin[kz (0)c − sgn(kx )ωQ1D t],
~
~
(2–18)
where ωQ1D =
vF eBc
~
sin(θ). Thus, this group velocity is oscillatory, and is charac-
terized by the frequency ωQ1D . This frequency is related not only to the magnetic
25
field B and the band parameter (Fermi velocity vF ), but also the lattice constant c
which is associated with the corrugation on the FS.
Like the CR, the conductivity is easily calculated. The real part of σzz is given
by,
Re σzz (ω, B, θ) =
σdc
1
1
[
+
],
2 1 + (ω + ωQ1D )2 τ 2 1 + (ω − ωQ1D )2 τ 2
(2–19)
which gives rise to a POR at ω = ±ωQ1D , i.e., the resonance condition is given by
ω=
vF eBc
| sin(θ)|.
~
(2–20)
Thus, one can measure the Q1D POR at different frequencies ω, magnetic fields B
and angles θ. By investigating the position of the POR, one can determine vF from
Eq. 2–20, and one can also determine the scattering time τ by analyzing the shape
of the POR.
The FS corrugation pattern of real Q1D conductors can be much more
complicated, particularly for materials with low-symmetry crystal structures,
i.e., monoclinic, triclinic or rhombohedral. As a result, the hopping energy to 2nd
and higher-ordered nearest neighbors becomes considerable. We can see such an
example in the study of the organic conductor α-(ET)2 KHg(SCN)4 by Kovalev
et al. [46]. The FS of α-(ET)2 KHg(SCN)4 , below 6 K, is shown in Fig. 2–5. At
8 K, α-(ET)2 KHg(SCN)4 undergoes a phase transition into a charge-density-wave
(CDW) ground state. The FS in the CDW state is characterized by a wave vector
Q. The FS consists of both Q2D and Q1D sections. In the experiments by Kovalev
et al., the Q1D POR was observed by sweeping the magnetic field and the angle
of the magnetic field. The observed POR contain many harmonic resonances, as
seen in Fig. 2–5(b) and (c). The existence of the harmonics implies that the Q1D
section is rather two-dimensional, as can be seen in Fig. 2–5(a). In the case of
α-(ET)2 KHg(SCN)4 , the Q1D section is not a simple flat sheet. Since the Q1D
26
(b)
(a)
z
x
(c)
Q1D FS Q2D FS
(d)
Figure 2–5. POR in α-(ET)2 KHg(SCN)4 . (a) FS in α-(ET)2 KHg(SCN)4 below 6
K. α-(ET)2 KHg(SCN)4 exhibits a phase transition at 8 K into a CDW
state. This state is characterized by the wave vector Q. As a result,
α-(ET)2 KHg(SCN)4 has Q2D and Q1D FS sections. These Q2D and
Q1D sections originate from the high temperature Q2D FS. Thus, the
Q1D FS is strongly warped (Reprinted figure with permission from
Kovalev et al. [46]. Copyright 2002 by the American Physical Society.).
(b) Microwave absorption as a function of the magnetic field. Many
harmonic POR are seen (Reprinted figure with permission from Kovalev et al. [46]. Copyright 2002 by the American Physical Society.).
(c) Microwave absorption as a function of the angle of the magnetic
field. The data exhibit many harmonic resonances which imply that
the warping pattern is more two-dimensional (Reprinted figure with
permission from Kovalev et al. [46]. Copyright 2002 by the American
Physical Society.). (d) Summary of the POR in α-(ET)2 KHg(SCN)4
(Reprinted figure with permission from Kovalev et al. [46]. Copyright
2002 by the American Physical Society.).
27
section is reconstructed from a Q2D section belonging to the high temperature FS
(not shown), the Q1D FS sheet still retains a two-dimensional shape, so that it is
strongly corrugated. Such a Q1D FS can be represented by finite higher-ordered
transfer integrals which produce the harmonic resonances. In contrast, the Q1D
conductor (TMTSF)2 ClO4 has a more one-dimensional FS. Experimental results for
the Q1D POR in (TMTSF)2 ClO4 are shown in Chap. 4.
2.4
Periodic Orbit Resonance for a Quasi-two-dimensional Fermi
Surface
As shown in the previous section, the POR effect is associated with the
warping of a FS. Therefore, the POR is also observed for a warped 2D FS, i.e., a
quasi-two-dimensional (Q2D) FS. In the case of many two-dimensional conductors,
the conductivity along the z-axis (the least conducting direction), σzz , is non-zero,
e.g., for κ-(ET)2 Cu(NCS)2 , σxx /σzz ∼ 1000 [3], for Bi2 Sr2 Can−1 Cun Oy , σxx /σzz ∼
10000 [47], for YBa2 Cu3 Oy , σxx /σzz ∼ 100 [47] and for Sr2 RuO4 , σxx /σzz ∼ 4004000 [48]. These are so-called Q2D conductors. We now show that σzz can also
have a resonance, the so-called periodic orbit resonance (POR). Here we consider a
Q2D electron model, i.e., the energy dispersion is represented by
E(k) =
~2 2
(k + ky2 ) − 2tz cos(kz c),
2m∗ x
(2–21)
where c is the interlayer spacing. The bandwidth in the z-direction, tz , is much
p
smaller than the Fermi energy, i.e., EF /tz ∼ σxx /σzz ∼ 30 for κ-(ET)2 Cu(NCS)2 .
As a result, the shape of the FS is similar to the 2D electron model, but the FS has
a slight corrugation along the z-direction. The magnitude of the small corrugation
of this cylinder corresponds to the bandwidth 4tz along the z-direction. Such a FS
is shown in Fig. 2–6(a). We now consider the motion of electrons on this FS under
a dc magnetic field similar to the case of the 2D electron model. The dc magnetic
field is applied at an arbitrary tilted angle, i.e., B = (B sin θ, 0, B cos θ). The
28
(a)
z
B
B z
θ
z
Oscillatory vz
y
x
Trajectory
Fermi
velocity
Fermi
velocity
Fermi
surface
(b)
Trajectory
B
B
No oscillatory vz
Figure 2–6. Oscillatory group velocity vz for the Q2D POR. (a) The trajectory
of electrons on the Q2D FS in the case of a tilted magnetic field. The
corrugation is caused by the transfer integral along the z-direction.
The z component of the group velocity becomes oscillatory. This oscillatory group velocity causes the resonance in the conductivity σzz , the
so-called periodic-orbit resonance. The right figure depicts the change
of vz . The trajectory on the FS is represented by a solid line. The
Fermi velocity is shown by dash arrows, which is always perpendicular
to the FS. vz is represented by the red solid arrows. (b) The trajectory
of electrons on the Q2D FS in the case of a magnetic field along the
z-axis. vz is not oscillatory in this case, so that no POR is seen.
29
motion is given by the following equations,
eB cos θ
k˙x = −
ky (t),
m∗
eBtz c sin θ
eB cos θ
k˙y = −
cos[kz (t)c] +
kx (t),
~
m∗
eB sin θ
ky (t),
k˙z =
m∗
(2–22a)
(2–22b)
(2–22c)
and
~
kx (t),
m∗
~
vy (t) = ∗ ky (t),
m
2tz c
vz (t) =
sin[kz (t)c].
~
vx (t) =
Because
eBtz c
~
¿
eB
k
m∗
(2–23a)
(2–23b)
(2–23c)
(except when the field is oriented close to xy-plane, i.e., θ ∼
90), the equation (2–22b) may be approximated as
eB cos θ
kx (t),
k˙y ∼
m∗
(2–24)
so that the motion can be calculated in the same way as the 2D electron model.
The resultant motion therefore becomes, kx (t) = k cos[ω2D t + φ(k)], ky (t) =
k sin[ω2D t + φ(k)], and kz (t) = k tan θ cos[ω2D t + φ(k)]; vx = v cos[ω2D t + φ(k)],
vy = v sin[ω2D t + φ(k)] and vz (t)= 2t~z c sin[kc tan θ cos{ω2D t + φ(k)}]. The resonance
frequency is the same as Eq. 2–12. The z-component of the group velocity, vz , is
now non-zero. This gives finite conductivity along the z-direction, σzz , and to a
resonances. Using the Boltzmann equation, the conductivity, σzz , is given by
∞
J0 (γ tan θ) X
1
1
σzz (ω, B) ∝
+
Jn (γ tan θ)[
+
],
2
2
2
2
2τ 2
1+ω τ
1
+
(ω
−
nω
)
τ
1
+
(ω
+
nω
)
2D
2D
n=0
(2–25)
where Jn (x) is the n-th order Bessel function, γ = kF c, σxx and σyy are given by
the same expressions as Eq. 2–7b. The conductivity therefore has resonances at
30
frequencies,
ω = ±nω2D =
neB
| cos θ|.
m∗
(2–26)
Thus, multiple resonances in σzz are predicted instead of the single resonance in σxx
and σyy found in the previous discussion for the 2D case. By investigating the POR
at different ω, B and θ, one can determine the effective mass, m∗ . By analyzing
the shape of the POR, one can also determine the scattering time τ , since the
half-width of the POR is ∼ 2/τ .
Although the motion of an electron along the z-direction gives resonances
in the conductivity, σzz , with similar resonance conditions, the picture of the
motion is quite different from the motion in the xy-plane. Fig. 2–6 shows a typical
trajectory of electrons on the FS. In Fig. 2–6(a), the applied magnetic field is tilted
from the z-axis, so that the trajectory on the FS is also tilted. As shown in the
right panel in Fig. 2–6(a), the group velocity vz becomes periodic because of the
corrugation along the z-direction. This oscillatory vz brings a resonance effect in
the conductivity, σzz . On the other hand, when the applied magnetic field is along
the corrugation axis (the z-axis in the present example), the group velocity vz is
not oscillatory. In this case, no POR is observed, as shown in the right panel in
Fig. 2–6(b). In the case of a low-symmetry crystal structure, the corrugation axis
can be different from the z-axis. We see such an example in Chap. 6.
2.5
POR and Angle-dependent Magnetoresistance Oscillations
The angle-dependent microwave conductivity measurements we have discussed
here can also be performed via dc measurements. In particular, in the case of
many clean low dimensional conductors, one can observe strong angle-dependent
magnetoresistance oscillations (AMRO) in the dc conductivity (or resistivity) by
rotating a magnetic field of a fixed strength relative to the sample. The AMRO
observed for rotation in different crystallographic planes in Q2D and Q1D FSs are
named differently. Fig. 2–7 and Fig. 2–8 illustrate each case. Fig. 2–7 explains the
31
z
Yamaji Oscillations in Sr2RuO4
θ B
y
φ
x
Q2D FS
Figure 2–7. AMRO for Q2D conductors. In the case of a Q2D FS, Yamaji oscillations may be observed by varying the field orientation from the z-axis
to the xy-plane. Here, the conducting plane is the xy-plane. θ represents the angle between the z-axis and xy-plane. φ represents the angle
from the x-axis in the xy-plane. The Yamaji oscillations in Sr2 RuO4
are shown. The strongest peak in the resistivity ρc is seen at around
θ = 35◦ , and many oscillations are seen at higher angles (Reprinted
figure with permission from Ohmichi et al. [49]. Copyright 1999 by the
American Physical Society.).
32
AMRO effect for a Q2D FS, so-called Yamaji oscillations. For the observation of
the Yamaji oscillations, a fixed magnetic field is rotated between the z-axis and
the xy-plane. The experimental result of the Yamaji oscillations is shown in the
right panel of Fig. 2–7. Sr2 RuO4 shows the strongest resistivity peak at around
θ = 35◦ , and many oscillations are also seen at higher angles. On the other hand,
in the case of a Q1D FS, three types of the AMRO effects are named: Lebed (z − y
rotation) [26, 27, 51], Danner-Kang-Chaikin (DKC) (z − x rotation) [28] and 3rd
angular effects (x − y rotation) [29], as shown in Fig. 2–8. In the case of the DKC
effect in (TMTSF)2 ClO4 , the most pronounced peaks are seen at around θ = 85◦
and 95◦ , and smaller oscillations are also seen at angles between θ = 85◦ and 95◦ .
The strength of the peak depends on the magnetic field strength. The oscillations
become smaller when the magnetic field is weaker. This is because of the lower
product of ωQ1D τ . In the case of the Lebed effect, many minima are seen. These
angles are the so-called Lebed magic angles. The 3rd angular effect is represented
by two minima in the resistance.
Since the AMRO is nothing more than dc POR, they are easily described by
simply applying ω = 0 in the ac conductivity. For example, in the case of the Lebed
effect, one can use Eq. 2–19 with ω = 0,
Re σzz (ω, B, θ) =
σdc
.
2
1 + ωQ1D
τ2
(2–27)
Fig. 2–9 plots the inverse conductivity as a function of the angle θ. The plot shows
minima, which correspond to the POR at ω 6= 0, and the AMRO at ω = 0. The
positions of the minima shift continuously by changing the frequency. The positions
of the minima at ω = 0 represent the directions of the warping. The Lebed effect in
(TMTSF)2 ClO4 is presented in Chap. 4.
Next, we consider the case of a Q2D FS (i.e., Yamaji oscillations). Like
the Q1D AMRO, by applying ω = 0 in Eq. 2–25, one obtains the following
33
Lebed effect in (TMTSF)2PF6
DKC effect in (TMTSF)2ClO4
Q1D FS
z
DKC
Lebed
B
y
x
3rd angular
conducting axis
3rd angular effect in (TMTSF)2ClO4
Figure 2–8. AMRO for Q1D conductors. In all cases, Rzz is plotted. In the case
of a Q1D FS, three AMRO effects can be observed: Lebed (y − z rotation), DKC (x − z rotation) and the 3rd angular effect (x − y rotation).
In the case of the DKC effect in (TMTSF)2 ClO4 , the most pronounced
peak is seen around θ = 85◦ and 95◦ , and smaller oscillations are also
seen at angles between θ
=
85◦ and 95◦ . The strength of the peak
depends on the magnetic field (Reprinted figure with permission from
Danner et al. [28]. Copyright 1994 by the American Physical Society.). In the case of the Lebed effect, many minima are seen. These
resonance angles are the so-called Lebed magic angles (Reprinted from
Kang et al. [50], Copyright 2003, with permission from Elsevier.). In
the case of the 3rd angular effect, two minima are seen in the resistance
(Reprinted figure with permission from Osada et al. [29]. Copyright
1996 by the American Physical Society.).
34
1/σzz
z
θ
ωcτ = 2
POR
B
ω = 0.5 ωc
ω = 0.25 ωc
AMRO
-180
-90
ω=0
0
90
180
θ (degrees)
Figure 2–9. The Lebed effect in both the dc and ac conductivity. The inverse
conductivity as a function of the angle θ is simulated using Eq. 2–19.
The inverse conductivity has two minima at ω 6= 0. These minima are
the POR. The inverse conductivity has one minimum at ω = 0, the
so-called AMRO.
conductivity,
∞
X
2Jn (γ tan θ)
σzz (B, θ) ∝ J0 (γ tan θ) +
,
2τ 2
1
+
(nω
)
2D
n=1
(2–28)
where ω2D = ωc cos θ and ωc = eB/m∗ . Fig. 2–10(a) plots the conductivity given
by Eq. 2–28. Strong oscillations are clearly seen in Fig. 2–10, the so-called Yamaji
oscillations. Just like the POR effect, the amplitude of these oscillations depends
on the product of ωc τ . When ωc τ is high, the oscillations are well-pronounced. On
the other hand, they are smeared when ωc τ is small. However, the positions of the
maxima and minima of the oscillations are independent of the product ωc τ . The
position of the dc resonance depends only on γ = kF c; the figure shows AMRO for
two values of γ = kF c, i.e., γ = 2 and γ = 3, at ωc τ = 3. Thus one can determine
kF c and, eventually, kF from the Yamaji oscillations.
In Fig. 2–11, we show the frequency dependence of the Yamaji oscillations.
The calculation for the Boltzmann equation Eq. 2–25 was performed by numerically
35
σzz(B)
θ
z
B
γ=kFc=3
ωcτ=0.5
ωcτ=1
ωcτ=3
γ=kFc=2
-60
30
-30
0
Angle θ(degrees)
60
Figure 2–10. Yamaji oscillations. The conductivity from Eq. 2–28 is shown as a
function of angle between the magnetic field and the least conducting
direction (the z-axis). The plot shows strong oscillations, the so-called
Yamaji oscillations. The oscillations are pronounced due to the large
product ωc τ . The position of the oscillations depends on γ = kF c, but
is independent of ωc τ .
solving the differential equations in Eq. 2–22 and Eq. 2–23 with the Q2D energy
dispersion E(k) =
~2
(kx2
2m∗
+ ky2 ) − 2tz cos(kz c). The calculation covers ν =dc
to 500 GHz and θ =0 to 90 degrees. Fig. 2–11 shows clearly that peaks in the
conductivity shift to higher angles when the frequency becomes higher. In the case
of ac conductivity measurements, the maxima are called POR.
We introduced the AMRO effect using a semiclassical description in this
section. However the origin of the Lebed effect is still an open question although
many theories have been proposed. The semiclassical description we used in this
chapter is suitable for some theories, but not for others. This difference may be
more explicit when the Lebed effect is considered at dc to microwave frequencies.
Using the POR effect, we recently tested whether the semiclassical description is
applicable to explain the Lebed effect. We will have this discussion in Chap. 4.
36
σzz (unit arb.)
(a)
z)
GH
y(
nc
ue
eq
Fr
500
400
300
200
100
Angle (degrees)
(b)
0
30
s)
egree
d
(
e
l
Ang
60
0 90
90
60
30
0
0
100
200
300
400
500
Frequency (GHz)
Figure 2–11. Numerical calculation of the conductivity for a Q2D FS at different
frequencies and orientations of the magnetic field. The parameters
were chosen for the κ-(ET)2 X salt: a = 16.3 Å, b = 8.4 Å, c = 13.1 Å,
α, β and γ = 90 degrees, m∗ = 3.9me and tz = 1 meV, τ = 4 ps
and B = 30 tesla. (a) 3D plots of the conductivity. The Yamaji oscillations are seen at ω = 0. In the case of the ac conductivity, similar
traces are shown. However the maxima of the ac conductivity are the
so-called POR. (b) Contour plot of the conductivity. The maxima
are represented by the red color. It is seen that the maxima of the
conductivity are shifted to higher angles by increasing the frequency.
37
2.6
Quantum Effects in the Conductivity
In a quantum mechanical picture, the energy dispersion becomes quantized
under magnetic fields, as does the FS. Such a quantized FS gives rise to important
magnetic quantum oscillation effects in clean compounds and at low temperature,
i.e., ~τ −1 < ~ωc , e.g., the de Haas-van Alphen (dHvA) and Shubnikov-de Haas
(SdH) effects. These can also provide useful information about the FS. Moreover,
both the POR and the SdH phenomena appear in the electronic conductivity,
so that both phenomena can be observed at the same time. In fact, as seen in
Chap. 6, we observed both effects in the experiments for κ-(ET)2 X. In this section,
we introduce magnetic quantum oscillations briefly. We also introduce the angledependence of the frequency of the SdH effect since the angle dependence played an
important role in Chap. 6. A more detailed description of the quantum mechanical
effects can be found in many books, e.g., by Shöenberg [25] and Abrikosov [21].
We consider the case of Q2D conductors, i.e., the energy dispersion is modeled
by Eq. 2–21. A magnetic field is now applied along the z-axis. Considering the
gauge transformation, H → H 0 = |p − qA|2 + qφ with the vector potential
A = (0, Bx, 0) and φ = 0, one can get the following energy dispersion,
1
En (kz ) = ~ωc (n + ) − 2tz cos(kz c),
2
(2–29)
where ωc (B) = eB/m∗ is the cyclotron frequency again and n = integer (0, 1, 2...).
The energy dispersion is therefore quantized. Each energy is called a Landau level.
The separation of the energy levels depends on the strength of the magnetic field.
This means that the strength of the magnetic field changes the number of filled
Landau levels below the Fermi energy EF . Fig. 2–12 shows changes of the Landau
tubes with magnetic field. If B=0, the energy spectrum is, of course, continuous. If
B = finite, the energy spectrum becomes quantized. When n=20, 20 Landau tubes
intersect the FS. Upon increasing B, the number of Landau tubes within the FS
38
B=0
Strength of B
n=∞
(continuous)
n=20
n=8
n=4
Figure 2–12. Landau tubes in a magnetic field. If B = 0, the energy spectrum
is continuous. If B = finite, the energy spectrum becomes discrete
(quantized). Upon increasing the magnetic field, the number of Landau tubes within the FS becomes smaller because the momentum
separation becomes greater.
becomes smaller because the momentum separation kxy becomes greater. In the
quantum mechanical picture, the CR and POR phenomena are represented by a
quantum transition between Landau levels, e.g., n to n+1.
The quantized energy dispersion, Eq. 2–29, brings about radical changes in
the density of states (DOS). Since the Landau levels are discrete, this change leads
to a degeneracy in each Landau level, and the DOS is then also discrete. The
degeneracy is proportional to the strength of the magnetic field. The discrete DOS
in a non-interacting system should be a delta-function, but in a more realistic
system, the DOS has some width due to scattering of carriers. Fig. 2–13 shows a
typical example of the DOS under the magnetic fields in the Q2D system. The
DOS consists of states of each Landau band (n = 0, 1, 2...). Each state is given
by δ(E − E(n, kz )) ∼
1
(E−E(n,kz ))2 +(~τ −1 )2
where τ is the scattering time. As a
result, each DOS has a finite width. The DOS due to each Landau tube therefore
superimpose upon each other. Moreover the overlap of each DOS depends on τ so
that the shape of the DOS varies with material, temperature and so on. Because
39
EF
Lower B
Ν(Ε)
Higher B
ħτ -1
1/2
3/2
5/2
7/2
9/2
E/ħωc
Figure 2–13. DOS of a Q2D conductor in a magnetic field in units of ~ωc . The
DOS
P in a Q2D1 conductor is given by, N (E) = Σn,kz δ(E − E(n, kz )) ∼
n,kz (E−E(n,kz ))2 +(~/2τ )2 . The dotted line represents the DOS for
each Landau level. The solid line is the sum of the dotted lines. The
dashed line represents Fermi energy. When the magnetic field increases, the Fermi energy line moves to left in the figure and vice
versa. Thus, the DOS at the Fermi energy EF is varied by the magnetic field. This is a the origin of the SdH effect.
40
of the superposition of states of equally spaced Landau levels, the DOS becomes
oscillatory as a function of the energy. The period of the oscillation of the DOS
depends on the strength of the magnetic field. When a higher magnetic field is
applied, the period becomes greater and vice versa. In Fig. 2–13, the DOS is
represented in units of ~ωc . When the magnetic field increases, the unit of the
x-axis in Fig. 2–13 becomes larger so that the Fermi energy shifts to left. i.e., the
dashed line representing the Fermi energy shifts to left when the magnetic field
increases, and to right when it decreases. Thus, N (EF ) traces the oscillatory
DOS when changing the magnetic field. Recalling Eq. 2–6b, one can see that the
conductivity is proportional to N (EF ). This is the origin of the SdH effect. On the
other hand, the POR comes from the oscillatory vi (k, t). Thus these phenomena
have completely different origins.
As shown in Fig. 2–13, the period of the quantum oscillations is given by
∆(EF /~ωc ) = 1. Using the Fermi energy EF = ~2 (kF,x (kz )2 + kF,y (kz )2 )/2m∗ and
the cross sectional area of the FS A = π(kF,x (kz )2 + kF,y (kz )2 ) = 2πm∗ EF /~2 at
each kz , the period of the oscillation as a function of the magnetic field can also be
written,
1
~e
2πe
∆( ) =
=
.
∗
B
EF m
~A
(2–30)
Furthermore, when the magnetic field is tilted, the period is given by
1
~e
2πe
∆( ) =
=
.
∗
B
EF m
~A cos θ
(2–31)
Thus one can measure the cross sectional area of the FS by studying the magnetic
quantum oscillations. The effective mass can be estimated by studying temperature
dependence and magnetic field dependence of magnetic quantum oscillations. The
book by Shöenberg gives the details [25].
41
2.7
Summary
In this chapter, we introduced the theoretical basis and usage of the POR
phenomena. The physical aspects of the POR are similar to CR and AMRO since
all of them can be explained by the same magneto-transport model. On the other
hand, the SdH and dHvA effects are purely quantum phenomena.
CHAPTER 3
EXPERIMENTAL SETUP
The major results presented in this chapter can be found in the article entitled
Rotating cavity for high-field angle-dependent microwave spectroscopy of lowdimensional conductors and magnets, S. Takahashi and S. Hill, Review of Scientific
Instruments 76 023114 (2005).
