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ANGLEDEPENDENT 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 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 alvb i 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 ,i:. 1 i. us 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 equipment for our experiments at the NHMFL. I also had the great fortune to have discussions with and receive ii:. 1 i ,s 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 BetancurRodiguez, Saiti Datta, SungSu Kim, Dan Ben jamin, Emmitt Thompson, Costel Rotundu, Tara Dhakal, N i,.. i: 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. TABLE OF CONTENTS page ACKNOWLEDGMENTS ................... .......... iv LIST OF TABLES ................... .............. ix LIST OF FIGURES ................... ............. x ABSTRACT ................... ................. xiii CHAPTER 1 INTRODUCTION ................... .. ..... .. 1 1.1 Overview of Low Dimensional Systems .............. 1 1.2 Fermi Surfaces of Low Dimensional Conductors ......... 2 1.3 Quasionedimensional and Quasitwodimensional Materials 2 1.3.1 The Quasionedimensional Conductor (TMTSF)2C104 . 4 1.3.2 The Quasitwodimensional Conductor K(ET)2X . 4 1.4 Instability in Low Dimensional Conductors . . . 6 1.5 Superconductivity in Low Dimensional Materials . ..... 11 1.6 Impurity Effect on the Superconductivity .. . . 12 2 PERIODIC ORBIT RESONANCE . . . . ... 14 2.1 Experimental Techniques to Study Fermi Surfaces . ... 14 2.2 Cyclotron Resonance . . . ....... . 15 2.3 Cyclotron Resonance Involving an Open Fermi Surface: Periodic Orbit Resonance . . . . . .. 22 2.4 Periodic Orbit Resonance for a Quasitwodimensional Fermi Sur face ...... ........ . .. .. ............. 27 2.5 POR and Angledependent Magnetoresistance Oscillations . 30 2.6 Quantum Effects in the Conductivity . . ..... 37 2.7 Summary .. . . . .. . . .... 41 3 EXPERIMENTAL SETUP .. . . . .. ..... 42 3.1 Overview of Microwave Magnetooptics . . ....... 42 3.2 Experimental Setup ... . . .. . .... 45 3.3 Rotating Cavity .... . . .. . .... 53 3.4 Model of the Resonant Cavity . . . . .... 65 3.5 Positioning Low Dimensional Conductors and Superconductors in the Cylindrical Cavity . . . . . 69 3.5.1 Inplane Measurements. .................. 69 3.5.2 Interlayer Measurements. ................. ..70 3.5.3 Configuration for Interlayer Measurements Using the Mag netic Component of the Microwaves . . ... 71 3.6 Microwave Response of Low dimensional Conductors and Super conductors. .................. ........ 72 3.6.1 Skin Depth Regime .................. .... 72 3.6.2 Metallic Depolarization Regime .............. ..77 3.7 Measurement of the C!:i ,i,'. of the Complex Impedance Z . 77 3.7.1 Frequencylock Method ................ .. 79 3.7.2 Phaselock Method ................ . .80 3.8 Summary ............... ........... .. 81 4 PERIODIC ORBIT RESONANCES IN QUASIONEDIMENSIONAL CONDUCTORS ................ ............. 82 4.1 The Quasionedimensional Conductor, (T\ TSF)2C104 . 82 4.2 Semiclassical Description of the Periodic Orbit Resonance and the Lebed Effect .......... ............ 89 4.3 Observation of the Periodic Orbit Resonances in (T\ iTSF)2C104 93 4.4 Summary .................. ............ 102 5 NONMAGNETIC IMPURITY EFFECTS ON THE SUPERCONDUC TIVITY IN (T\iTSF)2C104 .................. .... 103 5.1 Overview of the Superconductivity in (TMTSF)2C104 ...... .103 5.2 DC (u w 0) Transport Measurements for Different Cooling Rates 106 5.3 Study of the Periodic Orbit Resonance at Different Cooling Rates 109 5.4 All i i of the Scattering Rate F . . . 110 5.5 Relation Between T, and the Pair Breaking Strength a ..... ..116 5.6 Summary ............... ........... .. 117 6 PERIODICORBIT RESONANCE IN QUASITWODIMENSIONAL CONDUCTORS ............. ............. 118 6.1 The Quasitwodimensional Conductors K(ET)2X . ... 118 6.2 Periodicorbit Resonance in K(ET)2X . . 121 6.3 Experiments for K(ET)2Cu(NCS)2 ............. 125 6.4 Experiments for K(ET)23 .... . .. . 129 6.5 Angleresolved Mapping of Fermi Velocity: A Proposed Experi ment for Nodal Q2D Superconductors ............. ..132 6.6 Summary .................. ............ 139 7 SUMMARY ................... ..... ........ 140 APPENDIX SEMICLASSICAL CALCULATION OF THE ELECTRICAL CONDUCTIVITY ................... ....... 144 A.1 A Simple Quasionedimensional Model ....... ........ 144 A.2 A General Quasionedimensional Model ...... ....... 147 A.3 A Simple Quasitwodimensional Model ....... ........ 149 REFERENCES ...................... ........... 152 BIOGRAPHICAL SKETCH ................... ........ 160 LIST OF TABLES Table page 31 Available magnet systems at UF and the NHMFL. ......... ..48 32 Probes used for the cavity perturbation technique. ......... .51 33 Resonance parameters for several different cavity modes . .... 61 41 Lattice parameters and the AMRO notations for the nth nearest neigh bors. ................... ................ 92 61 Unit cell parameters for K(ET)2X. .................. 119 LIST OF FIGURES Figure page 11 Illustration of the FS by varying the bandwidths tb and t. . 3 12 Illustration of the < i  .1 structure of (T:\ TSF)2C104 and the T\ TSF molecule ................ ........... 5 13 Illustration of the < i 1 structure of K(ET)2Cu(NCS)2 and the ET molecule ................ ........... 6 14 Illustration of the Peierls instability in a 1D system. . . ... 8 15 Illustration of the confinement effect on the trajectories of electrons in a magnetic field .................. ....... 9 16 TH phase diagram for (TA\iTSF)2C14. ................ 10 21 The Fermi surface for the twodimensional electron system. ..... .. 16 22 Real part of a,, as a function of frequency for various values of cwU. 19 23 The Fermi surface for the twodimensional electron system with an arbitrary direction of the magnetic field. ............ 21 24 Oscillatory group velocity v for the Q1D POR. ........... .23 25 POR in a(ET)2KHg(SCN)4 ................ .... 26 26 Oscillatory group velocity v for Q2D POR. ............. .28 27 AMRO for Q2D conductors. ............. .... 31 28 AMRO for Q1D conductors. ............. .... 33 29 The Lebed effect in both the dc and ac conductivity. . ... 34 210 Ye,"i i ii oscillations. ............... ...... 35 211 Numerical calculation of the conductivity for a Q2D FS. ...... ..36 212 Landau tube with a magnetic field. .................... .. 38 213 DOS of a Q2D conductor in a magnetic field in units of hc. . 39 31 Frequency range for the Q2D POR in magnetic fields accessible at the NHMFL. ................. ... ... ....... 45 lds accessible at the NHMFL. .... . . . .......... 45 32 Overview of the experimental setup. .................... .. 46 33 A schematic diagram of the 3He probe. ................ 50 34 Vertical temperature distribution in the cryostat for the QD PPMS 7 T and Oxford Instruments 17 T magnets. . . 52 35 A schematic diagram of the rotating cavity system. . .... 55 36 Photographs of the rotating cavity system. ............. 57 37 Schematic diagrams showing various different sample mounting con figurations. .................. .. ........ 60 38 Angledependence of the cavity resonant properties. . .... 63 39 A simple description of the resonant cavity. ............. ..67 310 Schematic diagram illustrating the various possibilities for exciting inplane and interlayer currents in a Q2D platelike sample. . 70 311 Positioning of the sample in the TEO11 mode. ........... ..73 312 Typical changes in the amplitude and phase of the microwaves trans mitted through the cavity. .................. .... 79 41 Electronic properties of the (T:\ITSF)2X. .... . ... 83 42 Illustration of i i1I 1 axes for (T\ TSF)2C104. ............ .86 43 The dc AMRO experiment in (T\ITSF)2C1O4 ........... 88 44 The oblique realspace crystal lattice. ................ 90 45 Resonance conditions and the POR by sweeping the angle and mag netic field. . . . . . .. . 94 46 Overview of the orientations in the experiments. .......... ..96 47 Microwave absorption as a function of the magnetic field. ....... 97 48 Angle dependence of the quantity v/Bs... .............. 99 49 Angle sweep and field sweep measurements for (T\ITSF)2C104. .. 100 410 Summary of the p/q =0, 1 and 1 POR data for sample C. ..... ..101 51 Illustration of the C104 anion and the crystal structure of (T\ iTSF)2C104 below TAO . . . . . . .. . 104 52 dc transport measurements at different cooling rates. . ... 107 53 Summary of the dc transport measurements. . . 108 54 Microwave absorption as a function of magnetic field at different rates. 110 55 Leastsquare fit to microwave absorption . . ..... 112 56 Cooling rate and temperature rate dependence of the scattering rate. 113 57 Comparison between the resistance R,, and the scattering rate F as a function of the cooling rate. . . . .. .. 115 58 T, vs. the scattering rate . . . . .. 116 61 Phase diagram for the organic conductors K(ET)2X. . . 119 62 Fermi surface (FS) and trajectories of an electron under a magnetic field for K(ET)2Cu(NCS)2. . . . .... .. 120 63 Warping on a Q2D FS. . . . . .. . 121 64 Numerical calculation of the ac conductivity for a Q2D FS. . 124 65 Overview of the orientations in the experiments on K(ET)2X. . 125 66 Experimental data for K(ET)2Cu(NCS)2. . . . 128 67 Angledependence of the POR and the SdH oscillations for two kinds of rotations in K(ET)2Cu(NCS)2. . . . ....... 129 68 Experimental data for c(ET)23. . . . .. . 130 69 Angledependence of the POR in c(ET)213 . . .. 131 610 Selfcrossing orbits and open trajectories. . . . 134 611 Angleresolved mapping of vp. . . . ..... .. 137 Al Representation of rotation of a magnetic field relative to a Q1D FS. 145 A2 Representation of rotation of the magnetic field for a Q2D FS. . 150 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 ANGLEDEPENDENT HIGH MAGNETIC FIELD MICROWAVE SPECTROSCOPY OF LOW DIMENSIONAL CONDUCTORS AND SUPERCONDUCTORS By Susumu Takahashi December 2005 C(! nI: Stephen O. Hill Major Department: Physics This dissertation presents studies of angledependent highfield 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 socalled periodic orbit resonance (POR). In C!i lpter 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 quasitwodimensional (Q2D) FS, but also to a quasionedimensional (Q1D) FS. In C!i ipter 3, our experimental techniques are presented. We outline a rotating cylindrical cavity, which enables angledependent cavity perturbation measurements in ultrahighfield magnets, and twoaxis rotation capabilities in standard highfield superconducting splitpair magnets. In C'!i ipters 4 and 5, the results of studies of the Q1D conductor (TA [TSF)2C104 are shown. Using the POR, we determined the Fermi velocity vp and revealed new information concerning the nature of the socalled Lebed effect in ('!i Ipter 4. In C'!i lpter 5, we studied the nonmagnetic impurity effect and its influence on the possible spintriplet superconductivity in (TA[TSF)2CO14. In C'!i lpter 6, measurements of the POR are performed in the Q2D conductors K(ET)2X [XCu(NCS)2 and I3]. In X=I3, POR involving the magnetic breakdown effect was observed for the first time. 