# Microwave properties of potassium caused by the charge-density-wave broken symmetry

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Complies with University regulations and m eets the standards of the Graduate School for originality and quality Doctor o f Philosophy For the degree of Signed by the final examining committee: , chair . 5 ^ s. Approved by: v Department Head D ate U jL if □ iIS This thesis S is not to be regarded a s confidential. (LtOvCt (0 Major Professor Format Approved by: Chair, Final Examining Committee / Thesis Format Adviser Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. / Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. MICROWAVE PROPERTIES OF POTASSIUM CAUSED BY THE CHARGE-DENSITY-WAVE BROKEN SYMMETRY A Thesis Submitted to the Faculty of Purdue University by Mi-Ae Park In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 1996 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9713578 UMI Microform 9713578 Copyright 1997, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To the memory of my mother Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS I would like to thank Professor Albert W. Overhauser for his assistance in the preparation of this thesis and for his patience, encouragement, and guidance throughout the course of this work. His endless power of thinking on one subject made the thesis more perfect than I planned. I thank Professor Ronald G. Reifenberger, Professor Sherwin T. Love, and Professor Stephen M. Durbin for being on my committee. I am particularly indebted to professor James G. Mullen. His encouragement, counseling, and friendship have truly been invaluable in making the past few years at Purdue pleasant for me. I am proud to acknowledge the loving support of my family whose encour agement contributed significantly to the completion of this work. Cheers and encouragement from my father were essential. My sister and her husband were most helpful financially and mentally. Especial appreciation goes to Dr. Yong-Jihn Kim. As a colleague and husband, he has been very supportive. The one person I should thank most is my late mother. She was modest, but she had vision. Her wisdom gave me strength to aspire for a Ph. D. Without her confidence in my ability, I would not have pursued graduate study. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. iv TABLE OF CONTENTS Page LIST OF TABLES.................................................................................... v LIST OF FIG U R E S................................................................................. vi A BSTRA CT............................................................................................. ix 1. INTRODUCTION............................................................................... 1 2. SURFACE RESISTANCE IN A PERPENDICULARMAGNETIC FIELD 2.1 2.2 2.3 2.4 2.5 7 Introduction..................................................................................... 7 Effect of the heterodyne g a p s ......................................................... 10 Resonance from the Fermi-surface cylinder.................................... 16 Conclusions..................................................................................... 23 Reconciliation of the surface resistance and microwavetransmission 25 3. MICROWAVE TRANSMISSION IN A PERPENDICULAR MAG NETIC FIELD....................................................................................... 33 3.1 3.2 Gantmakher-Kaner oscillations ...................................................... High-frequency oscillations.............................................................. 4. THEORY OF THE CYCLOTRON RESONANCETRANSMISSION 4.1 4.2 4.3 4.4 4.5 34 35 . . 42 Introduction..................................................................................... Microwave transmission in an anisotropic, non-localmedium . . . . Effect of minigaps on microwave transm ission.............................. Conductivity tensor from a tilted Fermi-surfacecylinder................. Conclusions..................................................................................... BIBLIOGRAPHY..................................................................................... 42 46 52 57 61 66 APPENDICES Appendix A: Appendix B: Calculation of the conductivity ..................................... Polarization of the field inside an anisotropic metal . . . V I T A ......................................................................................................... Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 74 80 LIST OF TABLES Page Table 1.1 2.1 Unexpected phenomena for a nearly-free electron model of potas sium : all of them require the existence of a CDW broken symmetry. 5 Calculated values, from Ref. 8, of the first five minigaps and het erodyne gaps for K. The main CDW gap was taken to be 0.62 eV and the zone-boundary energy gap was 0.40 eV................................ 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vi LIST OF FIGURES Figure Page 1.1 The Brillouin zone of potassium on an (001) plane in A-space. The angular tilt, relative to [110], of the CDW wave vector, Q, has been exaggerated for clarity. The minigaps and heterodyne gaps are asso ciated with the periodicities of Eqs. (1.3) and (1.4). The horizontal line is a [110] direction which is perpendicular to a smooth potas sium surface. The Fermi sphere for a free-electron model is also shown.................................................................................................. 6 2.1 Surface resistance of potassium versus magnetic field (u/c = eH/m*c). The data, due to C. C. Grimes (1969), is for T=2.5K, and circularly polarized radiation at u/2-ir = 23.9GHz. The dips near ±0.77 are due to particles of CiiSO^-hH^O, embedded in the cavity walls during fabrication. The cyclotron resonance, at uic/u> = —1, occurs when H = 1.03T. The small resonance, at ujc/ u) = 1 is caused by a small admixture of the opposite polarization. The theoretical curve is for a purely spherical Fermi surface, which potassium would have in the absence of a CDW broken symmetry. .......................... 26 2.2 The Brillouin zone of potassium on an (001) plane in fc-space. The angular tilt, relative to [110], of the CDW wave vector, Q, has been exaggerated for clarity. The minigaps and heterodyne gaps are as sociated with the periodicities of Eqs. (2.1) and (2.2). The shaded areas are the two halves of the Fermi-surface cylinder, which form between the CDW gap and the first minigap. The axis of the cylin der is Guo —Q, which is also the direction of the heterodyne-gap vectors. The dc magnetic field, H, is applied parallel to [110], which is the habitual texture direction, perpendicular to smooth potassium surfaces. The (ideal) Fermi sphere is also shown............................... 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vii Figure Page 2.3 Theoretical surface resistance for a Fermi sphere having only heterodyne-gap intersections. The parameters of Eq. (2.6), which quantify the loss in effective cyclotron motion on equatorial orbits, are / = 0.8 and /? = 20. The drop in R for \H\ > Hc increases with decreasing / . The steepness of the decline increases with increas ing /?. The magnetic-breakdown field is Ho = 4T. The electron scattering time corresponds to wr = 30............................................. 28 2.4 Theoretical surface resistance of conduction electrons having ut = 30 on the Fermi sphere and u/rc = 150 on the Fermi-surface cylinder (containing ~ 4 x 10-4 electrons per atom). The axis of the cylinder is, here, parallel to H, and the heterodyne gaps (intersecting the sphere) are ignored. A 4 : 1 ratio of left to right circular polarization is assumed.......................................................................................... 29 2.5 Theoretical R(H) for potassium based on the heterodyne-gap pa rameters of Fig. 2.3 and the Fermi-surface cylinder model of Fig. 2.4, except that the cylinder’s axis is tilted 45° from [110]. (The tilt is required to m inim ize the elastic stress of the periodic lattice dis tortion needed to neutralize the electronic CDW.) This calculated behavior should be compared with C. C. Grimes’ data in Fig. 2.1. . 30 2.6 Theoretical R(H) for potassium based on the Fermi surface model of Fig. 3.4. The Q-domain size assumed for the specimen mea sured by C. C. Grimes is D=1.5/im. The other parameters used are enumerated in Sec. 2.5....................................................................... 31 3.1 Rectangular microwave cavities used for the observations. Small fused quartz windows are placed on both sides of the sample........... 38 3.2 Microwave transmission signal vs. H through a potassium slab in a perpendicular magnetic field. (H = 3.42T at u>c/ uj = 1) The microwave frequency is 79.18 GHz and the temperature is 1.3K. The field at ujJ oj = 1 is 3.42T. The phase of the microwave reference was adjusted so that the cyclotron resonance is symmetric. The slab thickness is L = 85fim. The data, provided by G. L. Dunifer, were obtained from sample K4, one of fifteen samples listed in Ref. 1. . . 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. viii Figure 3.3 Landau-level oscillations near uc/u = 1.2 and 0.6. The periodicity in (a) requires a cylinder radius, kc ~ kpf 8, corresponding to the middle cylinder of Fig. 3.4. The periodicity in (b) requires a cylinder radius, kc ~ fcp/16, appropriate to the smallest cylinder in Fig. 3.4. Page 40 3.4 Fermi surface cylinders of potassium. The horizontal axis is parallel to [110] and to the dc magnetic field. The shaded cylinders are cre ated by the CDW gap and the first three minigaps. The thicknesses of the cylinders have been exaggerated by a factor of ten. Each of the half-cylinders shown is joined to a partner on the opposite side of the Fermi surface by Bragg reflection at the energy-gap planes. The complete Brillouin zone is shown in the inset.................................41 4.1 Theoretical microwave transmission signal versus uic/ uj for potas sium if the Fermi surface is spherical. Only Gantmakher-Kaner oscillations appear (since the electron-spin magnetic moment is ne glected). The sample parameters are ujt0 = 150 and L = 85/zm. . . 63 4.2 Theoretical transmission signal when interruption of cyclotron mo tion by the CDW minigaps is modeled by the vz dependent relax ation time, Eq. (4.4), and with magnetic breakdown of the minigaps described by Eq. (4.5). The parameters, y0 = 21 and H0 = 5.8T, were adjusted so that the GK amplitude at uc/u = 1.47 agrees with the (calibrated) data from Fig. 3.2 and with its observed growth by a factor of five from low to high fields................................................... 64 4.3 Theoretical transmission signal when the non-local conductivity of the largest Fermi surface cylinder, shown in Fig. 3.4, is added to the main Fermi surface conductivity (employed in Fig. 4.2). The tilt, 6 — 50°, of the cylinder’s axis was adjusted so that ratio of the main CR to the first subharmonic (at u)c/u = 0.5) is ~ 10, consistent with Fig. 3.2. The cylinder radius, ke = 3kp/S, in the (110) plane was adjusted so that the ratio of the CR to the high-field GK amplitude agrees with that observed in Fig. 3.2. ujtq = 150 and L = 85(im. . . 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT Park, Mi-Ae. Ph.D., Purdue University, August 1996. Microwave Properties of Potassium Caused By the Charge-Density-Wave Broken Symmetry. Major Profes sor: Albert W. Overhauser. The microwave surface resistance of potassium in a perpendicular magnetic field, measured by C. C. Grimes in 1969, has never been completely explained. The sharp cyclotron resonance peak (at a magnetic field Hc) is caused by a small cylindrical section of Fermi surface created by the charge-density wave (CDW) m in igaps, having periodicities Kn = (n + 1)Q —nG110. The shape of the observed resonance requires a tilt of the CDW vector Q away from [110], predicted by Giuliani and Overhauser in 1979. An abrupt drop of the surface resistance for \H\ > |Bc| is caused by the heterodyne gaps, which have periodicities Kn = n(Guo —Q). These very small gaps, which begin to undergo magnetic breakdown for fields H > IT, interrupt the cyclotron motion of equatorial orbits. The abrupt drop in surface resistance for |B| > |BC| is caused by a partial loss of carrier effectiveness for electrons having velocities nearly parallel to the surface. Microwave transmission through potassium by Dunifer et al. shows five sig nals. They are Gantmakher-Kaner (GK) oscillations, conduction-electron spin res onance, high-frequency oscillations, cyclotron resonance, and cyclotron-resonance subharmonics. Only the spin resonance has been successfully explained using a free electron model. However, such a model predicts GK oscillations which are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. too large by several orders of magnitude. Lacueva and Overhauser have shown that CDW energy gaps which cut through the Fermi surface reduce the GK signal. The high-frequency oscillations were shown to result from Landau-level quantiza tion in a Fermi-surface cylinder created by the CDW. In this study we show that oscillatory motion, parallel to the field, of electrons in a tilted cylinder cause the cyclotron-resonance tr ansm ission. This signal and its subharmonics would be com pletely absent without the tilt. Consequently, four of the five transmission signals require a CDW broken symmetry. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1. INTRODUCTION The crystalline potential of potassium acting on the conduction electrons is so small that the degenerate Fermi sea is free-electron-like. Therefore the Fermi surface of a potassium should be almost spherical, and simply connected. De Haasvan Alphen measurements were thought to show that the Fermi surface deviates from a perfect sphere by only one or two parts per thousand.1 During the last thirty years a variety of magnetoconductivity anomalies have been discovered in potassium, the simplest monovalent metal.2,3 Unlike Li and Na, which undergo a crystallographic transformation to the 9R structure4 when cooled to low temperature, a single crystal of K is not destroyed by cooling. Without a charge-density wave (CDW) broken symmetry, the bcc lattice of K would support a spherical conduction-electron Fermi surface. Low-temperature transport anomalies could not then arise. Nevertheless, in dc experiments extraordinary phenomena occur which require the Fermi surface to be multiply connected.3 Unexpected phenomena also appear in the microwave properties of potassium.5, 6 Thirty two anomalous examples are listed in Table 1.1. Most of these data require that the translation symmetry of the b.c.c. lattice be broken by an incommensurate CDW structure. The Brillouin zone of an alkali metal has twelve congruent faces, each perpendicular to a [110] reciprocal lattice vector. The distance of each face from k = 0 is 14% larger than kp- The energy gaps at the zone faces, caused by the ionic Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 2 potential, are about OAeV in potassium and leave the Fermi surface undistorted from its spherical shape. However, the discovery by Mayer and El Naby of an intense optical absorption with a threshold of Q.6eV created a theoretical crisis.7 This optical anomaly indicates that there is another periodic potential with a wave vector Q nearly parallel to one of the [110] axes, for which the reciprocal lattice vector is G. The angle between Q and G is expected to be only about one degree. The modulated electron charge density is partially neutralized by a sinusoidal de formation of the positive ion background. This lattice deformation contributes extra diffraction peaks in x-ray or neutron-scattering experiments. CDW satellite reflections have been observed by Giebultowicz et al.8, 9, and the CDW wave vector was determined accurately: O7r Q = (0.995,0.975,0.015)— , a (1.1) where a is a lattice constant. |Q| differs by only 1.5% from |G| To find the energy spectrum for conduction electrons in a CDW, one has to solve a Schrodinger equation having potential terms: V{r) = 0.4cos(G • f) -1- 0.6cos(Q • f). (1.2) The first term on the right arises from the lattice and the second from the CDW instability. The solution of Schrodinger’s equation with this potential shows that the Fermi surface of potassium is very anisotropic and multiply connected,10 i.e., the Fermi surface suffers a distortion, and is sliced into several pieces by extra energy gaps. An approximate shape of the Fermi surface is shown in Fig. 1.1. The horizontal axis is parallel to [110] and perpendicular to a smooth potassium surface. The Fermi surface is pierced by two families of extra energy gaps: minigaps and heterodyne gaps. Minigaps, shown by the solid lines, are caused by higher order Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 perturbations having periodicities: Kn = (n + 1)Q —riGno> n = 1,2,... (1.3) A periodicity with n = 0 in this equation gives rise to the main CDW gaps. The influence of the minigaps has been noticed in a study of the GantmakherKaner oscillations.11 Several small cylindrical sheets of Fermi surface are created by the minigaps. One such cylinder explains both the cyclotron resonance in the perpendicular-field surface resistance12 and the Landau-level oscillations in the microwave transmission.13 Heterodyne gaps, shown by the dashed lines in Fig. 1.1, are caused by pertur bations with periodicities: Kn = n(Guo - Q ) , n = 1,2,... (1.4) The occurrence of open orbits caused by the heterodyne gaps14 and minigaps causes the open-orbit magnetoresistance peaks observed by Coulter and Datars.15,16 The purpose of this study is to investigate the microwave properties that result from the CDW in potassium. In Chapter 2 we study the surface resistance of potassium when a dc magnetic field is applied perpendicular to the surface. Grimes found a sharp resonance in the surface resistance at |/T| = Hc followed by a drop for \H\ > Hc.5 (Hc = m’u/c/e.) Both of these anomalies have been unexplained for twenty five years. In the experiment, circularly polarized microwaves are incident on the potassium surface. Electrons having velocities nearly parallel to the surface dominate the surface resistance. Because these electrons encounter the heterodyne gaps, their cyclotron motion is interrupted. However, when H becomes large, the electrons will undergo magnetic breakdown and act as if they were free. We model the consequences of the heterodyne gaps, and include a reduction of their influence Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 at high fields from magnetic breakdown. The sharp resonance is shown to arise from the minigaps, which create a tilted Fermi-surface cylinder. The tilt of the cylinder is essential for understanding the shape of the resonance. Microwave transmission through a potassium slab will be studied in Chapters 3 and 4. The data reported by Dunifer et. al. show signals that cannot be explained using a free-electron model.6 Linearly polarized microwaves are incident on a potassium slab with a dc magnetic field perpendicular to the slab. For microwave transmission, electrons having large velocity components parallel to the field are most important. Such electrons encounter the minigaps. We introduce magnetic breakdown of the minigaps to explain the field-dependence of the GantmakherKaner signal. A Fermi-surface cylinder created by the CDW plays a dominant role in the transmission. Because the cylinder is tilted with respect to the dc magnetic field, electrons in the cylinder have a velocity component parallel to the field. In chapter 4, we show that the cyclotron resonance transmission together with its subharmonics are caused by this longitudinal oscillation of the electrons in the tilted cylinder. Therefore, in this thesis, we explain four of the anomalous phenomena from Table 1.1, 7, 12, 28, and 31, by taking account of the CDW broken symmetry. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 Table 1.1 Unexpected phenomena for a nearly-free electron model of potassium : all of them require the existence of a CDW broken symmetry. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 1963 1963 1966 1968 1968 1968 1969 1969 1971 1971 1973 1974 1975 1976 1977 1978 1980 1980 1981 1982 1982 1983 1983 1985 1986 1987 1988 1989 1990 1992 1993 1996 Mayer-El Naby optical anomaly. Optical anisotropy. Conduction-electron spin-resonance splitting. Non-saturating transverse magnetoresistance. Kohler-slope variability. Doppler-shifted cyclotron-resonance discrepancy. Perpendicular-field cyclotron resonance. Longitudinal magnetoresistance. Four-peaked induced-torque anisotropy. Residual-resistance variability. Hall-coefficient discrepancy. Cyclotron-resonance transmission. Oil drop effect. Deviations from Matthiessen’s rule. Residual-resistivity anisotropy. Low-temperature phason resistivity. Open-orbit magnetoresistance resonances. Phason peak in point-contact spectroscopy. Variability of electron-electron scattering resistivity. Phason heat capacity peak. Temperature-dependence of the surface impedance. Field-dependence of the residual-resistance anisotropy. Four-peaked phase anomalies. Fermi-energy photoemission peak. Diffraction satellites. Magneto-serpentine effect. Splitting of paramagnetic-spin-wave sidebands. Subharmonic cyclotron-resonance transmission. Infrared inverse-photoemission (across minigaps). Landau-level oscillations from the cylindrical Fermi surface. Gantmakher-Kaner oscillations (too small by 104). Anomalous CESR of K-Fe bilayers. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 CDW GAP F.S. SPHERE [110] HETERODYNE GAPS GAPS BZ Figure 1.1 The Brillouin zone of potassium on an (001) plane in A:-space. The angular tilt, relative to [110], of the CDW wave vector, Q, has been exaggerated for clarity. The m inigaps and heterodyne gaps are associated with the periodicities of Eqs. (1.3) and (1.4). The horizontal line is a [110] direction which is perpendicular to a smooth potassium surface. The Fermi sphere for a free-electron model is also shown. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 2. SURFACE RESISTANCE IN A PERPENDICULAR MAGNETIC FIELD 2.1 Introduction Cyclotron resonance of the conduction electrons in potassium was first observed by Grimes and Kip1 using the Azbel-Kaner configuration,2 for which the dc mag netic field H is parallel to the metal’s surface. The effective mass was found to be m* = 1.21m. Resonant peaks in the (microwave) surface resistance also occur at subharmonic values,3 He/ n , n = 2,3,4,..., in addition to the fundamental res onance which occurs at Hc = m'uc/e. For conduction electrons having an energy spectrum E(k) that is spherically symmetric, a resonance in the surface resistance should never occur if H is perpendicular to the surface.4 Nevertheless Grimes, using a perpendicular-field configuration, found a sharp fundamental resonance in the surface resistance of potassium.5 His data are shown in Fig. 2.1 together with the theoretical R(H), which has no resonant structure at all. The magnetic-field sweep, expressed as cjc/ uj (where u//27r is the microwave fre quency, 23.9GHz, and uc = eH/m’c), includes both positive and negative values because the microwave field was circularly polarized. The sharp cyclotron reso nance, at u!c/u> = —1, corresponds to Hc = 1.03T. Not only was the existence of the resonance unexpected, but the sharp drop of R(H) for \H| > Hc has remained unexplained for twenty five years. The reason why a resonance is not expected in a perpendicular field is easily understood. The skin depth is ~ 10~4cm and the Fermi velocity is ~ 108cm/s. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 8 Accordingly, the time an electron (having the Fermi velocity) remains in the mi crowave field (~ 10"12sec) is an order of magnitude shorter than the microwave period. (Electrons do not return periodically to the skin depth in a perpendicular field, as occurs if the parallel-field configuration is employed.) The resonance cannot be an Azbel-Kaner signal from an oblique surface patch (at the sample’s edge) since there are no subharmonics. Neither can the resonance be attributed to electrons in a (110) surface-state band, since the bottom of such a band lies ~ 0.45eV above the Fermi level. The only satisfactory explanation of the resonance in a perpendicular field is based on the charge density wave (CDW) broken symmetry of potassium.6 Many anomalous properties (now numbering more than thirty) require the presence of a CDW,7 which causes two sequences of small energy gaps to cut the Fermi sur face,8 as illustrated in Fig. 2.2. The “minigaps” are higher-order gaps created by periodicities: Kn = (n + 1)Q —tiGno, (n = l,2 ,...), (2.1) where Q is the CDW wave vector and Guo is the (110) reciprocal lattice vector parallel to H. (It is known from optical properties that Q and one of the {110} reciprocal lattice vectors are nearly perpendicular to a smooth potassium surface.9) The calculated values8 of the first five minigaps are given in Table 2.1. In Fig. 2.2 the black regions outline a small cylindrical section of Fermi surface formed by the CDW energy gap and the first minigap. Only a small fraction, tj ~ 4 x 10-4, of the conduction electrons are enclosed by this Fermi-surface cylinder. Nevertheless, these electrons are responsible for the cyclotron-resonance structure in the surface resistance.10 Landau-level oscillations caused by the cylinder have been observed in microwave transmission.11 The periodicity of the oscillations Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 9 (versus 1/H) indicates that the cylinder radius is fcf/8.12 The small velocities of the cylinder electrons enable them to remain in the microwave skin region and to exhibit a sharp resonance absorption. The prior treatment of this resonance succeeded in identifying the cylindrical Fermi-surface component as its cause.10 However two puzzles remained: The cal culated shape of the resonance was antisymmetric rather than (nearly) symmetric. It was possible to “fix” this problem by mixing almost equal amounts of surface reactance and surface resistance. A small amount of such mixing could be toler ated experimentally,13 but the required mixing angle of ~ 47° seems excessive. In Sec. 2.3 we will show that this problem disappears when one recognizes that the cylinder’s axis is ~ 45° from the [110] (and H) direction. This axis tilt is required theoretically,14 and has been verified experimentally by the location of the CDW diffraction satellites.15 (The cylinder’s axis is parallel to Guo —Q, which is tilted ~ 45° when Q is only ~ 1° away from [HO].8) The experimental resonance shape can then be ascribed to the surface resistance alone. The second puzzle is the sharp drop in R{H) for \H\ > Hc, mentioned above. In the following section, we will show that this effect arises from the “heterodyne” gaps, created by the periodicities, Kn = n(G110 - <?), (n = 1,2,...). (2.2) The energy-gap planes of this family are shown by the dashed lines in Fig. 2.2, which cut at an angle, ~ 45°, through the central region of the Fermi “sphere”. The calculated values8 of the first five heterodyne gaps are given in Table 2.1. Cyclotron orbits for which kz is near zero can be “Bragg” reflected by the periodic potentials associated with {Kn}i E<1- (2.2). When such reflections occur, the elec trons become “ineffective” with regard to their cyclotron rotation. A quantitative Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 model for this phenomenon is presented in Sec. 2.2; and the observed behavior of R{H) when \H\ > |# c| is explained. 2.2 Effect of the heterodyne gaps In this section we will develop a model to account for the disruption of cyclotron motion caused by the heterodyne gaps, which cut through the central section of the Fermi sphere, as shown by the dashed lines in Fig. 2.2. (The dc magnetic field H is parallel to the horizontal, z axis.) The main contribution to the surface resistance R(H) arises from electrons having velocities nearly parallel to the surface; so these electrons (with kz ~ 0) necessarily encounter the heterodyne gaps. An electron which meets a heterodyne gap during its cyclotron motion can suffer a momentum transfer ±hK n, given by Eq. (2.2). The result is a disruption of its cyclotron motion (in the xy plane); and the change in z component of its velocity can cause it to rapidly leave the microwave skin depth, so its cyclotron motion is no longer fully effective. We introduce a factor, / < 1, which describes the probability that the electron behaves “effectively”, i.e., as if there were no gaps. An electron encountering a small energy gap can also continue on its path in ib-space, as it would if the gap were not present. This phenomenon is called “magnetic breakdown”. The breakdown probability P depends exponentially on H:ls (2.3) The parameter Hq depends critically on the energy gap Eg and the orbit geometry: (2.4) where H is a unit vector parallel to H, and v is the electron’s velocity at the energy-gap plane (if Eg were zero). This invariant form17 for Ho is equivalent to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 the result derived by Blount.16 It is clear from Fig. 2.2 that an electron with kz ~ 0 will encounter several heterodyne gaps. For simplicity, we will still employ Eq. (2.3) to describe the net result of all such encounters. The effective fraction, on taking into account magnetic breakdown, is then = 0) = / + (1 - / )e -3 “. (2.5) At very high fields, when magnetic breakdown is complete, / e// = 1, i.e., the electrons behave as they would without a CDW. For small H, / ./ / = / , the parameter we introduced above. /, a constant, will be adjusted to fit the data, (f is not zero because electrons with kz ~ 0 sustain part of their cyclotron motion.) On account of the complexity, the breakdown parameter reliably; but we have estimated it to be: H q~ H q cannot be calculated 4T . Equation (2.5) applies only to orbits for which fc* ~ 0; so we must generalize the effective fraction for all kz. Electrons having a rapid speed along z don’t remain in the skin layer very long anyway, so the interruption of their xy motion by the heterodyne gaps is of little consequence. Thus, their effectiveness will approach unity as |fcz| increases. This behavior can be described heuristically by, — fra g i — • ( 2 -6 ) The constant 0 will be adjusted to fit the surface-resistance data. The fitted values are / = 0.8 and 0 = 20. It is clear that f ejj approaches unity rapidly as kz becomes appreciable; and (of course) / e// equals Eq. (2.5) when kz = 0. The foregoing ideas are needed to correct the theoretical electron-gas conduc tivity, aap(q,uj), which is obtained by solving the Boltzmann transport equation. For an isotropic, free-electron metal the solution is standard. However, we display Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 12 axx and c^y, the components derived from Eqs. (12) and (13) of Ref. 10: + ixt + 1 — ixtt]' axs = ~ i L * (1 ~ e)[l - i aia+ ++ - i ta_ a. + + ix (?xy — 3lCT° f 1 dt( 1 - t2)[------- ------------------- -------- ], 8 J-1 v 1 —ta+ + ixt 1 -ia_. — + + ix ixtth (2.7) K 1 where a+ = ne2r m* (a; + a;c)r, a_ = (a; —a>c)r, x =ql = qvpT, t t is the scattering time, s kF (2.8) and the magnetic field, H (parallel to £), appears linearly in a;c, the cyclotron frequency, eH/m*c. The cartesian components of a are displayed here, instead of the circularly polarized ones, to anticipate the requirements of Sec. 2.3. Notice that the factor (1 —£2) in the integrand of Eq. (2.7) is proportional to the cross-sectional area of the Fermi surface for t = kz/k F, i.e., to the number of electrons in the slice of width dt. However, as argued above, the heterodyne gaps reduce the effective number by the factor, Eq. (2.6). Consequently, we must replace: (1 - i2)- (1 - i2) /« //( U (2.9) when the integrals are evaluated. Fortunately these integrals can be found analyti cally because, as is evident in what follows, the surface resistance involves a further integration over the wave vector q, which can only be carried out numerically. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 a.na.lyt.ir expressions for (Txx and <r*y which incorporate the substitution, Eq. (2.9), are given in Appendix A. Now, the surface impedance Z for an isotropic metal, having an xy surface at z = 0, is defined by 2 = kT3x{z)dz 3 v °k - (210) With the use of Stoke’s theorem for a circuit in the yz plane and the two Maxwell curl equations, £ (0 ) = ^ f i ° M z ) d z . The p rim p (2.11) indicates d/dz, and the time dependence of the fields is taken as exp(—iu/t). It follows that, _ 4ttzu; £*(0) c2 £'x(0)' , } {' } Solution of Maxwell’s equations in the metal with specular boundary conditions at z = 0 can be found in Kittel,18 whom we follow. For the a = x,y components of polarization, <p£a(z) u)2 . . . 47riu/ fcW ... , /n *o\ (2.13) Solution of this equation may be obtained by Fourier transform. It has been shown experimentally19 that conduction electrons are specularly reflected from shiny potassium surfaces. Under these conditions, one can treat the metal as infinite, instead of semi-infinite, provided £(z) is extended symmetrically to the region z < 0.This means that at z = 0, S' must undergo a jump from- £ '( 0) to S'{0). Accordingly integration by parts gives f ° S'e-iqzdz = ( f~° + [°°)£'e~iqzdz = -2£'(0) - q2E(q). 7 —oo 7+o 7 —oo Reproduced with permission of the copyright owner. Further reproduction prohibited without permission (2.14) 14 The Fourier transform of Eq. (2.13) is then ( V + ~ ) E a{q) = <2-l5> where for each component, a = x,y, E (q) = - £ / > * * * ■ S(z) = -i= r E{q)&**dq, V i'K J —oo j(z) = - L P J(q)c*dq. (2.16) VZ7T •/—oo Equation (2.15) is actually a pair of coupled equations because the conductivity tensor, (2.7), has off-diagonal components. On using <Xy to eliminate Ja{q), Eq. (2.15) becomes, !><,(»,«)£,(«) = - ^ 3 ( 0 ) , (2.17) where ( q2 _ ^ _ * ™ axx Dij(q, uj) = I V ~ ^ a xy 2 “ ^ o -y x q2 - ^ r ~ \ . j • “p ^ y y J (2 -18) For a spherical Fermi surface, axx = <Tyy and ayx = —<Jxy Eq.(2.17) can then be solved: _ , , [2 - g - * g V „)3 (0 ) + (2 , 9) We now introduce circularly polarized waves accordingly to the convention: 4 (z) = (x ± iy)f±(0)e‘(9z- w<). (2.20) 3 ( 0) = ± 13 (0). ( 2.21) It follows that Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 This relation allows one to solve Eq. (2.19) for Ex(q)/S'X(Q). Subsequently, the third relation of Eq. (2.16), with z=0, can be used to find £s(0)/£'x(0), which is all one needs to evaluate the surface impedance (2.12). The final result is, after restricting the integration to positive q, Z±{H) = - 8t u f o ^ q2 _ u 2 _ A^ iu ^ xx±iaxyy (2.22) (That the integrand is even in q follows from the symmetry of £(z) mentioned above.) The integration in dq must be carried out numerically with the expressions for oxx and from Appendix A. It was found sufficient to sum from q = 0 to 500,000 in 50,000 steps. (Doubling the range or reducing the step size by 10 did not alter the output noticeably.) Inspection of the experimental data of Fig. 2.1 reveals that the cavity was not driven in a pure ” mode. Accordingly, we have calculated the surface resistance given by, R(H) = Real[0 .8 Z.(H ) + 0.2Z+(H)]. (2.23) The residual-resistance ratio of potassium, p(Z00K)/p(4K), is typically ~ 5000. This value implies a scattering time, r ~ 2 x 10- 10sec. For 23.9GHz, ojt = 30. R(H) calculated from Eq. (2.23) is shown in Fig. 2.3. The heterodyne gaps cause the surface resistance to decrease when \H\ > Hc and to level off near \ucfu\ ~ 2. Not shown is the eventual recovery of R(H) to the ideal Fermi-sphere result for \u ju \ > 3. The rate of this high-field approach to the ideal R(H) depends on the magnetic-breakdown parameter, Ho’, so Hq can in principle be estimated by studying R(H) in the high-field regime. Baraff has reported20 that unpublished data of Grimes does indeed show the recovery of R(H) just described. (We have not seen this particular data.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 Interruption of the cyclotron motion for electrons having kz ~ 0, caused by the heterodyne gaps, reproduces the observed behavior of R(H) when |f/| > He. The sharp peaks at cyclotron resonance, however, are caused by the cylindrical section of Fermi surface shown in Fig. 2.2, and will be explained below. The observed resonance dips near uijw — ±0.77 have nothing to do with the potassium sample. They are caused by embedded particles of Cu^SO^ • 5 H2 O in the cavity walls created during fabrication.13 2.3 Resonance from the Fermi-surface cylinder The minigaps, shown by the short, solid lines in Fig. 2.2, correspond to the periodicities of Eq. (2.1). The sizes of the first few minigaps,8 tabulated in Table 2.1, are substantial. The two black patches in Fig. 2.2 represent a small Fermisurface cylinder which forms between the first minigap and the main CDW gap (having periodicity Q). It has already been shown10 that such a cylinder can explain the occurrence of the sharp cyclotron resonance observed by Grimes, and reproduced in Fig. 2.1. The size of the resonance requires the volume of the cylinder (pieced together from the two halves) to be a very small fraction, 77 ~ 4 x 10~4, of the Fermisphere-volume. It is noteworthy that this volume fraction agrees with the value calculated from the product of the cylinder’s length and its cross-sectional area. The former is obtained from the neutron-diffiraction measurement of Q,15 and the latter from the periodicity of the Landau-level oscillations,12 observed in microwave transmission.11 The cylinder’s radius is kc = kp/8, and its length (projected along [110]) is 0.015(?no. Although Q is tilted from [110] by about 1°, the cylinder’s Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission 17 axis, Q' = Guo —Q, is tilted about 45° from [110]. O' « (0.025,0.015,0.005)— . a (2.24) Its cross section is approximately circular in a plane perpendicular to [110]. It is of interest to calculate first the surface