3.1
Overview of Microwave Magneto-optics
In recent years, microwave (millimeter and sub-millimeter wave) technologies,
covering frequencies from 10 GHz to 10 THz (0.33 − 330 cm−1 ), have become the
focus of intensive efforts in many fields of research. In engineering and medicine,
THz imaging represents one of the next-generation technologies, enabling nondestructive materials inspection, chemical composition analysis [52, 53, 54], and
medical diagnoses [52, 53, 55]. In the fundamental sciences, physics, chemistry and
biology, microwave spectroscopy is also very useful for investigating the physical
properties of a material. This is particularly true for the sub-field of condensed
matter physics, where the millimeter and sub-millimeter spectral range can provide extremely rich information concerning the basic electronic characteristics of
a material [56, 57, 58, 59]. Furthermore, combining microwave techniques and
high magnetic fields (microwave magneto-optics), allows many more possibilities,
including: cyclotron resonance (CR) [32, 33, 36, 39, 43, 46, 60, 61]; electron paramagnetic resonance (EPR) [62, 63, 64, 65]; antiferromagnetic resonance (AFMR) [1];
Josephson Plasma Resonance (JPR) measurements of layered superconductors [66, 67, 68, 69]; and many others. In each of these examples, the magnetic
field influences the dynamics of electrons at frequencies spanning the microwave
42
43
spectral range. Another very important aspect of microwave magneto-optical investigations is the possibility to study angle dependent effects by controlling the
angle between the sample and the microwave and dc magnetic fields. For instance,
through studies of the angle dependence of CR amplitudes, one can extract detailed
information concerning the Fermi surface (FS) topology of a conductor [46, 61, 70].
Consequently, angle-dependent microwave spectroscopy has been widely used
in recent years to study highly anisotropic magnetic and conducting materials.
Problems which have been addressed using these methods include: high-Tc superconductivity [66], and other low dimensional superconductors, e.g., organic
conductors [32, 33, 43, 46, 61, 68, 69], Sr2 RuO4 [60] etc.; the quantum and fractional quantum Hall effects [71, 72]; and low dimensional magnets, including
single-molecule magnets (SMMs) [63, 64, 65, 73].
Unfortunately, the microwave spectral range presents many technical challenges, particularly when trying to study very tiny (¿ 1 mm3 ) single-crystal samples within the restricted space inside the bore of a large high-field magnet system
(either resistive or superconducting [74].) Problems associated with the propagation
system stem from standing waves and/or losses [75]. Several methods have been
well documented for alleviating some of these issues, including the use of fundamental TE and TM mode rectangular metallic waveguides, low-loss cylindrical
corrugated HE waveguides [76], quasi-optical propagation systems [76], and in-situ
generation and detection of the microwaves [77]. Standing waves are particularly
problematic in the case of broadband spectroscopies, e.g., time-domain [77] and
fourier transform techniques [78], as well as for frequency sweepable monochromatic
sources [79, 80]. In these instances, the optical properties are usually deduced via
reflectivity or transmission measurements, requiring a large well-defined (i.e., flat)
sample surface area (>λ2 ). For cases in which large samples are not available
(note: λ spans from 3.34 cm at 10 GHz to 0.33 mm at 1 THz), resonant techniques
44
become necessary, e.g., cavity perturbation [56, 57, 58, 62, 75]. This unfortunately
limits measurements to the modes of the cavity. In addition, making absolute
measurements of the optical constants of a sample, as a function of frequency, is
extremely difficult to achieve using cavity perturbation because of its narrow-band
nature [56, 57, 58]. However, the cavity perturbation technique is ideally suited for
fixed-frequency, magnetic resonance (magnetic field-domain) measurements [75].
Furthermore, as we have recently shown, it is possible to make measurements at
many different frequencies by working on higher order modes of the cavity [75]. To
date, measurements in enclosed cylindrical copper cavities have been possible at
frequencies up to 350 GHz (see Takahashi et al. [81]).
The frequency range in the magnetic field-domain technique depends on the
nature of the magnetic resonance. For example, in the case of the Q2D periodic
orbit resonance (POR), the frequency range depends on the effective mass because
the resonance condition for the Q2D POR is given by Eq. 2–26, νc = eB/(2πm∗ )
where νc is the cyclotron frequency, B is the magnetic field strength and m∗ is
the effective mass. Fig. 3–1 shows the frequency range for the Q2D POR. In the
upper panel, the resonance is shown for different frequencies. In the lower panel,
the frequency ranges are given for different m∗ . As seen in the figure, the magnetic
field-domain technique works very well in the case of small effective mass. For
instance, in the case of m∗ = me where me is electron mass, the field range from
0-15 tesla, which is relatively easy to obtain in a laboratory, corresponds to 0420 GHz. This wide frequency range is therefore available to increase ωτ . The
upper panel in Fig. 3–1 illustrates this point. ωτ can be enhanced from 1 to 4
by changing the frequency. However, in the case of a higher effective mass, the
corresponding frequency range is narrower. In this case, one may need to employ
a high magnetic field facility. Fields in the range of 0 - 45 T at the National High
Magnetic Field Laboratory (NHMFL) in Tallahassee FL, USA corresponds to 0
45
Re σ (ω ,B)
ν=νc=50 GHz, ωτ=1
ν=νc=100 GHz, ωτ=2
ν=νc=200 GHz, ωτ=4
50 GHz
(1.67 cm-1)
100 GHz
(3.34 cm-1)
200 GHz
(6.67 cm-1)
5 tesla
10 T
20 T
20 T
40 T
30 T
300 GHz
(10.0 cm-1)
10 tesla
400 GHz
(13.3 cm-1)
500 GHz
(16.7 cm-1)
15 tesla
m*=me
40 T
m*=5 me
m*=10 me
Figure 3–1. Frequency range for the Q2D POR in magnetic fields accessible at the
National High Magnetic Field Laboratory (NHMFL) in Tallahassee
FL, USA. Since νc
=
eB/(2πm∗ ), the frequency range depends on
the effective mass. For example, in the case of a small effective mass,
the frequency range is very wide, e.g., the fields in the range of 0 45 T correspond to 0 - 1.26 THz (0 - 42.08 cm−1 ) for m∗ = me . For a
higher effective mass, the frequency range is narrower, e.g., the fields
in the range of 0 - 45 T corresponds to 0 - 252 GHz (0 - 8.42 cm−1 ) for
m∗ = 5me .
- 252 GHz (0 - 8.42 cm−1 ) for m∗ = 5me , and 0 - 126 GHz (0 - 4.21 cm−1 ) for
m∗ = 10me .
3.2
Experimental Setup
Fig. 3–2 shows an overview of our setup for the cavity perturbation technique.
The sample is mounted inside a cylindrical cavity which is positioned in the
magnetic field center. The microwave signal coming from the generator (not shown)
is transmitted through the incident waveguide into cavity and couples to the
sample, and is then returned through a second transmission waveguide. The probe,
which consists of the cavity, waveguides, electronics (thermometer, heater etc.) and
vacuum jacket, is placed inside the cryostat with a small mount of exchange gas.
In the experiment, we study the microwave response of the sample by changing:
the microwave frequency; the strength of the magnetic field; the orientation of the
magnetic field; the temperature etc. This is achieved through a combination of
many instruments.
46
Magnetic Field
for Axial Magnet
(Oxford 17T,
NHMFL 33T)
B
Magnetic Field
for Transverse Magnet
(QD PPMS 7T)
B
Figure 3–2. Overview of the experimental setup. The probe is inserted vertically
in the center of the Dewar. The direction of the magnetic field is shown
in the figure. For the case of an axial magnet, the direction is vertical
in the figure and parallel to the probe axis. For the case of a transverse
magnet, the field is horizontal in the figure and perpendicular to the
probe axis.
47
As a microwave source and detector, we employ a Millimetre Vector Network
Analyzer (MVNA) with an External Source Association (ESA) option (not shown)
and several Schottky diodes manufactured by AB millimetre [82]. The source
frequency of the MVNA is tunable in the range of F1 = 8-18.5 GHz. By feeding
the source frequency to a Schottky diode, which is a passive non-linear device,
harmonic components of the source (Fmm = N × F1 , N = integer) are produced
and transmitted to the waveguide probe. The Schottky diode that act as the source
is called the harmonic generator (HG). The optimized harmonics depend on the
type of Schottky diode. Several Schottky diodes are available [K-band (N = 2,
3, ν = 18-40 GHz), V-band (N = 3 and 4, ν = 48-72 GHz), W-band (N = 5
and 6, ν = 72-110 GHz), and two sets of D-band diodes (N ≥ 6, ν ≥ 110 GHz)].
For detection, the microwave signal (Fmm ) returning to the MVNA is mixed
with a second microwave signal (F2 ) at a second Schottky diode, the so-called
harmonic mixer (HM). The beat signal (Fbeat = N × F1 ± N 0 × F2 ) and the phase
(φbeat = N × φ1 ± N 0 × φ2 ) is then sent to a heterodyne vector reciever (VR)
in the MVNA. By choosing appropriate HG and HM so that N = N 0 , and by
locking the phases φ1 = φ2 , the noise associated with the phase is cancelled, and
a low noise level is achieved. The detection of the signal is performed on the MHz
component in the beat signal, i.e., FM Hz = N × (F1 − F2 ). The MVNA employs
FM Hz = 9.01048828125 or 34.01048828125 MHz for the vector measurement.
Although the D-band Schottky diode can produce a fairly powerful microwave
signal on harmonics N = 6 and 7, the power diminishes for N ≥ 8. For higher
frequencies (ν ≥ 170 GHz), we usually use the ESA option which consists of a
Gunn diode, a directional coupler-harmonic mixer and a multi-harmonic multiplier.
With this option, the multiplier is fed by a more powerful higher frequency
microwave signal (PF 1 ∼ 30 mW, F1 = 69-82.3 and 82.5-102.7 GHz) from a Gunn
diode. Since the Gunn source frequency is much higher than the internal source in
48
Table 3–1. Available magnet systems at the University of Florida (UF) and the
NHMFL. The table lists the field geometry and magnet type, the maximum available field Bmax , the probe length, the available temperature
(T) range, and the outer diameter of the cavity probe. The Quantum
Design (QD) Physical Property Measurement System (PPMS) allows
two-axis rotation.
Magnet
45 T (NHMFL)
33 T (NHMFL)
25 T (NHMFL)
Oxford Inst. (UF)
QD PPMS (UF)
Bmax (T)
45
33
25
17
7
Type
Axial hybrid
Axial resistive
Axial resistive
Axial SC
Transverse split-coil SC
length
1.67 m
1.45 m
1.6 m
1.9 m
1.15 m
T (kelvin)
1.4 - 300
1.4 - 300
0.5 - 3001
0.5 - 300a
1.7 - 400
Probe dia.
3/4”
3/4”
1”
1” (7/8”a )
1”
the MVNA, one can achieve high frequencies on lower harmonics, thereby enabling
measurements to much higher frequencies with the ESA option. For more details
concerning the MVNA and the ESA option, see Goy and Gross [79] and Mola et
al. [75]. With this setup, we can work in a wide frequency range from 8-700 GHz
(0.27-23.3 cm−1 ).
Since we wish to investigate the microwave response for a wide variety electronic and magnetic systems, many magnet systems are needed. Two magnet
systems are available at UF, one produced by Oxford Instruments [83] and one
by Quantum Design [84], and we can also use the magnets at the NHMFL in
Tallahassee, FL, USA [74]. The various magnet systems which are compatible
with the instrumentation described here, are listed in TABLE 3–1. All systems
are standard type magnets with vertical access for measurements (not horizontal access magneto-optical magnets). The standard cryostats designed for these
magnets are all 4 He based (either bath or flow cryostats). However the base temperature of the 4 He based cryostat is limited, i.e., to roughly 1.4 K. Because of
the demand to work at lower temperatures, we have constructed a simple 3 He
refrigerator which is compatible with the 17 tesla Oxford Instruments superconducting magnet at UF, and the 25 T resistive magnet at the NHMFL, as listed
49
in TABLE 3–1. A schematic of this refrigerator is shown in Fig. 3–3. The 1.9 m
long waveguide/cavity probe is inserted directly into the 3 He space, which is constructed from a 7/8” (= 22.2 mm) outer diameter stainless steel tube with a 0.010”
(= 0.25 mm) wall thickness. The lower 254 mm of this tube is double jacketed
with a 1.00” (= 25.4 mm) outer diameter. The volume between the two tubes is
vacuum sealed in order to provide thermal isolation between the 3 He liquid and
the surrounding 4 He vapor. The 3 He condenses by means of heat exchange with
the walls of the 7/8” tube (above the double jacketed region) which is inserted
into the Oxford Instruments 4 He flow cryostat operating at its base temperature
of ∼ 1.4 K. After condensation of the full charge of 3 He (5 liters at STP), subkelvin temperatures are achieved by pumping directly on the liquid by means of
an external sealed rotary pump. The refrigerator operates in single-shot mode,
i.e., the 3 He is returned to a room temperature vessel, where it is stored until the
next cooling cycle. A simple gas handling system controls the condensation of 3 He
gas, and the subsequent pumping of the gas back to the storage vessel. The 3 He
tube and gas handling system is checked for leaks prior to each cool down from
room temperature. Although this design is simple, it has the disadvantage that the
microwave probe comes into direct contact with the 3 He vapor, thus potentially
affecting the tuning of the cavity, as well as the phase of the microwaves reaching
the cavity via over 3.8 m of waveguide; such phase fluctuations can cause drifts in
signal intensity due to unavoidable standing waves in the waveguide. However, we
have found that these problems are minimal when operating at the base 3 He vapor
pressure (0.15 torr). The temperature of the sample is then controlled by supplying
heat to the copper cavity, which acts as an excellent heat reservoir, i.e., it ensures
good thermal stability. The base temperature of the 3 He refrigerator is 500 mK and
it provides hold times of up to 2 hours.
50
Waveguides
Pumping Line
Fins
1K Pot
Cavity
Liquid 3He
Figure 3–3. A schematic diagram of the 3 He probe used for sub-kelvin experiments
in the Oxford Instruments 17 T superconducting magnet. See main
text for a detailed description of its construction (Reused with permission from Takahashi et al. [81]. Copyright 2005, American Institute of
Physics.).
51
Table 3–2. Probes used for the cavity perturbation technique. The coaxial cable,
K- and V-band waveguide probes were built in-house. The corrugated
waveguide (HE mode) probe is produced by Thomas Keating Ltd. [85].
V and R denote vertical and rotating cylindrical cavities respectively. D
and H represents the diameter and height of the cavities respectively.
Probe
Coaxial cable
ν range (GHz)
8-18
Harmonic N
1
K-band (WR-42) waveguide
18-40
2, 3
V-band (WR-15) waveguide
48-350
3-25
Corrugated waveguide
170-700
15-30
Cavity size (D × H)
0.70” × 0.50” (V, R)
+ dielectric material (ε=15)
0.70” × 0.50” (V)
0.52” × 0.52” (V)
0.30” × 0.30” (V, R)
0.40” × 0.30” (V)
0.25” × 0.25” (V)
No cavity (flat end plate)
In order to make the best use of the wide frequency range provided by the
MVNA with the ESA option, we have developed several cavity perturbation probes
which have different optimized frequencies. We list the probes in TABLE 3–2. The
coax cable, K- and V-band probes are home-made, and the corrugated probe is
fabricated by Thomas Keating Ltd. [85]. We have also constructed several sizes of
cylindrical cavities for each probe, and rotating cavities for angle-dependent studies. All cavities are made of copper. The details of the vertical cavity are explained
in our previous paper [75]. The details of the rotating cavity will be introduced in
Sec. 3.3. Probes have been designed separately for each magnet system. For the
sake of good temperature control, the probes are made from a combination of high
and low conductivity materials, i.e., copper waveguide and stainless steel (S.S.)
waveguide. The length of each waveguide section was determined by the temperature profile in the cryostat. Fig. 3–4 shows the temperature profile for the QD
PPMS 7 T and Oxford Instruments 17 T magnet systems. The S.S. waveguides are
used where the temperature change is large, and the copper waveguides are used
for the reminder. See Mola et al. [75] for further details concerning the microwave
probe design.
52
(a)
(b)
Distance (in.)
0.0 3.9 7.9 11.8 15.7 19.7 23.6 27.6 31.5
0.0
S.S. waveguide
60
Copper
waveguide
50
40
30
20
T=2 K
T=10 K
T=30 K
10
0
10 20 30 40 50 60 70 80
Distance from the magnet center (cm)
S.S. waveguide
60
Temperature (K)
Temperature (kelvin)
Distance (in.)
15.7 23.6 31.5
39.4
47.2
70
70
0
7.9
T=3 K
T=10 K
T=20 K
50
40
S.S. waveguide
Copper
waveguide
30
20
10
0
0
20
40
60
80
100 120
Distance from the magnet center (cm)
Figure 3–4. Vertical temperature distribution in the cryostat for the QD PPMS
7 T and Oxford Instruments 17 T magnets. (a) Vertical temperature
distribution in the cryostat for the QD PPMS 7 T magnet. The temperature distribution was measured by three different cryostat system
temperatures Tcryo =2, 10, 30 K. The length of the waveguide sections
was determined by this temperature distribution. (b) Vertical temperature distribution in the cryostat for the Oxford Instruments 17 T
magnet. The temperature distribution was measured by three different
cryostat system temperatures Tcryo =3, 10, 20 K.
53
3.3
Rotating Cavity
The standard approach for studying angle-dependent effects using the cavity
perturbation technique is to use a split-pair magnet and/or goniometers, and is
widely used in lower frequency commercial EPR instruments, e.g., X-band, K-band
and Q-band [36, 86]. In the case of the split-pair approach, the DC magnetic field
is rotated with respect to a static waveguide/cavity assembly. However, those
approaches are usually limited in terms of the strength of the magnetic fields
obtained. In this section, we outline a method for in-situ rotation of part of a
cylindrical resonator which we developed recently, thus enabling angle-dependent
cavity perturbation measurements in ultra-high-field magnets, and two-axis
rotation capabilities in standard high-field superconducting split-pair magnets.
Details of the rotating cavity have been published in Takahashi et al. [81]. As we
shall outline, the rotation mechanism preserves the cylindrical symmetry of the
measurement, thereby ensuring that the electromagnetic coupling to the microwave
fields does not change upon rotating the sample. This is particularly important
for studies of low dimensional conductors, where sample rotation alone (as in the
case of a goniometer) would lead to unwanted instrumental artifacts associated
with incommensurate symmetries of the sample and cavity. The rotating cavity
described here is compatible with all magnet systems listed in TABLE 3–1.
We note that a rotating cavity has previously been developed by Schrama
et al. [87], at the University of Oxford, also for high-field microwave studies. As
we will demonstrate, the cylindrical geometry offers many advantages over the
rectangular design implemented by the Oxford group. For example, the waveguides
are coupled rigidly to the cylindrical body of the cavity in our design (only the
end-plate rotates), whereas the coupling is varied upon rotation in the Oxford
version, resulting in effective “blind-spots”, i.e., angles where the microwave fields
in the waveguides do not couple to the cavity; in contrast, the cylindrical version
54
offers full 360◦ + rotation. Furthermore, the rigid design offers greater mechanical
stability and, therefore, less microwave leakage from the cavity, resulting in
improved signal-to-noise characteristics [75]. The TE01n cylindrical modes also
offer the advantage that no AC currents flow between the curved surfaces and flat
rotating end-plate of the cylindrical resonator. Consequently, the moving part
of the cavity does not compromise the exceptionally high quality (Q−) factors
associated with these modes. Indeed, Q−factors for the first few TE01n modes
vary from 10, 000 to 25, 000 (at low temperatures), as opposed to just 500 for the
rectangular cavities. This order of magnitude improvement translates into vastly
increased sensitivity, enabling e.g., , EPR studies of extremely small single crystal
samples. Finally, the cylindrical cavity is machined entirely from copper, and held
rigidly together entirely by screws. This all-copper construction results in negligible
field sensitivity, i.e., the field dependence of the cavity parameters is essentially flat
and, most importantly, the cavities do not contain any paramagnetic impurities
that could give rise to spurious magnetic resonance signals.
The configuration of the rotating cavity is shown in Fig. 3–5. The principal
components consist of the open-ended cylindrical resonator, the cavity end-plate
and worm gear, a worm drive for turning the end-plate, and a wedge which facilitates external clamping and un-clamping of the cavity and end-plate. The cavity
assembly is mounted on the under-side of a stage (not shown in Fig. 3–5), and
the end-plate is centered on the axis of the cavity by means of a centering-plate.
The upper part of the wedge is threaded, and passes through a threaded channel
in the stage so that its vertical position can be finely controlled via rotation from
above. Likewise, the worm drive is rotated from above, and accurately aligned
with the worm gear via an unthreaded channel in the stage. Finally, the wormdrive and centering plate are additionally constrained laterally by means of an
end cap and spindle (see below) which attaches to the under side of the resonator.
55
Worm Gear
Cavity
End PlateWorm Resonator
Wedge
Coupling Hole
CenteringPlate
End Cap
Spindle
Figure 3–5. A schematic diagram of the various components that make up the
rotating cavity system. The sample may be placed on the end plate,
which can then be rotated via an externally controlled worm drive.
The wedge is used to clamp and un-clamp the end plate to/from the
main resonator. See main text for a detailed description of the assembly and operation of the rotating cavity (Reused with permission from
Takahashi et al. [81]. Copyright 2005, American Institute of Physics.).
56
The cavity, end plate, and stage are each machined from copper, thus ensuring
excellent thermal stability of the environment surrounding the sample; the heater
and thermometry are permanently contacted directly to the stage. The remaining
components shown in Fig. 3–5 are made from brass [88].
The internal diameter of the cavity (7.62 mm) is slightly less than the diameter of the end-plate, which is free to rotate within a small recess machined into the
opening of the resonator. On its rear side, the copper end-plate mates with a brass
gear which, in turn, rotates on an axis which is fixed by the centering-plate. As
mentioned above, rotation of the worm gear and end-plate is achieved by turning
the worm drive with the wedge disengaged from the end-plate. During experiments,
a good reproducible contact between the end-plate and the main body of the cavity
is essential for attaining the highest resonance Q-factors. This is achieved by engaging the wedge through a vertical channel in the centering-plate, where it transfers
pressure along the axis of the end-plate. Fig. 3–6 displays labeled photographs of
the 1st and 2nd generation rotating cavity assemblies. The 2nd generation version
employs a smaller home-built worm drive, reducing the overall diameter of the
probe to slightly below 3/4” (= 19.1 mm), which enables its use in the highest field
magnets at the NHMFL; the 1st generation probe has an outer diameter of just
under 1”.
The wedge and worm gear are driven by stainless steel rods (Diameter = 1/16”, or 1.59 mm) which pass through vacuum tight ‘o’-ring seals at the
top of the waveguide probe. Small set screws are used to fix the steel control rods
into the worm drive and wedge (Fig. 3–5), and to fix the end-plate within the
worm gear. Rotation of the worm gear is monitored via a simple turn-counting dial
mounted at the top of the probe, having a readout resolution of 1/100th of a turn.
Different worm drive/gear combinations are employed in the 3/4” and 1” diameter
probes (see Fig. 3–6), with 1/41 and 1/20 gear ratios, respectively. Thus, the angle
57
(a)
Waveguides
1/16" S.S.
Rod
Worm
Gear
Heater
1"
Wedge
Worm
(b)
Stage
End Plate
CenteringPlate
3/4"
Cavity
End Cap
Spindle
Figure 3–6. Photographs of the rotating cavity system; part of the cavity has been
disassembled (end-plate and centering-plate) in order to view the inside of the cylindrical resonator. (a) shows the 1st generation rotating
cavity, which fits into a 1” thin-walled stainless steel tube; this cavity is
compatible with the 25 T magnets at the NHMFL and the QD PPMS
and Oxford Instrument magnets at UF (see Table 3–1). (b) shows the
2nd generation rotating cavity, which fits into a 3/4” thin-walled tube;
this cavity is compatible with the highest field (45 T) resistive magnets
at the NHMFL, as well as the 3 He probe designed for the Oxford Instruments magnet at UF (see Fig. 3–7). (Reused with permission from
Takahashi et al. [81]. Copyright 2005, American Institute of Physics.)
58
resolution on the dial redout corresponding to the actual sample orientation is
approximately 0.09◦ for the 3/4” probe, and 0.18◦ for the 1” probe. Although
both probes exhibit considerable backlash (∼ 1◦ ), this is easily avoided by consistently varying the sample orientation in either a clockwise or counter-clockwise
sense. High resolution EPR measurements on single-molecule magnets (reported in
Takahashi et al. [81]) have confirmed the angle resolution figures stated above.
The stage also performs the task of clamping the V-band waveguides into
position directly above the cavity coupling holes. As with previous cavity designs,
a small channel [0.02” (= 0.51 mm) wide and 0.02” deep] is machined between
the waveguides on the under side of the stage; this channel mates with a similarly
sized ridge located in between the coupling holes on the upper surface of the
cavity housing (see Fig. 3–5). Our previous studies have demonstrated that
this arrangement is extremely effective at minimizing any direct microwave leak
between the incident and transmission waveguides and is, therefore, incorporated
into all of our cavity designs [75]. A direct leak signal can be extremely detrimental
to cavity perturbation measurements, causing a significant reduction in the useful
dynamic range, and to uncontrollable phase and amplitude mixing, as explained in
our previous paper [75].