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 exter nal 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 quasitwo dimensional (Q2D) or quasionedimensional (Q1D). Recently, the study of these systems has been attractive because many interesting phenomena have been dis covered in them, including unconventional superconductivity, metalinsulator tran sitions, antiferromagnetism, spindensitywaves (SDW) and chargedensitywaves (CDW). In order to understand these low dimensional systems, it is important to study their detailed electronic properties, as well as their superconducting proper ties. 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 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 t, : ty : tz = 1 : 1 : 1. In general, the shape of the FS can be complicated, even when the conducting properties are threedimensional and free electronlike. However, we here consider a perfectly spherical FS as the simplest case, as shown in the upper left picture of Fig. 11. Starting with this FS, we change the anisotropy. When the bandwidth along the zdirection becomes smaller, e.g., t, : ty : tz = 1 : 1 : 1/2, the FS sphere is stretched along the z direction. The smaller the zaxis band width, the more the zdirection of the FS is stretched, as shown in the upper middle figure. Eventually the zcomponent of the FS connects with the FS in the next Brillouin zone, and then the FS becomes open along the zdirection, namely a cylinderlike 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., t, : t, : tz = 1 : 1 : 1/100. Such an anisotropic FS is said to be Q2D. Examples of materials with Q2D FSs are K(ET)2Cu(NCS)2 and K(ET)213 (See O'i p1 6.). Next, we change the band width along the ydirection. While the band width is reduced, the ydirection of the FS tube is stretched step by step, as seen in the lower left figure. Eventually, the FS becomes a couple of planelike sheets with small corrugations, as shown in the lower right panel of Fig. 11. These corrugations are related to the small transfer energies along the y and zdirections, e.g., t, : ty : t, = 1 : 1/100 : 1/100, representing a Q1D FS. This is the case for (T [\TSF)2C104 described in C'i p1 4. 1.3 Quasionedimensional and Quasitwodimensional Materials Here we introduce examples of low dimensional materials which have the FSs described in the previous section. 1.3.1 The Quasionedimensional Conductor (TMTSF)2C104 The organic metal (TMTSF)2C104 belongs to the family of quasione dimensional (Q1D) Bechgaard salts [2], having the common formula (T\ TSF)2X. TMTSF is an abbreviation for tetramethyltetraselenafulvalene, and the anion X is AsF6, C1O4, PF6, ReO4, etc. The (TMITSF)2X compounds are the socalled 2:1 charge transfer salts which transfer one electron from two TMTSF molecules to one X anion. The < iI 1 structures of the (TSITSF)2X series are similar. Fig. 1 2(a) shows the < i, 1 structure of (T\ITSF)2C104. As shown to the left of Fig. 1 2(a), the planar T\ [TSF molecules are stacked along the adirection. The conducting properties come from the overlap of rorbitals on the T\ TSF molecules, as shown in Fig. 1 2(b). Since the rorbitals are oriented perpendicular to the T\ TSF molecule, the molecules couple strongly along the adirection via the overlap of partially occupied rorbitals so that this direction becomes the most conducting direction. On the other hand, the overlap along the b and cdirections is weak. In particular, the overlap is extremely weak along the cdirection because the coupling between T\ TSF molecules is hindered by the insulating CO14 anion sheets, as shown in the right panel in Fig. 12(a). It turns out that the conducting proper ties are extremely onedimensional. In the case of (T\ iTSF)2C104, the transfer integrals are approximately ta : tb : t = 250 meV : 20 meV : 1 meV [2]. This common ( i I 1 structure is the origin of the Q1D conducting properties for all of the (TMTSF)2X compounds. Thus, (TMTSF)2X, including X C14, PF6 etc., are good examples for studying Q1D FSs. 1.3.2 The Quasitwodimensional Conductor K(ET)2X The organic superconductors K(ET)2X (X = Cu(NCS)2, 13 etc.) belong to the ET family of chargetransfer salts (CTS), where ET represents bisethylenedithio tetrathiafulvalene [(CH2) 22C6Ss (alternatively denoted BEDTTTF). In contrast to the planar T\ iTSF molecule (shown in Fig. 12), the ET molecule shown in 6 (b) (a) t,, S ET molecule b ET molecule Figure 13. Illustration of the crystal structure of K(ET)2Cu(NCS)2 and the ET molecule. (a) Structure of the ET molecule. In contrast to the T\ iTSF molecule, the ET molecule is not planar. (b) The crystal structure of K(ET)2Cu(NCS)2 viewed along the a*axis. Fig. 1 3(a) is not exactly planar. This nonplanar nature of the ET molecule complicates the morphology of the ET crystal structures compared with other 2:1 (TMTSF)2X salts. For instance, there are four kinds of iv I I morphologies in the case of (ET)213, i.e., the a, 3, 0 and Kphases, where a, 3, 0 and K denote different crystal structures. Fig. 1 3(b) shows the crystal structure viewed along the a*axis for the Kphase of (ET)2Cu(NCS)2. For the case of K(ET)2Cu(NCS)2, the ET molecules dimerize in a face to face arrangement. The dimer pairs then pack orthogonally so that the overlap of the rorbitals of the ET molecules is fairly isotropic in the beplane. On the other hand, the overlap along the adirection is much smaller because it is prevented by the insulating Cu(NCS)2 anion l .r1. It turns out that the conducting properties in K(ET)2Cu(NCS)2 are highly two dimensional, or quasitwodimensional (Q2D), i.e., oll/oL ~ 1000 or tll/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. Examples include the Peierls instabilities, chargedensitywave (CDW) states, spin Peierls instabilities, antiferromagnetic (AF) states and spindensitywave (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 socalled nesting vector. The correlation of electrons on the FS becomes divergently strong at Qx = 2kg. This is the socalled Kohn anomaly. As a result, the electron phonon interaction becomes divergently strong at the nesting vector Q, = 2kF with decreasing temperature. Therefore, the phonon mode at Q, = 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 Q, = 2kg, the socalled Peierls distortion. This Peierls transition can be described by a mean field theory, and the transition temperature Tp is given by a BCStype gap equation [2]. This distortion also affects the electronic state. An example for a halffilled 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 Q, = 2kF = 7/2a becoming dimerized. This changes the halffilling to full filling. As a result, electrons cannot move to other sites, and the system becomes an insulator. This metalinsulator 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 electronelectron 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 (T\ilTSF)2X and ETsalts [2]. (a) D 1DFS kky /Q = 2kF High T itinerant 2D 2 k metallic state 44444444~ a Low T (< T) no hopping insulating state   i~i 2a n = halffilling Figure 14. 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 Q, = 2kF, the socalled 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 disper sion becomes zero at Q = 2kF. This causes a static lattice distortion, the socalled 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 electronphonon interaction becomes divergently strong, so that it causes a lattice dis tortion, the socalled Peierls distortion. For the case of halffilling, the system dimerizes below the transition temperature Tp due to the Peierls distortion. This affects the electronic structure. The distor tion 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 metalinsulator transition. a Figure 15. 9 B =W 4tbb/hwo oc 1/B r ^^ 2 W B Low B Low B High W Large W Small 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 in crease in the onedimensional 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)2X. This phase is the socalled fieldinduced SDW (FISDW) state. For (T\ ITSF)2X, the transfer energy along the bdirection, tb, is significant, tb/t ~ 0.1, so that the motion of electrons is somewhat two dimensional. Application of a magnetic field leads to confinement of the 2D motion. Fig. 15 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 bdirection. Then the motion of the electrons is effectively onedimensional. In the case of (T\ TSF)2X, this dimensional crossover effect is related to the confinement in the abplane, so that the FISDW phase has a minimum critical field when the field is perpendicular to the abplane. In the case of (TMTSF)2C104 [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]. 0 5 10 15 20 25 30 Field (T) Figure 16. TH phase diagram for (T:iTSF)2C104. The FISDW phase is ob served 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 abplane. The another phase transition was also proposed at higher magnetic fields using normal and Hall resis tance 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.). 7 6 2 5 4 E 2 1 0 1.5 Superconductivity in Low Dimensional Materials As mentioned, exotic superconductivity has been observed in irn il: low di mensional materials, e.g., the HTSC, K(ET)2X, Sr2RuO4 and (T\ITSF)2X. These exotica may be represented by anisotropic superconducting energy gaps, spintriplet Cooper pairs, non electronphonon interaction mediated Cooper pairing mecha nisms, coexistence of superconductivity and magnetism. For example, (T\ TSF)2X has recently been considered to be a spintriplet superconductor since it shows no 7Se Knight shift through the superconducting transition temperature [9], exceeding the Pauli paramagnetic limit in the upper critical field [10], and nonmagnetic impurity effects [11]. K(ET)2Cu(NCS)2 is considered to be a dwave superconduc tor because it shows a fourfold symmetry in the magnetothermal 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 super conductors, their superconducting properties are often anisotropic. For the case of the quasitwodimensional (or 1 li, 1, ,) superconductors, the superconducting anisotropy parameter 7 is represented by the ratio between the interlayer and inplane penetration depth or coherence length, i.e., 7 = AI/AI or 7' = /~1_L re spectively. In the extreme cases, 7 can be several hundred: e.g., Bi2Sr2CaCu208+y, 7 ~ 50 200 [15]; and K(ET)2Cu(NCS)2, 7 100 200 [16]. This high degree of anisotropy is related to a weak Josephson coupling between the superconducting liv. The characteristic interlayer Josephson plasma frequency is often in the range of microwave frequencies. This can be seen as a Josephson plasma resonance (JPR). We have studied the JPR phenomena for the Q2D organic supercon ductor K(ET)2Cu(NCS)2. The results were reported in Kovalev et al. [17] and B(: i r,' ,i*, et al. [18]. 1.6 Impurity Effect on the Superconductivity For unconventional superconductors, the nonmagnetic impurity effect is one of the main evidences for unconventional superconductivity. We have therefore performed an experiment to study the nonmagnetic impurity effect in (TMTSF)2C104. This result is described in C'!h 1 5. The original work on the impurity effect was developed for magnetic impuri ties in conventional superconductors in the 1950's and 1960's. In 1959, Anderson extended the BardeenCooperSchreiffer (BCS) theory [19] for very dirty supercon ductors, where elastic scattering from nonmagnetic impurities is large compared with the superconducting energy gap [20]. In this theory, Anderson showed that the Cooper pair with the momentum (k T, k 1) is immune to nonmagnetic scat tering. 