resistance, R(H), caused by a cylinder having its axis parallel to the magnetic field, H. On account of the cylinder’s small size, electron velocities on the Fermi surface of the cylinder are also small. Accordingly we will use a local conductivity tensor for the cylinder. The dc conductivity in the xy plane is t/<toc and crzz = 0, where <jQc is ne2rc/m m. The sharpness of the observed resonance corresponds to cjtc ~ 150. That rc (on the cylinder) should be ~ 5 times larger than r on the main Fermi surface is reasonable because of the smaller velocities of the cylinder electrons. The cylinder’s conductivity tensor is then: ( I - i0JTc a cyl _ _________ ^ O c (1 - iurc)2 + (ujctc)2 LJCTC 0 —coct c 0\ 1 —iUTc 0 0 (2.25) 0J For this exercise we will neglect the effect of the heterodyne gaps. Consequently ocvt, Eq. (2.25), is added to the conductivity, Eq. (2.7), for an ideal Fermi sphere. The surface impedance is still given by Eq. (2.22), and R(H) for 80% circular polarization is obtained from (2.23). The result is shown in Fig. 2.4 with utc = 150. A sharp cyclotron resonance is obtained but, unlike the data of Fig. 2.1, the shape is asymmetric. The sharp, asymmetric resonance shown in Fig. 2.4 was obtained previously,10 but the remedy attempted then involved introduction of a more than 50-50 admix ture of surface reactance and surface resistance. However, a remedy not involving such an admixture is possible. Since the cylinder’s axis must, theoretically, be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 tilted ~ 45° from [110],M, an angle confirmed by neutron diffraction,15 we now study the effect of such a tilt on the resonance shape. The equation for a cylindrical surface of constant energy, e = Ep, having an axis at an angle 6 relative to the direction of H, and with k relative to the cylinder’s center, is € —€0 = tan e)2 + (2-26) This cylinder has a circular cross section in the xy plane. Consequently the cy clotron frequency, with H along z, is unchanged. (For the cylinder of interest here, € —eo = Ep/64.) On account of its small size, as already discussed, the electron velocities on this surface are ~ Vp / S . We will therefore employ local equations of motion to find the tilted cylinder’s conductivity tensor, cr0*1. The Lorentz equation for motion in the electric and magnetic fields is 'v = —eM~l (k)[E + - v x H] — C Tc (2.27) where hv = Vfce(^), and the effective mass tensor is, (2-28> Then Vx — ■■(^>x fcjtan^), m Vy * — +ky, m* vz — — ~ (k x — kz tan 0) tan 6. m* (2.29) Equations (2.26)-(2.29) can now be used to find the conductivity of the cylinder. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 19 W dc tf* = (1 / + (wcrc) 2 1— iu)Te iuTc ) 2 ucTe —< jJctc —tan 0(1 —iu>Tc) ^ 1 —iuTc - tan 9(ojcrc) v—tan 0(1 —iurc) tan 6((Jctc) (2.30) 2 tan2 0(1 —ia/rc) / After compaLiring this tensor with Eq. (2.25), for which 0 = 0, it is clear that the electric field may now have a longitudinal, z component. Jx and Jy of Eq. (2.15) now involve Ez because crxz and ayz are no longer zero. However, we can express Ez in terms of Ex and £y by using the requirement that the total longitudinal current Jz be zero everywhere. Accordingly, (2.31) The longitudinal conductivity of the spherical portion of the Fermi surface must be calculated nonlocally using the Boltzmann transport equation. For a longitudinal electric field, proportional to <zzl h = e t d t \ — r + 2 J —i 1 — iujr v ixt ’ 3<70 [2x —r + 2urp + uj2t 2t + i(p —2u)tx + 2utt —u»2r 2p)], (2.32) 2x* with P 2 1 + (x + a;r)2 r = tan-1 (x —ur) + tan-1 (x + vr). (2.33) Equation (2.31) together with Eq. (2.15) changes Eq. (2.17) as follows: r2 u2 Airiuj , I? ~ ^2 4iriu , ^ 2 .2 ~ + [? ~~ 471i u , l x u2 g xy V = - ^ ( 0), Amu , g a y y \E y 2 — y 7r y Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. (2.34) 20 where ovy = a i f 1+ a^j1. These two equations can be expressed compactly: Ai(?, «)£<(») = (2-35) with £>«(,,*) = ■*. 4*«>/ _ / 1 , _2 ?~°V* T . or 4 4tiw _ / -^ -°v y . 1 • (2-36) Here, 4 = ** - (2-37) O-zz i and j denote x or y. The difference between Eq. (2.36) and Eq. (2.18) of Sec. 2.2 is that all transverse conductivities, cr^, in Eq. (2.18) are replaced by in Eq. (2.36). For example, a'xx includes axz, azx1 and azz as well as The longitudinal motion of electrons in the cylinder leads to creation of an electric field in the z direction. The Sz which arises (to preserve charge neutrality) plays a role in producing the transverse currents j x and j y due to the nonzero values of axz and ayz. (The tilted Fermi-surface cylinder mixes the transverse and longitudinal motions.) Even though the number of electrons in the cylinder is small, this mixing causes a large change in the surface impedance. The total conductivity tensor has the following properties: (Tyx = ~ 0 ’x yi &zy = ~ & y zi Accordingly, from Eq. (2.37), <7^ = — *^ E ( ) v^ a zx ~ &xz- (2.38) Equation (2.35) may now be solved: - 4.niu}<T,yy)£x(0 ) + V 7r {q2c? —ui2 —A.'Kiuia'xx){q2(? — a/2 — Akujcx^) + {kxiwo'^ ) 2 ’ _ f2 P (q2 c? - J 2 A‘KiuioJXySx(0) + (gV - u 2 - 4iriuja,xx)£y(0) V 7T(q2c* —u}2—47riwa'I )(g 2c 2 — w2 — iiriujcr^) -f (Airiua!^ )2 (2.39) Reproduced with permission of ,„ e COpyrigtl, ow„ er. Fui1her reproduc(|on prombjted ^ 21 Using Eq. (2.11), we express £•(0) in terms of the total current density J,-, Ji= f Jo ji(z)dz. (2.40) The third equation of (2.16), together with (2.38)-(2.40), give the electric field at z=0: m r00, ________ (gV - u? - 4xiu}<T,yy)Jx -f 4'Kiua/:cyJy________ X h ^(g2^ — oj2 —A^iuia'xx){q2(^ —u 2 — 47riu;o^y) + (47rio;cT^y)2 ’ £*«>) = - f /•“ Jo , _______ -iwiu)<j,xyJx + ( g V — u ) 2 - 4itiua/xx)Jv_______ ^(q2c? —uj2 —47riu/<r'3.)(g2c2 —to2 —47riu;a^y) 4 - {i.iriua,xy)2' (2.41) These expressions can be written compactly: £r(0) = ZXXJX + ZxyJy, £y(0) = —ZxyJx + Zyyjyi (2-42) which by inspection of (2.41) defines the four components of ZQ$, the surface impedance tensor. It is clear fromEqs. (2.40) and (2.41) that JQ (a = x,y) depend intricately on the bulkelectricfields. Anisotropy caused by thecylinder’s tilt causes Ja to be a complicated function of the conductivity components. This asymmetry also prevents the field from having perfect circular polarization. This behavior is studied in Appendix B. Nevertheless, on account of the small size of the cylinder, the electric field polarization is almost circular. Accordingly, Hy(Q) « ±iHx(0), Jy « ±iJx. Reproduced with pemrission o f the copyright owher. Further reproduction prohibited without permission (2.43) 22 The electric field at the surface will be Sx = Soe- *"* and £v = i£x, which corre sponds to right circular polarization. (So is real.) Then from Eq.(B.25), J ' — c^ Dr ~ iwt _ 2ir ’ J» = (2,44) The power absorbed per unit area per unit time is S, = ^-{& [£(0)) x Jie[£(0)]}„ 47T = ^ { B e [^ (0)]Be[H,(0)] - J8e[^(0)]He[«,(0)]}, = Be[£«(0)]fle[JJ + Re[£,(0)\Re{J,\. (2.45) We now separate Zap [defined by (2.41) and (2.42)] into their real and imaginary parts, i.e., Zap = Ra0 + iIQ0, (2.46) where R a 0 is the real part of Z a 0 and I a 0 its imaginary part. It follows that, /» Re(Jx) = —- cos(a;t), 27T Re(Jy) = ~siii(w i), Re(Sx) = -r-^[i2xx cos(wt) + Ixx sin(ut) 4- iZ™sin(ut) - 1 ^ cos(u>£)], 2/ir cSn Re(Sy) = ~ [ R n sin(a;t) —Im cos(a;i) —i?™cos(ut) —Ixy sin(cjt)]. (2.47) 27T By using these expressions in Eq. (2.45) and averaging over time, we find the absorbed power. Sz = Re[Sx(Q)]Re[Jx] + iZe[£v(0)]/2e[Jv], i C2^ 2 87T2' ( R x x ■+" R y y ~ 2/ry)» z"2/*2 1 ^ . * e [ i ( Z xx + Z„) + iZxy\. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.48) 23 The effective surface resistance is therefore, R = Re[-(Zxx + Zn ) + iZxy]. (2-49) From Eqs. (2.41) and (2.42), and £y = i£z for right circular polarization, the surface impedance is, ZR = -(■£«: + Zyy) + iZxy, dq{[q2<? - u P - 2 'Kiu(o'xx + a'm)\ - 4?r = X{q2<P—up —4iriuj<T/xx)(q2<P—up —47rio;<7£v) -+• {Aitiuia'^)2' For a left circularly polarized wave on the front surface, i.e., £y = —i£x, the surface impedance is, Z l = —( Z x x + Z y y ) — i Z x y , = JQ dq{[q c ? -U 2 2 - 2 niuj(</xx + < T'yy)] + 4Ku><T,xy} (q2P —up- — 4iziuja'xx){q2c2 — up — Aniua'yy) 4- (Airtua'^)2 ^ ^ Eqs. (2.50) and (2.51) must be evaluated numerically, as in Sec. 2.2. The effective surface resistance applicable to the experiment, for which the polarization was about a 4:1 admixture of L and R, is now R(H) = Real[0.8ZL{H) + 0.2ZR(H)]. (2.52) The theoretical R{H), which includes effects from both the tilted cylinder and the heterodyne gaps, is shown in Fig. 2.5. The agreement with the experimental data of Fig. 2.1 is remarkable. 2.4 Conclusions Inspection of Figs. 2.1-2.5 allows one to recognize that the CDW in potas sium6, 7>8 has profound consequences in studies of the perpendicular-field cyclotron Reproduced with permission o fth e copyright owner. Further reproduction prohibited without permission 24 resonance. The fact that cyclotron resonance even exists (in the surface resistance, R vs. H) attests to the presence of the small Fermi-surface cylinder (the dark areas of Fig. 2.2), created by the CDW gap and the first minigap. A theory based on only a spherical Fermi surface does not allow any structure near a/c = a/, as shown by the top curve of Fig. 2.1. The shape of the R{H) resonance (compare Figs. 2.4 and 2.5) reveals that the cylinder’s axis is tilted away from [110] (the field direction) by ~ 45°, as was found theoretically.15, 8 (The reason for the tilt is to minimize the elastic-stress energy involved in creating the periodic lattice distortion, of wave vector Guo —Q, needed to screen the electronic CDW.14) The drop in R for \H\ > |# c|, see Fig. 2.3 and the experimental data of Fig. 2.1, arises from the heterodyne gaps (Fig. 2.2), which interrupt the cyclotron motion of equatorial orbits, and cause a partial loss in carrier effectiveness. The volume of the Fermi-surface cylinder (corresponding to i) = 4 x 10~4 electrons/atom) was determined from the size of the resonance relative to R(HC) — R{O).10 The fact that this volume equals the product of the cylinder’s length (along [110]), determined from Q (observed in neutron diffraction15) and the cylinder’s cross section (perpendicular to [110]), defined by the periodicity of Landau-level os cillations observed in microwave transmission,12 indicates a compelling consistency among relevant phenomena. Fracture of potassium’s Fermi surface by CDW minigaps and heterodyne gaps, Fig. 2.2, is not only evident in the surface resistance anomalies studied here, but is the cause of many other magnetotransport effects, the most spectacular of which are the multitudinous open-orbit resonances21 created by the minigaps and heterodyne gaps. These open-orbit spectra have been explained within the same Reproduced with permission o fth e copyright owner. Further reproduction prohibited without permission. 25 framework employed here.17 Without a broken symmetry, potassium would be the simplest metal of all since, unlike Li22 or Na, it would retain its cubic symmetry to helium temperature. However, as a consequence of its CDW, potassium has provided (during the last thirty three years) a veritable universe of unanticipated behavior - a challenge to all who seek to understand electrons in metals. 2.5 Reconciliation of the surface resistance and microwave transmission In Chap. 4 it will be found that a theory of the CR transmission and the CR subharmonics requires the presence of a cylindrical Fermi surface having a radius in the xy plane, k& ~ 3fcf/8. The question arises, of course, how this additional cylinder affects the surface resistance behavior. This complication has been resolved by taking into account the size of Q-domains, which are likely to be small in Grimes’s specimens, since they were rolled (in oil) between mylar films, and so would be expected to have small-scale undulations. Only cyclotron orbits which do not intersect Q-domain boundaries would con tribute to the surface-resistance cyclotron resonance. The effective fraction of cylin der electrons is then, 77^ [l —(2RifD)], where Ri is the cyclotron radius (i=l,2,3), hckd/eH, and D is the width of the (laminar) Q-domains. (If the factor in square brackets is negative, it must be set equal to zero.) A modified theory of the surface resistance which includes all three cylinders in Fig. 3.4 is shown in Fig. 2.6. The Q-domain size is D=1.5/xm, 6 — 50°, yo = 3, u/rc = 150, Hq = 4, f=0.78, and (3 = 20. In this fashion both the CR transmission and the surface resistance can be understood quantitatively utilizing the same Fermi surface model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 CO c Theory 3 CO DPPH Marker o S: X DC Experiment - 1.5 1 - 0.5 o 0.5 1 1.5 ooc / c o Figure 2.1 Surface resistance of potassium versus magnetic field (loc = eH/m*c). The data, due to C. C. Grimes (1969), is for T=2.5K, and circularly polarized radiation at v/2 ir = 23.9GHz. The dips near ±0.77 are due to particles of C 11SO 4 • 5H2 0, embedded in the cavity walls during fabrication. The cyclotron resonance, at u)c/u> = —1, occurs when H = 1.03T. The small resonance, at (jjJoj — 1 is caused by a small admixture of the opposite polarization. The theoretical curve is for a purely spherical Fermi surface, which potassium would have in the absence of a CDW broken symmetry. Reproduced with permission o fth e copyright owner. Further reproduction prohibited without permission. 