The internal dimensions of the cylindrical resonator are 7.62 mm × 7.62 mm
(diameter × length). The center frequency (f◦ ) of the TE011 mode of the unloaded
cavity is 51.863 GHz, with the possibility to work on higher-order modes as
well. Using the ESA option, we have been able to conduct measurements up to
350 GHz [89]. While the modes above about 150 GHz are not well characterized,
they do provide many of the advantages of the well-defined lower frequency
modes, e.g., enhanced sensitivity, control over the electromagnetic environment
(i.e., E vs. H field) at the location of the sample, and some immunity to standing
waves. Table 3–3 shows resonance parameters for several unloaded cavity modes
59
(there are many others which are not listed). The TE01n (n = positive integer)
modes are probably the most important for the rotating cavity design, because
their symmetry is axial. Thus, rotation of the end-plate not only preserves the
cylindrical symmetry, but also ensures that the sample remains in exactly the same
electromagnetic field environment, i.e., upon rotation, the polarization remains in
a fixed geometry relative to the crystal. A sample is typically placed in one of two
positions within the cavity: i) directly on the end-plate; and ii) suspended along
the axis of the cavity by means of a quartz pillar (diameter = 0.75 mm) which is
mounted in a small hole drilled into the center of the end-plate. These geometries
are depicted in Fig. 3–7 for the TE011 mode (see Sec. 3.5 for details), and for
the two dc magnetic field geometries, i.e., axial and transverse. Each geometry
possesses certain advantages for a particular type of experiment. For example, the
quartz pillar configuration is particularly useful for EPR experiments in the axial
high-field magnets [Fig. 3–7(b)] since, for the TE01n modes, the sample sits in a
microwave ac field (H̃1 ) which is always transverse to the dc magnetic field (B◦ ).
We discuss this in more detail in Mola et al. [75] and Takahashi et al. [81].
Another advantage of the TE01n modes is the fact that no microwave currents flow between the end-plate and the main body of the resonator. Thus, the
Q-factors of these modes are high, and essentially insensitive to the mechanical contact made with the wedge. Table 3–3 lists the key resonance parameters associated
with several modes. For the case of the TE011 mode, the low temperature (∼ 2 K)
Q-factor is ∼ 21, 600, and the contrast between the amplitude on resonance [A(f◦ )]
and the amplitude far from resonance (leak amplitude, Al ) is 31.7dB. These parameters are essentially the same as the optimum values reported for the fixed
cylindrical geometry in our earlier paper [75], thus confirming the suitability of this
new rotating design for cavity perturbation studies of small single-crystal samples,
both insulating and conducting. We note that the Q-value for the non-cylindrically
60
(a)
End-plate configuration
Probe
Axial
B
Sample
Trans.
B
End Plate
(b) Quartz pillar configuration
Axial
B
φ rotation
Trans.
B
Quartz pillar
θ rotation
Figure 3–7. Schematic diagrams showing various different sample mounting configurations for both axial and transverse magnetic field geometries,
including the two-axis rotation capabilities (Reused with permission
from Takahashi et al. [81]. Copyright 2005, American Institute of
Physics.).
61
Table 3–3. Resonance parameters for several different cavity modes. The first column indicates the given mode. The second column lists the resonance
frequencies (f◦ ). The third column lists the Q-factors. The final column
lists the contrast in dB, i.e., A(f◦ )−Al , where A(f◦ ) is the transmission
amplitude at the resonance frequency, f◦ , and Al is the leak amplitude
(Reused with permission from Takahashi et al. [81]. Copyright 2005,
American Institute of Physics.).
Mode
TE011
TE212
TE012
TE015
f0 (GHz)
Q
51.863 21,600
54.774
3,300
62.030 16,600
109.035 8,600
A(f0 )-Al (dB)
31.7
26.5
25.2
19.0
symmetric mode in Table 3–3 (TE212) is almost an order of magnitude lower than
that of the TE011 mode. As noted above, this is due to the flow of microwave currents associated with the TE212 mode across the mechanical connection between
the main body of the resonator and the end-plate.
As discussed earlier, even though the end-plate can be rotated, the waveguides are coupled absolutely rigidly to the cavity via the stage. As with earlier
designs [75], the microwave fields in the waveguides are coupled into the resonator
by means of small circular coupling holes which are drilled through the sidewalls
of the cavity. The sizes of these coupling holes [diameter = 0.038” (=0.97 mm), or
∼ λ/6] have been optimized for the V-band, and the cavity side-wall was milled
down to a thickness of 0.015” (=0.38 mm, or ∼ λ/15) at the location of these holes.
Once again, these numbers are essentially the same as those reported in Mola et
al. [75]. The key point here is that this coupling never changes during rotation.
Therefore, the full 360◦ angle range may be explored, and with excellent mechanical
stability. Fig. 3–8 shows the typical random fluctuations in the cavity resonance
parameters for the TE01n mode during a complete 360◦ rotation of the end-plate:
the center frequency varies by no more than ±120 kHz (∼ 3 ppm, or 5% of the
62
resonance width); the Q-factor is essentially constant, to within ±1%; and the contrast, [A(f◦ )−Al ], fluctuates between 29 dB to 33 dB, corresponding respectively
to leak amplitudes of 3.5% and 2.2% of A(f◦ ). Another important consequence
of coupling through the side walls of the cavity involves selection rules for TM
cylindrical cavity modes. At the lowest frequencies, only the TE01 mode of the
V-band waveguide propagates. Thus, the microwave H̃1 field in the waveguide is
polarized parallel to the cavity axis, i.e., it is incompatible with the symmetry associated with the H̃1 patterns of the TM modes. Consequently, we do not observe, for
example, the TM111 mode. This offers added benefit, since the TM11n modes are
ordinarily degenerate with the TE01n modes, and steps have to be taken to either
lift these degeneracies, or to suppress the TM modes all together [75]. Here, we
simply do not couple to these modes at the lowest frequencies. Even at the highest
frequencies, we do not expect coupling to TM modes, provided the polarization of
the microwave sources is maintained throughout the waveguide.
The main design challenge in the development of a useful rotating cavity
system concerned the space constraints imposed by the high-field magnets at
the NHMFL; the specifications of each magnet system are listed in Table 3–1. A
prototype configuration was first developed, based on a 1” outer tube diameter.
This prototype was subsequently implemented in both magnet systems at UF,
and in the 25 T, 50 mm bore resistive magnets at the NHMFL. A picture of this
cavity system, which remains in use, is shown in Fig. 3–6(a). The major reason
for the large size of the cavity assembly is the large worm drive diameter (3/8” =
9.525 mm), which is determined by the smallest readily available commercial
components [90]; this 1st generation worm drive is made of nylon. A more compact
2nd generation rotating cavity was developed by machining a considerably smaller
custom worm drive (and gear) in-house. This cavity assembly, which is shown in
Fig. 3–6(b), is small enough to fit into a 3/4” diameter thin-walled stainless steel
63
(a)
Phase
90
120
60
30
150
180
0
330
210
300
240
270
A(f0 )-A ∞(dB)
Q(%)
f0(ppm)
(b)
3
f 0 =51.86335(GHz)
0
2
Q=21600
1
0
-1
33
32
31
30
29
0
50
•
•
100 150 200 250
Angle(degree)
300
Figure 3–8. Angle-dependence of the cavity resonant properties. (a) A polar plot
of the complex signal transmitted through the cavity as the frequency
is swept through the TE011 mode; the polar coordinate represents the
phase of the transmitted signal, while the radial coordinate corresponds
to the linear amplitude. The Lorentzian resonance is observed as a circle in the complex plane. Points close to the origin correspond to frequencies far from resonance, while the resonance frequency corresponds
to the point on the circle furthest from the origin. The resonance parameters are obtained from fits (solid line) to the data (open circles)
in the complex plane (see Mola et al. [75] for a detailed explanation of
this procedure). The average center frequency, f◦ , is 51.863 GHz, and
the Q factor is 21600. (b) Full 360◦ angle-dependence of the fluctuations in the resonance parameters for the TE011 mode: upper panel
− f◦ ; center panel − Q; and lower panel − difference between the amplitude on resonance [A(f◦ )] and the amplitude far from resonance
(the leak amplitude, Al ). These data illustrate the excellent reproducibility of the resonance parameters upon un-clamping, rotating,
and re-clamping the cavity (Reused with permission from Takahashi et
al. [81]. Copyright 2005, American Institute of Physics.).
64
tube, thus enabling measurements in the highest field 45 T hybrid magnet at the
NHMFL. Furthermore, this cavity is compatible with the 3 He probe constructed
for the 17 T Oxford Instruments superconducting magnet (and the 25 T resistive
magnet at the NHMFL), allowing for experiments at temperatures down to
500 mK. We note that the 1st and 2nd generation cavity assemblies may be
transferred relatively easily from one particular waveguide probe to another,
requiring only that the thermometry be unglued and re-glued to the stage using
GE varnish; all other connections are made with screws. In addition, we have
constructed extra parts for both designs, including several cavities and end-plates.
This enables preparation of a new experiment while an existing experiment is in
progress.
Finally, we discuss the two-axis rotation capabilities made available via a 7 T
Quantum Design (QD) magnet (see Table 3–1). The 7 T transverse QD system is
outfitted with a rotation stage at the neck of the dewar. A collar clamped around
the top of the waveguide probe mounts onto this rotation stage when the probe
is inserted into the PPMS flow-cryostat. The rotator is driven by a computer
controlled stepper motor, with 0.01◦ angle resolution. The motor control has
the advantage that it can be automated and, therefore, programmed to perform
measurements at many angles over an extended period of time without supervision.
In the 50 − 250 GHz range, the compact Schottky diodes can be used for microwave
generation and detection. These devices are mounted directly to the probe, and are
linked to the MVNA via flexible coaxial cables (feeding the diodes with a signal in
the 8 − 18 GHz range). Therefore, the waveguide probe can rotate with the source
and detector rigidly connected, while the vacuum integrity of the flow cryostat
is maintained via two sliding ‘o’-ring seals at the top of the dewar. In fact, this
arrangement rotates so smoothly that it is possible to perform fixed-field cavity
perturbation measurements as a function of the field orientation, as has recently
65
been demonstrated for the organic conductors α-(BEDT-TTF)2 KHg(SCN)4 [46]
and (TMTSF)2 ClO4 . (See Chap. 4.) We generally use the stepper-motor to control
the polar coordinate, while mechanical control of the cavity end-plate is used to
vary the plane of rotation, i.e., the azimuthal coordinate.
Due to the extremely precise control over both angles, and because of the need
for such precision in recent experiments on single-molecule magnets which exhibit
remarkable sensitivity to the field orientation [73], we have found it necessary
to make two modifications to the 3/4” rotating cavity probe for the purposes of
two-axis rotation experiments. The first involves constraining the cavity assembly
and the waveguides within a 3/4” thin-walled stainless steel tube which is rigidly
connected to the top of the probe. This tube reduces any possible effects caused by
magnetic torque about the probe axis, which could mis-align the cavity relative to
the rotator. The second modification involves attaching a spindle on the under-side
of the end-cap. This spindle locates into a centering ring attached to the bottom of
the PPMS flow-cryostat, thus preventing the waveguide probe from rotating off-axis
(note that the inner diameter of the cryostat is 1.10” as opposed to the 3/4” outer
diameter of the probe).
3.4
Model of the Resonant Cavity
The characteristics of a cavity resonator are affected by many factors,
e.g., shape, dimensions and material of the cavity. In the ideal case in which
the cavity is made by a piece of perfect conducting material and filled with a
lossless dielectric with µ and ², the cavity modes are perfectly discrete. The shape
of each resonance is characterized by the delta function, i.e., infinite height and
no width. The resonant frequency for each mode is easily calculated by solving
Maxwell’s equations with boundary conditions for the geometry of the cavity. For
the cylindrical cavity we use, the resonance conditions are given by the following
66
equation [62],
fmnp
ωmnp
1
=
= √
2π
2π µ²
r
(
xmn 2
pπ
) + ( )2 ,
R
d
(3–1)
where m, n and p are integers (n and p >0). R is the radius and d is the height
of the cylindrical cavity. xmn is the n-th root of the 1st derivative of the Bessel
0
function, Jm
(x), and the n-th root of the Bessel function, Jm (x), for the TEmnp
and the TMmnp modes respectively. The electromagnetic field configurations for the
TEmnp modes are given below.
pπH0
pπ
0 xmn
(
r) cos(mφ) cos( z),
√ Jm
ωmnp d µ²
R
d
(3–2a)
r
)
mpπH0 Jm ( xmn
pπ
R
=
sin(mφ) cos( z),
√
xmn r
ωmnp d µ²
d
R
(3–2b)
Hrmnp (r, φ, z) =
Hφmnp (r, φ, z)
xmn H0
xmn
pπ
r) cos(mφ) sin( z),
√ Jm (
ωmnp R µ²
R
d
r
r
)
µ Jm ( xmn
pπ
Ermnp (r, φ, z) = −m
H0 xmnRr sin(mφ) sin( z),
²
d
R
r
µ
pπ
0 xmn r
Eφmnp (r, φ, z) = −
H 0 Jm
(
) cos(mφ) sin( z),
²
R
d
(3–2d)
Ezmnp (r, φ, z) = 0.
(3–2f)
Hzmnp (r, φ, z) =
(3–2c)
(3–2e)
In reality, the cavity is not perfect. For instance, the cavity will be made from
several pieces of finitely conducting materials. In particular, the contact between
the various pieces will have resistive, capacitive and inductive losses. The cavity
also need to have coupling holes to connect to the microwave source and detector.
This breaks the symmetry of the cavity modes. All of these factors influence the
characteristics of the cavity resonator. It is too complicated to model a realistic
cavity. Therefore, in order to simplify the model, an equivalent model is often used.
Fig. 3–9(a) shows the equivalent RLC circuit. The impedance of the circuit is given
by [91],
i
ω2
Ze = R − iωL +
= R − iωL(1 − 02 ),
ωC
ω
(3–3)
67
(b)
(a)
R
Pav(ω)/Pav(ω0)
1.0
L
C
0.5
Γ
0
ω0
ω
Figure 3–9. A simple description of the resonant cavity. (a) An equivalent RLC
circuit representing the cavity. (b) A Lorentzian resonance produced by
the RLC circuit; ω0 denotes the resonance frequency; Γ denotes the full
width at half maximum of the resonance.
where the complex quantity Ze is the so-called impedance, the real quantities
R, L and C are the resistance, inductance and capacitance respectively, and
√
ω0 = 1/ LC. The circuit with the impedance given above has a resonant peak in
the average absorbed power Pave by the resistance R, which is given by,
2
Pave = Irms
R=
2
Vrms R
Vrms
Rω 2
=
.
e2
Rω 2 + L2 (ω 2 − ω02 )
|Z|
(3–4)
√
Thus ω0 = 2πf0 = 1/ LC is the resonance frequency. In the case of ω0 À |ω − ω0 |,
Eq. 3–3 is written,
Ze ' −2iL(ω − ω
f0 ),
(3–5a)
where the complex resonant frequency ω
f0 is defined as,
Γ
ω
f0 = ω0 − i ,
2
(3–5b)
and Γ = R/L. Using the above impedance, the power is given by
Pave =
2
(R/L2 )
P0
Vrms
= Γ 2
.
Γ 2
2
( 2 ) + (ω − ω0 )
( 2 ) + (ω − ω0 )2
Fig. 3–9(b) shows the power as a function of frequency. The resonance peak
is clearly seen at the resonant frequency ω0 . Γ represents the full width of the
(3–6)
68
resonance at half-maximum (FWHM) shown in Fig. 3–9(b). The FWHM is often
expressed by using the quality factor (Q), defined by,
Q=
ω0
ω0 L
=
.
Γ
R
(3–7)
In our setup, the Q-factor for an unloaded cavity can be up to 25000 [75].
When a small (perturbative) specimen is introduced into the cavity, the
characteristics of the resonance change slightly. This may be expressed by a shift
of the resonant frequency from ω0 to ωs , and a change of Γ from Γ0 to Γs (or
the Q-factor from Q0 to Qs ). The complex impedance is therefore changed from
ω
f0 to ωes . By using Eq. 3–5, the difference in the real and imaginary part of the
complex impedance, which represents the dissipative and dispersive response of the
specimen, is given by,
fs − Z
f0 ≈ ig0 L∆ωes
∆Ze = Z
= ig0 L(ωs − ωs −
iω0 1
1
(
−
))
2 Qs Q0
i
= ig0 L(∆ω − ∆Γ),
2
(3–8)
where we introduced a geometrical factor g0 which depends on the experimental
geometry (the location and size of the sample in the cavity) [56, 91]. Defining
∆Ze = R + iX (R = sample resistance and X = sample reactance), the sample
response is given by,
∆Γ
ω0 1
1
R
= (
−
)=
,
2
2 Qs Q0
g0 L
(3–9a)
and
∆ω = ωs − ω0 =
X
.
g0 L
(3–9b)
69
3.5
Positioning Low Dimensional Conductors and Superconductors in
the Cylindrical Cavity
The position of the specimen placed in the cavity is determined by several
factors. These include: the shape and size of the sample; the coupling strength with
the microwaves; favorable polarization of the microwaves for a given experiment;
the choice of whether to use the electric or magnetic component of the microwaves;
the orientation of the sample with respect to a dc magnetic field for magnetooptics etc... This is particulary important when studying anisotropic materials.
In the case of the cavity perturbation technique, the polarization of the induced
currents is often different from the polarization of the microwaves. In this section,
considering the case of Q2D materials with a plate-like shape, we briefly introduce
two cases, in-plane and interlayer measurements. During this discussion, it is
assumed that the least conducting direction of the sample is perpendicular to
sample platelet, and that both skin depths δk and δ⊥ are smaller than the sample
dimensions (skip depth regime) where δ⊥ and δk represent skin depth for the least
and the good conducting direction respectively. More details are found in Hill [33].
3.5.1
In-plane Measurements.
In order to measure the in-plane properties, the induced current has to be
polarized along only the in-plane direction, otherwise the interlayer component of
the current dominates the microwave response, as explained in the next subsection.
Fig. 3–10(1) a) illustrates the case of a measurement using the electric component
of the microwaves, with the ac electric field applied in-plane. However, the current
is induced not only along the in-plane direction, but also the interlayer direction.
This is because the current flow is limited to the surface, i.e., within the skin
(penetration) depth δ (λ) of the perimeter of the sample, and this flow is disrupted
in order to screen the electric field within the interior of the sample. Thus, this
configuration is not ideal for in-plane measurements. Another possibility is to
70
z
(2) Inter layer measurements
(1) In-plane measurements
y
a)
x
b)
Figure 3–10. Schematic diagram illustrating the various possibilities for exciting
in-plane and interlayer currents in a Q2D plate-like sample. (1) a)
In-plane currents induced by an ac electric field (Reprinted figure with
permission from Hill [33]. Copyright 1997 by the American Physical
Society.). b) In-plane currents induced by an ac magnetic field. We
use the latter configuration for the in-plane measurements (Reprinted
figure with permission from Hill [33]. Copyright 1997 by the American
Physical Society.). (2) Interlayer measurements: a) current induced
by an ac electric field (Reprinted figure with permission from Hill [33].
Copyright 1997 by the American Physical Society.). b) current induced by an ac magnetic field. This latter is the most widely used
configuration (Reprinted figure with permission from Hill [33]. Copyright 1997 by the American Physical Society.).
use the magnetic component of the microwaves, as shown in Fig. 3–10(1) b). In
this case, only current parallel to the conducting xy-plane is induced. Thus, it is
possible to perform the in-plane measurement with this configuration. However it is
impossible to distinguish between σxx and σyy , only the average is measured.
3.5.2
Interlayer Measurements.
This is the most relevant case for our experiments. Since the skin depth or
penetration depth for low dimensional conductors and superconductors is highly
anisotropic, the absorption of microwaves is also highly anisotropic. As a result,
the microwave response can be dominated by the absorption along the interlayer
direction. Let’s examine example cases for anisotropic conductors. For the skin
depth regime, the microwave absorption is simply related to the surface area and
71
skin (penetration) depth [92], for currents flowing across faces of the sample which
are parallel (k) and perpendicular (⊥) direction to the in-plane direction, i.e.,
P⊥
δ⊥ a⊥
=
,
Pk
δk ak
(3–10)
where a⊥ and ak are the surface areas parallel to the least and the good conducting
directions, respectively. In the case of the Q1D conductor, (TMTSF)2 ClO4 ,
sample crystals are typically needle shaped so that ak ∼ 0.4 mm2 and a⊥ ∼ 0.1
mm2 . The skin depth is estimated to be δ⊥ ∼ 100 µm and δk ∼ 1 µm at Helium
temperatures at a microwave frequency. As a result, P⊥ /Pk =20. In the case of
the Q2D conductor, κ-(ET)2 Cu(NCS)2 , P⊥ /Pk is typically 6-100 at a microwave
frequency. Thus, the energy dissipation P⊥ along the least conducting direction is
dominant in both cases. One can therefore choose various configurations for the
polarization of the incident microwaves. Here we illustrate two cases which are
likely to maximize the interlayer current. Fig. 3–10(2) a) shows the situation in
which the electric component is applied perpendicular to the in-plane direction.
The current is induced from one face of the sample platelet to the other via
the edge of the sample. Fig. 3–10(2) b) shows magnetic field excitation. The ac
magnetic component is applied along the in-plane direction in this case. The
current is induced over the sample surface again. However, there is no displacement
current in this configuration. In fact, this configuration is the most widely used for
angle-dependent magneto-optical studies of low dimensional materials.
3.5.3
Configuration for Interlayer Measurements Using the Magnetic
Component of the Microwaves
Next we explain actual experimental geometries for angle-dependent magnetooptical studies of Q2D metals. Configurations are specified for interlayer measurements using the magnetic component of the microwaves.
72
For the case of the TE011 mode in the cylindrical cavity, which we use most
often, two different examples of the position of the sample in the cavity are shown
in Fig. 3–11. In the case of TE011 mode, the antinodes of the magnetic component
of the microwaves are located at the center of the cavity, in the middle of the side
wall, and at the halfway point between the center and edge of the end plate, as
shown in Fig. 3–11. Thus, in order to have a strong coupling, we usually place the
sample on a quartz pillar [Fig. 3–11(a)], on the end plate [Fig. 3–11(b)], or on the
side wall (not shown). These configurations are different in terms of the angledependent magneto-optical studies. For example, we can consider the combination
of a vertical magnetic field and a transverse cavity, which is the case for the
rotating cavity with an axial magnet. In the case of the quartz pillar configuration,
as shown in Fig. 3–11(a), the dc magnetic field can be applied from parallel to
perpendicular to the sample platelet while the sample is rotated along the quartz
rod direction using the rotating cavity. On the other hand, in the case of the end
plate configuration, the magnetic field is always along the sample platelet while the
sample is rotated.
3.6
Microwave Response of Low dimensional Conductors and
Superconductors
In this section, we derive an expression for the microwave responses due to
the interlayer properties of the sample. i.e., sample impedance, resistance and
e and H
e with
reactance. The problem is mainly to solve Maxwell’s equations for E
proper boundary conditions for the sample. There are many papers which discuss
different cases [56, 91, 93, 94]. Here we introduce two important cases, i.e., skin
depth regime and depolarization regime.
3.6.1
Skin Depth Regime
Here we consider good conductors and superconductors, where the skin
(penetration) depth is smaller than the thickness of the sample, are said to be in
73
(b)
(a)
Coupling hole
microwaves
Sample
R
~
Eac
|H(r)|
~
R
0
Hac
sin(π z/d)
|H(z)|
d
J0'(x01 r/R)
B
d
0
Figure 3–11. Positioning of the sample in the TE011 cavity mode. The dotted and
dash lines represent the electric and magnetic components of the mieac and H
e ac respectively. The sample is plate-like,
crowaves, denoted E
which is a typical shape for Q2D materials. The magnetic component
of the microwaves has maxima in the center of the cylindrical cavity,
at the halfway point between the center and the edge of the end plate,
and at the middle of the side wall. In these diagrams, the sample is
mounted at the antinode of the ac magnetic field, and the ac field
is parallel to the plane of the sample platelet. We consider namely
the case where the dc magnetic field is perpendicular to the cavity
axis. This configuration corresponds to employing the rotating cavity
with the axial magnet. In (a) the sample is positioned at the center
of the cavity using a quartz pillar. Using the rotating cavity, the dc
magnetic field can be rotated between the interlayer and the in-plane
directions. In (b) the sample is positioned on the end plate. Using
the rotating cavity, the dc magnetic field can be rotated within the
in-plane directions.