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 pre dicts 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 timereversal symmetry. For instance, in the case of BCStype pairs isotropicc gaps with the momentum (k T, k 1), the scattering by nonmagnetic impurities, which acts on the elec tric 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 spin and ultimately flip one of the spins. As a result, the scattering changes the pair to a parallel spin alignment (the timereversal symmetry is broken), and the Cooper pair is destroyed. Furthermore, extensive studies show that the broken timereversal 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 T, is simply related to the scattering time TK of the destructive perturbation. The function to express the relation is the socalled universal function given by the following equation. In( )= +), (11) Tco 2 2 27TkBTC where Teo = Tc(O) is the superconducting critical temperature without impurities, (z) = P'(z)/F(z) is the digamma function, and a is the pairbreaking strength, 2a =, (12) TK where TK is the scattering time for depairing. Impurity effects were observed not only in the BCStype superconductors, but also in many exotic superconductors. In particular, in the case of the unconven tional superconductors, effects were observed even due to nonmagnetic impurities. That is why this nonmagnetic impurity effect is considered to be an evidence of unconventional superconductivity. For example, the spintriplet superconductor Sr2RuO4 shows nonmagnetic 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. C'i lpter 3 shows our experimental setup. Chapter 4 and 5 explain the study of the FS and superconductivity for the Q1D material (T i[TSF)2C104. C'!i pter 6 show the study of the FS for the Q2D materials K(ET)2X. CHAPTER 2 PERIODIC ORBIT RESONANCE 2.1 Experimental Techniques to Study Fermi Surfaces Since the Fermi surface (FS) can explain many aspects of conductors, su perconductors and itinerant magnetic systems, many techniques have been developed to study FS topology. Examples include: angleresolved photoemis sion spectroscopy (ARPES) [24], the deHaasvan Alphen effect (dHvA) [25], the Shubnikovde Haas effect (SdH) [25], angledependent magnetoresistance oscilla tions (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 mea suring inplane band structures of quasitwodimensional (Q2D) systems because of the resolution of interlayer scattering processes. The SdH and dHvA effects probe the effective mass, m*, and the quasiparticle lifetime, 7r. However these quantities are averaged over the FS. The SdH and dHvA effects are also limited to threedimensional (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., kBT < hwuc, where uc is the cyclotron frequency. AMRO can provide additional topographic information in reciprocal space such as the Fermi wave vectors kF. AMRO is also applicable in the case of quasionedimensional (Q1D) systems. CR can be used to determine the averaged m* and r over the FS. Since CR requires closed orbits, the technique is limited to 3D and Q2D systems. Our new experimental technique, the socalled 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 T, as shown in C'!h 1 4. In the case of Q2D systems, the POR probes both the average m* and the kdependent Fermi velocity VF(k), as well as r(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 .1Iv,, e.g., the twodimensional 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 twodimensional (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. 21, and an energy dispersion represented by E(k) =2* (k + k ), (21) where m* is the effective mass. We now consider a dc magnetic field applied along the zaxis perpendicular to the conducting plane, i.e., B = (0, 0, B), as shown in Fig. 21. 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 the equation of motion for the Lorentz force, hk e(v x B), (22) and the definition of the group velocity is Vg Vk[E(k)]. (23) In the present case, using the energy dispersion in Eq. 21, the explicit expressions corresponding to the above equations are, eB m = ky,(t), (24a) m* eB ky kx (t), (24b) m* k = 0, (24c) and vX(t) k(t). (25a) VY M h ky M (25b) v, = 0. (25c) Thus, solving the differential equations 24 and 25, one can see that the resultant motion is oscillato,;, i.e., kx(t) = kcos(uett+(k)) and ky(t) = ksin(ct+O(k)); the oscillatory velocities, v, = v cos(ut + 0(k)) and v, = vsin(uct + O(k)); and k, = 0 and v, = 0, where ,c = eB/m* is the characteristic frequency of the oscillatory motion, the socalled I ;.. u./,..; f,,'.;". ;./ ;; The cyclotron frequency, vc = uc/27r 28 (GHz/T) x 'B, 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* ~ O0.1m 10,. where m, is the electron mass. There is no kinetic energy change, because the motion is caused by the magnetic field. This oscillatory motion is illustrated in Fig. 21. 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. 21, is albvl perpendicular to the FS. As a result, the group velocity rotates a full 3600 as the electron rotates around the surface, as shown in Fig. 21. 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], 2 2 afO k 0 16a) Vq(ua, B) dk [ ]vp(k, 0) dtvq(k,t) exp( zi)t, (26a) V fd[ Eoo or an alternative expression is, ,(, B) = dE[ f(E)]N(E) dk2v(k, 0)vq,(, k), (26b) where Tp(w,k) = dtvp(k,t) exp( iw)t, (26c) where p and q are indices of a cartesian coordinate system, i.e., x, y or z. In Eq. 26c, exp(iut) is a oscillatory term to give arise the resonance, and exp(1/r) is a damping term to dephase the oscillatory motion. In the low temperature limit, O f 6 k kF) or f( f( 6(E EF), so that the motion of electrons affecting the conductivity is restricted to the FS. The conductivity along the xaxis is then given by the following expression, (See Appendix A.1 for details of the calculation.) 2eC2 0 17a) 7(wa), B) 2 d2Sv(k, 0) dtvx(k, t) exp( i)t, (27a) 2e2v 1 1 V [ (+ ] (2 7b) V i(w + ) + i(w o) + ) i((a +o +) 19 ar=3 o. =2 oT=0.5 0 1 2 3 4 5 Figure 22. Real part of ax as a function of frequency for various values of cur. The peak in the conductivity represents CR. The resonance condition is w = wu. For cur >1, the CR is wellpronounced. However, the CR becomes too weak to observed for Uwr <1. Thus uwr has to be greater than 1 in order to observe CR. Thus, the real part of r,, is given by adc 1 1 Re a1, (, B) dc t + W (28) 2 1+(1 t + L')2c2 2 1 )2' where ard is the dc (w = 0) conductivity and r is the relaxation time of the electrons on the FS. In the present case, ao, = ayY and az, = 0. The real part of the conductivity, Re aog, is plotted in Fig. 22 as a function of wu/uc. There is a clear Lorentzianlike peak at the resonance condition, u/wc = 1 (or wu= wc). This peak in Re ar, is caused by the cyclotron motion. This is ;/. 1.Itron resonance. Since c = eB/m*, in the present case, wc goes to zero when B 0. The conductivity at B = 0 is therefore given by Re a (, B = 0) (29) 1+((2 9) Eq. 29 is of course the ac Drude conductivity which has a peak at w 0 [31]. Thus, the peak corresponding to CR is the same as that of the ac Drude conductiv ity. The width and height of the CR peak depend on the parameter, Uwr, as shown in Fig. 22. The peak is wellpronounced if uwr >1. However the peak becomes smeared out if uwr <1. As with any resonance phenomena, the wur condition is re lated to phase memory, or dephasing. The wur < 1, eletrons dephase faster than the time taken to complete an orbit. Thus, the oscillatory motion is highly damped and the exponentiol dominates Eq. 26c. On the other hand, if uwr > 1, electrons may execute many orbits before d. pl. 'iw and the oscillatory term dominates Eq. 26c, resulting in the resonance. Thus, the key to observing CR is an oscillatory 1... H:1 and uwr > 1. We can now consider the case of a tilted magnetic field, e.g., B = (B sin 0, 0, B cos 0). (See Fig. 23.) The equation of motion and the group velocity are now given by eB cos O k1 n* k9(t), (210a) m* k cos kM(), (2O1b) m* eB sin 0 M k =es k,(t), (210c) m* and ,(t) k(t) (211a) m Vw(t= hkw(t), (2lib) v, = 0. (21 c) Similarly, the resultant motion therefore follows an oscillatory tr r i* /.. ; kx(t) = k cos(c2Dt + 0(k)), ky(t) = k sin()2Dt + 0(k)), and kz(t) = k tan 0 cos(c2Dt + Q(k)); the oscillatory velocities, v, = v cos(u2Dt + 0(k)) and v, = vsin(w2Dt + 0(k)); and v, = 0. The cyclotron frequency is given by eB cos 0 W2D = (212) m* which now depends on the angle 0. Again a,, = ayy, similar to Eq. (27b), and a,, = 0 with the resonance condition w = W2D, or, eB w I cos60. (213) m* 2.3 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 socalled periodic orbit resonance (POR) [33], or also the FermiHai {f..: traversal resonances (FTR) by Ardavan et al. [32]. The POR was originally predicted by Osada et al. [42] for a quasionedimensional (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 quasionedimensional (Q1D) FS which consists of completely open FS sheets. The highly anisotropic energy dispersion can be written in the form E(k) = hvp\(I k kF) 2ty cos(kyb) 2t, cos(k,c), (214) where EF > ty and tz, vp is the Fermi velocity, and b and c are lattice constants along the y and zdirections respectively. t, and tz are the transfer energy associ ated with the lattice vectors R, and R,. 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 (a) Q1D Fermi sur (b) 23 Fermi velocity CorIvrugation z face B(O) oscillatory velocity B(0) Figure 24. Oscillatory group velocity v, 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 zdirection. The z component of the group velocity becomes oscillatory. This oscil latory group velocity causes the resonance in the conductivity za,. The periodicity of the POR is angledependent. (b) Trajectory of electrons on the Q1D FS in the case of a magnetic field along the zaxis. v, is not oscillatory in this case, so that no POR is seen. Q1D organic conductor, (T:\iTSF)2C104 is t : ty : t 250 : 20 : 1 meV [2]. Fig. 24 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. 24(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. 24.) is alvwb 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 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. 24(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 zaxis. The applied magnetic field is expressed by B = (0, B sin 0, B cos 0) where 0 is a angle between the magnetic field and the zaxis. Recalling the equation of motion, Eq. (22), S2tbbBos() sin[k(t) b] + sin( sin[k, (t)c], (215a) vFy B cos(0)(2b) k, = sgn(k,) (215b) k = sgn(k,) B sin() (215c) and v, = sgn(k,)vF, (216a) 2t b v,(t) = 2t sin[k,(t)b], (216b) 2t c v (t) = sin[k (t)c]. (216c) Solving the above equations, the zcomponent of the trajectory and velocity is given by vFeB k, = sgn(k)[ sin(O)]t + k,(0), (217) and 2t c vFeBc v, (t) = sin[k,(0)c sgn(k,) sin(U)t] 2tecsin[k,(0)c sgn(k,)wQlDt], h h (218) where WQID = pB, sin(0). Thus, this group velocity is oscillatory, and is charac terized by the frequency WQ1D. This frequency is related not only to the magnetic by the frequency WQ1D. This frequency is related not only to the magnetic field B and the band parameter (Fermi velocity vp), 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 a,, is given by, adc 1 1 Re azz (, B,0) = c[ + 1], (219) 2 1 + (u + WQ1D)2 1+ ( Q1 UQD)2 which gives rise to a POR at w = UWQID, i.e., the resonance condition is given by w BeI sin(eO). (220) Thus, one can measure the Q1D POR at different frequencies w, magnetic fields B and angles 0. By investigating the position of the POR, one can determine vp from Eq. 220, and one can also determine the scattering time r by analyzing the shape of the POR. The FS corrugation pattern of real Q1D conductors can be much more complicated, particularly for materials with lowsymmetry (i ~ I 1 structures, i.e., monoclinic, triclinic or rhombohedral. As a result, the hopping energy to 2nd and higherordered nearest neighbors becomes considerable. We can see such an example in the study of the organic conductor a(ET)2KHg(SCN)4 by Kovalev et al. [46]. The FS of a(ET)2KHg(SCN)4, below 6 K, is shown in Fig. 25. At 8 K, a(ET)2KHg(SCN)4 undergoes a phase transition into a chargedensitywave (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. 