27 CDW GAP r* F.S. SPHERE [1 1 0 ] HETERODYNE GAPS GAPS BZ Figure 2.2 The Brillouin zone of potassium on an (001) plane in fc-space. The angular tilt, relative to [110], of the CDW wave vector, Q, has been exaggerated for clarity. The minigaps and heterodyne gaps are associated with the periodicities of Eqs. (2.1) and (2.2). The shaded areas are the two halves of the Fermi-surface cylinder, which form between the CDW gap and the first minigap. The axis of the cylinder is Guo— Q, which is also the direction ofthe heterodyne-gap vectors. The dc magnetic field, H, is applied parallel to [110], which is the habitual texture direction, perpendicular to smooth potassium surfaces. The (ideal) Fermi sphere is also shown. Reproduced with permission o fth e copyright owher. Further reproduction prohibited without permission. 28 1.04 o £ i tt 0.96 1 0 1 C0 c / CO Figure 2.3 Theoretical surface resistance for a Fermi sphere having only heterodyne-gap in tersections. The parameters of Eq. (2.6), which quantify the loss in effective cyclotron motion on equatorial orbits, are / = 0.8 and 0 = 20. The drop in R for |F | > Hc increases with decreasing /. The steepness of the decline increases with increasing 0. The magnetic-breakdown field is Hq = 4T. The electron scattering time corresponds to u t = 30. Reproduced with permission o fth e copyright owner. Further reproduction prohibited without permission. 29 O cc x oc 0.9 1 0 1 G)c / CO Figure 2.4 Theoretical surface resistance of conduction electrons having u;r = 30 on the Fermi sphere and ujtc = 150 on the Fermi-surface cylinder (containing ~ 4x 10-4 electrons per atom). The axis of the cylinder is, here, parallel to H, and the heterodyne gaps (intersecting the sphere) are ignored. A 4 : 1 ratio of left to right circular polarization is assumed. R eproduced will, perm ission o fth e copyright owher. Further reproduction prohibited without perm ission. 30 1.04 © CC I CC 0.96 1 0 1 (0 C / G) Figure 2.5 Theoretical R (ff) for potassium based on the heterodyne-gap parameters of Fig. 2.3 and the Fermi-surface cylinder model of Fig. 2.4, except that the cylinder’s axis is tilted 45° from [110]. (The tilt is required to minimize the elastic stress of the periodic lattice distortion needed to neutralize the electronic CDW.) This calculated behavior should be compared with C. C. Grimes’ data in Fig. 2.1. Reproduced with permission o fth e copyright owner. Further reproduction prohibited without permission 31 1.04 © £ x £ 0.96 1 0 1 (Dc / 0) Figure 2.6 Theoretical R(H) for potassium based on the Fermi surface model of Fig. 3.4. The Q-domain size assumed for the specimen measured by C. C. Grimes is D=1.5/zm. The other parameters used are enumerated in Sec. 2.5. Reproduced with permission o fth e copyright owner. Further reproduction prohibited without permission. 32 Table 2.1 Calculated values, from Ref. 8, of the first five minigaps and heterodyne gaps for K. The main CDW gap was taken to be 0.62 eV and the zone-boundary energy gap was 0.40 eV. n Minigap Heterodyne gap 1 90 meV 16 meV 2 67 14 3 51 12 4 34 8 5 15 3 Reproduced w ill, permission o fth e copyright owner. Further reproduction prohibited without permission. 3. MICROWAVE TRANSMISSION IN A PERPENDICULAR MAGNETIC FIELD. The cyclotron resonance is measured using a microwave transmission technique. The metallic sample forms a common wall between the transmit and receive mi crowave cavities. The microwave cavities used are rectangular and are excited in the TE101 mode. As a consequence the driving fields at the surface of the sample are linearly polarized, and the receiving cavity accepts only the linearly polarized field having the same orientation as the driving field. As shown in Fig. 3.1 the potassium sample is compressed between two quartz windows. To improve the surface quality of the samples, the compression was carried out in vacuum. The small area of the quartz window relative to the sample area provides microwave isolation between the send and receive cavities. A dc magnetic field is oriented perpendicular to the plane of the sample. The transmitted signal is detected and displayed by conventional techniques. Dunifer, Sambles, and Mace reported transmission data on 15 potassium sam ples.1 Figure 3.2 shows the transmission signal in sample K-4 as the magnetic field is swept from 0 to 5.5kG at T = 1.3K. The field is expressed in terms of the dimensionless quantity uc/u> where o/c = eH/m*c is the cyclotron frequency. The effective mass m* is 1.21m.2 The microwave frequency is 79.18GHz. Therefore the field at u)c/u> = 1 is 3.42T. The conduction-electron spin resonance (CESR) and associated spin wave modes lie in the region 0.8 < ujc/ oj < 0.9. For display pur- with permission o fth e copyright owner. Further reproduction prohibited without permission. 34 poses, the gain in this region has been reduced by a factor of five on account of the strength of the spin resonance. Close to uc/ uj = 1 there is a strong transmission signal and is, of course, the cyclotron resonance (CR). The general oscillatory sig nal seen over most of the field sweep is the Gantmakher-Kaner (GK) oscillations. There are high-frequency (HF) oscillations near u/c/u/ = 1.2 and 0.6. Finally, super imposed on the GK oscillations can be seen subharmonics of the CR at u c = u/2 and w/3. Therefore there are five signals which appear in the microwave transmis sion of potassium, (a) CESR, (b) CR, (c) GK oscillations, (d) HF oscillations, and (e) CR subharmonics. The study of the CESR and the spin wave side bands is well understood.3 This is the only signal that can be explained using a free-electron model. Even for this signal, there are splittings of some spin-wave side bands which can’t occur if the Fermi surface is spherical. The splittings were explained by in volving CDW domains.4 Among the four non-spin related signals only two have been successfully explained.They are the GK oscillations and the HF oscillations. In the following sections we briefly review these two signals.In Chap. IV, we will study the (until now) unexplained signals, CR and CR subharmonics. 3.1 Gantmakher-Kaner oscillations The oscillatory signal seen throughout the field sweep are the GantmakherKaner (GK) oscillations.5 We can clearly see them near o/c/o/ = 1.3 in Fig. 3.2. When a magnetic field is applied perpendicular to the surface, the electrons with the fastest velocity parallel to the field cause the GK oscillations. Suppose that an electron travels for a time equal to an integer number of cyclotron periods : _ 27tm*c t = n T = ti— — . eH ,. (3.1) We suppose that during this time the electron travels from one surface of the slab R eproduced with perm ission o fth e copyright owner. Further reproduction prohibited without permission 35 to the other. This condition requires, 2ttm*c vFt = n— tz—vf = L, eJi , . (3.2) n = n eL- - H. 2ftcm*vp (3.3) whereupon The oscillations are therefore periodic in the magnetic field. If we rewrite this equation in terms of a/c and a/, uc/u> = —— n. Loj (3.4) For An = 1, A(wc/w) = (3.5) The periodicity of the GK oscillations is determined by the Fermi velocity of the conduction electrons and the sample thickness only. Therefore, with L = vp = 7.36 x 107cm/sec, and u = 79GHz, we obtain A(a/c/u;) for each GK oscilla tion to be ~ 0.1. This agrees with the data in Fig. 3.2. Note that the amplitude of the GK oscillations increases as the field increases in Fig. 3.2. In Chapter IV, we will explain this behavior using a magnetic breakdown effect at the minigaps. 3.2 High-frequency oscillations There are two sets of high-frequency oscillations in Fig. 3.2, one near uc/u = 0.6 and the other near u/c/u/ = 1.2. We show enlarged views of those regions in Fig. 3.3. Lacueva and Overhauser studied the data in Fig. 3.3(a) to discern the origin of the oscillations.6 They found that the periodicity of the oscillations is linear in 1/H. This is the signature of Landau-level oscillations. Landau-level quantization causes a periodic variation of any physical property that depends R eproduced wim perrrission C h e copyright owher. Further reproduction prohibited without pem rission 36 on conduction-electron response.7 A property A will then acquire an oscillatory component, n = integer, A ~ cos(2irn), (3.6) where » - (*£>/<*■>• chfcj? ~2eH' F H' _ “ (3.7) F is the de Haas-van Alphen frequency, = F 27re (3-8) A is the extremal area in k space (perpendicular to H ) of the Fermi surface in volved. For a free-electron Fermi sphere A = nkp = 1.74 x 1016cm-2 for potassium with a lattice constant, a = 5.2295A. Then the de Haas-van Alphen frequency for a Fermi sphere is F0 = 1.828 x 104T. (3.9) The de Haas-van Alphen frequency from the data is Fi = 266T. (3.10) The ratio between F0 and Fi is, Cl = 69. (3.11) Because the frequency is proportional to the extremal area as in Eq. (3.8), we cam estimated the cross-sectional area which causes the signal. £ = | = 69. Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. (3.12) 37 Therefore the equivalent radius of the new Fermi surface is, ki kp/%, (3.13) The only possible Fermi surface (in addition to a sphere) is a cylinder created by the CDW as shown in Fig. 3.4. As one can see in Fig. 3.3(b), there are also high-frequency oscillations near ujJ oj = 0.6. From Eq. (3.7), one can find the periodicity of these oscillations versus field. |A"I = P -14) The interval between oscillations is then, * * « IS - (3i5) If the oscillations near ojc/ u> = 0.6 were caused by the same cylinder as for uc/u = 1.2, the spacing between them would be reduced by a factor of four. However, the periods of both oscillations are about the same, so the Fermi surface area causing the oscillations in Fig. 3.3(b) is four times smaller than the Fermi surface area causing the oscillations in Fig. 3.3(a). Accordingly, These Fermi-surface cylinders are the two smallest ones shown in Fig. 3.4. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission 38 From Klystron (79 GHz) To Detector (0 - 5.5 T) Potassium slab (L = 85 urn) Figure 3.1 Rectangular microwave cavities used for the observations. Small fused quartz windows are placed on both sides of the sample. Reproduced with p e n s i o n ot the copyright owner. Further r e p r o d u c e prohibhed without permission 39 Cyclotron Resonance CESR Landau-Level Oscillations from FS cylinder \ / Cyclotron Resonance Subharmonics 0 Gantmakher-Kaner Oscillations 1 0 .5 1.5 G)c / CO Figure 3.2 Microwave transmission signal vs. H through a potassium slab in a perpendicular magnetic field. (H = 3.42T at cjJoj = 1) The microwave frequency is 79.18 GHz and the temperature is 1.3K. The field at u)c/u) = 1 is 3.42T. The phase of the microwave reference was adjusted so that the cyclotron resonance is symmetric. The slab thickness is L = 85pm. The data, provided by G. L. Dunifer, were obtained from sample K4, one of fifteen samples listed in Ref. 1. Reproduced with p e n s i o n o , the copyright owner. Further reproduction prohibhed without p e n s i o n 40 W c 3 ■ JD a. <0 -I < z o CO 1.1 1.15 1.2 (Dc t 1.25 1.3 0.65 0.7 CO w c 3 n>_ <0 -i < z C5 CO 0.5 0.55 0.6 COc /( D Figure 3.3 Landau-level oscillations near a/c/cu = 1.2 and 0.6. The periodicity in (a) requires a cylinder radius, kc ~ kp/8, corresponding to the middle cylinder of Fig. 3.4. The periodicity in (b) requires a cylinder radius, ke ~ k p /16, appropriate to the smallest cylinder in Fig. 3.4. Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 41 Figure 3.4 Fermi surface cylinders of potassium. The horizontal axis is parallel to [110] and to the dc magnetic field. The shaded cylinders are created by the CDW gap and the first three minigaps. The thicknesses of the cylinders have been exaggerated by a factor of ten. Each of the half-cylinders shown is joined to a partner on the opposite side of the Fermi surface by Bragg reflection at the energy-gap planes. The complete Brillouin zone is shown in the inset. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 4. THEORY OF THE CYCLOTRON RESONANCE TRANSMISSION 4.1 Introduction During the last thirty years a variety of magnetoconductivity anomalies have been discovered in potassium, the simplest monovalent metal.1, 2 Unlike Li and Na, which undergo a crystallographic transformation to the 9R structure3 when cooled to low temperature, a single crystal of K is not destroyed by cooling. Without a charge-density wave (CDW) broken symmetry, the bcc lattice of K would support a spherical conduction-electron Fermi surface. Low-temperature transport anomalies could not then arise. Nevertheless, in dc experiments extraordinary phenomena occur which require the Fermi surface to be multiply connected. Examples are the four-peaked induced-torque patterns of single-crystal spheres,4i 5 the many open-orbit resonances,6, 7 and the magnetoserpentine effect8. It is not surprising, therefore, that unexpected phenomena also appear in the microwave properties of K. Fig. 3.2 shows the microwave transmission signal (at a frequency, u;/27r = 79.18GHz) through a K slab in a perpendicular magnetic field. Dunifer et al.9 studied this phenomenon in fifteen samples at T = 1.3K. The data shown (from sample K-4) were kindly selected by G. L. Dunifer, since it revealed clearly all five transmission signals. The horizontal axis of Fig. 3.2 is ujc/ u , which is proportional to the external magnetic field, H, since ojc = eH/m*c. The cyclotron mass is m* = 1.21m;10 so the field for cyclotron resonance, ujc/ oj = 1, is Hc = 3.42T. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission 43 The signal at and near the conduction-electron spin resonance (CESR) has been explored extensively.11 It is the only feature in Fig. 3.2 that can be explained by a free-electron model. Gantmakher-Kaner (GK) oscillations should also appear, but their amplitude should be a hundred times larger.12 The cyclotron resonance (CR), its subharmonics, and the rapid oscillations shown near u c/ui = 1.2 should not even exist (without a CDW). The purpose of this study is to show that the CDW broken symmetry of potas sium explains CR transmission and CR subharmonics. One must, of course, solve self-consistently a Boltzmann transport equation (for conduction electrons) to gether with Maxwell’s equations. Of crucial importance is the influence of CDW energy gaps on the Fermi-surface topology. A schematic illustration of the com plexity introduced by the CDW is shown in Fig. 3.4. It is known from observed optical anisotropy13 that K has a single CDW. The CDW wave vector Q is tilted about a degree from a [110] direction. From detection of neutron diffraction satellites,14,15 it was found that, -• 27T Q = (0.995,0.975,0.015)— . a (4.1) The magnitude of Q is 1.5% smaller than that of the smallest reciprocal lattice vector, Guo. The phonon mode which screens the electronic CDW has wave vector, Q ' = G u o - <?• (4.2) Minimization of the elastic energy required to neutralize the electronic CDW leads to a tilt of Q' about 45° away from [110].16 The “heterodyne gaps”, shown by the dashed lines of the inset in Fig. 3.4, are created by the periodicities, nQ\ n = 1,2,... They are important in explaining the open-orbit spectra7 and in un derstanding the shape of the CR signal observed in perpendicular-field microwave Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission 44 surface resistance.17 The heterodyne gaps do not significantly affect the microwave transmission, so we shall ignore them in what follows. The horizontal axis of Fig. 3.4 is parallel to [110], parallel to the magnetic field, and perpendicular to the surface of the K slab. Although such samples are polycrystalline, they have recrystallized in contact with the smooth, amorphousquartz plates used to hold the K slab in the window between the transmit and receive cavities.9 It is known from low-energy electron diffraction that thin alkalimetal samples, so deposited (or recrystallized), are epitaxially oriented with closepacked (110) planes parallel to the surface. Furthermore, such surfaces are smooth enough for conduction electrons to be specularly reflected.18 Interfacial energy will be optimized when Q is also perpendicular to the surface and, therefore, nearly parallel to the [110] surface normal. The important energy gaps for the present study are the main CDW gap and the sequence of “minigaps”, shown in Fig. 3.4. The wave vectors that describe these small gaps are: K = (n + l)Q —tzGho- (4-3) The sizes of these gaps have been estimated theoretically,19 and are listed in Table 2.1. The minigaps create several cylindrical sheets of Fermi surface. Each of the three shaded surfaces shown in Fig. 3.4, formed by the CDW gap and the first three minigaps are joined (by Bragg reflection) to equivalent surfaces on the opposite side of the Brillouin zone. Accordingly, each cylinder has twice the length that appears in the figure. The cross section of each cylinder in a plane perpendicular to [110], i.e., not perpendicular to the cylinder’s axis, is circular. The rapid oscillations near uic/u = 1.2, shown in Fig. 3.2, have been found to be periodic in l/H to very high precision.20 The periodicity corresponds to a Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. cross-sectional area 69 times smaller than irkp, the extremal area of the ideal Fermi sphere. Consequently this Landau-level oscillation pattern arises from a cylinder with a radius (in a plane perpendicular to [110]) equal to kp/8. We believe this cylinder is the one formed by the first and second minigaps. The reason for this assignment is that the analogous, rapid oscillations near v c/u = 0.6 corresponds to a cylinder with a radius, ~ kpj 16. This cylinder is likely the one formed by the CDW gap and the first minigap. The third cylindrical surface, shown in Fig. 3.4, is the one to which we will attribute the CR transmission and the CR subharmonics. Its radius is estimated in what follows to be ~ Zkp/8. The influence of the minigaps has also been noticed in a study of the GK oscillations.12 If a free-electron Fermi-surface sphere is employed, the transmitted power ratio will be 10-18 instead of the value, 10-22, observed in sample K-4. Since the microwave transmission signal is carried primarily by electrons having a rapid velocity, u2, parallel to [110], they are the ones most affected as they encounter the minigaps while undergoing cyclotron rotation in the xy plane. The n’th GK oscillation occurs when the time, L/vp, to traverse the slab (of thickness L) equals n cyclotron periods. Accordingly, the GK oscillations are periodic in H, as shown by the transmitted signal amplitude in Fig. 4.1. This signal was calculated from a free electron model (without spin). Not only are the features (described above) of the observed signal absent, but the gradual growth from small to large uc of the GK oscillations (a factor of five in Fig. 3.2) does not appear. The effect of the minigaps on the cyclotron motion of electrons with large |uz| can be modeled by a v2-dependent scattering time.12 r(vz) = -----—r, 1 + y\v»/vF\ (4.4) where r0 is the scattering time attributable to impurities, and y is an adjustable permission o f the copyright owner. Further reproduction prohibited without permission. 46 parameter intended to account for the interruption of cyclotron motion by mini gaps. Magnetic breakdown of electron trajectories at the small minigaps implies that y will be a function of H. The simplest way to model this effect is to let, y(H) = y0[l - exp(-H 0/H)], (4.5) where y0 is a constant and Ho is a magnetic-breakdown field. Incorporation of (4.4) and (4.5) in the Boltzmann equation allows one to reduce the GK signal by the required two orders of magnitude and to fit its observed field dependence. This consequence of the minigaps is treated in Sec. 4.3. The major challenge of this study, however, is to include the influence of a tilted Fermi-surface cylinder in the Boltzmann transport theory, solved self-consistently with Maxwell’s equations. Complexity arises from the lack of axial symmetry about the magnetic field direction. The general theory is developed in Sec. 4.2 and is applied to the microwave transmission of K in Sec. 4.4. 4.2 Microwave transmission in an anisotropic, non-local medium Microwave propagation (which we take to be along z, perpendicular to the metal surface) is governed of course by Maxwell’s equations : *7 ld S c V x £ - _ _ T-i 4x-? 1 d£ V xH = —j + . c c at (4.6) These six equations can be reduced to three by taking the fields proportional to e-iwt and eliminating H : dP6a{z) 4niuj - gzT ~ + . 47riu; d2 . u)2 „ - g - h + ~^;£z - a = *,y> „ 0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ., (4.7) 47 The current density j and the electric field £ depend only on z. However, since the conduction-electron mean free path, I = vpr, in potassium near T=0 is typically ~ 10_2cm, whereas the microwave skin depth is approximately 2 x 10~5cm, the relation between j and £ is nonlocal: M Z) = J 2 [ Klm(z,z',u)£m(z')dz'. m=lJo (4.8) Kim{z,z’1u) is the (non-local) conductivity tensor, and L is the thickness of the metal slab. Specular reflection of conduction electrons at smooth K surfaces18 allows a simplification of Eq. (4.8) if the Fermi surface has axial symmetry about the surface normal (and the dc field H). The integration limits can be extended to infinity provided the microwave field is described by a Fourier cosine series (so that £ and j are symmetric about z=0, L).21 Accordingly, jiiz) = Y l [ Ktm{z - z\u})£m{z')dz'. (4.9) m = l J ~ °° The Fourier expansions for £ and j are : £*{z) = £ £ £ c o s ( g nz), n=0 Jaiz) = 53 n=0 C0S(?n2), (4-10) n = integer. (4-11) where T17T qn = -p ; . The fundamentalreason Eq. (4.9) is allowed arises from the fact that on specular reflection vz changes sign but not its magnitude. The reflection symmetry of £a(z), Eq. (4.10), implies that the past history of an electron approaching 2 = 0from the right is the sameas that imputed for an electron approaching from theleft along a path which joins that of the original reflection. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 48 In general Eq. (4.9) would not be valid for a Fermi surface lacking axial sym metry, e.g., an ellipsoid having its axis tilted from the surface normal. However, for the special case of an anisotropic conductivity caused by a tilted, cylindrical Fermi surface, Eq. (4.9) is still valid because vz retains its magnitude upon reflection. The formalism developed here anticipates that our application in Sec. 4.4 will be for a Fermi surface having both a spherical and a (tilted) cylindrical piece, created by the CDW broken symmetry. Nevertheless, there is a hidden approximation which is discussed at the end of Sec. 4.3. The Fourier coefficients in Eq. (4.10) are obtained by multiplying each equation by cos qmz and integrating over the interval (0, L). E% = - - -/ ° m [ L £a(z)cos(qmz)dz, JO ^°™ JQ Ja(z) cos(qmz)dz. (4.12) Furthermore, from Eqs. (4.9) and (4.10), the Fourier components of the conduc tivity tensor, crjm(<7„,o/), can be related to the coefficients in (4.12). J? = ffim(9n,w)££, (4.13) All nine components of <Jim will be non-zero when, in Sec. 4.4, we include the effects of a tilted cylinder. The electric-field components E£ are obtained by taking the Fourier cosine transform of Eq. (4.7), («; - + = ! ^ S [ £ J ( £ ) ( - 1) * - S ( 0)], (4.14) = ^ K ( £ ) ( - 1 )* -5 (0 )], (4.15) = 0. (4.16) Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission 49 Since S'a(L) is 10 orders of magnitude smaller than £4(0) in a slab 10/xm thick, we can neglect S’JJL). We next use Eq. (4.13) to express in terms of Bp. The longitudinal field can then be related to the transverse field from Eqs. (4.16) and (4.13), + CTzVB^ (4.17) 1+ With the help of these substitutions Eqs. (4.14) and (4.15) may be written in a -?~&xX 4-ru.j _ / —3~ °»x „2 9 1 xy cl u>2 'E £ to 1 ixiuJ_/ 4jrtri; _ / 1 fci2 4»i«j _./ w -l A i r -2 9 r coupled format: L (4.18) . ^ ( 0). where &CLZ&Z& a a0 ~ ~ (4.19) The solution for the x component of the electric field is, for each Fourier component, E* = <?(2 - <Son) L (c2^ -up- - £TTuoolyy)£,x{Q) 4- 47Tia;g4y£ '( 0 ) _________ {<?q2 - u P - 4Trioja'xx)((?q2 - up- - 4x*w<r^) - (4iriu)2cr'iy<7'yx ’ (4.20) At this point we must specify the boundary conditions at the front surface of the sample, so that ££(0) and £'(0) in Eq. (4.20) can be determined. Rectangular microwave cavities, excited in a T E m mode, were used.9 Consequently, from Eq. (4.6), uo (4.21) The magnetic field Hx{0) is zero, so that £1(0) = ~ f l , ( o ) = o. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission (4.22) 50 Accordingly the solution, given by (4.20) is completely specified. We need only to evaluate SX{L) to find the transmitted signal. From an analysis given by Cochran22 and by Lacueva and Overhauser12, the transmitted electric field just outside the rear surface is twice the incident field at L in the infinite medium. From Eqs. (4.10) and (4.20), the electric field at z = L is therefore, _ .... 