74
the so-called skin depth regime. As discussed in Sec. 3.5, the main contribution
to the absorption comes from the induced currents along the interlayer direction.
We can simplify the problem by considering the electrodynamics of a semi-infinite
medium. The flat conducting sample (the xz-plane) is set at y = 0 so that the
medium for y >0 has the dielectric constant ²1 , permeability µ1 and conductivity
σ1 , and the medium for y <0 is vacuum. The sample surface is placed at an
antinode of the ac magnetic field. We now calculate the magnetic field inside
the sample, assuming the magnetic field is along the x-axis and has harmonic
iωt
g
time dependence, i.e., (H
. The magnetic field inside the sample is
ac )x = H0 e
calculated by Maxwell’s equations,
∇ · D = ρ,
(3–11a)
∂H
= J,
∂t
∂B
∇×E+
= 0,
∂t
∇×H−
(3–11b)
(3–11c)
∇ · B = 0,
(3–11d)
J = σ1 E.
(3–12)
and Ohm’s law,
Using the above equations with ρ = 0, one obtains the following differential
equation,
∇2 H − µ1 ²1
∂ 2H
∂H
− µ1 σ 1
,
2
∂t
∂t
(3–13)
g
and its solution, with the boundary condition H(y = 0) = H
ac , is given by
e t) = H0 eiqey eiωt x
b,
H(y,
(3–14)
where the complex wavevector is given by
r
qe = ω
µ1 σ1
√
² 1 µ1 − i
= ω µ1
ω
r
²1 − i
σ1
.
ω
(3–15)
75
For convenience, the following definition of the complex dielectric constant e
² and
conductivity σ
e are also used frequently:
e
² = ²1 + i
σ1
≡ ²1 + i²2 ,
ω
(3–16a)
and the complex conductivity σ
e is defined as
σ
e
e
² ≡ ²0 + i .
ω
(3–16b)
Therefore, the real and imaginary parts of the conductivity become,
σ1 = ω²2 ,
(3–16c)
σ2 = ω(²0 − ²1 ).
(3–16d)
and
Additionally, the complex wavevector qe is also expressed as,
e = ω( n + i k ),
qe = ω N
c
c
(3–17)
e is the complex refractive index, n is the real refractive index and k is the
where N
√
imaginary refractive index. c is the speed of light c = 1/ µ0 ²0 .
The induced electric field is calculated easily using Eq. 3–11b,
e t) = − µ1 ω H0 eiqey eiωt y
b.
E(y,
qe
fs ≡ E/
e H,
e
Thus one can obtain the impedance of the sample, Z
s
µ
ω
i2 µ1
1
fs = −
=
.
Z
qe
²1 − i σω1
(3–18)
(3–19)
Assuming ²1 À ²0 , which is often the case for good conducting materials, we obtain
the well-known expression for the surface impedance,
r
iµ1 ω
fs ∼
= Rs + iXs .
Z
=
σ1 − iσ2
(3–20)
76
As discussed in Sec. 3.4, the microwave response is related to Rs and Xs , so that
we can extract both σ1 and σ2 by using the conversions,
Rs Xs
,
(Rs2 + Xs2 )2
(3–21a)
Xs2 − Rs2
.
(Rs2 + Xs2 )2
(3–21b)
σ1 = 2µ1 ω
and
σ2 = µ1 ω
Moreover, rewriting the above equations, one can obtain,
sp
r
σ12 + σ22 − σ2
ωµ1
∆Γ ∝ Rs =
,
2
σ12 + σ22
r
∆ω ∝ Xs = −
ωµ1
2
(3–22a)
sp
σ12 + σ22 + σ2
.
σ12 + σ22
(3–22b)
where ∆Γ = 1/Qs −1/Q0 is related to the change of the Q-factor, and ∆ω = ωs −ω0
is the change of the cavity resonant frequency.
In the case of superconducting materials, we need to consider the response
of the normal quasi-particles and the Cooper-pairs. This is often treated by using
a general two fluid model, in which σ
e represents the conductivity due to normal
quasiparticles, and σes is the superfluid contribution to the conductivity.
σ
e = σ1 − iσ2 + σ1S − iσ2S
= σ1 − iσ2 +
πns e2
ns e2
δ(ω)
−
i
,
m∗
m∗ ω
(3–23)
where ns is superfluid density, and the penetration depth λ is also used for
ns e2 /m∗ = (µ1 ωλ2 )−1 . In a simple case (T ∼ 0 and a s-wave superconductor),
at microwave frequencies, σS1 = 0 so that the complex conductivity often becomes,
σ
e w σ1 − σ2S =
i
,
µ1 ωλ2
(3–24)
77
which assumes σ2S À σ1 . Thus, the microwave response is given by the following
expression,
r
∆Γ ∝ Rs w
s
µ1 ω
2
and
r
∆ω ∝ Xs w
3.6.2
1
σ12
= µ21 ω 2 σ1 λ3 ,
3
2σ2S
2
(3–25a)
µ1 ω
= µ1 ωλ.
σ2S
(3–25b)
Metallic Depolarization Regime
When the skin depth is larger than the thickness of the sample (a), the
electrodynamics are said to be in the so-called metallic depolarization regime
represented by qea À 1 [56]. This regime can be understood easily because the
transmitted microwaves induce homogenous currents inside the sample, so that
the microwave loss can be calculated in the quasi-static limit. As a result, the
conductivity is simply related to the sample resistance, namely,
σ1 w
1
1
∝ .
ρ1
R
(3–26)
Thus, using Eq. 3–9a, one can obtain the following expressions,
∆Γ ∝ R ∝
3.7
1
.
σ1
(3–27)
Measurement of the Change of the Complex Impedance Ze
In the experiments discussed here, we focus on investigating relative changes
of the sample properties by changing external conditions, e.g., magnetic field,
the orientation of the magnetic field, temperature and frequency. In our setup,
experiments are performed using the following three methods: 1) measuring
the cavity center resonance frequency ω0 and Γ (or Q-factor); 2) measuring the
amplitude and phase of the transmitted signal continuously at a fixed frequency,
the so-called frequency-lock method; 3) and measuring the amplitude and frequency
continuously at a fixed phase, the so-called phase-lock method. Method 1) is the
78
e since ∆Γ and
most straightforward for determining the complex impedance Z,
∆ω are measured directly. However the estimation of ∆Γ and ∆ω is a rather
time-consuming process, which involves sweeping the frequency, subtracting a
background and fitting the cavity resonance. Therefore, we rarely use method 1).
Instead, we mainly use methods 2) and 3), since they allow a much faster data
acquisition rate.
The cavity resonance can be represented by the Lorentzian function seen
in Sec. 3.4. Similar to Eq. 3–6, the amplitude and phase are expressed by the
following equations,
A(ω) = A0 q
1
ω0 2
) + (ω − ω0 )2
( 2Q
1
= A0 q
( Γ2 )2
+ (ω − ω0
,
(3–28a)
)2
and,
tan φ(ω) = −
ω − ω0
ω − ω0
=− Γ ,
ω0
( 2Q )
(2)
(3–28b)
where Γ = ω0 /Q, ω0 is the cavity resonance frequency, and Q is the quality-factor.
The external conditions are now changed, e.g., by applying a magnetic field. The
changes of the microwave response by the sample are then represented by shifts of
Γ and the resonance frequency ω0 , i.e., Γ → Γ + ∆Γ and ω0 → ω0 + ∆ω0 . Thus the
amplitude and phase of the microwave response also changes,
A(ω) → A(ω) = A0 q
1
)2
( Γ+∆Γ
2
+ (ω − ω0 − ∆ω0
,
(3–29a)
)2
and,
tan φ(ω) → tan φ(ω) = −
ω − ω0 − ∆ω0
.
( Γ+∆Γ
)
2
(3–29b)
Fig. 3–12 shows a schematic view of these changes. The amplitude at the resonance
frequency becomes smaller because Γ → Γ + ∆Γ. The change also appeared in the
phase, i.e., the position at φ(ω) = 0 is shifted to at φ(ω0 + ∆ω0 ) = 0.
79
Α(ω)
(a)
φ(ω)
ω0 + ∆ω0
(b)
Γ
ω0
0
Γ + ∆Γ
ω0
ω0 + ∆ω0
Figure 3–12. Typical changes in the amplitude and phase of the microwaves transmitted through the cavity. When the sample properties change by
varying the external conditions, changes appear in the microwave
response of the cavity. As a result, the amplitude and phase are
changed: (a) Typical changes in the amplitude of the microwaves,
represented by ∆Γ and ∆ω0 ; (b) typical changes in the phase of the
microwaves, represented by ∆Γ and ∆ω0 .
3.7.1
Frequency-lock Method
We perform the frequency-lock using a Phase Matrix/EIP 575B frequency
counter [95]. Replacing ω = ω0 in Eq. 3–29a and Eq. 3–29b, the amplitude and
phase during the frequency-lock are given by
1
A(ω0 ) = A0 q
)2 + (∆ω0 )2
( Γ+∆Γ
2
,
(3–30a)
and,
tan φ(ω0 ) = −
∆ω0
.
Γ+∆Γ
( 2 )
(3–30b)
Thus, rewriting the above equations, the microwave response is given by
∆Γ =
2A0
1
p
− Γ,
2
A(ω0 ) tan φ(ω0 ) + 1
(3–31a)
and
∆ω0 =
A0
q
A(ω0 )
1
1
tan2 φ(ω0 )
+1
.
(3–31b)
When the phase shift is small, tan φ → 0,
∆Γ w
2A0
− Γ,
A(ω0 )
(3–32a)
80
and
∆ω0 w
A0
φ(ω0 ).
A(ω0 )
(3–32b)
Thus, when the microwave response has small phase shifts, ∆Γ depends only on
a change of the amplitude. Eq. 3–32a also shows the change of the amplitude is
linear, i.e., A(ω0 ) = A0 + δA so [A(ω0 )]−1 = 1/A0 [1 − δA/A0 ]. The frequency
shift is therefore proportional to φ(ω0 ). On the other hand, this method can be
problematic when the phase shift is huge, because the frequency becomes offresonant, and ∆Γ starts to depend on both A(ω0 ) and φ(ω0 ). Thus the phase and
amplitude become mixed, i.e., dissipation and dispersion are mixed. Empirically,
the frequency-lock is not appropriate when the phase shift is more than 10 degrees.
However we can correct this problem by performing a vector analysis with Eq. 3–31
or a vector fit [96]. Furthermore we can also avoid the problem using the phase-lock
method.
3.7.2
Phase-lock Method
The phase-lock method involves stabilization of the frequency using the phase
of the microwaves transmitted through the cavity. If the stabilization is perfect,
and the chosen cavity mode is well-separated from the other cavity modes, standing
waves and other sources of dφ/dν, this method is the most useful. However, the
phase-lock is less stable than the frequency lock so that the signal we obtain by this
method often has a smaller signal-to-noise ratio. Thus, we usually begin by using
the frequency lock. Then we perform the phase-lock measurement if we encounter
problems discussed in the previous subsection. As seen in Fig. 3–12, the phase lock
always keeps φ(ω0 + ∆ω0 )=0. Thus ω=ω0 +∆ω0 . Putting φ = 0 and ω=ω0 +∆ω0
into Eq. 3–29a, the amplitude becomes,
A(ω) = A0
2
.
Γ + ∆Γ
(3–33)
81
Thus, ∆Γ is given by,
∆Γ =
2A0
− Γ.
A(ω)
(3–34)
Therefore, this expression is the same as Eq. 3–32a for the frequency-lock. By
recording the frequency with a frequency counter, we can also measure the frequency shift ω + ∆ω.
3.8
Summary
In this chapter, we explained our experimental techniques. In particular, for
the purpose of angle-dependent high-field microwave spectroscopy, we developed
a rotating cavity which is compatible with several high-field facilities. In the
later part of the chapter, we explained the electrodynamics of low dimensional
conductors and superconductors appropriate to our experimental configuration.
CHAPTER 4
PERIODIC ORBIT RESONANCES IN QUASI-ONE-DIMENSIONAL
CONDUCTORS
The results presented in this chapter can be found in the articles, Periodicorbit resonance in the quasi-one-dimensional organic superconductor (TMTSF)2 ClO4 ,
S. Takahashi, S. Hill, S. Takasaki, J. Yamada and H. Anzai, Physical Review B 72
024540 (2005), and Are Lebed’s Magic Angles Truly Magic?, S. Takahashi, A.
Betancur-Rodiguez, S. Hill, S. Takasaki, J. Yamada and H. Anzai, submitted to the
ISCOM conference proceedings.
4.1
The Quasi-one-dimensional Conductor, (TMTSF)2 ClO4
The organic metal (TMTSF)2 ClO4 belongs to the family of quasi-onedimensional (Q1D) Bechgaard salts [2], having the common formula (TMTSF)2 X.
TMTSF is an abbreviation for tetramethyl-tetraselenafulvalene, and the anion X
is AsF6 , ClO4 , PF6 , ReO4 etc. The (TMTSF)2 X series have been the most extensively studied organic materials. The reason may be the extremely rich phases
for (TMTSF)2 X. This includes unconventional metallic states, spin-density-wave
(SDW) states, field-induced spin-density-wave (FISDW) states, possible spin-triplet
superconducting states and so on. Unlike other one-dimensional materials, The
(TMTSF)2 X compounds tend to be good conductors at room temperature although the (TMTSF)2 X eventually meet the metal-insulator transition at a low
temperatures, except for (TMTSF)2 ClO4 , which becomes a superconductor. The
metal-insulator transition in the (TMTSF)2 X compounds is seen in the resistivity
of Fig. 4–1(a). This transition is explained by the nesting of the FS. In the case
of the (TMTSF)2 X compounds, the transfer energy to the b-direction is finite,
so that the FS is not a perfect flat plane, and the nesting is not as strong as the
82
83
a)
b)
LL
2D FL
Temperature (kelvin)
100
X = PF6
ClO4
10
3D FL
1
SDWI
SC
0.1
Pressure
Temperature dependence of
resistivity in (TMTSF)2X
5 kbar
Phase diagram for (TMTSF)2X
Figure 4–1. Electronic properties of the (TMTSF)2 X. (a) Temperature dependence of the resistance in (TMTSF)2 X. With the exception of
(TMTSF)2 ClO4 , all samples undergo a metal-insulator transition
at low temperatures (Reprinted from Bechgaard et al. [97], Copyright 1980 with permission from Elsevier). (b) The phase diagram
of (TMTSF)2 X as a function of the pressure. The pressure may be
controlled by replacing the anion X and applying external pressure.
LL denotes a Luttinger liquid regime, 2D (3D) FL denotes a 2- (3-)
dimensional Fermi liquid regime, SDWI represents spin-density-wave
insulating states and, SC is a superconducting state. Varying pressure, high temperature and ground states are changed. For example,
(TMTSF)2 PF6 exhibits a SDWI ground state at ambient pressure.
However, under 6 kbar, the ground state of (TMTSF)2 PF6 becomes
superconducting (See Jérome [98], Moser et al. [99] and Dressel [100]
for the details of the phase diagram.).
84
1D system as we saw in Sec. 1.4. This imperfectly nested FS therefore shifts the
instability to a lower temperature. In the case of the (TMTSF)2 X compounds, this
nesting causes the SDW instability, so that the ground state is the SDW insulating
state. Moreover, the anion X affects the nature of the SDW instability, since it
plays a role in determining the spacing of the TMTSF molecules. By replacing the
anion X, one can therefore change the spacing, and then control the coupling of the
TMTSF molecules along the b and c-directions, i.e., one can change the transfer
energies tb and tc . This change may be small compared with the Fermi energy EF ,
but it is significant enough to shift the critical temperature TM I for the SDW state.
This is actually seen in Fig. 4–1(a). By replacing the anion X=BF4 to PF6 , TM I
shifts from 80 to 15 K. This may be understood in terms of a change of the dimensionality, and can be probed by detailed investigation of the FS. More importantly,
by applying pressure, one can control the dimensionality more systematically. For
instance, Fig 4–1(b) shows a proposed phase diagram of (TMTSF)2 X as a function
of the pressure. The pressure may be controlled by replacing the anion X and applying external pressure. Varying pressure, high temperature and ground states are
changed. For instance, The TM I in (TMTSF)2 PF6 decreases rapidly under pressure.
Above 8 kbar, the SDW insulating phase disappears, and then the superconducting
phases appears as a ground state. In fact, this was the first superconductivity
discovered in the organic systems in 1979 [101].
Among the Bechgaard salts, (TMTSF)2 ClO4 is the only material which
has a superconducting state under an ambient pressure. (TMTSF)2 ClO4 has a
structural phase transition of the ClO4 anions at 24 K. This due to the ordering
of the non-centrosymmetric tetrahedral ClO4 anion, so-called anion ordering. The
anion ordering is extremely sensitive to the cooling process of the crystal. X-ray
scattering studies reveal that a superlattice structure with Q = (0, 1/2, 0) is
formed at the anion ordering temperature (TAO = 24 K)[102, 103]. This occurs
85
when (TMTSF)2 ClO4 is cooled slowly at around 24 K (< 0.1 K/min). Then
(TMTSF)2 ClO4 has a metallic and superconducting state in the low temperature
regime, the so-called relaxed state. On the other hand, when (TMTSF)2 ClO4
is cooled rapidly (> 50 K/min), the anion is disordered. This is the so-called
quenched state. We have investigated the cooling process dependence of the
electronic properties of (TMTSF)2 ClO4 , especially in terms of its influence on the
superconductivity. This investigation is introduced in Chap 5.
The crystal structure of (TMTSF)2 ClO4 shown in Fig. 1–2 is triclinic. The
lattice parameters in the relaxed state are a = 7.083 Å, b = 15.334 Å, c = 13.182 Å,
α = 84.40◦ , β = 87.62◦ and γ = 69.00◦ . In general, the shape of the (TMTSF)2 ClO4
single crystal is needle-like, as shown in Fig. 4–2(a). The most conducting direction
is the needle direction (the a-axis). For convenience, the b0 , c0 and c∗ axes are often
employed. The b0 (c0 ) axis is defined as the projection of the b- (c-) axis onto the
plane perpendicular to the a-axis, i.e., b0 = b sin γ and c0 = c sin β. The c∗ -direction
is defined as the direction perpendicular to the ab-plane, i.e., c∗ = c0 sin α0 .
Similarly, The b∗ -direction is defined as the direction perpendicular to the ac-plane,
i.e., b∗ = b0 sin α0 where α0 is the angle between b0 and c0 . α∗ is also often used for
the angle between b∗ and c∗ . These angles α∗ and α0 are simply expressed by
α∗ = π − α0 = cos−1 [
cos β cos γ − cos α
].
sin β sin γ
(4–1)
(TMTSF)2 ClO4 is a 3/4-filled system similar to other (TMTSF)2 X. The Fermi
surface estimated by band calculations reveals that the FS consists of four sheets,
as shown in Fig 4–2(b), because of the superlattice structure caused by the anion
ordering. The planar FS has a small warping due to small transfer energies along
the b0 and c0 -directions.
As shown in Sec. 1.4, (TMTSF)2 ClO4 has a FISDW phase in high magnetic
field. Since this FISDW phase is caused by the enhancement of the nesting
86
b) Fermi Surface
a) Sample View c*
c*
c‘
c
b'
α'
β
Conducting
axis
a
b'
γ
b
(TMTSF)2ClO4 crystal
a
Figure 4–2. Illustration of crystal axes for (TMTSF)2 ClO4 . (a) View of a single crystal sample of (TMTSF)2 ClO4 . The shape of the sample is
needle-like. The most conducting direction is the needle direction. The
b0 (c0 )-direction is defined as a projection of the b (c)-axis onto the
plane perpendicular to the a-axis. The c∗ -direction is defined as the
direction perpendicular to the ab-plane. Similarly, The b∗ -direction is
defined as the direction perpendicular to the ac-plane. (b) The FS of
(TMTSF)2 ClO4 . Because of the superlattice structure, the FS consists
of four planar sheets instead of two. This sheet has a small warping
due to the small transfer energies in the b- and c-directions.
87
property along the ab-plane (See Sec. 1.4), the critical field of the FISDW BF ISDW
is simply expressed by
1
BF ISDW
∝ | cos θ|,
(4–2)
where θ is an angle between the c∗ -axis and the magnetic field.
Like (TMTSF)2 PF6 , (TMTSF)2 ClO4 shows unusual dc transport properties
in the metallic state via angle-dependent magnetoresistance oscillations (AMRO),
e.g., the resistance shows sharp dips when the field is applied along the so-called
Lebed magic-angles [26, 27, 51]. Moreover, recent studies have shown that the
Nernst effect also becomes anomalous at these angles [104]. Although many
theories for Lebed’s magic-angle effect have been proposed, the subject remains
controversial. These theories include the FISDW instability [51], proximity effects
of the FISDW phase [105], commensurability effects on the FS [42, 106], electronelectron interactions [107], field induced confinements [108], hot spot effects [109]
etc. Lebed’s original theory predicts resistance resonances at field-independent
angles, θpq , related to the crystal structure,
tan θpq =
pb0
− cot α∗ ,
qc∗
(4–3)
where p and q are integers, b0 , c∗ and α∗ are lattice constants. These θpq are the
so-called Lebed magic angles. Fig. 4–3 shows the AMRO data taken by Kang et
al. [50]. The dips indicated by arrows correspond to the Lebed magic angles. The
index number of the dips denotes the value of p/q. One of the most controversial
issues concerns whether the Lebed magic angles are truly magic, or in other words,
whether the Lebed effect is a fixed angle effect. In order to answer this question,
we have employed a high-frequency (microwave) magneto-transport technique
to study the Lebed effect [70]. The reason for doing so is because semiclassical
considerations of the Lebed effect, which we employed in Chap. 2, show that the
AMRO condition differs from Eq. 4–3 when the measurement frequency becomes
88
8.0 T
7.5 T
7.0 T
6.0 T
5.0 T
4.0 T
Figure 4–3. The dc AMRO experiment in (TMTSF)2 ClO4 . The figure was provided by W. Kang. See Kang et al. [50] for the detail of the experiment. The measurements were carried out at fixed magnetic fields, by
changing the orientation of the magnetic field in the c∗ b0 -plane. The
traces from the highest to the lowest resistance represent data at 8 T,
7.5 T and then 7 T to 1 T in 1 T steps. The arrows denote the positions of the dips in the resistance which correspond to the Lebed magic
angles. The notation is explained in TABLE 4–1.
89
comparable to the typical frequency with which quasiparticles cross the Brillouin
zone under the influence of an applied magnetic field.
4.2
Semiclassical Description of the Periodic Orbit Resonance and the
Lebed Effect
In Sec. 2.3, we explained the periodic orbit resonance (POR) for a simple
Q1D FS. The POR in this case was a single resonance in the conductivity because of the simple sinusoidal corrugation. However this may not be the case for
(TMTSF)2 ClO4 , since the conducting properties are slightly two-dimensional,
i.e., tb and tc are finite. The FS of (TMTSF)2 ClO4 consists of multiple corrugations. In this case, the observed POR may consist of multiple resonances, as
we saw in the example of α-(BEDT-TTF)2 KHg(SCN)4 in Chap. 2 [46]. Here we
consider a more general expression of the resonance condition for the POR using an
appropriate model for (TMTSF)2 ClO4 .
(TMTSF)2 ClO4 has a low symmetry (triclinic) crystal structure. For convenience, in order to describe the corrugation on the FS, we consider an oblique
lattice for the b0 c∗ -plane, as shown in Fig 4–4. The conducting a-direction is perpendicular to the oblique lattice. The lattice vectors on the b0 c∗ -plane may be
represented by Rpq =pbb0 +q cb0 =(pb0 +qd)b
y+qc∗b
z, where p and q are integer. b0 , c0 and
c∗ are lattice constants, as explained in Fig. 4–2(a). d is defined as d = c0 cos α0 .
In order to treat the oblique lattice model, we use the following energy dispersion.
This energy dispersion was also employed by several other authors [42, 46, 106].
E(k) = ~vF (|kx | − kF ) −
X
tpq cos((pb0 + qd)ky + qc∗ kz ),
(4–4)
p,q
where tpq is the transfer energy associated with the lattice vector Rpq . This energy
dispersion describes a general Q1D FS. The 1st term of the energy dispersion
is responsible for a flat shape of the FS, and the 2nd term is responsible for the
warping along the lattice vector Rpq . The physical meaning of each transfer energy
90
z B
p=0
p=1
p=2
p=3
q=4
θ
Rpq
q=3
θ−θpq
d
α'
α'
θpq
c*
q=2
c´
q=1
b´
y
q=0
Figure 4–4. The oblique real-space crystal lattice. Rpq = (0, pb0 + qd, qc∗ ) are
lattice vectors. The energy dispersion for the oblique lattice is given
by Eq. (4–4). The Fourier components of the corrugation of the FS are
represented by tpq in Eq. (4–4).