25(b) and (c). The existence of the harmonics implies that the Q1D section is rather twodimensional, as can be seen in Fig. 25(a). In the case of a(ET)2KHg(SCN)4, the Q1D section is not a simple flat sheet. Since the Q1D mple flat sheet. Since the Q1D 0 1 2 3 4 5 6 Magnetic field (tesla) m=2 m=l m=o m=1 3T 4T 20 40 60 80 100 120 140 160 180 200 220 0 (degrees) 60 40 20 0 20 40 60 0 (degrees) Figure 25. POR in a(ET)2KHg(SCN)4. (a) FS in a(ET)2KHg(SCN)4 below 6 K. a(ET)2KHg(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, a(ET)2KHg(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 Ko valev 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 twodimensional (Reprinted figure with permission from Kovalev et al. [46]. Copyright 2002 by the American Physical Society.). (d) Summary of the POR in a(ET)2KHg(SCN)4 (Reprinted figure with permission from Kovalev et al. [46]. Copyright 2002 by the American Physical Society.). section is reconstructed from a Q2D section belonging to the high temperature FS (not shown), the Q1D FS sheet still retains a twodimensional shape, so that it is strongly corrugated. Such a Q1D FS can be represented by finite higherordered transfer integrals which produce the harmonic resonances. In contrast, the Q1D conductor (TM TSF)2C104 has a more onedimensional FS. Experimental results for the Q1D POR in (TMTSF)2C104 are shown in C'!h 1 4. 2.4 Periodic Orbit Resonance for a Quasitwodimensional 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 quasitwodimensional (Q2D) FS. In the case of many twodimensional conductors, the conductivity along the zaxis (the least conducting direction), Uaz, is nonzero, e.g., for K(ET)2Cu(NCS)2, axx/a t 1000 [3], for Bi2Sr2Ca~_CunOy, aX,/azz 10000 [47], for YBa2Cu30y, a7x/a,, ~ 100 [47] and for Sr2RuO4, a1xx1/az 400 4000 [48]. These are socalled Q2D conductors. We now show that a,, can also have a resonance, the socalled periodic orbit resonance (POR). Here we consider a Q2D electron model, i.e., the energy dispersion is represented by E(k) 2= (k2 + k2) 2tz cos(kc), (221) where c is the interlayer spacing. The bandwidth in the zdirection, t,, is much smaller than the Fermi energy, i.e., Ep/tz ~ V/ z/a ~ 30 for K(ET)2Cu(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 zdirection. The magnitude of the small corrugation of this cylinder corresponds to the bandwidth 4tz along the zdirection. Such a FS is shown in Fig. 26(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,0, B cos0). The motion is given by the following equations, X eBcosk(, (222a) m* Ky eBtc sin 0 eB cos 0 kh (,,[. (t)c] + k* (t), (2 22b) h m eB sin 0 kz k, (t)B (222c) m* and vX (t) kh (t), (223a) m v9(t)  k(t), (2 23b) m 2t c (t) sin[k (t)c]. (223c) Because eBt < $Bk (except when the field is oriented close to ,'plane, i.e., 0 ~ 90), the equation (222b) may be approximated as eB cos 0 ky k(t), (224) m* so that the motion can be calculated in the same way as the 2D electron model. The resultant motion therefore becomes, kx(t) = kcos[2Dt + 0(k)], ky(t) k sin[wDt + 0(k)], and kz(t) = ktan 0cos[w2Dt + O(k)]; vx = vcos[w2Dt + (k)], v = vsin[2Dt + Q(k)] and v (t)= 2 sin[kctan 0cos{w2Dt + Q(k)}]. The resonance frequency is the same as Eq. 212. The zcomponent of the group velocity, vz, is now nonzero. This gives finite conductivity along the zdirection, a, and to a resonances. Using the Boltzmann equation, the conductivity, a is given by Jo(7 tanO) 0 1 1 , zz (, B) o tan + ( nt2tan 0)2 + ,( + 1 1 2r2 +Z _t c n lW2D 272 t (; 2D2 (225) where Jn(x) is the nth order Bessel function, = kpc, crx and ay are given by the same expressions as Eq. 27b. The conductivity therefore has resonances at frequencies, neB ) = nw2D = cos 01. (226) m* Thus, multiple resonances in ua, are predicted instead of the single resonance in a,, and ~yy found in the previous discussion for the 2D case. By investigating the POR at different c, B and 0, one can determine the effective mass, m*. By analyzing the shape of the POR, one can also determine the scattering time r, since the halfwidth of the POR is ~ 2/r. Although the motion of an electron along the zdirection gives resonances in the conductivity, a z, with similar resonance conditions, the picture of the motion is quite different from the motion in the 'plane. Fig. 26 shows a typical trajectory of electrons on the FS. In Fig. 26(a), the applied magnetic field is tilted from the zaxis, so that the trajectory on the FS is also tilted. As shown in the right panel in Fig. 26(a), the group velocity v, becomes periodic because of the corrugation along the zdirection. This oscillatory v, brings a resonance effect in the conductivity, za,. On the other hand, when the applied magnetic field is along the corrugation axis (the zaxis in the present example), the group velocity v, is not oscillatory. In this case, no POR is observed, as shown in the right panel in Fig. 26(b). In the case of a lowsymmetry crystal structure, the corrugation axis can be different from the zaxis. We see such an example in ('! 11' 6. 2.5 POR and Angledependent Magnetoresistance Oscillations The angledependent 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 angledependent 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 ( i I 11. ..raphic planes in Q2D and Q1D FSs are named differently. Fig. 27 and Fig. 28 illustrate each case. Fig. 27 explains the Yamaji Oscillations in Sr2RuO4 Q2D FS Figure 27. 90 06 0 8 (degree) AMRO for Q2D conductors. In the case of a Q2D FS, Yo.i1 .ii oscilla tions may be observed by varying the field orientation from the zaxis to the ';plane. Here, the conducting plane is the ';plane. 0 repre sents the angle between the zaxis and ';plane. Q represents the angle from the xaxis in the I;plane. The Yei" iii oscillations in Sr2RuO4 are shown. The strongest peak in the resistivity p, is seen at around 0 = 350, and many oscillations are seen at higher angles (Reprinted figure with permission from Ohmichi et al. [49]. Copyright 1999 by the American Physical Society.). AMRO effect for a Q2D FS, socalled Yi,, i ii oscillations. For the observation of the Ya,, ii oscillations, a fixed magnetic field is rotated between the zaxis and the 'iplane. The experimental result of the Yei"" iii oscillations is shown in the right panel of Fig. 27. Sr2RuO4 shows the strongest resistivity peak at around 0 = 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], DannerKangC'l 1d:!:1. (DKC) (z x rotation) [28] and 3rd angular effects (x y rotation) [29], as shown in Fig. 28. In the case of the DKC effect in (T\iTSF)2C104, the most pronounced peaks are seen at around 0 = 850 and 950, and smaller oscillations are also seen at angles between 0 = 850 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 WJQlDT. In the case of the Lebed effect, many minima are seen. These angles are the socalled 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 u = 0 in the ac conductivity. For example, in the case of the Lebed effect, one can use Eq. 219 with w = 0, Re ,(w, B,0) = . (227) t + L;2 T2 Fig. 29 plots the inverse conductivity as a function of the angle 0. The plot shows minima, which correspond to the POR at w / 0, and the AMRO at w = 0. The positions of the minima shift continuously by changing the frequency. The positions of the minima at w = 0 represent the directions of the warping. The Lebed effect in (TMTSF)2C104 is presented in C'!i p1 4. Next, we consider the case of a Q2D FS (i.e., Yei", iii oscillations). Like the Q1D AMRO, by applying a = 0 in Eq. 225, one obtains the following DKC effect in (TMTSF)2C104 ?5 / T2T 20 10 10 2C ,(degree) Q1DFS z x conducting ax i Lebed effect in (TMTSF)2PF6 (TMTSF)FPFe T=1.55K P=7.9kbar S4 2 c 3 120 } 2 80 c i o b' ; 1 b1. 0.01~ ~ ^1 ^ B 90 60 30 0 30 60 90 Angle (T ) y B=12T B=9T S(TMTSF)2CIO4 " T=1.7K 0.01 90 60 30 0 30 60 90 3rd angular effect in (TMTSF)2C104 Figure 28. 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 rota tion), DKC (x z rotation) and the 3rd angular effect (x y rotation). In the case of the DKC effect in (TM iTSF)2C104, the most pronounced peak is seen around 0 = 850 and 950, and smaller oscillations are also seen at angles between 0 = 850 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 Soci ety.). In the case of the Lebed effect, many minima are seen. These resonance angles are the socalled 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.). 1/Gzz Figure 29. B 0.5 0c 0.25 (0c / AMR O \ 0) = 0 180 90 0 90 180 0 (degrees) The Lebed effect in both the dc and ac conductivity. The inverse conductivity as a function of the angle 0 is simulated using Eq. 219. The inverse conductivity has two minima at w / 0. These minima are the POR. The inverse conductivity has one minimum at w = 0, the socalled AMRO. conductivity, 2J a, (B, 0) o Jo(y tan 0) + (7tan) (228) S1+ (nUw2272 where w2D u cos 0 and c = eB/m*. Fig. 210(a) plots the conductivity given by Eq. 228. Strong oscillations are clearly seen in Fig. 210, the socalled Yoii i: ii oscillations. Just like the POR effect, the amplitude of these oscillations depends on the product of wc,. When wuT is high, the oscillations are wellpronounced. On the other hand, they are smeared when wr is small. However, the positions of the maxima and minima of the oscillations are independent of the product wc. The position of the dc resonance depends only on 7 = kFc; the figure shows AMRO for two values of 7 = kpc, i.e., 7 2 and 7 = 3, at wT = 3. Thus one can determine kFc and, eventually, kF from the Yi,"" iii oscillations. In Fig. 211, we show the frequency dependence of the Ye i iii oscillations. The calculation for the Boltzmann equation Eq. 225 was performed by numerically Figure 210. )cj=0.5 (0)c=l 0c1=3 B *4 V ykFc 2 \' 60 30 0 30 60 Angle O(degrees) Yelin iii oscillations. The conductivity from Eq. 228 is shown as a function of angle between the magnetic field and the least conducting direction (the zaxis). The plot shows strong oscillations, the socalled Yeon" i ii oscillations. The oscillations are pronounced due to the large product wc. The position of the oscillations depends on 7 = kpc, but is independent of uwcT. solving the differential equations in Eq. 222 and Eq. 223 with the Q2D energy dispersion E(k) = (k2 + k ) 2tz cos(kzc). The calculation covers v =dc to 500 GHz and 0 =0 to 90 degrees. Fig. 211 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 CI '1. 4. Gzz(B) 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., hr1 < hj,,c e.g., the de Haasvan Alphen (dHvA) and Shubnikovde 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 C'! '1' 6, we observed both effects in the experiments for K(ET)2X. In this section, we introduce magnetic quantum oscillations briefly. We also introduce the angle dependence of the frequency of the SdH effect since the angle dependence ilp i, d an important role in C'! 