2cu/t__ £X(L) — — — Hy(0) ^QnXc2^ - u 2- foiujcr'yy) cos(nTr)____________ X b o (c2^ - W2 - 4xiua'xx) (c2?2 - w 2- 47riuja^) - (±iriujya'xya'yx' (4.23) y ____________(2 - We now replace {gn} with their values from Eq. (4.11), so that the integers {n} appear. C7T £ 5 1"2 - ( £ ) 2 - (“ g V J l " * - (“§)! (4.24) Since the sum in Eq. (4.24) converges very slowly, one needs to include terms to n = 107 to obtain reliable values for SX{L). Now, for large n, the a[m terms are much smaller than n2, so we can take advantage of this disparity as follows.21 2w£* rr /rt\ TO ^ /yjr2 *' ' ‘ n=A f+1 ("I)" «2 U f __________(2 - & .) ( - l) V - (ffi2 - d * P K .)__________ I"2- O 2- (^KJ[»a- (#)2- (*gV„] - (^ir)2^ (4.25) The break point, M, is large. The final sum on the right hand side can be reex Reproduced with perm ission o f the copyright owner. Further reproduction prohibited without permission. 51 pressed by using the exact value, £ (-1 )” _ V ( - 1)” , V (_1)“ ^ ~ - £ “ + J +1~ — s- M-fil (4-26) Therefore the transmitted field is ^ ) = M 1 f v(ok£ + 2 E ^ _ f __________ (2 - f t.) ( - l) V h [n2 - - (at)* - ( » g V j ___________ c-^-yjin2 ~(t )2 ~( tS tK J - (*£)2 - ' (4.27) On account of the large reflectivity, the field at the front surface, Hy{0), is twice the incident field H*.12 In the vacuum, just outside of the metal, £* — Hy. Therefore the signal, defined to be the ratio of the transmitted electric field (just beyond z = L) to the incident field at z=0, is „ 5,(5) 5,(1) 25,(L) s ~ - ~ H f ~ -W M ' (4'28) Prom Eqs. (4.27) and (4.28), the signal is, C7T2 _ f 6 “ n2 __________ (2 - f t .) ( - i ) V - ( g )2 - ( ^ V j ___________ , „.o [n2 - ( f )2 - ( “ ^ f e K - ( ^ ) 2 - ( * $ V J - ( ^ ) 2< t ^ (4.29) We have taken the upper limit of n to be M=50 000, since doubling M doesn’t change S by more than one part in 105. Eq. (4.29) can be evaluated once the conductivity tensor cr/m(gn,a»), defined by Eqs. (4.13) and (4.19), is specified. In the following section we shall determine a[m for the main (spherical) part of potassium’s Fermi surface, including the influence of the minigaps on the scattering time r(vz), discussed in the introduction. In Reproduced with permission o f ,ire copyright owner. Further reproduction prohibited without permission Sec. 4.4 the contribution of the tilted cylindrical part of the Fermi surface will be incorporated. 4.3 Effect of minigaps on microwave transmission The conductivity tensor derived in this section will be that for a spherical Fermi surface. However, the effect of the minigaps will be modeled by a -dependent relaxation time described by Eqs. (4.4) and (4.5). The Boltzmann transport equation for the electron distribution function f(k, r, t) is, (4.30) where fo(k) is the equilibrium distribution. For linear response one takes, /(£, r, £) = /o(fc) + fi{k, f, £), (4.31) with the understanding that fi is first order in the microwave field. Accordingly for each Fourier component, £(r, t) = E(q,uj) exp{iq- f - iut), f i ( k ,f ,t ) = fq(k) exp(iq- r —iujt). (4.32) The linearized transport equation is then (4.33) It is convenient to change coordinates from k to e, kz, <f>,where <j> = tan l {kv/k x), (4.34) permission of the copyright owner. Further reproduction prohibited without permission. 53 <f> is the azimuthal angle of an electron in its cyclotron orbit. On introduction of these variables, the last term on the left-hand side of Eq. (4.33) reduces to uc(.dfq/d<i>), so [1 + i(q ■v - w)r]/f + = erE • v ^ - . (4.35) The velocity in Eq. (4.35) is = m (4 3 6 ) Accordingly, U h 2m*e vx = — kx = —7J — 2 - - k 2z cos4>, 771* 771* V h n h / 2m* . v, = A * ,. m* (4.37) Only propagation vectors, q = qz, are of interest. Since we shall assume here that vz is independent of <f>, Eq. (4.35) is easily solved: U = Uc OS rJ—oo « S ■" « p { t l -+-iY(gl,;-~ ‘j)1( ^ - « } . UJCT (4.38) The Fourier component of the current density is, naturally, A M = -J ffi f (4.39) and the relation between current density and the electric field is 3 J{ = £ crimEm. (4.40) m =l The tensor components of conductivity are then O’t'm — ^2 x I* # W ( ^ M e x p {(1 + <T^ J —oo c Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. MD 54 Here the volume element, dzk, in Eq. (4.39) has been transformed to (m* ffi2)dedkzd<p, appropriate for the cylindrical coordinates of Eq. (4.34). Three of the four integrations in (4.41) can be evaluated after use of Eqs. (4.37) and (4.4). All nine components of the conductivity tensor, <rim(g,u;), can be identified 1 3a° f dt( l~e l~* 1 8 7-i 1 + y|£| —ia+ + ixt 1 + y|£| —ia_ + ixt 3ia0 r' 1 - t 2_____________ l - t 2 8 7-i 1 + y|£| —ia+ + ixt 1 + y|£| —ia_ + ixt CTxy 2 7-1 Cyx — 1 —iwTQ v, + ixt ’ &xyi ®yy — &xxi CTxz = CTzi = (Tyx = <Tzy = 0 . ( 4 -4 2 ) where t a± = it kF' = (u/± (jc)tq, x = qvFTQ, a0 ne2r° . = —— m a(4.43) (The lower limit of the integration in d$, which is the first to be executed, con tributes nil on account of the exponential factor.) Notice that azz depends only on To, the impurity-scattering relaxation time, since we have allowed the minigaps to interrupt only the cyclotron motion, as given by Eq. (4.4). The major influence of the minigaps on azz and the nil components of a in (4.42) will be treated in Sec. IV, where the existence and tilt of a Fermi surface cylinder will be incorporated. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 It is fortunate that the remaining integrals in (4.42) can be evaluated analyt ically since the transmitted signal, Eq. (4.29), requires a further sum over the allowed values of q. The integrations are rather tedious; the final expressions are: = x l ^ ? ^ - 7 T ? +/+p+" J+r+ + /;i/+ " s'+r; + / - p- - g - r - + fjp L - 9-7- + i[f+T+ + 9+P+ + /+*+ + 9+P+ + f - T- = ^ { 2(^ (v g; % — + ' - r - + *-p- + + u - - -9+P+ - f+r+ ~ 9+P+ + *[f+P+ - 9 +r+ + /+p'+ - g+r'+ - /_p_ +9 - r - - fLpL + plr'_]}, (4.44) where a:3) 4- 2(a± + yx){3y2x —x3)]/(x2 4- p2)3, f ± = [(1 —a | —p2 + x2)(3y2x — g± = [2(a± + px)(p3 - 3yx2) + (1 - a | - p2 + x2)(3p2x - x3)]/(x2 + y2)3, 1 .(a± —x)2 + (p + l)2, p± = 2 -------(T T 4 )------- ]’ r± = tan"1! -Q^ ± ------- ]• p + 1 + a± —xa± (4.45) Furthermore, f ±(x) = /± (-x ), g'±(x) = g±(-x), p'±(x) = p±{-x), and r'±(x) = r±(—x), and 3cr0 azz 2p)], zz = ~ ^ [ 2 x - r + 2wT0p + u 2TqT + i(p - 2u;r0x + 2u/r0r - o;2r0 2x? (4.46) where n - 1 inr1 + (J - u;ro)2l P ~ 2 1 + (x + a;ro)2 r = tan-1(x —wr0) + tan-1(x + cdr0). (4-47) The results must now be incorporated in Eq. (4.29) to obtain S. The trans mitted signal shown in Fig. 4.1 is obtained by setting p = 0, in Eqs. (4.4), (4.44), Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 and (4.45), i.e., by neglecting all interruptions of the cyclotron motion by the minigaps. The impurity scattering rate corresponds to ujt0 = 150. The observed GK oscillations in Fig. 3.2 are smaller in amplitude by two to three orders of mag nitude.12 What is perhaps more puzzling is the growth of the GK amplitude by about a factor of five between uc/w = 0.7 and 1.5. We have adjusted y0 = 21 and Ho = 5.8T in Eq. (4.5) to model the effect of magnetic breakdown of the minigaps. The revised transmission signal is shown in Fig. 4.2. The gradual growth of S with increasing uc/v , as shown in the figure, is typical for samples having a thickness L = 85(j.m (or larger). At very high fields the GK oscillations may decrease,9 as might be anticipated if there are slight variations in L across the sample surface. We have explored the consequences of adding to the conductivity a contribution from a small Fermi-surface cylinder having its axis parallel to the applied field H. The calculated signal was indistinguishable from the curve shown in Fig. 4.2. This result justifies an approximation which we have not yet discussed. When an electron in the cylinder is reflected from the surface, vx changes sign as well as vz (but vy does not). Consequently the xy cyclotron motion is interrupted, and Eq. (4.9) does not strictly apply. However, since the current in the xy plane arising from the cylinder is too small to affect the transmission signal, as we have just mentioned, the error which has been tolerated is minimal. The CR transmission signal is caused by the z component of the cylinder’s current, as shown in the following section. Since the sign reversal of vz on reflection is treated correctly by Eq. (4.9), the non-local theory for the cylinder’s conductivity, given below, should be adequate for microwave transmission (but not for a theory of the surface impedance). Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 57 4.4 Conductivity tensor from a tilted Fermi-surface cylinder A striking consequence of the minigaps, which correspond to the periodicities of Eq. (4.3), is the creation of small Fermi-surface cylinders, shown in Fig. 3.4. Giuliani and Overhauser showed that the CDW wave vector, Q, is rotated from the [110] direction by a small angle, ~ 1°. The cause of this rotation is the need to minimize the elastic stress energy of the positive-ion lattice distortion which arises to screen the charge of the CDW. The wave vector, Q' = Guo ~ Q> °f the phonon mode involved is ~ 45° from the [110]. The cylindrical Fermi surfaces created by the minigaps have axes parallel to O'.19 We will find below that a quantitative fit for the cyclotron resonance signals in Fig. 3.2 requires the tilt angle of Q' to be 9 ~ 50°. It is necessary to obtain the (non-local) conductivity of a tilted cylinder by solving once again the Boltzmann transport equation (4.30). The cross section of a tilted cylinder in a plane perpendicular to [110] is circular.19 Accordingly the conduction electron energy spectrum for the cylinder is, £ = 2m* ~ kz tai10)2 + *9' (4-48) It is again appropriate to change notation to cylindrical coordinates : e, fcz, and (f>. (h = tan-1( —------ ). , [2mfe i . »x = y —^5“ cos^ + «2ta n 0, K = y "T7” Sin<£. (4.49) The velocity components required for the transport equation are : h ,, = ^ h I2m*e x~ z ta n i f cos 01 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. v m*V *v h m* t/* = — (—tan0)(kx —kz tan0) = — (—tanfl)</- ^ - cos<ft. m* to* Since vz is here a function of V * (4.50) solution of the differential equation (4.35) for this case is, instead of (4.38), fs = — ^ Wc f d<f>'E • v exp[ia(<£' —<f>)+ i/5(sin <£' - sin <f>)], J—o o C7£ (4.51) where, a = .1 - iwTo VcTq —t -------------- , (3 = (—qh tan 6/ m*u;c) ^ ^ 2 ~• (4-52) The relations, (4.39) and (4.40), between current density and electric field can be used to identify the tensor components of the cylinder’s conductivity. ^ = “4 ? / x/ r<t> J~ OO r d(f>'vm(e, fc2,0;) exp[ia(<£' —<p) + i/5(sin<£' —sin 0)]- (4.53) The integration in dfi is enabled by expanding exp (i/3 sin 4>) in a Bessel function series.23 etfsi»*= £ Jm(/3)eif^ . m=— oo (4.54) We provide details only for crzx. Ott. = 2e2 m* — / (2tr)3fi2,a;, j / * ,/« , de J where, /« f 2Td<f>cos0e-,a^ = m f) Jn(-0)eT+ n= —00 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.55) 59 x [* i # c x f e * * - i y j Z 00 m= —00 W )J : 3) <'(m + a)T JM e™ (4.56) Using the relations,23 ■W-0) = (-1 )m^n(/3), J_m(/3) = ( - l ) roJm(^), andletting 6 bethe length of the cylinder along kz (i.e.,twice the length in Fig. 3.4), (4.57) shown and withkc the radius of the circular cross section (in the plane perpendicular to z), we may evaluate Eq. (4.55). where, /3p = (—qh tan. 6/m*u}c)kc. (4.59) Let 77 be the fraction of the conduction electrons contained in the cylinder, i.e., ’ - f t - (4-60) Eqs. (4.52), (4.58), and (4.60) then determine the component, <rXI, of the cylinder’s conductivity tensor : _ axx ne2r0 ^ V rn* 2 , 1 - i{u - m uc)T0 "* F (cjcT0)2 + [ l - t ’(a;-m a;c)ro]2' In a similar fashion the eight remaining components can be found : _ CXV ne2r0 ^ 2 . _________ - uctq________ V m’ mioo m F (wcr0)2 + [1 - i(w - mcjc)r0]5 &XZ = Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. (4.61) 60 CXyx — &xv> °yy = &xxi O’yz — (Tjytill 0 , axx = —axxtan.0, o’zy = o'Xy tan 0, (TZz — &xx tan20. (4.62) The transmission signal, S, including the effect of the tilted cylinder, can be calculated from Eq. (4.29) by adding the conductivity tensor for the main Fermisurface, Eqs. (4.42)-(4.47), to the tensor just derived for the tilted cylinder, Eqs. (4.61) and (4.62). The microwave detector is sensitive to the phase of a reference signal from the klystron; and this phase can be adjusted (as was done for the data in Fig. 3.2) so that the main cyclotron resonance appears symmetric.9 Accordingly, the calculated electric field, ET, of the transmitted signal will depend on a phase X : E t = Re(S) cos x + Im(S) sin x- (4.63) The calculated cyclotron resonance signal, shown in Fig. 4.3, is symmetric with X = 280°. The experimental9 value for r0 corresponds to wr0 = 150. A crucial feature of Eqs. (4.61) and (4.62) is the resonance denominator which becomes small whenever ujc = u}/{m +1), m = 0,1,2... This feature is responsible for the occurrence of the cyclotron-resonance subharmonics (as well as the main resonance for m=0), and is similar to the Azbel-Kaner oscillations24 which occur in parallel-field surface resistance studies.25 The parameters associated with the cylinder’s geometry are 6, kc, and 6. The cylinder’s length is determined by the CDW diffraction satellites,14, 15 and Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission 61 is 0.015 [Guo | - The ratio of the main cyclotron resonance to the first subharmonic (at ucf<j) = 0.5) is sensitive to the tilt angle 9. The observed ratio, ~ 10, in Fig. 3.2 is reproduced by the calculated ratio in Fig. 4.3 with 6 ~ 50°. The absolute size of the main resonance is sensitive to fcc, which we find to be ~ Zkp/8. This value, together with 6, indicates that the conduction-electron fraction of the third and largest cylinder shown in Fig. 3.4 is tj ~ 0.004. (The absolute size of the main CR is based on the amplitude of the GK oscillations near uc/ui = 1.5. The GK amplitude was determined in Ref. 12 from the original data and instrumental calibrations kindly provided by G. L. Dunifer.) Only the largest cylinder, shown in Fig. 3.4, was included in the foregoing calculation. We have found that the influence of the smaller cylinders on the height of the main CR is ~ 7%. How ever, their presence is necessary to account for the Landau-level oscillations near £jc/u;=0.6 and 1.2. 