91
tpq is therefore a Fourier component of the warping. In the case of (TMTSF)2 ClO4 ,
t01 = tc ∼ 1 meV, t10 = tb ∼ 20 meV and higher order transfer energies tpq are also
expected to be finite. The t10 = tb component dominates the FS shape. However,
this component has no contribution to the z-axis conductivity because the group
velocity is always perpendicular to the z-axis when q = 0 in Eq. 4–4.
We now consider the POR with the above energy dispersion. A magnetic
field is applied along an arbitrary direction, B = (B⊥ , Bk sin θ, Bk cos θ) =
(B sin φ, B cos φ sin θ, B cos φ cos θ). Solving the equation of motion (Eq. 2–2) using
Eq. 2–3 for the group velocity with the energy dispersion above, one can obtain the
following group velocity,
vz (t) =
X
(qc∗ )tpq
p,q
vF eBk
sin[
{(pb0 + qd) cos θ − qc∗ sin θ}t + {(pb0 + qd)ky (0) + qc∗ kz (0)}],
~
(4–5a)
or
vz (t) =
X
pq
t + {(pb0 + qd)ky (0) + qc∗ kz (0)}].
(qc∗ tpq ) sin[ωQ1D
(4–5b)
p,q
The period of the POR is given by
pq
=
ωQ1D
vF eBk
[(pb0 + qd) cos θ − qc∗ sin θ].
~
(4–6a)
The above equation can also be simplified in the following way with Rpq =
p
(pb0 + qd)2 + (qc∗ )2 ,
pq
ωQ1D
=
vF eBk Rpq
vF eBRpq
sin(θ − θpq ) =
cos φ sin(θ − θpq ),
~
~
(4–6b)
where θpq is Lebed’s magic angle corresponding to the angle of the lattice vector
(Rpq ). Using Eq. 4–5 and the Boltzmann equation (Eq. 2–6a), the real part of the
92
Table 4–1. Lattice parameters and the AMRO notations for the n-th nearest
neighbors.
n-th
1
2
3
4
5
6
7
p q
0 1
1 -1
1 1
0 2
-1 2
-2 1
1 2
AMRO notations
p/q=0
p/q=-1
p/q=1
p/q=0
p/q=-1/2
p/q=-2
p/q=1/2
Rpq (Å)
13.2
18.6
20.3
26.3
28.4
30.4
31.1
θpq (degrees)
5.1
-45.1
49.7
5.1
-24.6
-64.5
32.4
conductivity is obtained by
Re σzz (ω, B, θ) ∝
X
(qc∗ tpq )2 [
p,q
1
1
+
],
pq
pq
1 + (ω + ωQ1D )2 τ 2 1 + (ω − ωQ1D
)2 τ 2
(4–7)
pq
is given by
Thus the resonance condition ω = ±ωQ1D
vF eBRpq
| cos φ sin(θ − θpq )|,
~
(4–8a)
ν
vF eRpq
=
| cos φ sin(θ − θpq )|.
B
~
(4–8b)
ω=
or
It is therefore expected that the POR in (TMTSF)2 ClO4 has not only a single
resonance, but also multiple harmonic resonances corresponding to the integers p
and q. This resonance condition (Eq. 4–8b) depends on the frequency (ν), magnetic
field (B) and angle (θ), which implies that the POR can be observed by sweeping
the magnetic field, angle or frequency. On the other hand, at ν = 0, the resonance
condition reproduces Lebed’s magnetic angles θ = θpq (Eq. 4–3) which are fieldindependent. The resonance at ν = 0 can therefore be observed by sweeping only
the angle. TABLE 4–1 shows the lattice parameters (Rpq and θpq ) for the n-th
(n = 1 ∼7) nearest neighbor hopping terms in (TMTSF)2 ClO4 .
93
In the dc case (ν = 0), the description we use in this section corresponds to the
commensurability effect on the FS. As seen from the resonance condition (Eq. 4–
8b), the position of the resonance at microwave frequencies becomes different from
the Lebed magic angles. Thus, based on the commensurability arguments, the
Lebed effect is not a fixed angle effect (not magic). Fig. 4–5 shows the resonance
conditions and the POR obtained by sweeping the angle and magnetic field.
Fig. 4–5(a) represents the case of an angle sweep. The resonance condition is
shown as a function of the angle in the upper panel in Fig. 4–5(a). The condition
is plotted for three different lattice vectors, i.e., p/q = 0, 1 and -1. The inverse
conductivity is plotted for the dc and ac cases in the lower panel. In the ac case,
two PORs are seen for each p/q = 0, 1 and -1 lattice vectors, and the position
of the POR is shifted by changing the value of ν/B. The fundamental p/q PORs
are so pronounced that the harmonic p/q = 1 and -1 PORs becomes invisible
when these resonances are close to the p/q resonance. In the dc limit, these two
resonances become one resonance, which is the Lebed resonance, and they appear
at Lebed’s magic angles. The POR can also be observed by sweeping the magnetic
field. Fig. 4–5(b) shows the case for a field sweep. As shown in the plot of the
conductivity in the lower panel of Fig. 4–5(b), the position of the POR is shifted
by changing the angle. In contrast, Lebed’s original theory predicts that the
resonance condition given by Eq. 4–3 should always hold at any frequencies. In this
case, the resonance can only be observed by sweeping the field orientation, not its
magnitude.
4.3
Observation of the Periodic Orbit Resonances in (TMTSF)2 ClO4
Here we report the observation of periodic orbit resonances in (TMTSF)2 ClO4 .
Microwave measurements were carried out using a millimeter-wave vector network
analyzer and a high sensitivity cavity perturbation technique; this instrumentation
is described in detail elsewhere [75]. In order to enable in-situ rotation of the
94
(b) Field sweep
(a) Angle sweep
θ (degrees)
θ (degrees)
-180
-90
p/q = 0
0
90
-180
180
ν/Β
ν/Β
90
180
p/q = 1
p/q = −1
p/q = 0
p/q = 1
p/q = −1
0
-90
B Sweep, ω ≠ 0
ω≠0
θ sweep
Incease
ν/Β
0
0
Shift θ
p/q = 0
σzz
1/σzz
dc
shift
θ
p/q = 0
increase
ν/Β
p/q = 1
p/q = -1
-180
-90
0
90
180
p/q = 1
0
2
4
6
8
10
12
θ (degrees)
B (tesla)
Angle sweep
Field sweep
14
Figure 4–5. Resonance conditions and the POR by sweeping the angle and magnetic field. (a) Angle sweep: the upper panel shows the resonance
conditions for p/q
=
0, 1 and -1; the inverse conductivity is plotted in the lower panel. The resonances are indicated by the dashed
lines for the eye. For the simulation, θpq are taken in TABLE 4–1,
t20,1 : t21,−1 : t21,1 = 1 : 0.3 : 0.1 and ωc τ = 2 at 7 T. (b) Angle sweep: the
upper panel shows the resonance conditions for p/q = 0, 1 and -1. The
conductivity is plotted in the lower panel. The resonances are indicated
by the dashed lines for the eye. For the simulation, θpq are taken from
TABLE 4–1, t20,1 : t21,−1 : t21,1 = 1 : 0.3 : 0.1 and ωτ = 3 at 100 GHz.
95
sample relative to the applied magnetic field, we employed two methods. The
first involved a split-pair magnet with a 7 T horizontal field and a vertical access.
Smooth rotation of the entire rigid microwave probe, relative to the fixed field, was
achieved via a room temperature stepper motor (with 0.1◦ resolution). The second
method involved in-situ rotation of the end-plate of a cylindrical cavity, mounted
with its axis transverse to a 17 T superconducting solenoid or the NHMFL 33 T
axial resistive magnets. Details concerning this cavity, which provides an angle
resolution of 0.18◦ , have been published elsewhere [81]. Several needle shaped
samples were separately investigated by placing them in one of two geometries
within the cylindrical TE011 cavities, enabling (i) field rotation in the b0 c∗ -plane,
and (ii) rotation away from the b0 c∗ -plane towards the a-axis. Each sample was
slowly cooled (0.01 ∼ 0.1 K/min between 32 K and 17 K) through the anion
ordering transition at 24 K to obtain the low temperature metallic state [2].
Experiments were performed at 2.5 - 1.7 K, and data for three of the samples are
presented in this chapter (labeled A, B and C, dimensions ∼ 1 × 0.2 × 0.1 mm3 ).
In order to verify that the relaxed state is reproducibly achieved, we can observe
signatures of the FISDW phase transition in the microwave response (see Fig. 4–
7) for experiments conducted in the NHMFL 33 T magnet. Studies of the angle
dependence of the FISDW transition also allow us to determine the orientations of
the b0 and c∗ directions in-situ, i.e., simultaneous to the POR measurements.
Fig. 4–6 explains two rotations performed in the experiment. In case 1, the
magnetic field is rotated in the b0 c∗ -plane. In case 2, the magnetic field is rotated
from the b0 c∗ -plane to the a-direction. For the sake of the angle dependent study,
we set the z-direction as an arbitrary direction within the b0 c∗ -plane. We label
the angle between the z-axis and the magnetic field on the yz-plane as θyz , and
the angle between the yz-plane and a magnetic field as θxz . We now rewrite the
resonance condition using θyz and θxz . Case 1 corresponds to φ = 0, θ = θyz in
96
b)
a)
Case2
Bxz
θxz
z
Byz
z
Bxz
Case1
θyz
x
Byz
θxz
θyz
Rpq
y
x
y
Figure 4–6. Overview of the orientations in the experiments. The sample was rotated in the yz and xz-planes, denoted case 1 and 2 respectively. (a)
The orientation in terms of the sample view. The needle direction corresponds to the a-axis. (b) The orientation in terms of the FS. For the
yz-rotation, B is rotated in a plane parallel to the FS.
Eq. 4–8b, therefore,
ν
evF Rpq
=
| sin(θyz − θpq )|.
B
h
(4–9a)
ν
evF Rpq
=
| cos(θxz ) sin(θ0 )|.
B
h
(4–9b)
Similarly, for case 2,
where θ0 is an arbitrary direction from the z-axis on the b0 c∗ -plane which depends
on how the sample is mounted.
In Fig. 4–7, we present the microwave absorption taken by a field sweep at
ν = 51.9 GHz and T = 1.7 K for (TMTSF)2 ClO4 sample A. The data were taken
by sweeping magnetic fields between 0 and 32 T at the NHMFL. The sample was
rotated in the yz-plane from -35 to 175 degrees using the rotating cavity. The data
show extremely rich behavior. In low fields, the POR is clearly seen, as indicated
by the dashed line. The FISDW phase starts from 7.5T which is seen as a step-like
shape in the microwave absorption. As shown in Eq. 4–2, the critical field of the
FISDW phase, BF ISDW , is minimum when the magnetic field is applied along the
97
ν=51.9 GHz, T=1.7K
POR
Absorption (arb. unit-offset)
-35º
c*
FISDW
0
5
10
15
20
25
Magnetic field (tesla)
QO
30
175º
Figure 4–7. Microwave absorption as a function of the magnetic field for sample A.
The data were obtained in the 33 T resistive magnet at the NHMFL
at ν=51.9 GHz and T=1.8 K. The data show extremely rich behavior
including POR, the FISDW phase, and quantum oscillations (QO).
The dashed lines are guides for the eye.
c∗ -axis. Using this fact, one can easily identify the direction of the c∗ -axis. In high
fields, quantum oscillations (QO) were also observed.
During the experiment, the (TMTSF)2 ClO4 samples showed huge phase shifts
so that some data were taken by the phase-lock method and other data taken by
the frequency-lock method were analyzed by the vector analysis using Eq. 3–31,
and the position of the POR (Bres ) is then determined from the position of the
peak in ∆Γ. The approximately Lorentzian lineshape of the POR is characteristic
of the bulk conductivity, which indicates that the sample is in the depolarization
regime in the microwave frequency range, i.e., ∆Γ = 1/σ. From the data, we
obtained τ ∼ 7.3 ps and 7.0 ps at T = 2.5 K for sample A and B respectively.
As clearly seen in the data in Fig. 4–7, the position of the POR depends on
the orientation of the magnetic field which, in turn, reveals the direction of
98
the warping. This observation of the POR by the field sweep, and its angledependence, suggest that the observed POR is semiclassical, as we explained in the
previous section.
We then studied the angle dependence of the POR effect and the FISDW
transition. Fig. 4–8 shows the angle dependence of the ratio of the frequency
(ν) and the position of the POR, i.e., ν/Bres . The angle dependence of ν/Bres
for case 1 is plotted in Fig. 4–8(a), including data from two samples (sample A
and B) and two measurements (A1 and A2). A fit to Eq. 4–9a is applied to the
angle dependence of the data. The agreement between fit and the data is very
good. Fig. 4–8(a) also shows the angle dependence of the FISDW phase boundary
plotted as 1/BF ISDW . The fit to Eq. 4–2 also agrees well. We determined the b0
and c∗ -directions from this plot. Then, using the fit to Eq. 4–9a, we found θpq ∼
=5
degrees, which corresponds to p = 0, i.e., the c0 -direction. Fig. 4–8(b) shows the
ratio ν/Bres for sample A, for the xz-plane rotation. The angle dependence for
this plane of rotation is straightforward. We found the maximum ratio when the
magnetic field is parallel to the yz-plane, and the minimum when the magnetic
field is along the x-direction. The data fit to Eq. 4–9b again agrees well. The
amplitude of the ratio ν/Bres is smaller than that in Fig. 4–8(a). This implies that
the sample was mounted with some angle between the plane of rotation and the
c∗ -axis, i.e., θ0 ∼ 70 degrees.
From these studies, we obtained the prefactor in Eq. (4–9a) evF Rpq /h=24
(GHz/Tesla). Since the direction of the warping is the c0 -direction, Rpq = R01 =13.1
Å, as shown in TABLE 4–1. This gives a Fermi velocity vF = 0.76 × 105 m/s.
This value is about half the value obtained from heat capacity measurements,
vF = 1.4 × 105 m/s [110]. However it is close to the value obtained by recent dc
AMRO measurements with pulsed electric fields vF = 3 − 6 × 104 m/s [111].
99
25
20
a)
B
y
15
A1
A2
A2
B
ν/Bres (GHz/tesla)
10
5
θyz z
c*
b’
c’
0
20
b)
z
θxz
15
B
10
x
x
z
5
0
BFISDW
c*
-90
0
90
180
Angle (degrees)
Figure 4–8. Angle dependence of the quantity ν/Bres for (a) b0 c∗ -plane rotations
(Eq. 4–9a) (Case 1), and (b) rotations away from the b0 c∗ -plane towards a (Eq. 4–9b) (Case 2); the inserts depict the experimental geometries. Data were obtained for two samples (A and B); A1 and A2
denote successive cool downs for sample A. The solid curves are fits
to Eq. 4–9a, and the directions of high symmetry are indicated in the
figure. The experiments were all performed at 2.5 K, and at frequencies
ranging from 45 to 69 GHz (not labeled). The quantity ν/Bres is frequency independent to within the scatter of the data. In Fig. 4–8(a),
the angle dependence of ν/BFISDW is also shown (filled squares), where
ν = 61.8 GHz; these data have been fit to an inverse cosine function
(dashed curve), enabling an accurate determination of the b0 and c∗ directions (Reprinted figure with permission from Takahashi et al. [70].
Copyright 2005 by the American Physical Society.).
100
(b)
ν = 52.1 GHz
6.7 T
6.5 T
T = 2.0 K
p/q = 1
c'
p/q = -1
5.0 T
p/q=0
4.0 T
3.0 T
c*
0
p/q= 1
p/q=0
-135.9˚
6.0 T
Absorption (arb. Unit-offset)
Microwave transmission (abs. unit)
(a)
p/q= 1
b'
90
Angle (degrees)
p/q=0
180
0
-250.9˚
2
4
Magnetic field (tesla)
6
p/q=0
Figure 4–9. Angle sweep and field sweep measurements for (TMTSF)2 ClO4 . (a)
Microwave transmission as a function of the field orientation for sample
A, for different field strengths. The field is rotated in the b0 c∗ -plane,
and the angle θpq is measured relative to c∗ . The minima correspond
to the POR. (b) Magnetic field swept absorption data for sample C at
different field orientations. Here, the maxima correspond to the POR.
The angle step θpq = 5◦ , the measurement frequency ν = 75.5 GHz and
T = 2 K. The right panel shows the enlargement of small resonances
denoted as p/q = 1.
Moreover, we have recently studied the POR by sweeping both the angle
and field, as shown in Fig. 4–9. Microwave transmission is shown for sample A
in Fig. 4–9(a), as a function of the field orientation relative to the c∗ -axis, for
different magnetic field strengths between 3 and 6.7 T. The minima correspond
to the POR. As seen in the figure, several resonances are observed (indicated by
dashed curves), which are one kind of pronounced resonances (denoted as p/q = 0)
and two kinds of small resonances (denoted as p/q = 1 and -1). All resonances
shift in angle upon changing the magnetic field strength. This again suggests
that the observed resonances are semiclassical in origin, i.e., they obey Eq. 4–8b.
Fig. 4–9(b) shows another field sweep of the microwave absorption as a function of
magnetic field strength for sample C, for different field orientations from -135.9 to
-250.9 degrees. From the data, we obtained τ ∼ 6.2 ps at T = 2.5 K for sample
C. The quality of sample A, B and C is therefore similar. In this case, the maxima
101
p/q = 1
p/q = − 1
p/q = 0
50
45
ν/Bres (GHz/tesla)
40
35
30
25
20
15
10
p/q = 1
5
p/q = −1
0
-225
-180
-135
-90
-45
0
Angle (degrees)
45
p/q = 0
90
Figure 4–10. Summary of the p/q = 0, 1 and -1 POR data for sample C. The solid
lines indicate the best fits to Eq. 4–8b.
correspond to POR. We observed two kinds of the POR, which are one kind of
pronounced resonances (denoted as p/q = 0) and another kind of small resonances
(denoted as p/q = 1). In the right panel of Fig. 4–9(b), the enlarged figure is
shown. Comparing with the pronounced peaks, the small peaks are very small.
Since the resonances can be observed in a magnetic field swept experiment, this
again suggests that the resonance condition is given by Eq. 4–8b.
A summary of the data for sample C is compiled in Fig. 4–10, where the
scaled resonance positions Bres are plotted versus the field orientation θ. Similar
results were obtained for sample A. The solid curves correspond to fits to the data,
using Eq. 4–8b, for different p/q ratios. As seen from Eq. 4–8b, the zeros in ν/Bres
should correspond to Lebed’s magic angles found from dc AMRO measurements
[Eq. 4–3]. We find that the extrapolation of our high-frequency measurements to
zero-frequency is in excellent agreement with published AMRO data [26, 27]. From
these extrapolations, we can identify the POR corresponding to the p/q = 0, 1
102
and -1 Lebed resonances. The fundamental resonance (p/q = 0) is the most
pronounced, while the other two are weak but highly reproducible. Overall, the
relative intensities of the POR are quite similar to the AMRO minima observed
at comparable temperatures and field strengths, as shown in Fig. 4–3 and Fig. 4–
9. We therefore conclude that the Lebed resonances and the POR represent the
same effect, and we argue that the former may be understood simply in terms of
semiclassical arguments. In other words, the Lebed magic angles are not a fixed
angle effect (not “magic”) at microwave frequencies.
4.4
Summary
In this chapter, we reported the observation of the Q1D POR for (TMTSF)2 ClO4 .
We estimated the Fermi velocity vF using the Q1D POR [70]. Additionally, We
proved that the periodic orbit resonances correspond to the same Lebed resonances
observed in dc AMRO experiments. However, we conclude that the Lebed magic
angles are not “magic” at frequencies on the order of the cyclotron frequency.
CHAPTER 5
NON-MAGNETIC IMPURITY EFFECTS ON THE SUPERCONDUCTIVITY IN
(TMTSF)2 ClO4
The results presented in this chapter can be found in the article entitled, Study
of Non-Magnetic Impurity Effects of the Organic Superconductor (TMTSF)2 ClO4 ,
S. Takahashi, S. Hill, S. Takasaki, J. Yamada and H. Anzai, submitted to the LT24
conference proceedings.
5.1
Overview of the Superconductivity in (TMTSF)2 ClO4
The nature of the superconductivity for the quasi-one-dimensional (Q1D)
superconductor (TMTSF)2 X, (X = ClO4 , PF6 etc.) is still an open-question, even
though it was first discovered 26 years ago [2]. It was believed for a long time
that (TMTSF)2 X materials were conventional superconductors. However recent
experiments have shown evidence of unconventional spin-triplet superconductivity.
For example, Lee et al. demonstrate that Hc2 exceeds the Pauli paramagnetic limit
in (TMTSF)2 ClO4 and (TMTSF)2 PF6 [10, 112], and that there is no
77
Se NMR
Knight shift through Tc in (TMTSF)2 PF6 [10]. Moreover Joo et al. reported the
suppression of superconductivity in (TMTSF)2 (ClO4 )1−x (ReO4 )x by changing
the concentration ratio of ClO4 and ReO4 anions [11] suggesting a non-magnetic
impurity effect. We recently studied the non-magnetic impurity effect on the
superconductivity in (TMTSF)2 ClO4 via the anion ordering. This ordering involves
a well-known order-disorder transition of the tetrahedral non-magnetic ClO4 anions
at the anion-ordering temperature TAO = 24 K. Fig. 5–1(a) shows the ClO4 anion.
Above TAO , these anions have a degree of freedom for two orientations. Below TAO ,
the anions order regularly because it is entropically favorable. Fig. 5–1(b) shows
the crystal structure of (TMTSF)2 ClO4 after the anion ordering viewed along the
103
104
(b)
c
O
Cl
~3Å
ClO4
TM
TSF
(a)
b
Figure 5–1. Illustration of the ClO4 anion and the crystal structure of
(TMTSF)2 ClO4 below TAO . (a) Illustration of the ClO4 anion. The
size of the anion is about 3 Å. (b) Crystal structure of (TMTSF)2 ClO4
below TAO . The ClO4 anions are highlighted differently in terms of the
orientation of the anion. As seen in the figure, the ClO4 anion planes
are ordered alternately along the b-axis. On the other hand, the same
type of anions are ordered consecutively along the c-axis as well as
the a-axis. As a result, below TAO , (TMTSF)2 ClO4 has a dimerised
structure characterized by the wave number Q = (0, 1/2, 0).
105
a-axis. In the figure, the two kinds of the ClO4 anions are colored differently. As
seen in the figure, the ClO4 anion planes are ordered alternately along the b-axis.
On the other hand, the same type of the anions are ordered consecutively along the
c-axis as well as the a-axis. As a result, below TAO (TMTSF)2 ClO4 has a dimerised
structure characterized by the wave number Q = (0, 1/2, 0). This ordering in
(TMTSF)2 ClO4 is extremely sensitive to the cooling rate around TAO . The anions
are successfully ordered only when the sample crystal is cooled very slowly (cooling
rate <0.1 K/min). The state in which the anions are ordered by slow cooling is
the so-called relaxed state. In the relaxed state, the (TMTSF)2 ClO4 crystal is
metallic down to 1.2 K, and then becomes a superconductor at 1.2 K. Conversely,
the anions are disordered when a (TMTSF)2 ClO4 crystal is cooled down rapidly
(cooling rate >50 K/min). The state with the rapid cooling is in the so-called
quenched state. The ground state of a quenched sample is not superconducting,
but a spin-density-wave (SDW) insulator. The transition temperature of the
metal-insulator transition TM I is 6 K. When (TMTSF)2 ClO4 is cooled down
with the cooling rate between the relaxed and quenched state, the anions are
not ordered perfectly, so that some anions are ordered, but others are not. This
disorder affects the superconducting properties in (TMTSF)2 ClO4 as well as the
normal conducting properties. An earlier study shows that the superconducting
critical temperature Tc is strongly affected by the cooling rate around TAO [2]. We
speculate that these effects of the disorder on the superconductivity correspond
to the non-magnetic impurity effect, since the ClO4 anion itself is non-magnetic,
and the small concentration of the disordered anion does not affect electronic band
structure. Thus the discussion in this chapter is about this disorder effect.
Non-magnetic impurity effects are often investigated by using only dc transport measurements. However this method leaves some ambiguity since one has to
106
assume that the band parameters, e.g., the effective mass m∗ , and carrier concentration n, are unchanged. It is assumed that only the scattering time τ is changed
when the impurity concentration is changed. In order to settle this ambiguity, one
needs to study both the critical temperature Tc and the scattering time τ . The
periodic orbit resonance (POR) method we employ is appropriate for this purpose
since it can determine the band structure (i.e., Fermi velocity vF ), and the scattering time τ independently at the same time. Thus our investigation here focuses on
the cooling rate dependence of Tc by dc transport measurements, and the scattering
time τ by the POR independently. In this way, we can identify the effect of the
disordered anion, and reveal the origin of the suppression of the superconductivity
in (TMTSF)2 ClO4 .