1' 6. A more detailed description of the quantum mechanical effects can be found in many books, e.g., by Shoenberg [25] and Abrikosov [21]. We consider the case of Q2D conductors, i.e., the energy dispersion is modeled by Eq. 221. A magnetic field is now applied along the zaxis. Considering the gauge transformation, H H' = p qA 2 + qQ with the vector potential A = (0, Bx, 0) and = 0, one can get the following energy dispersion, E,,(k,) = h,(n + ) 2t, cos(kc), (229) where wu(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 Ep. Fig. 212 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 B=O Strength of B n=oc n=20 n=8 n=4 (continuous) Figure 212. 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 Lan dau 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. 229, 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 noninteracting system should be a deltafunction, but in a more realistic system, the DOS has some width due to scattering of carriers. Fig. 213 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 6(E E(n, kz)) (EE(n,k))2+i( 1)2 where 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 7 so that the shape of the DOS varies with material, temperature and so on. Because EF iu h.~ B  I Lower B 1/2 3/2 5/2 7/2 9/2 E/hcoc Figure 213 DOS of a Q2D conductor in a magnetic field in units of h',c. The DOS in a Q2D conductor is given by, N(E) E= n,k,(E E(n, k )) Zn,k:I (E(,k7~) (/2)2 The dotted line represents the DOS for each Landau level. Ti.! solid line is the sum of the dotted lines. The dashed line represents Fermi energy. When the magnetic field in creases, the Fermi ent line moves to left in the figure and vice versa. Thus, the DOS at the Fermi energy EF is varied by the mag netic field. Ti., is a the origin of the SdH effect. 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. 213, the DOS is represented in units of hu,. When the magnetic field increases, the unit of the xaxis in Fig. 213 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. 26b, 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 (k, t). Thus these phenomena have completely different origins. As shown in Fig. 213, the period of the quantum oscillations is given by A(EF/hc) = 1. Using the Fermi energy E = h2(kF,1(kz)2 + kF,y(kj)2)/2m* and the cross sectional area of the FS A = (kFx(kz)2 + kFy(kz)2) = 27 *Ep/h2 at each k,, the period of the oscillation as a function of the magnetic field can also be written, 1 he 2we A( E h (230) B EFm* A Furthermore, when the magnetic field is tilted, the period is given by A( h) (2 31) B EFm* hA cosO0 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 Shoenberg gives the details [25]. um oscillations. The book by Shoenberg 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 magnetotransport model. On the other hand, the SdH and dHvA effects are purely quantum phenomena. CHAPTER 3 EXPERIMENTAL SETUP The in i jr results presented in this chapter can be found in the article entitled Rotating ..:; H:; for highfield 'iu1'.dependent microwave spectroscopy of low dimensional conductors and magnets, S. Takahashi and S. Hill, Review of Scientific Instruments 76 023114 (2005). 3.1 Overview of Microwave Magnetooptics In recent years, microwave (millimeter and submillimeter 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 nextgeneration technologies, enabling non destructive 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 subfield of condensed matter physics, where the millimeter and submillimeter spectral range can pro vide extremely rich information concerning the basic electronic characteristics of a material [56, 57, 58, 59]. Furthermore, combining microwave techniques and high magnetic fields (microwave magnetooptics), allows many more possibilities, including: cyclotron resonance (CR) [32, 33, 36, 39, 43, 46, 60, 61]; electron param agnetic resonance (EPR) [62, 63, 64, 65]; antiferromagnetic resonance (AFMR) [1]; Josephson Plasma Resonance (JPR) measurements of li,. 1t superconduc tors [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 spectral range. Another very important aspect of microwave magnetooptical in vestigations 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, angledependent 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: highTc su perconductivity [66], and other low dimensional superconductors, e.g., organic conductors [32, 33, 43, 46, 61, 68, 69], Sr2RuO4 [60] etc.; the quantum and frac tional quantum Hall effects [71, 72]; and low dimensional magnets, including singlemolecule magnets (SMMs) [63, 64, 65, 73]. Unfortunately, the microwave spectral range presents many technical chal lenges, particularly when trying to study very tiny (< 1 mm3) singlecrystal sam ples within the restricted space inside the bore of a large highfield 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 funda mental TE and TM\ mode rectangular metallic waveguides, lowloss cylindrical corrugated HE waveguides [76], quasioptical propagation systems [76], and insitu generation and detection of the microwaves [77]. Standing waves are particularly problematic in the case of broadband spectroscopies, e.g., timedomain [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 welldefined (i.e., flat) sample surface area (>A2). For cases in which large samples are not available (note: A spans from 3.34 cm at 10 GHz to 0.33 mm at 1 THz), resonant techniques 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 narrowband nature [56, 57, 58]. However, the cavity perturbation technique is ideally suited for fixedfrequency, magnetic resonance (magnetic fielddomain) 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 fielddomain 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. 226, v, = eB/(27m*) where v, is the cyclotron frequency, B is the magnetic field strength and m* is the effective mass. Fig. 31 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 fielddomain 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 015 tesla, which is relatively easy to obtain in a laboratory, corresponds to 0 420 GHz. This wide frequency range is therefore available to increase wc. The upper panel in Fig. 31 illustrates this point. wc 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 v=Vc=50 GHz, or=l .1=Vc=100 GHz, or2=2 \ / ,v=Vc=200 GHz, ay=4 500 GHz (16.7 cm') m*Me m*=5 me m*=10 m, Figure 31. Frequency range for the Q2D POR in magnetic fields accessible at the National High Magnetic Field Laboratory (NHMFL) in Tallahassee FL, USA. Since v, = eB/(27rm*), 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 cm1) 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 cm1) for m* = *,  252 GHz (0 8.42 cm1) for m* m* = 10me. ".;,. and 0 126 GHz (0 4.21 cm1) for 3.2 Experimental Setup Fig. 32 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. As a microwave source and detector, we employ a Millimetre Vector Network Analyzer (\IVNA) 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 = 818.5 GHz. By feeding the source frequency to a Schottky diode, which is a passive nonlinear device, harmonic components of the source (Fmm = N x 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 [Kband (N = 2, 3, v = 1840 GHz), Vband (N = 3 and 4, v = 4872 GHz), Wband (N = 5 and 6, v = 72110 GHz), and two sets of Dband diodes (N > 6, v > 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 socalled harmonic mixer (HM). The beat signal (Fbeat N x F1 N' x F2) and the phase ( ,, = N x k1 N' x Q2) is then sent to a heterodyne vector reciever (VR) in the MVNA. By choosing appropriate HG and HM so that N = N', and by locking the phases Q1 = 2, 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., FMHz N x (F1 F2). The MVNA employs FMHz = 9.010 !;2S125 or 34.010 !;;2S125 MHz for the vector measurement. Although the Dband Schottky diode can produce a fairly powerful microwave signal on harmonics N = 6 and 7, the power diminishes for N > 8. For higher frequencies (v > 170 GHz), we usually use the ESA option which consists of a Gunn diode, a directional couplerharmonic mixer and a multiharmonic multiplier. With this option, the multiplier is fed by a more powerful higher frequency microwave signal (PF1 ~ 30 mW, Fi = 6982.3 and 82.5102.7 GHz) from a Gunn diode. Since the Gunn source frequency is much higher than the internal source in Table 31. Available magnet systems at the University of Florida (UF) and the NHMFL. The table lists the field geometry and magnet type, the max imum available field Bax, 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 twoaxis rotation. Magnet Bmax (T) Type length T (kelvin) Probe dia. 45 T (NHMFL) 45 Axial hybrid 1.67 m 1.4 300 3/4" 33 T (NHMFL) 33 Axial resistive 1.45 m 1.4 300 3/4" 25 T (NHMFL) 25 Axial resistive 1.6 m 0.5 3001 1" Oxford Inst. (UF) 17 Axial SC 1.9 m 0.5 300a 1" (7/8"a) QD PPMS (UF) 7 Transverse splitcoil SC 1.15 m 1.7 400 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 8700 GHz (0.2723.3 cm ). Since we wish to investigate the microwave response for a wide variety elec tronic 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 31. All systems are standard type magnets with vertical access for measurements (not horizon tal access magnetooptical magnets). The standard cryostats designed for these magnets are all 4He based (either bath or flow cryostats). However the base tem perature of the 4He 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 3He refrigerator which is compatible with the 17 tesla Oxford Instruments supercon ducting magnet at UF, and the 25 T resistive magnet at the NHMFL, as listed in TABLE 31. A schematic of this refrigerator is shown in Fig. 33. The 1.9 m long waveguide/cavity probe is inserted directly into the 3He space, which is con structed 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 3He liquid and the surrounding 4He vapor. The 3He 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 4He flow cryostat operating at its base temperature of ~ 1.4 K. After condensation of the full charge of 3He (5 liters at STP), sub kelvin temperatures are achieved by pumping directly on the liquid by means of an external sealed rotary pump. The refrigerator operates in singleshot mode, i.e., the 3He 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 3He gas, and the subsequent pumping of the gas back to the storage vessel. The 3He 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 3He 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 3He 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 THe refrigerator is 500 mK and it provides hold times of up to 2 hours. Waveguides jf Pumping Line Fins 1K Pot Cavity Liquid 3He Figure 33. A schematic diagram of the 3He probe used for subkelvin experiments in the Oxford Instruments 17 T superconducting magnet. See main text for a detailed description of its construction (Reused with permis sion from Takahashi et al. [81]. Copyright 2005, American Institute of Physics.). Table 32. Probes used for the cavity perturbation technique. The coaxial cable, K and Vband waveguide probes were built inhouse. 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 v range (GHz) Harmonic N Cavity size (D x H) Coaxial cable 818 1 0.70" x i .I, (V, R) + dielectric material (E=15) Kband (WR42) waveguide 1840 2, 3 0.70" x I .1 (V) 0.52" x 0.52" (V) Vband (WR15) waveguide 48350 325 I :I x I :II (V, R) 0. 