5.2
DC (ω ≈ 0) Transport Measurements for Different Cooling Rates
We first show the dc (ω ≈ 0) transport experiment. The interlayer resistance
Rzz was measured at various cooling rates employing a standard 4-wire technique.
The connection was made by 25 µm gold-wires and gold paste. The resistance
was measured by a low frequency (ν ∼ 17 Hz) ac technique with a Stanford
research SR830 lock-in amplifier [113]. Therefore the electrodynamical condition
for the measurement is approximately zero-frequency (dc). Our home-made 3 He
cryostat system was used in the experiments to measure the superconducting
critical temperature Tc for (TMTSF)2 ClO4 (Tc ∼ 1.2 K in the relaxed state) [81].
The temperature was measured with a calibrated Cernox thermometer [114] which
was placed on the sample puck. During the cooling process, the sample was cooled
down from 50 K to 15 K at different cooling rates, and then cooled down to 0.5 K
smoothly. Thus, the resistance was measured from 50 K to 0.5 K contiguously. We
achieved cooling rates from 0.03 - 28 K/min. The resistance at the different cooling
rates is shown in Fig. 5–2. Each trace has a kink at around 24 K which indicates
the anion ordering in (TMTSF)2 ClO4 . The resistance data for different cooling
107
(a)
Anion ordering
TAO = 24 K
0.4
Rzz (Ohm)
0.03K/min
Rzz (Ohm)
0.2
Completion
Temperature
0.4
0.0
(b)
5
10
15
20
25
0.8
30
T (K)
35
27.4 K/min
23.0 K/min
18.7 K/min
14.3 K/min
12.6 K/min
6.1 K/min
1.0 K/min
0.1 K/min
0.03 K/min
0.0
0.5
1.0
1.5
Temperature (kelvin)
Figure 5–2. dc (ν ∼ 17 Hz) transport measurements for samples cooled at different rates in the range 27.4 - 0.03 K/min from 50 K to 15 K. (a) The
resistance Rzz below 36 K. The anion ordering is observed as a kink at
TAO = 24 K. (b) Rzz below 1.5 K. Rzz becomes higher when the cooling rate increases. The superconducting transition is seen in the figure
which also shifts with cooling rate.
108
(b)
(a)
1.0
Tc(kelvin)
Tc (kelvin)
1.0
0.8
0.6
0.01
0.1
1
10
Cooling Rate (K/min)
0.8
0.6
0.00
0.02
0.04
Residual resistance (Ohm)
Figure 5–3. Summary of the dc transport measurements. (a) Tc measured at different cooling rates between 27.4 - 0.03 K/min from 50 K to 15 K. (b)
Tc as a function of the residual resistance.
rates had no systematic difference in the range from 50 K to TAO = 24 K, as seen
in Fig. 5–2(a). However they became distinguishable below TAO clearly showing
that the resistances measured at fast cooling rates are higher than those measured
at slow rates at the same temperature. This tendency was completely systematic
and reproducible for all cooling rates. As seen in Fig. 5–2(a), the temperature
dependence of Rzz below 10 K is nearly linear. Rzz also does not show saturation
within the metallic phase. These tendencies are also seen in Joo et al. [11]. We
determined Tc at the temperature where the maximum dRzz /dT and zero resistance
lines intersected, as shown in the inset of Fig. 5–2(a), the so-called completion
temperature. As shown in Fig. 5–2(b), we observed that the superconducting
transition Tc was affected by the cooling rate. Tc shifts to lower temperature as the
cooling rate increases. The transition width ∆Tc is about 0.1-0.3 K. Finally, Fig. 5–
3 shows a summary of the dc transport measurements. The Tc data are shown as
a function of the cooling rate and the residual resistance in Fig. 5–3(a) and (b)
respectively. The residual resistance is determined by extrapolation of a polynomial
fit of the resistance to T = 0. As clearly seen in Fig. 5–3, Tc is suppressed by
cooling the sample faster: Tc varies from 0.95 K to 0.65 K. One can also see that
Tc decreases with increasing residual resistance. Since the resistivity is given by
109
ρzz ∝ 1/σzz = m∗ /neτ = m∗ Γ/ne, the resistance is proportional to the scattering
rate Γ. However the resistance also depends on m∗ and n. Thus we next show the
relation between Rzz and Γ = 1/τ using the POR experiment.
5.3
Study of the Periodic Orbit Resonance at Different Cooling Rates
As in our previous study for the POR in (TMTSF)2 ClO4 seen in Chap. 4 and
Takahashi et al. [70], we performed cooling rate dependent POR measurements
using the same setup, i.e., we employed a millimeter vector network analyzer,
a waveguide probe, and a cavity perturbation technique etc. [75]. An Oxford
Instruments 17 T superconducting solenoid magnet was also used for the magnetic
field and the temperature control. The orientation of the dc magnetic field was
controlled by a rotating cavity [81]. The temperature was measured by a Cernox
thermometer [114] placed on the copper cavity block. We measured two samples
(A and B) separately for this experiment. Sample A and B are different from
the sample used in the dc transport measurement. The shape of both samples
were needle-like. The dimensions were about 1 × 0.2 × 0.1 mm3 . We placed
both samples at the antinodes of the TE011 mode, i.e., sample A was placed in
the center of the cavity using the quartz rod, and sample B was placed on the
end-plate of the cavity.
First, we checked the angle-dependence of the POR to identify the direction
of the warping on the Fermi surface, as shown in Chap. 4. This was done in the
relaxed state after slow cooling (0.1 K/min). Then, we studied the cooling rate
dependence of the POR at various cooling rates. During the cooling process, we
controlled the temperature in a similar way to the dc transport measurements.
The samples were cooled down from 50 K to 15 K at different cooling rates. The
range of the cooling rate that we could achieve with our cavity perturbation
system was from 0.01 - 37 K/min. While we studied the cooling rate dependence
of the POR, we also measured the POR at several angles to ensure that the angle
110
Absorption (arb. unit-offset)
ν=51.8 GHz
T=1.4 K
0.1 K/min
1.1 K/min
Bres
33.3 K/min
10.4 K/min
25.7 K/min
0
2
4
6
8
Magnetic field (tesla)
10
12
Figure 5–4. Microwave absorption as a function of magnetic field for sample B.
The measurements were performed on one sample cooled at different rates. The data were taken by sweeping the magnetic field at
T = 1.4 K and ν = 51.8 GHz. A Lorentzian-like peak at around 2 T
corresponds to the POR. The slow cooling gives sharper POR. However
the position of the POR is independent of the cooling rate.
dependence of the POR did not change. Fig. 5–4 shows the microwave absorption
as a function of magnetic field at different cooling rates. The data were taken by
sweeping the magnetic field at T = 1.4 K and ν = 51.8 GHz. The POR is seen as a
Lorentzian-like peak at around 2 T. As seen in the figure, the position of the POR
is independent of the cooling rate. We also ensured that the angle-dependence of
the POR did not change (not shown). These results demonstrate that the Fermi
velocity vF does not vary with the cooling rate. On the other hand, as seen in
Fig. 5–4, the POR peak becomes broader. This implies that the scattering rate
Γ = 1/τ is cooling rate-dependent. Thus, one can control only the scattering rate Γ
by changing the cooling rate around TAO .
5.4
Analysis of the Scattering Rate Γ
The estimate of the scattering rate Γ was made by fitting the POR data with
the conductivity function given by Eq. 2–8 (see Fig. 5–5). As discussed in Sec. 3.6,
111
the electrodynamics of the microwave response is related to the skin depth of the
sample. The skin depth along the c-direction is about 100 µm at 50 GHz [70] while
the dimensions of the samples are about 100 - 200µm. Therefore the condition of
the electrodynamics are intermediate between the depolarization and skin depth
regimes. However, since we observed a Lorentzian-like POR peaks, as shown in
Fig. 5–4, we assume the depolarization regime for this analysis. Recalling Eq. 3–27,
one notes that the microwave absorption is 1/∆Γ ∼ Re σ in the depolarization
regime. We performed a least-square fit with 1/∆Γ ∼ Re σ. Fig. 5–5(a) shows the
microwave absorption data, and the least square fits. These data were obtained
when the sample was cooled at a rate of 1.1 K/min. The data in Fig. 5–5(a) were
taken at several different temperatures between 12-1.4 K. As seen in the figure, the
fits to the data taken at high temperature are good. At the low temperature, on
the other hand, the fits become worse. We can analyze this behavior more carefully.
The upper panel of Fig. 5–5(b) plots the χ2 values obtained from the fit. The
plot clearly shows that the χ2 value becomes larger at low temperatures, which
indicates worse fits. We speculate that this behavior may be caused by the change
of the electrodynamics, i.e., the approximation of the depolarization regime is less
appropriate at low temperatures because the skin depth becomes smaller. The
middle panel of Fig. 5–5(b) plots the scattering rate Γ = 1/τ . Γ becomes smaller
when the temperature decreases. This confirms that the ωτ product becomes
larger, and that the POR is more pronounced at low temperatures. In the lower
panel of Fig. 5–5(b), the position of the POR is plotted. This result also confirms
that the resonance position does not vary with temperature.
We applied the same analysis to data at different cooling rates. The upper
panel of Fig. 5–6(a) shows the cooling rate dependence of the scattering rate Γ
obtained from the POR. The results were estimated from two samples (Sample
A and B). As shown in the figure, the scattering rate increases by increasing
112
Sample B
(a)
Cooling rate = 1.1 K/min, ν = 51.8 GHz
Absorption (arb. unit-offset)
= fit to Re σ
= data
T = 1.4 K
T = 2.5 K
T=4K
T=8K
T = 12 K
0
5
10
Magnetic field (tesla)
5.0x10
-4
Reduced χ2
0.0
Bres (telsa)
Γ (GHz)
Reduced χ2
(b)
Scattering rate
600
400
200
0
3
2
1
Resonance position
0
0
2
4
6
8
10
Temperature (kelvin)
12
Figure 5–5. Least-square fit to microwave absorption. (a) The microwave absorption and the least-square fit. Sample B was cooled by 1.1 K/min. The
data were taken at different temperatures between 12-1.4 K. (b) Parameters obtained from the least-square fit. The upper panel shows
the χ2 value of the least-square fit. The χ2 value becomes larger at low
temperature, which indicates that the fit is worse. The middle panel
shows the scattering rate Γ = 1/τ . Γ decreases when the cooling rate
increases. The lower panel shows the position of the POR, which is
independent of the cooling rate.
113
(b)
(a)
T = 2.5 K
0.1 K/min
1.1 K/min
25.7 K/min
33.3 K/min
800
250
200
Sample A
Sample B
150
100
3.0
Bres (tesla)
Scattering rate
Resonance position
T = 2.5 K
2.5
2.0
0.01
Sample A
Sample B
0.1
1
Scattering rate (GHz)
G (GHz)
300
600
400
200
Sample B
0
0
10
2
4
6
8
10
12
Temperature (kelvin)
Cooling rate (K/min)
Scattering rate (GHz)
(c)
300 Scattering rate
T = 2.5 K (Sample A)
T = 2.5 K (Sample B)
250
Residual (Sample B)
200
150
100
50
0
0.01
0.1
1
10
100
Cooling rate (K/min)
Figure 5–6. Cooling rate and temperature rate dependence of the scattering rate.
(a) The upper panel shows the scattering rate as function of the cooling rate. The results were obtained for Sample A and B. The scattering
rate increases when the cooling rate increases. The lower panel shows
the position of the POR as function of the cooling rate. The POR
position is independent of the cooling rate. (b) The temperature dependence of the scattering rate for different cooling rates. The fast
cooling rate data shows higher scattering rates between 0-12 K. The
residual scattering rate is extracted using a polynomial fit, as indicated
by the dashed lines. (c) The residual scattering rate as function of the
cooling rate. The scattering rate at T
= 2.5 K is also plotted. The
solid line represents a polynomial fit to the residual scattering rate.
The dashed line represents a polynomial fit to the scattering rate at
T = 2.5 K.
114
the cooling rate. This indicates that the anion disordering enhances electronic
scattering processes. We speculate that this enhancement is due to non-magnetic
impurity scattering. In the lower panel of Fig. 5–6(a), the position of the POR as
a function of the cooling rate is shown. As seen, the position does not change by
varying the cooling rate, i.e., the Fermi velocity is the same for different cooling
rates. We also measured the temperature dependence of the POR for Sample B.
Fig. 5–6(b) shows the scattering rate as function of the temperature for different
cooling rates. Although the dependence is not perfectly clear, one can see that
the temperature dependence is more or less the same, and the higher cooling rate
results have a higher residual scattering rate. We obtained the residual scattering
rate by extrapolation of a polynomial fit to T = 0. The fitting line is represented
by the dashed line in Fig. 5–6(b). Fig. 5–6(c) shows the residual scattering rate
as a function of the cooling rate. The scattering rate at T = 2.5 K is also plotted
in the same panel. The residual scattering rate increases when the cooling rate
increases. The fit was performed using a polynomial fit to both the residual and
T = 2.5 K scattering rate data. Both fits are very similar, with some offset between
the scattering rates. Thus, this strongly implies that the non-magnetic impurity
scattering in (TMTSF)2 ClO4 can be controlled by the cooling rate around TAO . We
use this curve (solid line) of the residual scattering rate to determine the scattering
rate for an arbitrary cooling rate in the final analysis.
Next we compare the cooling rate dependence between the scattering rate Γ
and the resistance Rzz in order to identify the effect of changing the cooling rate.
Fig. 5–7(a) shows the comparison between the scattering rate Γ and the resistance
Rzz obtained from T = 2.5 K data; the left y-axis plots resistance and the right axis
plots scattering rate. As seen in the figure, the cooling rate dependence of Rzz and
Γ is similar at slow cooling rates, but a difference becomes obvious at fast cooling
rates. Fig. 5–7(b) shows the comparison between the residual scattering rate Γ and
115
Rzz at t = 2.5 K (Ohm)
0.10
Rzz & Γ (T = 2.5 K)
400
Rzz at T = 2.5 K
0.05
200
Scattering rate
Sample A
Sample B
0.00
0.01
0.1
1
10
Scattering rate (GHz)
(a)
0
Cooling rate (K/min)
(b)
Residual Rzz & Γ
Residual Rzz (Ohm)
0.03
Residual Rzz
200
0.02
100
0.01
Scattering rate
Sample B
0.00
0.01
0.1
1
10
Scattering rate (GHz)
300
0.04
0
Cooling rate (K/min)
Figure 5–7. Comparison between the resistance Rzz and the scattering rate Γ as
a function of the cooling rate. (a) Comparison at T = 2.5 K. The left
y-axis represents the Rzz and the right axis represents Γ. The cooling
rate dependences of both Rzz and Γ are similar at slow cooling rates,
but they become different at fast cooling rates. (b) Comparison between the residual Rzz and the residual scattering rate. The difference
between them becomes larger than the difference of the Rzz and Γ at
T = 2.5 K.
116
Tc/Tc0
1.0
Tc0 = 1.26 K
0.5
0.0
0.00
0.05
0.10
α/2πkBTc0
Figure 5–8. Tc vs. the scattering rate Γ. Assuming Γ is proportional to the pair
breaking strength α in the AG theory, we can performed a fit using the
universal function (Eq. 1–1). The agreement is good. From the fit, we
obtained Tc0 = 1.26 K.
the residual resistance Rzz obtained by the extrapolation to T = 0 K. In this result,
one can see the different dependence of the cooling rate more clearly. These results
suggests that this change in Rzz is caused not only by the increase of Γ, but also
due to a change of the carrier concentration n.
5.5
Relation Between Tc and the Pair Breaking Strength α
Finally, we discuss the nonmagnetic impurity effect on the superconductivity
for (TMTSF)2 ClO4 . In Fig. 5–8, using the data from Fig. 5–3(b), we plot Tc as a
function of the pair breaking strength introduced in Eq. 1–2, 2α = ~/τK = ~ΓK .
Here we assume that the scattering rate Γ = 1/τ obtained from the POR data
is proportional to ΓK = 1/τK in the pair breaking strength. The fit to Tc was
performed by the universal function in the Abrikosov-Gor’kov (AG) theory given by
117
Eq. 1–1,
ln(
Tc
1
1
α
) = ψ( ) − ψ( +
).
Tc0
2
2 2πkB Tc
(5–1)
where Tc0 = Tc (0) is the superconducting critical temperature without impurities,
ψ(z) = Γ0 (z)/Γ(z) is the digamma function. As seen, the data and the fit agree
very well. As a result, we obtained Tc0 = 1.26 K and ΓK ' 1.3Γ. This Tc0 = 1.26
K for the completion temperature for Tc may correspond to Tc0 ∼ 1.4 for the on-set
Tc which is slightly higher than observed Tc , e.g., Tc ∼ 1.2 K [2]. However, we
speculate that the anion ordering is probably never perfect unless the sample is
cooled down for an infinitely long time. Thus, this Tc0 = 1.26 K is reasonable. With
this fit, we conclude that this suppression of the superconductivity is caused by
the nonmagnetic impurity effect in (TMTSF)2 ClO4 , and the results support the
scenario of unconventional spin-triplet superconductivity in (TMTSF)2 ClO4 .
5.6
Summary
In this chapter, we showed the suppression of the superconductivity in the
organic superconductor (TMTSF)2 ClO4 . By varying the cooling rate around the
anion ordering temperature TAO , we controlled the impurity concentration, and
determined Tc as a function of the cooling rate using dc transport measurements.
We also performed experiments using the POR to find the scattering rate Γ as a
function of the cooling rate. With these results, we found a simple relationship
between Tc and the scattering time τ . Using the result of Fig. 5–8 and Chap. 4, we
also found the l ∼ vF τ value at which Tc becomes zero is 3000 Å. This value fairly
agrees with the coherence length ξ ∼ 800 Å [2]. Thus, these result supports the
scenario for unconventional spin-triplet superconductivity in (TMTSF)2 ClO4 .
CHAPTER 6
PERIODIC-ORBIT RESONANCE IN QUASI-TWO-DIMENSIONAL
CONDUCTORS
The results presented in this chapter can be found in the articles, Fermi
Surface Studies of Quasi-1D and Quasi-2D Organic Superconductors Using Periodic
Orbit Resonance in High Magnetic Fields, S. Takahashi, A. E. Kovalev, S. Hill,
S. Takasaki, J. Yamada, H. Anzai, J. S. Qualls, K. Kawano, M. Tamura, T.
Naito, H. Kobayashi, International Journal of Modern Physics B 18 Nos. 27-29,
3499 (2004), and Angle-Resolved Mapping of the Fermi Velocity in Quasi-TwoDimensional Conductors and Superconductors: Probing Quasiparticles in Nodal
Superconductors, Susumu Takahashi and Stephen Hill, Journal of Applied Physics
97, 10B106 (2005).
6.1
The Quasi-two-dimensional Conductors κ-(ET)2 X
As introduced in Sec. 1.3.2, the organic superconductors κ-(ET)2 X (X =
Cu(NCS)2 , I3 etc.) belong to the family of a ET charge-transfer salts (CTS),
and have quasi-two-dimensional (Q2D) electronic properties. Among organic
superconductors, κ-(ET)2 X is one of the most extensively studied materials because they have the highest superconducting transition temperatures, e.g., the
X = Cu[N(CN)2 ]Br compound has Tc = 11.3 K, and Tc = 10.4 and 3.6 K for
X = Cu(NCS)2 and I3 respectively [2]. Moreover the κ-(ET)2 X salts have many
similar properties to high temperature superconductors (HTSC), such as highly
anisotropic superconducting properties [3], and a possible nodal superconducting energy gap [12]. Also, the phase diagram is similar to that of the HTSC,
which consists of an antiferromagnetic insulating phase in close proximity to a
superconducting phase, an unconventional metallic phase and so on [115]. This
118
119
Temperature (kelvin)
PMI
100
Unconventional
Metal
10
AFI
Conventional
Metal
SC
Cu[N(CN)2]Br
1
X = I3
Cu(NCS)2
0.1
Cu[N(CN)2]Cl
Pressure
1 kbar
Figure 6–1. Phase diagram for the organic conductors κ-(ET)2 X. Similar to the
HTSC, κ-(ET)2 X show Superconducting (SC), insulating antiferromagnetic (AFI), and paramagnetic phases (PMI). One can change
the ground state of κ-(ET)2 X by varying the anion X (and pressure)
while the ground state of the HTSC can be varied by a doping (See
McKenzie [115] and Dressel [100] for details).
Table 6–1. Unit cell parameters for κ-(ET)2 X. The crystal structure is monoclinic
(see Ishiguro et al. [2]).
X
Cu(NCS)2
I3
a (Å) b (Å) c (Å)
16.248 8.440 13.124
16.387 8.466 12.832
β (degree) V (Å3 )
110.30
1688.0
108.56
1687.6
phase diagram is shown in Fig. 6–1. The nature of the superconductivity in κ(ET)2 X is not understood as well as that of the HTSC. However, unconventional
superconductivity has been suggested experimentally [12, 116, 117, 118] and
theoretically [115, 119, 120, 121].
The crystal structure of κ-(ET)2 X [X = Cu(NCS)2 and I3 ] is monoclinic. The
unit cell parameters are listed in TABLE 6–1. Since β is not 90 degrees, the a-axis
is not perpendicular to the bc-plane. For convenience, the a∗ -axis is conventionally
defined along the direction perpendicular to the bc-plane; we use x, y and z- for
120
γ
Q1D FS
b (x)
B
c (y)
α
a* (z)
β
Q2D FS
1BZ
Figure 6–2. Fermi surface (FS) and trajectories of an electron under a magnetic
field for κ-(ET)2 Cu(NCS)2 . The Fermi surface is represented by the
black line. The FS consists of a Q2D FS and a Q1D FS. The trajectories are shown when the magnetic field is applied along the a∗ direction. α and γ are normal trajectories for the Q2D and Q1D FS
respectively. β is the magnetic breakdown trajectory, which is realized under a strong magnetic field (Reprinted with permission from
Takahashi et al. [122]. Copyright 2004, World Scientific Publishing.).
the b, c and a∗ -axes, respectively. According to band structure calculations using
a tight binding extended Hückel method [2], the FS of κ-(ET)2 Cu(NCS)2 and
κ-(ET)2 I3 are very similar. Fig. 6–2 shows the calculated FS within the good
conducting plane (the bc-plane). As seen in the figure, the calculated FS consist
of two quasi-one-dimensional (Q1D) electron sheets and a Q2D hole pocket. The
separation between the Q1D and Q2D FSs in reciprocal space is relatively small, so
that a strong magnetic field can enhance the tunnelling probability between the the
two FSs drastically. This is the so-called magnetic breakdown effect [25]. Fig. 6–2
also shows trajectories of electrons under a magnetic field applied perpendicular
to the bc-plane. The α- and γ-orbits are the regular trajectories for Q2D and
Q1D quasiparticles respectively. The β-orbit represents the magnetic breakdown
trajectory. The α and β trajectories have been verified by the observation of
magnetic quantum oscillations, i.e., de Haas-van Alphen (dHvA) and Shubnikov-de
Haas effects (SdH) [25]. According to SdH experiments, the magnetic breakdown
121
(a)
a* (z)
(b)
Rwarping
a* (z)
β
b (x)
b (x)
Figure 6–3. Warping on a Q2D FS. (a) Warping without tilting, which is the case
for cubic, tetragonal and orthorhombic systems. (b) Warping with tilting, which is the case for lower symmetry systems. κ-(ET)2 X, X = I3
and Cu(NCS)2 correspond to this case because of β 6= 0.
fields, BM B is 25-30 tesla for X=Cu(NCS)2 [123, 124] and 2.5 tesla for X = I3 [125],
when the magnetic field is applied perpendicular to the bc-plane. The X = I3 CTS
will, therefore, exhibit magnetic breakdown effects more significantly at moderate
magnetic field strengths, i.e., B = 2.5 − 25 tesla. The topology of the FSs along the
least conducting direction (the a-direction), is weakly corrugated due to the small
hopping energy along the least conducting direction. Similar to the case of the
Q1D POR, the period of the corrugation is related to the crystallographic lattice
constants.
6.2
Periodic-orbit Resonance in κ-(ET)2 X
In Sec. 2.4, we discussed the POR for a simple Q2D FS which was warped
along the z-direction. This will be case for a cubic, tetragonal or orthorhombic
system. In the case of κ-(ET)2 X, X = I3 and Cu(NCS)2 , the warping direction is
tilted from the z-axis because the nature of the warping is closely related to the
crystal structure and the small integer transfer energy. This difference is explained
in Fig. 6–3. [(a) corrugated FS without tilting. (b) corrugated FS with tilting.]