111 x I I (V) 0.25" x 0.25" (V) Corrugated waveguide 170700 1530 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 32. The coax cable, K and Vband probes are homemade, 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 angledependent stud ies. 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 temper ature profile in the cryostat. Fig. 34 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 ct al. [75] for further details concerning the microwave probe design. 3.3 Rotating Cavity The standard approach for studying angledependent effects using the cavity perturbation technique is to use a splitpair magnet and/or goniometers, and is widely used in lower frequency commercial EPR instruments, e.g., Xband, Kband and Qband [36, 86]. In the case of the splitpair 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 insitu rotation of part of a cylindrical resonator which we developed recently, thus enabling angledependent cavity perturbation measurements in ultrahighfield magnets, and twoaxis rotation capabilities in standard highfield superconducting splitpair 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 31. We note that a rotating cavity has previously been developed by Schrama et al. [87], at the University of Oxford, also for highfield 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 endplate rotates), whereas the coupling is varied upon rotation in the Oxford version, resulting in effective "blindl.. I i.e., angles where the microwave fields in the waveguides do not couple to the cavity; in contrast, the cylindrical version offers full 3600+ rotation. Furthermore, the rigid design offers greater mechanical stability and, therefore, less microwave leakage from the cavity, resulting in improved signaltonoise characteristics [75]. The TEOln cylindrical modes also offer the advantage that no AC currents flow between the curved surfaces and flat rotating endplate 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, Qfactors for the first few TEOln 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 i ,~I I1 samples. Finally, the cylindrical cavity is machined entirely from copper, and held rigidly together entirely by screws. This allcopper 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. 35. The principal components consist of the openended cylindrical resonator, the cavity endplate and worm gear, a worm drive for turning the endplate, and a wedge which facili tates external clamping and unclamping of the cavity and endplate. The cavity assembly is mounted on the underside of a stage (not shown in Fig. 35), and the endplate is centered on the axis of the cavity by means of a centeringplate. 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 worm drive 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. Cavity End Plateworm Resonator Wedge Coupling Hole Centering Plate End Cap C Spindle Figure 35. 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 unclamp the end plate to/from the main resonator. See main text for a detailed description of the assem bly and operation of the rotating cavity (Reused with permission from Takahashi et al. [81]. Copyright 2005, American Institute of Physics.). 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. 35 are made from brass [88]. The internal diameter of the cavity (7.62 mm) is slightly less than the diame ter of the endplate, which is free to rotate within a small recess machined into the opening of the resonator. On its rear side, the copper endplate mates with a brass gear which, in turn, rotates on an axis which is fixed by the centeringplate. As mentioned above, rotation of the worm gear and endplate is achieved by turning the worm drive with the wedge disengaged from the endplate. During experiments, a good reproducible contact between the endplate and the main body of the cavity is essential for attaining the highest resonance Qfactors. This is achieved by engag ing the wedge through a vertical channel in the centeringplate, where it transfers pressure along the axis of the endplate. Fig. 36 dip,~'i labeled photographs of the 1st and 2nd generation rotating cavity assemblies. The 2nd generation version employs a smaller homebuilt 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 (Diame ter = 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. 35), and to fix the endplate within the worm gear. Rotation of the worm gear is monitored via a simple turncounting dial mounted at the top of the probe, having a readout resolution of 1/100th of a turn. Different worm drive/gear combinations are employ, 1 in the 3/4" and 1" diameter probes (see Fig. 36), with 1/41 and 1/20 gear ratios, respectively. Thus, the angle (a) (b) Figure 36. Photographs of the rotating cavity system; part of the cavity has been disassembled (endplate and centeringplate) in order to view the in side of the cylindrical resonator. (a) shows the 1st generation rotating cavity, which fits into a 1" thinwalled 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 31). (b) shows the 2nd generation rotating cavity, which fits into a 3/4" thinwalled tube; this cavity is compatible with the highest field (45 T) resistive magnets at the NHMFL, as well as the 3He probe designed for the Oxford In struments magnet at UF (see Fig. 37). (Reused with permission from Takahashi et al. [81]. Copyright 2005, American Institute of Physics.) resolution on the dial redout corresponding to the actual sample orientation is approximately 0.090 for the 3/4" probe, and 0.180 for the 1" probe. Although both probes exhibit considerable backlash (~ 1), this is easily avoided by con sistently varying the sample orientation in either a clockwise or counterclockwise sense. High resolution EPR measurements on singlemolecule magnets (reported in Takahashi et al. [81]) have confirmed the angle resolution figures stated above. The stage also performs the task of clamping the Vband 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. 35). 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 x 7.62 mm (diameter x length). The center frequency (fo) of the TE011 mode of the unloaded cavity is 51.863 GHz, with the possibility to work on higherorder 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 welldefined 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 33 shows resonance parameters for several unloaded cavity modes (there are many others which are not listed). The TEOln (n = positive integer) modes are probably the most important for the rotating cavity design, because their symmetry is axial. Thus, rotation of the endplate 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 endplate; 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 endplate. These geometries are depicted in Fig. 37 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 highfield magnets [Fig. 37(b)] since, for the TEOln modes, the sample sits in a microwave ac field (H1) which is alv transverse to the dc magnetic field (Bo). We discuss this in more detail in Mola et al. [75] and Takahashi et al. [81]. Another advantage of the TEOln modes is the fact that no microwave cur rents flow between the endplate and the main body of the resonator. Thus, the Qfactors of these modes are high, and essentially insensitive to the mechanical con tact made with the wedge. Table 33 lists the key resonance parameters associated with several modes. For the case of the TE011 mode, the low temperature (~ 2 K) Qfactor is ~ 21,600, and the contrast between the amplitude on resonance [A(fo)] and the amplitude far from resonance (leak amplitude, Al) is 31.7dB. These pa rameters 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 singlecrystal samples, both insulating and conducting. We note that the Qvalue for the noncylindrically (a) Endplate configuration (b) Quartz pillar configuration 0 rotation Figure 37. Schematic diagrams showing various different sample mounting con figurations for both axial and transverse magnetic field geometries, including the twoaxis rotation capabilities (Reused with permission from Takahashi et al. [81]. Copyright 2005, American Institute of Physics.). Axial AB Trans. B Axial AB Trans. B Table 33. Resonance parameters for several different cavity modes. The first col umn indicates the given mode. The second column lists the resonance frequencies (fo). The third column lists the Qfactors. The final column lists the contrast in dB, i.e., A(fo)A1, where A(fo) is the transmission amplitude at the resonance frequency, fo, and A, is the leak amplitude (Reused with permission from Takahashi et al. [81]. Copyright 2005, American Institute of Physics.). Mode fo(GHz) Q A(fo)A1(dB) TE011 51.863 21,600 31.7 TE212 54.774 3,300 26.5 TE012 62.030 16,600 25.2 TE015 109.035 8,600 19.0 symmetric mode in Table 33 (TE212) is almost an order of magnitude lower than that of the TEO11 mode. As noted above, this is due to the flow of microwave cur rents associated with the TE212 mode across the mechanical connection between the main body of the resonator and the endplate. As discussed earlier, even though the endplate can be rotated, the waveg uides 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 ~ A/6] have been optimized for the Vband, and the cavity sidewall was milled down to a thickness of 0.015" (=0.38 mm, or ~ A/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 3600 angle range may be explored, and with excellent mechanical stability. Fig. 38 shows the typical random fluctuations in the cavity resonance parameters for the TEOln mode during a complete 3600 rotation of the endplate: the center frequency varies by no more than 120 kHz (~ 3 ppm, or 5' of the resonance width); the Qfactor is essentially constant, to within 1 ., and the con trast, [A(fo)A ], fluctuates between 29 dB to 33 dB, corresponding respectively to leak amplitudes of 3.5'. and 2.'". of A(fo). Another important consequence of coupling through the side walls of the cavity involves selection rules for T\ i cylindrical cavity modes. At the lowest frequencies, only the TE01 mode of the Vband waveguide propagates. Thus, the microwave H1 field in the waveguide is polarized parallel to the cavity axis, i.e., it is incompatible with the symmetry asso ciated with the H1 patterns of the T\ i modes. Consequently, we do not observe, for example, the TM111 mode. This offers added benefit, since the TMlln modes are ordinarily degenerate with the TEOln modes, and steps have to be taken to either lift these degeneracies, or to suppress the TAi 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 T\ i 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 highfield magnets at the NHMFL; the specifications of each magnet system are listed in Table 31. 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. 