We now consider the Q2D POR for the case with tilted warping, i.e., for X
= I3 and Cu(NCS)2 . Assuming that the warping direction is along the a-direction,
122
Rwarping = (0, a cos β, a sin β), the energy dispersion can be described by,
E(k) =
=
~2 kx2 ~2 ky2
+
− 2ta cos(k · Rwarping ),
2m∗x
2m∗y
~2 kx2 ~2 ky2
+
− 2ta cos(ky a cos β + kz a sin β),
2m∗x
2m∗y
(6–1)
where m∗x and m∗y are the x and y diagonal components of the effective mass
tensor, ta is the transfer energy along the a-direction, a is the lattice constant,
and β is the angle between the a and c-axes for X = I3 and Cu(NCS)2 . We
consider the motion of electrons on this FS under a dc magnetic field similar
to the case in Sec. 2.4. A dc magnetic field is applied along an arbitrary angle,
i.e., B = (B sin θ cos φ, B sin θ sin φ, B cos θ). Using Eq. 2–2, the equations of
motion are as follows.
eB cos θ
2tz eB sin β sin θ sin φ
k˙x = −
ky (t) +
sin[ky (t)a cos β + kz (t)a sin β], (6–2a)
∗
my
~2
eB cos θ
2tz eB sin β sin θ cos φ
sin[ky (t)a cos β + kz (t)a sin β],
k˙y =
kx (t) −
∗
mx
~2
eB sin θ sin φ
eB sin θ cos φ
k˙z = −
k
(t)
+
ky (t).
x
m∗x
m∗x
(6–2b)
(6–2c)
And the group velocity (Eq. 2–3) is given by
vx (t) =
vy (t) =
~
kx (t),
m∗
~
2tz a sin β
ky (t) +
sin[ky (t)a cos β + kz (t)a sin β],
∗
m
~
2tz a sin β
sin[ky (t)a cos β + kz (t)a sin β].
vz (t) =
~
(6–3a)
(6–3b)
(6–3c)
Using the same approximation as the case in Sec. 2.4, i.e., EF À tz and θ ¿ 90
123
degrees, one obtains the following resultant group velocity along the z-direction,
vz (k, φk , kz , t) =
2tz a sin β
sin[ka(cos β − sin β tan θ sin φ) sin(ω2D t + φk )
s~
m∗x
ka tan θ cos φ cos(ωc t + φk ) + kz a sin β],
−
m∗y
where ω2D = eB cos θ/m∗ and m∗ =
(6–4)
p
m∗x m∗y . This is the same as Eq. 2–12. Thus,
the POR for a FS with tilted warping still has the same resonance condition, i.e.,
ω = nω2D = n
eB cos θ
.
m∗
(6–5)
However, the angle-dependence of the POR intensity can have a slight difference. We performed numerical simulations of the POR by using Eq. 6–2, Eq. 6–3
and Eq. 2–3. The results are shown in Fig. 6–4, which displays the the calculated
conductivity for the Q2D FS without the tilted warping [Fig. 6–4(a)] and with
the tilted warping [Fig. 6–4(b)]. The conductivity shows the fundamental mode
of the POR (n = 1) indicated by up-arrows, as well as higher harmonic modes
(n > 1) indicated by down-arrows, as shown in the upper panel in Fig. 6–4. The
great difference between the two cases is the trace at θ = 0. In the case of the Q2D
FS without the tilted warping, the POR does not occur at θ = 0 since the group
velocity along the z-axis is not oscillatory. On the other hand, in the case of the
Q2D FS with the tilted warping, the POR is always visible even in θ = 0. This
difference is more clearly seen in the contour plot of the conductivity shown in the
lower panel of Fig. 6–4. The fundamental POR indicated by the yellow dashed line
in Fig. 6–4 is always present. This difference can be taken as evidence to indicate
the tilted warping. The contour plot in Fig. 6–4 also shows the ac conductivity
resonance when the magnetic field is applied along the xy-plane. This effect is
discussed in Sec. 6.5.
124
(a)
(b)
0 degrees
15
30
45
50
55
60
σzz(B)
σzz(B)
0 degrees
15
30
45
50
55
60
0
5
10
15
20
25
30
0
5
10
15
20
25
30
Magnetic field (tesla)
Magentic field (tesla)
POR (n=1)
30
25
20
15
10
20
15
10
5
5
0
25
0
-90
-60
-30
0
30
Angle (degrees)
60
90
-90
-60
-30
0
30
60
90
in-plane field POR
Magnetic field (tesla)
Magnetic field (tesla)
30
Angle (degrees)
Figure 6–4. Numerical calculation of the ac conductivity for a Q2D FS. (a) In the
case of a warped FS without the tilting. The upper panel represents
the trace of the conductivity. Many PORs are seen. The fundamental
POR is denoted by up-arrows. Harmonic POR are denoted downarrows. As seen in the figure, no POR is seen at θ = 0 because of no
oscillatory group velocity in the z-component. The lower panel represents the contour plot of the conductivity. The red areas are the
POR. The position of the fundamental POR is indicated by the yellow
dashed line. The peak at θ = 90◦ is the resonance when a magnetic
field is applied within the good conducting plane (the bc-plane), as
explained in Sec. 6.5. (b) In the case of a warped FS with tilting. The
POR is seen at θ = 0 in this case. Experimentally, the observation of
the POR at θ = 0 is an evidence for tilted warping. The parameters for
the simulation are the following: a = 16.248 Å, b = 8.440 Å, c = 13.124
Å, α = γ = 90 degrees, β = 110 degrees, mx = 5.5 me , my = 4.3 me ,
ν = 92 GHz, τ = 4 ps.
125
(b)
z
(a)
z a*
Bz1 θz1 θz2
1
c
1
2
Bz2
φoff
b
2
Figure 6–5. Overview of the orientations in the experiments on κ-(ET)2 X. (a)
The orientations in terms of the sample view. The typical shape of
κ-(ET)2 Cu(NCS)2 and κ-(ET)2 I3 is plate-like. The good conducting
plane corresponds to the plane of the sample. The sample was rotated within two perpendicular planes denoted as the z1 and z2-planes
(Reprinted with permission from Takahashi et al. [122]. Copyright
2004, World Scientific Publishing.). (b) The orientations in k-space.
The angle-dependence of the Q2D POR is the same for the two planes
of rotation. However the angle-dependence of the Q1D POR is different for the two rotation planes (Reprinted with permission from
Takahashi et al. [122]. Copyright 2004, World Scientific Publishing.).
6.3
Experiments for κ-(ET)2 Cu(NCS)2
The FS of κ-(ET)2 Cu(NCS)2 calculated by an extended tight-binding model
consists of Q1D and Q2D sections, as shown in Fig. 6–2. It is therefore possible
to see multiple PORs originating from Q1D and Q2D sections of the FS. In order
to distinguish the origin of the POR, it is important to investigate its angledependence. The sample shape for κ-(ET)2 Cu(NCS)2 and κ-(ET)2 I3 is plate-like,
as shown in Fig. 6–5(a). The good conducting plane corresponds to the plane of
the sample. Thus, we identified the conducting plane by the shape of the sample
in the beginning of the experiments. This was subsequently confirmed by the
angle-dependence of the Shubnikov-de Haas (SdH) oscillations.
In order to distinguish the Q1D and Q2D POR, one needs to rotate the sample
in at least two orthogonal planes. Fig. 6–5 shows the two rotations we chose in the
experiments. We always rotated in two orthogonal planes (θyz and θxz ) defined by
126
the crystal faces, as shown in Fig. 6–5. For convenience, we rewrite the resonance
conditions for these rotations. In the case of the Q2D POR, recalling Eq. 2–26, the
resonance condition is given by the following:
ν
ne
=
cos θ,
B
2πm∗
(6–6)
where θ simply denotes the angle between the magnetic field and the least conducting direction, so that θ = θz1 or θz2 in Fig. 6–5(b). Hence, to a good approximation,
the resonance frequency is insensitive to the plane of the rotation. In contrast, the
Q1D POR condition, given by Eq. 4–8b,
evF Rpq
ν
=
| sin(ψ − θpq )|,
Bk
h
(6–7a)
depends on the projection of the field (Bk ) onto the FS. Consequently, the following
transformations are necessary for arbitrary field rotations:
q
B|| = B
sin2 θ cos2 φof f + cos2 θ,
tan ψ = tan θ cos φof f .
(6–7b)
(6–7c)
Here, θ is the orientation of the field relative to the crystallographic a∗ -axis,
i.e., θ = θz1 or θz2 , and φof f is the angle between the plane of rotation and the
crystallographic a∗ c-plane respectively. The warping direction is given by θpq ,
which is the angle relative to the z-axis. Therefore, the Q1D and Q2D resonance
conditions are quite different, and angle-dependent POR measurements enable
clarification of the source of the POR.
Angle dependent POR measurements for κ-(ET)2 Cu(NCS)2 have been carried
out using the 33 tesla resistive magnets at the National High Magnetic Field
Laboratory (NHMFL). The high fields are needed in order to access the metallic
⊥
phase at low temperatures (∼ 1.5 K), since the superconducting critical field Bc2
k
is ∼ 5 tesla for fields perpendicular to the layers and Bc2 ∼ 30 tesla for fields
127
parallel to the layers. Measurements were performed on two samples, and for two
planes of rotation each, as shown in Fig. 6–5(b). Sample rotation was achieved
using a recently developed rotating cavity which allows in-situ rotation in the high
field magnets with an angle resolution of 0.18 degrees [81]. According to Dressel
et al. [58], the skin depth of κ-(ET)2 Cu(NCS)2 at low temperatures at 60 GHz is
σ⊥ ∼ 100 µm. This length is smaller than typical sample dimensions, e.g., typical
dimensions ∼ 0.5 × 0.5 × 0.1 mm3 . Thus, we speculate that the sample is in the
depolarization regime, i.e., ∆Γ ∝ 1/Re σ.
We show the experimental spectra obtained at different angles in Fig. 6–6.
Since the data had a relatively large phase-shift, the plotted data were obtained
by the phase-lock method, as we discussed in Sec. 3.7.2. For convenience, the
microwave absorption (∼ ∆Γ) is plotted. The data contain extremely rich information. These include two kinds of PORs, as indicated by dotted and dashed lines;
the SdH oscillations at high-fields; a Josephson plasma resonances (JPR) at low
fields; and the superconducting transition around 5 telsa at B ⊥ bc-plane. The
JPR in κ-(ET)2 Cu(NCS)2 has been investigated and is presented elsewhere [18].
The angle dependence of the POR positions [Fig. 6–6] and SdH oscillations
are plotted in Fig. 6–7. The position of the POR is determined by the position
of the peak in the absorption. The dominant period of the SdH oscillations is
∆(1/H)=16.7 (10−5 /kOe) at θ = 0. This corresponds to the closed α-orbit, whose
angle-dependence should be the same as that of the Q2D POR, as we discussed in
Sec. 2.6. However, the angle dependencies of the two PORs in Fig. 6–7 are clearly
quite different from that of the SdH oscillations. Furthermore, the behavior depends on the plane of rotation. Therefore, this implies that the two PORs originate
from the Q1D FS.
Fitting of the data in Fig. 6–7 to the resonance condition allows us to determine the orientations of the corrugation axes at θpq = −19.7 ± 0.5◦ and
Absorption (arb. unit-offset)
128
POR2
θz2= −68.5°
POR1
0
10
20
30
Magnetic Field (tesla)
89.6°
Figure 6–6. Experimental data for κ-(ET)2 Cu(NCS)2 . The frequency is 95.0 GHz,
the temperature is 1.4 K, and the angle step is 8.78 degrees. The data
reveal extremely rich microwave responses in κ-(ET)2 Cu(NCS)2 including two kinds of POR, SdH oscillations, the JPR, and the superconducting transition. The dashed lines are guides for the eye (Reprinted
with permission from Takahashi et al. [122]. Copyright 2004, World
Scientific Publishing.).
129
ν/Bres (GHz/Tesla)
POR1
POR2
SdH
2
20
(b) B
1
15
15
10
10
5
5
0
-100
-50
0
50
Angle θz2 (degrees)
100
0
2
20
20
1
POR1
POR2
SdH
20
15
15
10
10
5
5
0
-100
-50
0
50
100
ν/Bres (GHz/Tesla)
(a)
θz1 z
B
∆(1/H) (10-5/kOe)
z θz2
0
Angle θz1 (degrees)
Figure 6–7. Angle-dependence of the POR and the SdH oscillations for two kinds
of rotations in κ-(ET)2 Cu(NCS)2 . (a) Angle dependence of the POR
and the SdH data in the the yz-plane (Reprinted with permission from
Takahashi et al. [122]. Copyright 2004, World Scientific Publishing.).
(b) Angle dependence of the POR and the SdH data in the xz-plane.
In both cases, the PORs show different angle-dependence from that
of the SdH. Thus, the PORs must originate from the Q1D FS section
(Reprinted with permission from Takahashi et al. [122]. Copyright
2004, World Scientific Publishing.).
34.7 ± 1.5◦ away from the z (a∗ )-axis. These directions correspond to the crystallographic T10 and T11 directions, where Tpq is related to the primitive lattice
vectors, Tpq = pa + qc. This result agrees with a previous study by Edwards et
al. [126]. We also obtained a value for the Fermi velocity, vF = 3.3 × 104 m/s for
κ-(ET)2 Cu(NCS)2 .
6.4
Experiments for κ-(ET)2 I3
We also have recently studied the X = I3 crystal. The measurements were
performed at the University of Florida (UF) using Oxford Instruments 17 tesla
superconducting magnet and the 25 tesla resistive magnet at the NHMFL. Similar to the X = Cu(NCS)2 crystal, the X = I3 crystal has a plate-like shape. The
conducting plane corresponds to the the plate of the sample crystal. In the angledependent experiments, two planes of rotation were utilized as we explained in
Fig. 6–5(b). Experimental data for X = I3 are shown in Fig. 6–8. Data were ob-
130
T=1.5 K, ν=91.8 GHz
Absorption (arb. unit offset)
JPR
POR θz2 = −90°
SdH
90°
Magnetic field (tesla)
Figure 6–8. Experimental data for κ-(ET)2 I3 . The frequency is 91.5 GHz, the temperature is 1.5 K, and the angle step is 18.0 degrees. The data contain
extremely rich information, similar to κ-(ET)2 Cu(NCS)2 . These include the POR, SdH oscillations, and the JPR etc. The dashed line in
low field indicates the JPR. The dashed lines at high fields represent
the POR.
tained by the phase-lock method because of large phase-shifts. The data show rich
information in κ-(ET)2 I3 . The JPR is seen in low fields. A number of PORs are
also seen in high fields. Even though the the SdH oscillations are also observed, it
is clearly seen that the frequency of the SdH oscillations is much higher than in X
= Cu(NCS)2 . Fig. 6–9 shows the angle-dependence of the POR and the SdH oscillations. At first, using the angle dependence of the SdH data, we determined that
the dominant period of the SdH oscillations is ∆(1/H) = 2.60 (10−5 /kOe) at θ = 0.
This value is very close to a previous study where ∆(1/H) = 2.58 (10−5 /kOe) [2].
We therefore found that this SdH frequency corresponds to the magnetic breakdown β-orbit. The results also confirm the earlier SdH experiments introduced in
Sec. 6.1 which suggested that κ-(ET)2 I3 has a much smaller magnetic breakdown
field, i.e., BM B ∼ 2.5 T for X = I3 . The two kinds of angle dependence of the POR
are shown in the figure. Different from κ-(ET)2 Cu(NCS)2 , the angle dependence
of both orientations in κ-(ET)2 I3 is the same as the dependence of the SdH data.
Thus the observed POR is associated with the Q2D section of the FS. This may
131
z θz2
θz1 z
2
2
1
1
ν/Bres(GHz/tesla)
ν/Bres(GHz/tesla)
∆(1/H)(1/gauss)
θz2 (degrees)
θz1 (degrees)
Figure 6–9. Angle-dependence of the POR in κ-(ET)2 I3 . (a) Angle dependence
of the POR and the SdH data in the the yz-plane. The data were
taken at frequency of 51.8 GHz and 91.8 GHz. The temperature is 1.4
K. (b) Angle dependence of the POR and the SdH oscillations in the
xz-plane. The data were taken at frequency of 91.8 GHz. The temperature is 1.4 K. The angle-dependence of the POR is same as the SdH
effect in both cases. Thus, the PORs must originate from the Q2D FS
section.
132
not be so surprising. Since we found from the SdH data that trajectories in a high
field will be highly dominated by the Q2D β-orbit of the magnetic breakdown
effect, we speculate that the trajectories for the POR are also dominated by the
β-orbit. Furthermore we observed many harmonic resonances. This also supports
the Q2D POR picture. The angle dependence of the POR implies that the warping
of the Q2D FS is tilted away from z-axis because the POR was observed around
θ = 0 degrees as we explained in Sec. 6.2.
Finally, fitting the angle-dependence by Eq. 6–6, we obtained the effective
mass m∗ = 4.6 me where me is electron mass. This effective mass is different
from the value obtained by reflectance measurements (m∗ = 2.4 ∼ 3.0 me ) [127].
However it agrees fairly well with the value determined by the SdH for the β-orbit
(m∗ = 3.9 me ) [2]. We also point out that the POR with the magnetic breakdown
effect has not been observed previously.
According to Kohn’s theorem [128], which is widely believed to be applicable
for the CR of conventional metals and semiconductors, the cyclotron effective
mass does not include the effect of electron-electron interaction while the quantum
oscillations effects (SdH and dHvA effects) does. The effective mass we obtained
is slightly larger than the mass in the SdH although the system is considered to
have the strong electron-electron interaction. This result may be inconsistent with
Kohn’s theorem.
6.5
Angle-resolved Mapping of Fermi Velocity: A Proposed
Experiment for Nodal Q2D Superconductors
We describe a technique which enables angle-resolved mapping of the in-plane
Fermi velocity (vF ) and the relaxation time (τ ) for Q2D conductors. The method is
based on a magnetic resonance effect in the ac conductivity for in-plane magnetic
fields, which has recently been successfully applied to the Q2D organic conductor
κ-(ET)2 I3 [61]. In this section, we consider the possibility of using this method
133
to probe the normal quasiparticles in nodal superconductors with numerical
simulations.
As shown in Chap. 2, we explained the POR phenomena as an extension
of the dc AMRO technique to high frequencies, such that ω > 1/τ . However
the picture of those two phenomena can be different when the magnetic field is
applied along the in-plane direction. Theoretical studies of dc AMRO have shown
that the conductivity of a Q2D metallic system is dominated by a small fraction
of the states on the FS for the situation in which the sample is subjected to a
strong in-plane magnetic field; these states coincide with unusual trajectories
near the self-crossing orbit on the FS induced by the Lorentz force (see Fig. 6–
10(a) and Peschansky and Kartsovnik [129]). More importantly, we have recently
demonstrated that the situation for the ac conductivity is considerably simpler
and potentially more useful [61]. Most notably, the dominant quasiparticle states
correspond to vertical open trajectories on opposite (symmetry-equivalent) edges of
the FS, as depicted in Fig. 6–10(a). Thus, by rotating the applied field in the xyplane, one can potentially gain access to information in the normal state concerning
the in-plane momentum (kxy ) dependence of the interlayer hopping, quasiparticle
lifetime (τ ), and quasiparticle density of states (m∗ ). Even more intriguing is the
possibility that one might be able to probe the normal quasiparticles that exist
along the line-nodes in a non s-wave superconductor (e.g., p- or d-wave).
Recent zero-field ac conductivity measurements have shown that it is possible
to measure the frequency-dependent conductivity due to the nodal quasiparticles in
YBa2 Cu3 O6+x (YBCO) [131]. Our magneto-optical technique offers the advantage
that one can, in principle, determine m∗ (or the Fermi velocity, vF ) and τ without
a detailed understanding of the quasiparticle excitation spectrum. Analogous to
the POR phenomenon in Q1D conductors explained in Sec. 2.3 and Chap. 4, the ac
conductivity resonance condition depends simply on the semiclassical trajectories
134
open
trajectories
kz
(a)
self-crossing
orbit
2π/R||
B
Corrugation
Period
Q2D Fermi Surface
ky
(b)
v||
B
vF(φ)
φ
v (ψ)
ext
ψ
kx
Q2D FS
Figure 6–10. Self-crossing orbits and open trajectories. (a) Q2D FS and quasiparticle trajectories under a magnetic field oriented parallel to the
conducting plane. The small arrows on the trajectories indicate
the direction of the Fermi velocity. The corrugation period is defined as 2π/Rk , where Rk is the lattice constant associated with the
corrugation. The dc AMRO is dominated by trajectories near the
self-crossing orbits, whereas the new ac conductivity resonance is
dominated by the extremal open trajectories. (b) The ellipse represents vF (φ); the magnetic field is applied at an arbitrary direction (ψ)
within the conducting plane; the resonance frequency is dominated
ext
by the extremal perpendicular component of the velocity, v⊥
(ψ).
vF (φ) can then be calculated from Eq. 6–10 (Reused with permission
from Takahashi et al. [130]. Copyright 2005, American Institute of
Physics.).
135
of quasiparticles on the presumed FS, i.e., the only assumption is that a FS exists
along certain vertical open trajectories in k-space [61]. In this section, we briefly
review the theory behind a new magneto-optical resonance which we have observed
in the normal state of a Q2D organic conductor [61]. We then present numerical
calculations for the case of a Q2D d-wave superconductor and discuss the possible
application of this method to organic superconductors and high-Tc superconductors
(HTSC).
The open-orbit ac conductivity resonance appears when one aligns the
magnetic field within the layers of a Q2D conductor. In this case, the quasiparticle
motion is principally open and periodic (except for a small fraction of the total
electrons see Fig. 6–10(a)); this is due to the underlying periodicity of the crystal
which leads to the FS corrugation. Here we employ the same method to calculate
the ac interlayer conductivity as shown in our previous paper (Kovalev et al. [61]),
i.e., we consider the Boltzmann equation in the T = 0 limit, and assume coherent
interlayer transport, i.e.,
E(k) =
~2 kx2 ~2 ky2
+
− 2tz cos(kz a),
2mx
2my
(6–8)
where mx and my are the in-plane effective masses, tz is the transfer energy along
the z-direction, and a is the interlayer spacing. As a result, we can write the ac
conductivity as follows (see Appendix A.3.),
Z
2π
σzz ∝
dφ
0
1 − iωτ
,
(1 − iωτ )2 + (ωc τ )2
(6–9)
where ωc = eBavF (φ) sin |φ − ψ|/~ ≡ eBav⊥ (φ)/~ (see Fig. 6–10(b)). The period
therefore depends on the magnetic field strength, B, and on the velocity component
(v⊥ ) perpendicular to the field. Averaging over the FS leads to the result that
ext
the extremal perpendicular velocity (v⊥
) dominates the electrodynamic response,
giving rise to a resonance in the interlayer conductivity (σzz ) when the period of
136
the electromagnetic field matches the periodicity of the extremal quasiparticle
ext
trajectories, i.e., when ω = ωcext (≡ eBav⊥
/~). Measurement of ωcext , as a function
ext
of the field orientation within the xy-plane, yields a polar plot of v⊥
(ψ) [61]. The
procedure for mapping vF (φ) is then identical to that of reconstructing the FS for
a Q2D conductor from the measured periods of Yamaji oscillations (AMRO) [30].
ext
(ψ), it is possible to generate the Fermi
Analytically, assuming one can measure v⊥
velocity vF (φ) using the following transformations (see also Fig. 6–10(b)):
q
vF =
ext 2
(v⊥
) + vk2 ,
φ = ψ + arctan(
vk = −
ext
v⊥
),
vk
ext
dv⊥
.
dψ
(6–10a)
(6–10b)
(6–10c)
vF (φ) is of course directly related to the quasiparticle density of states N (φ). Furthermore, the resonance lineshape contains information concerning the quasiparticle
lifetime τ (φ).
In the case of the HTSC and other candidate Q2D d-wave superconductors, it
is generally accepted that the normal quasiparticles reside along line-nodes oriented
along the original approximately cylindrical high-temperature FS (see Fig. 6–11).
These quasiparticles will dominate the low temperature ac conductivity in the
GHz range (all other single-particle excitations are gapped) [131]. A magnetic
field applied parallel to the xy-plane (B(ψ)) preserves in-plane momentum.
Consequently, such a field will tend to drive quasiparticles along the vertical line
nodes, thus preserving the magnetic resonance effect as we described. What is
more, the nodal quasiparticles will tend to be even longer lived than in the normal
state due to the reduced phase space for scattering [131]. Therefore, this dominance
of the nodal regions of the FS suggests that it may be possible to directly probe the
quasiparticles in nodal superconductors via the open-orbit conductivity resonance.
137
90
ψ
vxm
x
vF(φ)
Q2D FS
120
60
180
6
Bres (tesla)
B
y
vym
Re σZZ (ω, B) (arb. unit)
(a)
9
3
180
3
6
Bres
0
5
10
ψ=0
15
Magnetic field (tesla)
30
150
0
330
210
9
240
9
120
300
270
vF
x
180
Bres (tesla)
y
Re σZZ (ω, B) (arb. unit)
90
(b)
0
5
10
ψ=0
15
Magnetic field (tesla)
30
150
3
180
3
6
Bres
dx2-y2-wave gaps
6
60
9
0
330
210
240
300
270
Figure 6–11. Angle-resolved mapping of vF . (a) The left figure shows the Fermi
velocity, vF (φ), represented by an ellipse with major axis vxm , and
minor axis vym , together with its geometric relation to the underlying
Q2D elliptical FS. The middle figure displays the results of numerical calculations of the real part of the ac conductivity for ψ = 0 to
180 degrees in 5 degree steps. The peaks in the figure correspond to
the resonance. The positions of the resonances are represented by
the resonance field, Bres . The right figure shows a polar plot of Bres
versus (b) Fermi surface and superconducting gap for a dx2 −y2 -wave
superconductor (left), along with the corresponding calculations of the
ac conductivity (middle) and Bres (ψ) (right). Because of the assumption to count only the contribution from normal quasiparticles along
the line-nodes, a discrete vF (φ) is considered. In both calculations,
vxm = 5 × 104 m/s, vym = 3 × 104 m/s, ν = ω/2π = 50 GHz, τ = 5
ps and a=13 Å(Reused with permission from Takahashi et al. [130].
Copyright 2005, American Institute of Physics.).
138
Fig. 6–11 shows models of the angle-dependent Fermi velocity, and numerical
calculations of the ac conductivity σzz (ω) [Eq. 6–9] for two cases: (a) the normal
and (b) superconducting states of a Q2D d-wave superconductor. In the case of (a),
the normal state, we consider an elliptic angle-dependent Fermi velocity,
vF2 y
vF2 x
+
= 1,
2
2
vxm
vym
(6–11)
vF x = vF (φ) cos φ
(6–12a)
vF y = vF (φ) sin φ,
(6–12b)
with
and
where vxm and vym are the major and minor axes of the ellipse respectively
(see the left diagram in Fig. 6–11(a)). Such a FS is representative of the Q2D
organic conductor κ-(ET)2 Cu(NCS)2 . The middle figure is a calculation of the
ac conductivity σzz (ω) as a function of the magnetic field strength, for different
field orientations in the xy-plane. The peaks in the conductivity correspond to
the resonance; actual data for a real Q2D conductor are published in Kovalev et
al. [61]. We determine the resonance fields, Bres , from the peak positions. Each
in-plane field POR trace has only a single peak, since the resonance is governed
ext
entirely by the extremal perpendicular velocity (v⊥
). The right figure shows
a polar plot of Bres versus ψ. Because of the two-fold symmetry of the angledependent Fermi velocity, Bres also shows a two-fold symmetry. If the angledependent Fermi velocity were four-fold, Bres would of course also show a four-fold
symmetry.
The middle and right panels of Fig. 6–11(b) show numerical calculations
for the case of a dx2 −y2 -wave superconductor; the FS and superconducting gap
are depicted in the left panel. Since the resonance is dominated by the normal
quasiparticles at the line-nodes, Bres shows a four-fold symmetry. Consequently
139
the ac conductivity σzz (ω) exhibits four series of resonance peaks, which come
from the two pairs of opposite line nodes, i.e., the conductivity is dominated by
the nodal quasiparticles, not the extremal trajectories on the full FS. In principle,
we can apply this ac conductivity method to any nodal superconductors which
satisfies ωτ > 1. Using this method, we predict that it may be possible to measure
both vF and τ associated with the normal quasiparticles at the nodes of a d-wave
superconductor such as YBCO [131] or κ-(ET)2 Cu(NCS)2 . Moreover, we may be
able to confirm the symmetry of the superconducting gaps.
6.6
Summary
In this chapter, we presented POR studies for the Q2D conductors κ(ET)2 Cu(NCS)2 and κ-(ET)2 I3 . We observed the Q1D POR in κ-(ET)2 Cu(NCS)2
and the Q2D POR in κ-(ET)2 I3 . The origin of the POR was identified using the
angle-dependence of the POR and the SdH effect. We also deduced the origin of
the warping by using the angle dependence for κ-(ET)2 Cu(NCS)2 . We could not
obtain the direction of the warping for κ-(ET)2 I3 . However we found that, similar
to κ-(ET)2 Cu(NCS)2 , the direction of the warping is tilted away from the z-axis
from the harmonic content of the POR and from the fact that the POR at θ = 0
were observed. In the last section, we explained a proposed experiment for Q2D
nodal superconductors. By using the method, one can measure the angle-resolved
mapping of the Fermi velocity (vF ) and the relaxation time (τ ) for normal quasiparticles on the nodes of the superconducting gaps. Moreover this can identify the
symmetry of the superconducting gap. The method may be applicable for Q2D
nodal superconductors such as YBCO [131] or κ-(ET)2 Cu(NCS)2 .
CHAPTER 7
SUMMARY
This chapter summarizes the results of the studies described in this thesis. This Ph.D. dissertation is devoted to studies of angle-dependent high-field
microwave spectroscopy of low dimensional conductors and superconductors.
Chapter 1 is an introduction to low dimensional conductors and superconductors which are represented by high-Tc cuprates, organic conductors and Sr2 RuO4 .
We showed various shapes of Fermi surface (FS) for 3-dimensional (3D), quasitwo-dimensional (Q2D), 2-dimensional (2D), quasi-one-dimensional (Q1D) and
1-dimensional (1D) conductors. For these low dimensional materials, the topology
of the FS, i.e., the nesting property, plays a crucial role for their ground states.
The examples are illustrated by the Peierls instability and the field-induced spindensity-wave (FISDW) state. The superconductivity in low dimensional materials
is often unconventional having, for example, an anisotropic superconducting energy gap, spin-triplet Cooper pairing and non electron-phonon mediated Cooper
pairing. One of the main evidences for unconventional superconductivity is the
non-magnetic impurity effect, which is well described by the Abrikosov-Gor’kov
(AG) theory.
Chapter 2 explains the periodic-orbit resonance (POR) phenomenon which
we use to probe the topology of a Fermi surface. The principle of the POR is
closely related to conventional cyclotron resonance (CR) observed in metals with
closed Fermi surfaces, because the resonance involves an oscillatory group velocity.
However, the POR can even be seen in a system with only open trajectories
such as a Q1D conductor. We showed that the POR effect appears in the ac
conductivity using a semiclassical description. For the Q1D POR, by studying
140
141
its angle dependence, one can determine the Fermi velocity, vF , the direction
of a warping on the FS and the scattering time, τ . For the Q2D POR, one can
determine the effective mass, vF , and the scattering time, τ . For a frequency ν ∼ 0,
the POR is the same as the so-called angle-dependent magnetoresistance oscillation
(AMRO) effect, which shows sharp resonances in the dc conductivity. For a Q1D
FS, three types of AMRO effects are named: Lebed (z − y rotation), Danner-KangChaikin (DKC) (z − x rotation), and the 3rd angular effect (x − y rotation), where
the x-axis is the conducting direction, the y-axis is the intermediate direction, and
the z-axis is the least conducting direction. For a Q2D FS, the AMRO is called the
Yamaji effect (z to xy-plane rotation) where the xy-plane is the conducting plane.
Chapter 3 introduced our experimental technique. We employ an angledependent high-field microwave spectroscopy technique. Using a millimeter
vector network analyzer (MVNA) with a cavity perturbation probe, we achieve
a wide range of frequencies (8 - 700 GHz). High field magnets at the University
of Florida (UF) (B < 17 T) and the National High Magnetic Field Laboratory
(NHMFL) (B < 45 T) are also employed. These setups work down to 0.5 K
with a home-made 3 He system. For the angle dependent study, we developed a
rotating cavity which is fully compatible with the magnets at UF and the NHMFL.
Using the rotating cavity, one can also have two axes of rotation using a split-pair
magnet with a transverse field. In the latter part of the chapter, we explained
the electrodynamics for the cavity perturbation technique for low dimensional
conductors and superconductors.
In Chapter 4, we presented a study of the POR in the Q1D conductor
(TMTSF)2 ClO4 . For (TMTSF)2 X, it is considered that the Lebed effect is unusual since the AMRO at so-called Lebed magic angles shows stronger resonances
in the dc conductivity than expected by the Boltzmann transport theory. Moreover recent studies show that Nernst effect measurements become unconventional
142
at magic angles. In the investigation of the angle dependence of the POR in
(TMTSF)2 ClO4 , we found that all POR evolve from the Lebed magic angles in the
AMRO. However the dependence of the POR is semiclassical. The result concludes
that these angles are no longer “magic” at frequencies on the order of the cyclotron
frequency. In the (TMTSF)2 X series, a pressurized (TMTSF)2 PF6 sample may
be a more controversial example for the Lebed magic angle effect. We have been
developing an angle-dependent high-pressure magneto-optical cell environment in
order to investigate the Lebed effect in (TMTSF)2 PF6 .
Chapter 5 shows the results of the non-magnetic impurity effect on the
superconductivity in (TMTSF)2 ClO4 . Since recent studies suggested spin-triplet
pairing in (TMTSF)2 ClO4 , the superconductivity may be sensitive to impurities.
Varying the rate at which samples are cooled through an anion-ordering transition
at 24 K, we can control the disorder in the sample: i.e., we can vary the scattering
time τ of the POR. The disorder associated with the non-magnetic ClO4 anions is
known to affect the superconducting transition temperature (Tc ). We find a simple
relationship between Tc and τ , strongly suggesting that the superconductivity is
suppressed by non-magnetic impurities.
In Chapter 6, we present studies of the Q2D conductors κ-(ET)2 X [X =
Cu(NCS)2 and I3 ]. According to a tight-binding calculation, these κ-(ET)2 X have a
very similar FS which consists of Q1D and Q2D sections. The POR was observed
for both the κ-(ET)2 X compounds. Studying the angle-dependence, we found that
the POR in κ-(ET)2 Cu(NCS)2 originates from the Q1D section of the FS so that
the Fermi velocity vF was obtained from the angle-dependence. On the other hand,
we found that the resonance in κ-(ET)2 I3 is the Q2D POR which originates from
a magnetic breakdown orbit on the FS. The POR due to the magnetic breakdown
effect has not been observed previously. We obtained the effective mass from the
angle-dependence of the POR. The latter part of the chapter discusses a proposed
143
experiment for nodal superconductors. This experiment will be useful to study
properties of normal quasiparticles and the symmetry of superconducting energy
gaps of clean nodal superconductors such as the high-Tc cuprates (YBa2 Cu3 O6+x ,
Tl2 Ba2 CuO6+x etc) and κ-(ET)2 Cu(NCS)2 .
APPENDIX
SEMICLASSICAL CALCULATION OF THE ELECTRICAL CONDUCTIVITY
In this appendix, we show an analytical calculation of the electrical conductivity using a semiclassical description.
A.1
A Simple Quasi-one-dimensional Model
As noted in Chap. 2, in order to describe a simple Fermi surface (FS) for a
quasi-one-dimensional (Q1D) conductor, the following energy dispersion is often
used,
E(k) = ~vF (|kx | − kF ) − 2ty cos(ky b) − 2tz cos(kz c),
(A–1)
where EF À ty and tz , vF is the Fermi velocity, and b and c are lattice constants
along the y and z-directions, respectively. ty and tz are the transfer energies
associated with the lattice vectors Ry and Rz . This energy dispersion describes
a warped Q1D FS. The 1st term of the energy dispersion is responsible for the
flat shape of the FS, and the 2nd and 3rd term are responsible for the warping
along the lattice vectors. The physical meaning of each transfer energy ty and tz is
therefore the Fourier component of the warping.
Now we consider the resonance condition for periodic orbit resonance (POR)
in the case of a Q1D FS with a magnetic field applied along an arbitrary direction.
The applied magnetic field may be expressed by B = (B sin θ cos φ, B sin θ sin φ, B cos θ)
where θ is the angle between the magnetic field and the z-axis, and φ is the angle
between the magnetic field and the x-axis in the xy-plane, as shown in Fig. A–1.
Recalling the equation of motion,
~k̇ = −e(v × B),
144
(A–2)
145
B
z
Q1D FS
θ
Rpq
x
φ
conducting
axis
y
Bxy
Figure A–1. Representation of rotation of a magnetic field relative to a Q1D FS.
one obtains
2tb beB
2tc ceB
k˙x = −
cos(θ)
sin[k
(t)b]
+
sin(θ) sin(φ) sin[kz (t)c],
y
~2
~2
2tc ceB
vF eB
k˙y = −
sin(θ) cos(φ) sin[kz (t)c] + sgn(kx )
cos(θ),
2
~
~
vF eB
2tb beB
k˙z = −sgn(kx )
sin(θ) sin(φ) +
sin(θ) cos(φ) sin[ky (t)b],
~
~2
(A–3a)
(A–3b)
(A–3c)
and, using the following equation, the group velocity is given by
vg =
1
∇k [E(k)].
~
(A–4)
One then obtains the following equations,
vx = sgn(kx )vF ,
2ty b
sin[ky (t)b],
~
2tz c
vz (t) =
sin[kz (t)c].
~
vy (t) =
(A–5a)
(A–5b)
(A–5c)
146
Here we calculate the conductivity for rotation of the magnetic field in the
Lebed effect. In this case, the field angle θ is varied and φ = 90◦ . Therefore, kz and
vF can be calculated easily from
vF eB
k˙z = −sgn(kx )
sin(θ),
~
(A–6)
and then
kz (t) = −sgn(kx )
vF eB
sin(θ)t + kz (0).
~
(A–7)
Thus,
vz (t) =
2tz c
sin[−sgn(kx )ωQ1D (B, θ)t + kz (0)c],
~
(A–8)
where ωQ1D (B, θ) = evF cB sin θ/~. Now we calculate the conductivity using a
expression of the Boltzmann equation:
2e2
σzz (ω, B) =
V
Z
∂f (E)
dE[−
]N (E)
∂E
where
Z
v z (ω, k) ≡
Z
dk 2 vz (k, 0)v z (ω, k),
(A–9a)
0
1
dtvz (k, t) exp( − iω)t.
τ
−∞
(A–9b)
Using the following result,
Z
0
1
τ
eik
e−ik
dt sin(ωc t + k) exp( − i ω)t = [
−
], (A–10)
τ
2i 1 − i(ω − ωc )τ
1 − i(ω + ωc )τ
−∞
v z is obtained by,
v z (ω, k) =
tc cτ
eikz (0)c
e−ikz (0)c
[
−
].
i~ 1 − i(ω + sgn(kx )ωQ1D )τ
1 − i(ω − sgn(kx )ωQ1D )τ
(A–11)
147
Using this v z , the conductivity is given by
Z
2e2 N (EF )
σzz (ω, B, θ) =
dkF2 vz (k, 0)v z (ω, k)
V
Z π/c
2 2 2 Z π/b
4e tc c τ
=
dky
dkz sin[kz (0)c] ×
iV ~2
−π/b
−π/c
eikz (0)c
e−ikz (0)c
eikz (0)c
e−ikz (0)c
[
−
+
−
]
1 − i(ω + ωQ1D )τ
1 − i(ω − ωQ1D )τ
1 − i(ω − ωQ1D )τ
1 − i(ω + ωQ1D )τ
16π 2 e2 c2 t2c N (EF )τ
1
1
=
[
+
]. (A–12)
2
V~ b
1 − i(ω − ωQ1D (B, θ))τ
1 − i(ω + ωQ1D (B, θ))τ
A.2
A General Quasi-one-dimensional Model
Next we show the calculation of the Lebed effect using the following realistic
model.
E(k) = ~vF (|kx | − kF ) −
X
tpq cos[(pb0 + qd)ky + (qc∗ )kz ]
(A–13)
p,q
where tpq is the transfer energy associated with the lattice vector Rpq , p and q
are integers, and b0 and c∗ represent lattice constants which are used to represent
(TMTSF)2 X. This energy dispersion describes a general Q1D FS. As we explained
in Chap. 4, the 1st term of the energy dispersion is responsible for the flat shape
of the FS, and the 2nd term is responsible for the warping along the lattice vector
Rpq . We again consider that the magnetic field is applied along an arbitrary
direction.
In the present case, the equation of motion and the group velocity are given by
the following:
eBvy (t)
eBvz (t)
k˙x = −
cos(θ) +
sin(θ) sin(φ),
~
~
eBvz (t)
eBvx
k˙y = −
sin(θ) cos(φ) +
cos(θ),
~
~
eBvx
eBvy (t)
k˙z = −
sin(θ) sin(φ) +
sin(θ) cos(φ),
~
~
(A–14a)
(A–14b)
(A–14c)
148
and,
vy (t) =
X (pb0 + qd)tpq
~
p,q
vz (t) =
vx = sgn(kx )vF ,
(A–15a)
sin[(pb0 + qd)ky (t) + (qc∗ )kz (t)],
(A–15b)
X (qc∗ )tpq
p,q
~
sin[(pb0 + qd)ky (t) + (qc∗ )kz (t)].
(A–15c)
In general, it is difficult to find an analytical solution to the above differential
equations Eq A–14 and Eq. A–15. However, in the case of the Lebed effect, one
can obtain an analytical solution easily. Taking φ = 90◦ , k˙y and k˙y are highly
simplified.
eBvF
k˙y ∼
cos(θ),
= sgn(kx )
~
eBvF
k˙z ∼
sin(θ).
= −sgn(kx )
~
(A–16a)
(A–16b)
Therefore, the solutions are given by,
ky (t) = sgn(kx )
eBvF
cos(θ)t + ky (0),
~
kz (t) = −sgn(kx )
(A–17a)
eBvF
sin(θ)t + kz (0).
~
(A–17b)
Using Eq. A–17, the z-component of the group velocity is written by
vz (t) =
X (qc∗ )tpq
p,q
~
pq
(B, θ) =
where ωQ1D
v z (ω, k) =
pq
sin[−sgn(kx )ωQ1D
(B, θ)t + (pb0 + qd)ky (0) + (qc∗ )kz (0)], (A–18)
eBvF
~
[(pb0 + qd) cos θ − (qc∗ ) sin θ]. Using Eq. A–10, one obtains
X (qc∗ )tpq τ
p,q
2i~
0
∗
0
∗
ei(pb +qd)ky (0)+i(qc )kz (0)
e−i(pb +qd)ky (0)−i(qc )kz (0)
[
−
].
pq
pq
1 − i(ω + sgn(kx )ωQ1D )τ
1 − i(ω − sgn(kx )ωQ1D
)τ
(A–19)
149
Similar to the previous case, with Eq. A–9 and Eq. A–19, the electrical conductivity σzz is given by the following:
4π 2 e2 N (EF )τ
σzz (ω, B, θ) =
×
V ~2
X
1
1
+
]. (A–20)
(qc∗ )2 t2pq [
pq
pq
1
−
i(ω
−
ω
(B,
θ))τ
1
−
i(ω
+
ω
(B,
θ))τ
Q1D
Q1D
p,q
A.3
A Simple Quasi-two-dimensional Model
Here we consider a Q2D FS. We derive the electrical conductivity when
a magnetic field is applied in the conducting plane of a Q2D FS, as shown in
Chap. 6. As a simple example, we consider the following energy dispersion.
E(k) =
~2 kx2 ~2 ky2
+
− 2tz cos(kz c),
2m∗x
2m∗y
(A–21)
where m∗x and m∗y are the xx and yy components of the effective mass tensor,
tz is the hopping energy to the least conducting z-direction, and c is the interlayer spacing. The magnetic field is applied along an arbitrary direction.
Similar to the case of the Q1D FS, the applied magnetic field is expressed by
B = (B sin θ cos ψ, B sin θ sin ψ, B cos θ) where θ is the angle between the magnetic
field and the z-axis, and ψ is the angle between the magnetic field and the x-axis in
the xy-plane, as shown in Fig. A–2.
In the present case, the equation of motion and the group velocity are given by
the following:
2tz ceBvz (t)
eB
sin(θ) sin(ψ) sin[kz (t)c],
k˙x = − ∗ cos(θ)ky (t) +
my
~2
(A–22a)
eB
eBceB
sin(θ)
cos(ψ)
sin[k
(t)c]
+
cos(θ)kx (t),
k˙y = −
z
~2
m∗x
(A–22b)
eB
eB
k˙z = − ∗ sin(θ) sin(ψ)kx (t) + ∗ sin(θ) cos(ψ)ky (t),
mx
my
(A–22c)
150
least conducting
axis z
B
θ
x
ψ
y
Bxy
Q2D FS
Figure A–2. Representation of rotation of the magnetic field for a Q2D FS.
and,
vx (t) =
~kx (t)
,
m∗x
(A–23a)
vy (t) =
~ky (t)
,
m∗y
(A–23b)
2tz c
sin[kz (t)a].
~
(A–23c)
vz (t) =
Until this point, the direction of the magnetic field is arbitrary. Now we set the
magnetic field along the in-plane direction so that θ = 90. kx and ky become,
2tz ceBvz (t)
k˙x =
sin(ψ) sin[kz (t)c],
~2
(A–24a)
eBceB
cos(ψ) sin[kz (t)c].
k˙y = −
~2
(A–24b)
Moreover, when EF À tz , kx and ky are almost constant. Therefore, kx and ky are
approximately given by,
kx = kx (0),
(A–25a)
ky = ky (0).
(A–25b)
151
Using Eq. A–25, kz (t) is therefore given by
kz (t) = −
eB
(vx sin ψ + vy cos ψ)t + kz (0),
~
(A–26)
where vx and vy are defined as vx = ~kx (0)/m∗x and vy = ~ky (0)/m∗y respectively.
Furthermore, using the relation for an ellipse, vx = vxm cos θk and vy = vym sin θk .
The z-component of the group velocity is simply given by,
vz (t) =
where ωc (B, ψ, θk ) =
eBc
(vxm
~
2tz c
sin[−ωc (B, ψ, θk )t + kz (0)c],
~
(A–27)
sin ψ cos θk − vym cos ψ sin θk ). Employing Eq. A–10, v z
is obtained by,
v z (ω, k) =
tz cτ sin(kz (0)c)(−iωτ ) + cos(kz (0)c)(ωc τ )
[
].
~
(1 − iωτ )2 + (ωc τ )2
(A–28)
Thus, the conductivity is calculated by the next equation,
8πe2 t2z c2 N (EF )τ
σzz (ω, B, ψ) =
(1 − iωτ )
V ~2
Z
2π
dθk
0
(1 −
iωτ )2
1
.
+ (ωc (B, ψ, θk )τ )2
(A–29)
In the present case (elliptic FS), this integral is solvable. Using the result of the
integration,
Z
2π
0
(1 −
ia)2
dθ
2π
1
p
=
,
2
+ (b cos θ − c sin θ)
1 − ia (1 − ia)2 + b2 + c2
(A–30)
the electrical conductivity is given by
σzz (ω, B, θ) =
16π 2 e2 t2z c2 N (EF )τ
1
].
[q
2
V~
2 (v 2 sin2 ψ + v 2 cos2 ψ)
(1 − iωτ )2 + ( eBcτ
)
ym
xm
~
(A–31)
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BIOGRAPHICAL SKETCH
Susumu Takahashi was born in Yokohama, Japan and grew up in Sakura,
Chiba prefecture, Japan. After graduating from Yachiyo high school, he entered
Aoyama-Gakuin University in Tokyo, where he received his bachelor’s and master’s
degrees in physics in 1995 and 1997. In his senior year he began studying the
pairing mechanism of low dimensional superconductors such as the high-T c
cuprates under Prof. Duk Joo Kim. At Aoyama-Gakuin University, he met Ryoko,
who is now his wife. They were married at the end of the 20th century.
From April, 1997, to June, 2001, Susumu worked for Hitachi, Ltd., Tokyo,
Japan. In Hitachi, he spent two years researching a high-speed, high-bandwidth
DRAM memory bus with crosstalk transfer logic (XTL) interface for the next
generation memory bus. Aiming for a 1 GHz clock frequency for the bus system, he
and his team fabricated a prototype of the bus system to test with the latest 0.15
µm process technology at that time. He designed the delay control circuit of the
bus interface, the layout and the mask pattern, and also evaluated the prototype
using an original test platform. Two years later, he developed the high-performance
microprocessor, SH3-DSP. This processor consists of several million transistors.
He designed several parts of the logic-circuit modules and built environments for
the assembly and verification of whole logic modules. The microprocessor has been
employed in many digital cameras and video products.
In August, 2001, Susumu joined the Ph.D. program in physics at the University of Florida. He has worked for Professor Stephen Hill since the summer
of 2002. By 2005, he performed studies of angle-dependent high-field microwave
spectroscopy for low dimensional conductors and superconductors. As a result, he
160
161
completed his Ph.D. dissertation. He received the Ph.D. degree in physics in 2005.
While he lived in Gainesville, FL, he extended his family. His sons, Kai and Riku,
were born in 2001 and 2003 respectively.
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