36(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) inhouse. This cavity assembly, which is shown in Fig. 36(b), is small enough to fit into a 3/4" diameter thinwalled stainless steel 90 Phase 120 60 150A 30 180 0 210 330 240 300 270 f,=51.86335(GHz) A 3A A A A A 0 A A A A Q=21600 32 30 21 U 0 50 100 150 200 250 300 Angle(degree) Figure 38. Angledependence of the cavity resonant properties. (a) A polar plot of the complex signal transmitted through mpe cavity as the frequency is swept through the TEO11 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 cir cle in the complex plane. Points close to the origin correspond to fre quencies far from resonance, while the resonance frequency corresponds to the point on the circle furthest from the origin. The resonance pa rameters 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, fo, is 51.863 GHz, and the Q factor is 21600. (b) Full 3600 angledependence of the fluctua tions in the resonance parameters for the TEO11 mode: upper panel  fo; center panel Q; and lower panel difference between the am plitude on resonance [A(fo)] and the amplitude far from resonance (the leak amplitude, Al). These data illustrate the excellent repro ducibility of the resonance parameters upon unclamping, rol ilin :7 and reclamping the cavity (Reused with permission from Takahashi et al. [81]. Copyright 2005, American Institute of Physics.). 2' S: of Physics.). 2' S: tube, thus enabling measurements in the highest field 45 T hybrid magnet at the NHMFL. Furthermore, this cavity is compatible with the '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 reglued 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 endplates. This enables preparation of a new experiment while an existing experiment is in progress. Finally, we discuss the twoaxis rotation capabilities made available via a 7 T Quantum Design (QD) magnet (see Table 31). 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 flowcryostat. The rotator is driven by a computer controlled stepper motor, with 0.010 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 fixedfield cavity perturbation measurements as a function of the field orientation, as has recently been demonstrated for the organic conductors a(BEDTTTF)2KHg(SCN)4 [46] and (TMTSF)2C104. (See (' 111 4.) We generally use the steppermotor to control the polar coordinate, while mechanical control of the cavity endplate 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 singlemolecule 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 twoaxis rotation experiments. The first involves constraining the cavity assembly and the waveguides within a 3/4" thinwalled 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 misalign the cavity relative to the rotator. The second modification involves attaching a spindle on the underside of the endcap. This spindle locates into a centering ring attached to the bottom of the PPMS flowcryostat, thus preventing the waveguide probe from rotating offaxis (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 p and c, 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 equation [62], fmnp np n 2T m27 27 ( )2 + ( d)2, (3 1) 2w 2wp R d where m, n and p are integers (n and p >0). R is the radius and d is the height of the cylindrical cavity. xm is the nth root of the 1st derivative of the Bessel function, J((x), and the nth root of the Bessel function, Jm(x), for the TEmnp and the TMmnp modes respectively. The electromagnetic field configurations for the TE mp modes are given below. H, (r,,z) pHo J( r)cosm) cos( ), (32a) Wmnpd /l^t R d np mpwHo Jm(x ) ( 2P H 7(r, z )= (~)r sin(mf) cos( z), (32b) Wmnpd^^/' R d R d CU xmn COS() sin( (3 2c) EmTp(r, ,z) =0. (32f) 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. 39(a) shows the equivalent RLC circuit. The impedance of the circuit is given by [91], 2L + i Z =R iwL+ = R iwL(1 ), (33) wC w2 2 Z = R iwL + R iL(1 ), (33) wC w2 (b) (a) 1.0 R L 3 0.5 CT F 0 )o 0 Figure 39. A simple description of the resonant cavity. (a) An equivalent RLC circuit representing the cavity. (b) A Lorentzian resonance produced by the RLC circuit; wo denotes the resonance frequency; F denotes the full width at half maximum of the resonance. where the complex quantity Z is the socalled impedance, the real quantities R, L and C are the resistance, inductance and capacitance respectively, and wo = 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, V R mV2 R2 Pave 2 1 R Vrms rVms2 (3 Z Rw+ (2 w(3 4) ae12 R l + Lf2 o_ 2) Thus uo = 27rfo = /t LC is the resonance frequency. In the case of Lwo > wIU U1o , Eq. 33 is written, Z 2iL(w o), (35a) where the complex resonant frequency wo is defined as, F. = Uo i, (35b) and F = R/L. Using the above impedance, the power is given by Pae V2 (R/L2) PO S(1)2 + (; )2 (11)2 ( )2 (3 ) Fig. 39(b) shows the power as a function of frequency. The resonance peak is clearly seen at the resonant frequency w0. F represents the full width of the resonance at halfmaximum (FWHM) shown in Fig. 39(b). The FWHM is often expressed by using the quality factor (Q), defined by, wo woL Q (37) P R In our setup, the Qfactor 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 wo to us, and a change of F from Fo to F, (or the Qfactor from Qo to Q,). The complex impedance is therefore changed from cj0 to uas. By using Eq. 35, 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, AZ Z, Zo igoLAJa, iwo 1 1 SigoL(wu s (7 )) 2 Qs Qo SigoL(Aw Ar), (38) 2 where we introduced a geometrical factor go which depends on the experimental geometry (the location and size of the sample in the cavity) [56, 91]. Defining AZ R + iX (R = sample resistance and X =sample reactance), the sample response is given by, AF wo 1 1 R ( = (39a) 2 2 Q, Qo goL' and X AU = Us UJo = (39b) go0L 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 magneto optics etc... This is particular 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 platelike shape, we briefly introduce two cases, inplane 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 611 and 6 are smaller than the sample dimensions (skip depth regime) where 6 and 611 represent skin depth for the least and the good conducting direction respectively. More details are found in Hill [33]. 3.5.1 Inplane Measurements. In order to measure the inplane properties, the induced current has to be polarized along only the inplane direction, otherwise the interlayer component of the current dominates the microwave response, as explained in the next subsection. Fig. 310(1) a) illustrates the case of a measurement using the electric component of the microwaves, with the ac electric field applied inplane. However, the current is induced not only along the inplane direction, but also the interlayer direction. This is because the current flow is limited to the surface, i.e., within the skin (penetration) depth 6 (A) 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 inplane measurements. Another possibility is to (2) Inter layer measurements (1) Inplane measurements a)  WI z ^"", ^ oi D t fL T.t 11' H a t T >l~ i~l_, ,. .+.+++ +.... . I P'lOil fl, c, i, __ ,______ Schematic diagram illustrating the various possibilities for exciting inplane and interlayer currents in a Q2D platelike sample. (1) a) Inplane currents induced by an ac electric field (Reprinted figure with permission from Hill [33]. Copyright 1997 by the American Physical Society.). b) Inplane currents induced by an ac magnetic field. We use the latter configuration for the inplane 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 in duced by an ac magnetic field. This latter is the most widely used configuration (Reprinted figure with permission from Hill [33]. Copy right 1997 by the American Physical Society.). use the magnetic component of the microwaves, as shown in Fig. 310(1) b). In this case, only current parallel to the conducting tplane is induced. Thus, it is possible to perform the inplane measurement with this configuration. However it is impossible to distinguish between crx and ayy, 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 b) Figure 3 Figure 310. L1~ill'r I J Ft skin (penetration) depth [92], for currents flowing across faces of the sample which are parallel () and perpendicular (1) direction to the inplane direction, i.e., P1 61a (310) P\\ 611all where a1 and all are the surface areas parallel to the least and the good conducting directions, respectively. In the case of the Q1D conductor, (TM [TSF)2C104, sample iv I 1 are typically needle shaped so that all ~ 0.4 mm2 and a_ ~ 0.1 mm2. The skin depth is estimated to be 6_ ~ 100 pm and 611 ~ 1 pm at Helium temperatures at a microwave frequency. As a result, P1/P= 20. In the case of the Q2D conductor, K(ET)2Cu(NCS)2, P/IPi is typically 6100 at a microwave frequency. Thus, the energy dissipation P1 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. 310(2) a) shows the situation in which the electric component is applied perpendicular to the inplane direction. The current is induced from one face of the sample platelet to the other via the edge of the sample. Fig. 310(2) b) shows magnetic field excitation. The ac magnetic component is applied along the inplane 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 angledependent magnetooptical 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 angledependent magneto optical studies of Q2D metals. Configurations are specified for interlayer measure ments using the magnetic component of the microwaves. 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. 311. In the case of TEO11 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. 311. Thus, in order to have a strong coupling, we usually place the sample on a quartz pillar [Fig. 311(a)], on the end plate [Fig. 311(b)], or on the side wall (not shown). These configurations are different in terms of the angle dependent magnetooptical 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. 311(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 .i. li, 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 reactance. The problem is mainly to solve Maxwell's equations for E and H with 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 the socalled 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 semiinfinite medium. The flat conducting sample (the xzplane) is set at y = 0 so that the medium for y >0 has the dielectric constant e1, permeability pi and conductivity aU, 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 xaxis and has harmonic time dependence, i.e., (Hac)x = i,,. The magnetic field inside the sample is calculated by Maxwell's equations, V D p, (3 a) aH V x H  J, (31b) aB Vx E + 0, (3 1c) V B 0, (3 11d) and Ohm's law, J = E. (312) Using the above equations with p = 0, one obtains the following differential equation, a2H OH V72H ~atl 2 t (3 13) and its solution, with the boundary condition H(y = 0) = Hac, is given by H(y, t) /,,. estS, (314) where the complex wavevector is given by q'= wlli 1 i91 '/ l0i (3 15) LL)LL For convenience, the following definition of the complex dielectric constant 7 and conductivity a are also used frequently: 1 '= 1 + i+ C e + i2, (316a) and the complex conductivity a is defined as 'e co +. (316b) Therefore, the real and imaginary parts of the conductivity become, "1 = W')C2, (316c) and 02 = co O C1). (316d) Additionally, the complex wavevector is also expressed as, n k u UN w( +i), (317) c c where V is the complex refractive index, n is the real refractive index and k is the imaginary refractive index. c is the speed of light c = 1/ Poco. The induced electric field is calculated easily using Eq. 31 b, E(y, t)= Hoe e y. (318) Thus one can obtain the impedance of the sample, Z = E/H, L) 1 i2pl S. (3 19) g   Assuming el > co, which is often the case for good conducting materials, we obtain the wellknown expression for the surface impedance, s = i R + iXs. (320) V71 i172 As discussed in Sec. 3.4, the microwave response is related to Rs and Xs, so that we can extract both a, and 02 by using the conversions, RsX, a, = 2p1w ^R<2, (321a) (R2 + X)2' and X2 R2 2 = /1 2 (321b) (R + X2)2 Moreover, rewriting the above equations, one can obtain, AF oc R +2 (3 22a) A X, 2 + 2 (322b) where AF = 1/Q 1/Qo is related to the change of the Qfactor, and Awo= asUo is the change of the cavity resonant frequency. In the case of superconducting materials, we need to consider the response of the normal quasiparticles and the Cooperpairs. This is often treated by using a general two fluid model, in which a represents the conductivity due to normal quasiparticles, and (a is the superfluid contribution to the conductivity. S= 1 2 + 91S ia2S Tnse2 e2 1i 7i 2 + J6(u) (323) m m*uj where n, is superfluid density, and the penetration depth A is also used for nse2/m* = (piwA 2)1. In a simple case (T ~ 0 and a swave superconductor), at microwave frequencies gssi = 0 so that the complex conductivity often becomes, a U1 72S w2 (324) piauj\ piauj\ which assumes 92S a1. Thus, the microwave response is given by the following expression, AF oc R [1 = p2 2 3, (325a) FL2_ 2a12 2p1 and Aw oc X, p wA. (325b) 3.6.2 Metallic Depolarization Regime When the skin depth is larger than the thickness of the sample (a), the electrodynamics are said to be in the socalled metallic depolarization regime soepresented by qa > 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 quasistatic limit. As a result, the conductivity is simply related to the sample resistance, namely, a oc (326) Pi R Thus, using Eq. 39a, one can obtain the following expressions, AF oc R oc (327) a1 3.7 Measurement of the Change of the Complex Impedance Z In the experiments discussed here, we focus on investigating relative li.i, 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 wo and F (or Qfactor); 2) measuring the amplitude and phase of the transmitted signal continuously at a fixed frequency, the socalled fre pii i' vlock method; 3) and measuring the amplitude and frequency continuously at a fixed phase, the socalled phaselock method. Method 1) is the continuously at a fixed phase, the socalled phaselock method. Method 1) is the most straightforward for determining the complex impedance Z, since AF and Aw are measured directly. However the estimation of AF and Aw is a rather timeconsuming 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. 36, the amplitude and phase are expressed by the following equations, 1 1 A(w) = Ao = Ao (328a) (V )2 + o)2 (2 +( co)2 and, tan )(co) c c o (328b) ( 2Q) ) 2) where F = o/Q, uo is the cavity resonance frequency, and Q is the qualityfactor. 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 F and the resonance frequency wo, i.e., F  + AF and uo i wo + Aw0. Thus the amplitude and phase of the microwave response also changes, 1 A(w) A(w) = A, (329a) ( )2 (a) w0 Ao)2 and, c co A aco tan (w)  tan (o) L (329b) ( 2 Fig. 312 shows a schematic view of these changes. The amplitude at the resonance frequency becomes smaller because F  + AF. The change also appeared in the phase, i.e., the position at (w) = 0 is shifted to at Q(wo + Awo) = 0. ((m) (b) 0 coo 0 + Am0o ~oo Figure 312. Typical changes in the amplitude and phase of the microwaves trans mitted 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 AF and Awo; (b) typical changes in the phase of the microwaves, represented by AF and Awo. 3.7.1 Frequencylock Method We perform the frequencylock using a Phase Matrix/EIP 575B frequency counter [95]. Replacing w = wo in Eq. 329a and Eq. 329b, the amplitude and phase during the freq1' L. .i slock are given by A(wo) Ao a (F4"'F)2 + (A wo)2 and, tan 0(wo) = r+Ar Thus, rewriting the above equations, the microwave response is given by Thus, rewriting the above equations, the microwave response is given by 2Ao 1 A A(wo) /tan2 o) + A Ao 1 A(wo) 1 +1 Vtan2 ( o) When the phase shift is small, tan + 0, AFr^ 2Ao A(wo) (330a) (330b) (331a) (331b) (332a) (a) F and Ao Awo Ao ( o). (332b) A( o) Thus, when the microwave response has small phase shifts, AF depends only on a change of the amplitude. Eq. 332a also shows the change of the amplitude is linear, i.e., A(wo) = Ao + 6A so [A(o)]1 = 1/Ao[1 6A/Ao]. The frequency shift is therefore proportional to (wou). On the other hand, this method can be problematic when the phase shift is huge, because the frequency becomes off resonant, and AF starts to depend on both A(wo) and (awo). Thus the phase and amplitude become mixed, i.e., dissipation and dispersion are mixed. Empirically, the frequii' i,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. 331 or a vector fit [96]. Furthermore we can also avoid the problem using the phaselock method. 3.7.2 Phaselock Method The phaselock 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 wellseparated from the other cavity modes, standing waves and other sources of df/dv, this method is the most useful. However, the phaselock is less stable than the frequency lock so that the signal we obtain by this method often has a smaller signaltonoise ratio. Thus, we usually begin by using the frequency lock. Then we perform the phaselock measurement if we encounter problems discussed in the previous subsection. As seen in Fig. 312, the phase lock alhv, keeps Q(wo + Aweo)0. Thus w~ o+Awo. Putting = 0 and w =wo+Awo into Eq. 329a, the amplitude becomes, A(w) Ao (333) F AF Thus, AF is given by, S A r. (3 34) Therefore, this expression is the same as Eq. 332a for the frc'i,. n' lock. By recording the frequency with a frequency counter, we can also measure the fre quency shift u + Aow. 3.8 Summary In this chapter, we explained our experimental techniques. In particular, for the purpose of angledependent highfield microwave spectroscopy, we developed a rotating cavity which is compatible with several highfield 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 QUASIONEDIMENSIONAL CONDUCTORS The results presented in this chapter can be found in the articles, Periodic orbit resonance in the quasionedimensional organic superconductor (T I 1.F)2 C04, 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. BetancurRodiguez, S. Hill, S. Takasaki, J. Yamada and H. Anzai, submitted to the ISCOM conference proceedings. 4.1 The Quasionedimensional Conductor, (TMTSF)2C104 The organic metal (TMTSF)2C104 belongs to the family of quasione dimensional (Q1D) Bechgaard salts [2], having the common formula (T ITSF)2X. TMTSF is an abbreviation for tetramethyltetraselenafulvalene, and the anion X is AsFe, C104, PF6, ReO4 etc. The (TA TSF)2X series have been the most ex tensively studied organic materials. The reason may be the extremely rich phases for (TA TSF)2X. This includes unconventional metallic states, spindensitywave (SDW) states, fieldinduced spindensitywave (FISDW) states, possible spintriplet superconducting states and so on. Unlike other onedimensional materials, The (TMTSF)2X compounds tend to be good conductors at room temperature al though the (TA TSF)2X eventually meet the metalinsulator transition at a low temperatures, except for (TMTSF)2C104, which becomes a superconductor. The metalinsulator transition in the (TA TSF)2X compounds is seen in the resistivity of Fig. 41(a). This transition is explained by the nesting of the FS. In the case of the (TA TSF)2X compounds, the transfer energy to the bdirection is finite, so that the FS is not a perfect flat plane, and the nesting is not as strong as the Pressure 5 kbar Temperature dependence of resistivity in (TMTSF)2X Figure 41. Phase diagram for (TMTSF)2X Electronic properties of the (T\ TSF)2X. (a) Temperature de pendence of the resistance in (TMTSF)2X. With the exception of (T\ iTSF)2C104, all samples undergo a metalinsulator transition at low temperatures (Reprinted from Bechgaard et al. [97], Copy right 1980 with permission from Elsevier). (b) The phase diagram of (T\ TSF)2X 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 spindensitywave insulating states and, SC is a superconducting state. Varying pres sure, high temperature and ground states are changed. For example, (T\ TSF)2PF6 exhibits a SDWI ground state at ambient pressure. However, under 6 kbar, the ground state of (T\ITSF)2PF6 becomes superconducting (See J6rome [98], Moser et al. [99] and Dressel [100] for the details of the phase diagram.). 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 (T\ TSF)2X 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 pl a role in determining the spacing of the T\ TSF 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 cdirections, 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 TMI for the SDW state. This is actually seen in Fig. 41(a). By replacing the anion X BF4 to PF6, TMI shifts from 80 to 15 K. This may be understood in terms of a change of the dimen sionality, 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 41(b) shows a proposed phase diagram of (T\ TSF)2X as a function of the pressure. The pressure may be controlled by replacing the anion X and ap plying external pressure. Varying pressure, high temperature and ground states are changed. For instance, The TMI in (T\ TSF)2PF6 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, (T\ iTSF)2C104 is the only material which has a superconducting state under an ambient pressure. (T\TSF)2C104 has a structural phase transition of the C104 anions at 24 K. This due to the ordering of the noncentrosymmetric tetrahedral C104 anion, socalled anion ordering. The anion ordering is extremely sensitive to the cooling process of the crystal. Xray 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 when (T\[TSF)2C104 is cooled slowly at around 24 K (< 0.1 K/min). Then (TMTSF)2CO14 has a metallic and superconducting state in the low temperature regime, the socalled relaxed state. On the other hand, when (TMTSF)2CI04 is cooled rapidly (> 50 K/min), the anion is disordered. This is the socalled quenched state. We have investigated the cooling process dependence of the electronic properties of (TA\ TSF)2C104, especially in terms of its influence on the superconductivity. This investigation is introduced in C'i p 5. The < ir .1 structure of (TMTSF)2C104 shown in Fig. 12 is triclinic. The lattice parameters in the relaxed state are a = 7.083 A, b = 15.334 A, c = 13.182 A, a = 84.400, 3 = 87.620 and 7 69.000. In general, the shape of the (T:\ TSF)2C104 single < il I 1 is needlelike, as shown in Fig. 42(a). The most conducting direction is the needle direction (the aaxis). For convenience, the b', c' and c* axes are often emploil 1 The b' (c') axis is defined as the projection of the b (c) axis onto the plane perpendicular to the aaxis, i.e., b' = b sin 7 and c' = csin3. The c*direction is defined as the direction perpendicular to the abplane, i.e., c* = c'sina'. Similarly, The b*direction is defined as the direction perpendicular to the acplane, i.e., b* = b'sin a' where a' is the angle between b' and c'. a* is also often used for the angle between b* and c*. These angles a* and a' are simply expressed by cos 3 cos 7 cos a a* = 7 ' cos 1 [ ].os (41) sin 3 sin 7 (TMTSF)2C104 is a 3/4filled system similar to other (TA\JTSF)2X. The Fermi surface estimated by band calculations reveals that the FS consists of four sheets, as shown in Fig 42(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 b' and c'directions. As shown in Sec. 1.4, (TAUTSF)2C1O4 has a FISDW phase in high magnetic field. Since this FISDW phase is caused by the enhancement of the nesting 