4.5 Conclusions The complex Fermi surface of potassium, illustrated schematically in Fig. 3.4, is based on energy-band calculations19 that incorporate the periodic potential of an incommensurate CDW having the wave vector Q, Eq. (4.1). The presence of the three Fermi-surface cylinders, shown shaded (and with thickness exaggerated by a factor of ten), are manifested by the transmission data of Fig. 3.2. The rapid oscillations near o;c/o; = 0.6, arise from Landau-level quantization of the smallest cylinder (having a radius ~ k p /16). The Landau-level oscillations near ujc/ ui = 1.2 arise from the next smallest cylinder, which has a radius ~ kp/8.20 Fig. 3.3 is an enlarged view of the data in Fig. 3.2 showing the Landau-level oscillations near ujc/ ui=Q.Q and 1.2. The largest cylinder shown in Fig. 3.4 has a radius ~ 3kp/8, a value estimated from the main CR amplitude. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 The shape of the main CR varies from sample to sample, as shown in Fig. 3 of Ref. 9. Resonances from thick samples, L > lOOfim, often exhibit structure. This feature is to be expected as a consequence of CDW domains.7 Four possible CDW Qs are nearly parallel to a [110] direction. They can be obtained by interchanging the first two components of Eq. (4.1) and by reversing the sign of the third component. It is likely that all four domains occur across the sample area. If the magnetic field is exactly parallel to [110], the cylinders of all four domains will have the same cyclotron frequency. However, if H deviates from [110] by even a fraction of a degree, each cylinder will have its own u>c: UJrcos 9: ^ = i = 1'2'3’4' '464> where 0, is the angle between the cylinder’s axis, Q'0 given by Eq. (4.2), and the projection of H on the plane containing [110] and Q'{. (9 is the angle each axis would have if H were exactly parallel to [110].) The CR line can be merely broadened if the deviations of 9{ from 9 are very small. Splitting of spin-wave sidebands of the spin resonance signal has also been attributed to CDW domain structure.26 The major conclusion of this study is, of course, that transmission CR and the associated CR subharmonics arise in potassium from a cylindrical section of Fermi surface created by CDW minigaps. Without the CDW broken sy m m e try the transmission signal would be the one shown in Fig. 4.1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 L U-40 CM -80 0 1 0.5 1.5 coc / c o Figure 4.1 Theoretical microwave transmission signal versus loJ uj for potassium if the Fermi surface is spherical. Only Gantmakher-Kaner oscillations appear (since the electron-spin magnetic moment is neglected). The sample parameters are u/7o = 150 and L = 85/zm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 0.2 0.1 - 0.1 - 0.2 - 0.3 0 1 0.5 1.5 C0C / CO Figure 4.2 Theoretical transmission signal when interruption of cyclotron motion by the CDW minigaps is modeled by the vz dependent relaxation time, Eq. (4.4), and with magnetic breakdown of the minigaps described by Eq. (4.5). The parameters, yQ= 21 and Hq = 5.8T, were adjusted so that the GK amplitude at ojJ oj = 1.47 agrees with the (calibrated) data from Fig. 3.2 and with its observed growth by a factor of five from low to high fields. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 0.5 0 1 0.5 1.5 G)c / CO Figure 4.3 Theoretical transmission signal when the non-local conductivity of the largest Fermi surface cylinder, shown in Fig. 3.4, is added to the main Fermi surface conductivity (employed in Fig. 4.2). The tilt, 6 = 50°, of the cylinder’s axis was adjusted so that ratio of the main CR to the first subharmonic (at u/c/u; = 0.5) is ~ 10, consistent with Fig. 3.2. The cylinder radius, kc = 3kp/8, in the (110) plane was adjusted so that the ratio of the CR to the high-field GK amplitude agrees with that observed in Fig. 3.2. a;r0 = 150 and L = 85/zm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. BIBLIOGRAPHY Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 BIBLIOGRAPHY C hapter 1 1. D. Shoenberg and P. J. Stiles, Proc. Roy. Soc. A281, 62 (1964). 2. A. W. Overhauser, in Highlights of Condensed-Matter Theory, Proceedings of the International School of Physics “Enrico Fermi”, Course LXXXDC, Varenna on Lake Como, 1983, edited by F. Bassani, F. Fumi, and M. P. Tosi (North Holland, Amsterdam, 1985), p. 194. 3. A. W. Overhauser, Adv. Phys. 27, 343 (1978). 4. A. W. Overhauser, Phys. Rev. Lett. 53, 64 (1984). 5. G. A. Baraff, C. C. Grimes, and P. M. Platzman, Phys. Rev. Lett. 22, 590 (1969). 6. G. L. Dunifer J. F. Sambles and D. A. H. Mace, J. Phys. Condens. Matter 1, 875 (1989). 7. H. Mayer and M. H. El Naby, Z. Phys. 174, 269 (1963). 8. T. M. Giebultowicz, A. W. Overhauser, and S. A. Werner, Phys. Rev. Lett. 56, 2228 (1986). 9. S. A. Werner, A. W. Overhauser, and T. M. Giebultowicz, Phys. Rev. B41, 12536 (1990). 10. Yong Gyoo Hwang and A. W. Overhauser. Phys. Rev. B39, 3037 (1989). 11. Graciela Lacueva and A. W. Overhauser, Phys. Rev. B48, 16935 (1994). 12. G. Lacueva and A. W. Overhauser, Phys. Rev. B33, 3765 (1986). 13. Graciela Lacueva and A. W. Overhauser, Phys. Rev. B46, 1273 (1992). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 14. F. W. Holroyd and W. R. Datars, Can. J. Phys. 53, 2517 (1975). 15. P. G. Coulter and W. R. Datars. Can. J. Phys. 63, 159 (1985). 16. M. Hubennan and A. W. Overhauser. Phys. Rev. B25, 2211 (1982). C hapter 2 and Appendices 1. C. C. Grimes and A. F. Kip, Phys. Rev. 132, 1991 (1963). 2. M.Ya. Azbel and E. A. Kaner, Zh. Eksp. Teor. Fiz. 32, 896(1957) [Sov.Phys.-JETP 5, 730 (1957)]. 3. D. C. Mattis and G. Dresselhaus, Phys. Rev. I l l , 403 (1958). 4. R. G. Chambers, Philos. Mag. 1, 459 (1965). 5. G.A. Baraff, C. C. Grimes, and P. M. Platzman, Phys. Rev. Lett. 22, 590 (1969). 6. A. W. Overhauser, Phys. Rev. 167, 691 (1968). 7. A. W. Overhauser, Adv. Phys. 27, 343 (1978); in Highlights of CondensedMatter Theory, Proceedings of the International School of Physics “Enrico Fermi”, Course LXXXDC, Varenna on Lake Como, 1983, edited by F. Bassani, F. Fumi, and M. P. Tosi (North Holland, Amsterdam, 1985), p. 194. 8. Yong Gyoo Hwang and A. W. Overhauser, Phys. Rev. B39, 3037 (1989). 9. A. W. Overhauser and N. R. Butler, Phys. Rev. B14, 3371 (1976). 10. G. Lacueva and A. W. Overhauser, Phys. Rev. B33, 3765 (1986). 11. G. L. Dunifer, J. F. Sambles, and D. A. H. Mace, J. Phys. Condens. Matter 1, 875 (1989). 12. Graciela Lacueva and A. W. Overhauser, Phys. Rev. B46, 1273 (1992). 13. C. C. Grimes, private communication. 14. G. F. Giuliani and A. W. Overhauser, Phys. Rev. B20, 1328 (1979). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 15. T. M. Giebultowicz, A. W. Overhauser, and S. A. Werner, Phys. Rev. Lett. 56, 2228 (1986); Phys. Rev. B41, 12536 (1990). 16. E. I. Blount, Phys. Rev. 126, 1636 (1962). 17. M. Huberman and A. W. Overhauser, Phys. Rev. B25, 2211 (1982). 18. C. Kittel, Quantum Theory of Sohds, (John Wiley and Sons, New York 1963), p. 313. 19. P. A. Penz and T. Kushida, Phys. Rev. 176, 804 (1968). 20. G. A. Baraff, Phys. Rev. 187, 851 (1969), first paragraph. 21. P. G. Coulter and W. R. Datars, Can. J. Phys. 63, 159 (1985). 22. A. W. Overhauser, Phys. Rev. Lett. 53, 64 (1984). Chapter 3 1. G. L. Dunifer, J. R. Sambles, and D. A. H. Mace, J. Phys. Condens. Matter 1, 875 (1989). 2. C. C. Grimes and A. F. Kip, Phys. Rev. 132, 1991 (1963). 3. D. A. H. Mace, G. L. Dunifer, and J. R. Sambles, J. Phys. F; Met. Phys. 14, 2105 (1984). 4. Yong Gyoo Hwang and A. W. Overhauser, Phys. Rev. B38, 9011 (1988). 5. F. Gantmakher and E. A. Kaner, Soviet Phys. JETP 21, 1053 (1965). 6. Graciela Lacueva and A. W. Overhauser, Phys. Rev. B46, 1273 (1992). 7. D. Shoenberg, Magnetic Oscillations in Metals (Cambridge University Press, London, 1984), Chap. 4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 C hapter 4 1. A. W. Overhauser, in Highlights of Condensed-Matter Theory, Proceedings of the International School of Physics “Enrico Fermi”, Course LXXXIX, Varenna on Lake Como, 1983, edited by F. Bassani, F. Fumi, and M. P. Tosi (North Holland, Amsterdam, 1985), p. 194. 2. A. W. Overhauser, Adv. Phys. 27, 343 (1978). 3. A. W. Overhauser, Phys. Rev. Lett. 53, 64 (1984). 4. J. A. Schaefer and J. A, Marcus, Phys. Rev. Lett. 27, 935 (1971). 5. F. W. Holroyd and W. R. Datars, Can. J. Phys. 53, 2517 (1975). 6. P. G. Coulter and W. R. Datars, Can. J. Phys. 63, 159 (1985). 7. M. Huberman and A. W. Overhauser, Phys. Rev. B25, 2211 (1982). 8. A. W. Overhauser, Phys. Rev. Lett. 59, 1966 (1987). 9. G. L. Dunifer, J. R. Sambles, and D. A. H. Mace, J. Phys. Condens. Matter 1, 875 (1989). 10. C. C. Grimes and A. F. Kip, Phys. Rev. 132, 1991 (1963). 11. D. A. H. Mace, G. L. Dunifer, and J. R. Sambles, J. Phys. F: Met. Phys. 14, 2105 (1984). 12. Graciela Lacueva and A. W. Overhauser, Phys. Rev. B48, 16935 (1993). 13. A. W. Overhauser and N. R. Butler, Phys. Rev. B14, 3371 (1976). 14. T. M. Giebultowicz, A. W. Overhauser, and S. A. Werner, Phys. Rev. Lett. 56, 2228 (1986). 15. S. A. Werner, A. W. Overhauser, and T. M. Giebultowicz, Phys. Rev. B41, 12536 (1990). 16. G. F. Giuliani and A. W. Overhauser, Phys. Rev. B20, 1328 (1979). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 17. Mi-Ae Park and A. W. Overhauser, Phys. Rev. B54, (1996). 18. P. A. Penz and T. Kushida, Phys. Rev. 176, 804 (1968). 19. Yong Gyoo Hwang and A. W. Overhauser, Phys. Rev. B39, 3037 (1989). 20. Graciela Lacueva and A. W. Overhauser, Phys. Rev. B46, 1273 (1992). 21. B. Urquhart and J. F. Cochran, Can. J. Phys. 64, 796 (1986). 22. J. F. Cochran, Can. J. Phys. 48, 370 (1970). 23. N. W. McLachlan, Bessel Functions for Engineers, (Oxford University Press), London 1941), p. 158. 24. M. Ya. Azbel and E. A. Kaner, Zh. Eksp. Teor. Fiz. 32, 896 (1957) [Sov.Phys.-JETP 5, 730 (1957)]. 25. D. C. Mattis and G. Dresselhaus, Phys. Rev. I l l , 403 (1958). 26. Yong Gyoo Hwang and A. W. Overhauser, Phys. Rev. B38, 9011 (1988). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDICES Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 Appendix A: Calculation of the conductivity Equation (2.6) can be rearranged as follows: i+m _n (1 - / ) ( « - » - ! ) i + m ' Prom Eqs. (2.7) and (2.9) we have expressions for *xx = (A1) 1 j and 1 + P\t\ * L1 —ia+ -+- ixt 1 —ia_ + ixt = + [(1 - f ) ^ ~ 1 )K ,, _ 3«ff0 . . . / + (l-/)e -> -l. - — y-l ‘“ ( 1 _ ? ) [ 1 + — rT 3 tj— 1 r 1 1 1 —ia+ + ixt 1 —ia_ + ixt i X [ -----------------------------------------------------------------------J , = < 3 + [ / + ( l - / ) < r # - l K >, (A.2) where / ** , ffxy = ^8 7-if1 i f 1 ~ (2 | 1 i 1 1 + P\t\ 1 —ia+ + ixt 1 —ia_ + ixt 3iao yi 1 - 12 . 1__________1 8 7-i l + B\t\ 1 —ia+ + ixt 1 —za_ + ixt and a™9 and a™9 are the same as and 1 in Eq. (2.7). These expressions were evaluated previously.10. a™9 = ^ -{ 2 a +p+ + 2a_p_ - 2 + r+(x2 + 1 —a+) + r_x2 + 1 - a2) +t[a+ + a_ + p+(x2 + 1 —a2) —p~(x2 + 1 —a2) —2a+r+ —2a_r_}, axy9 = ~ a + + P - f c 2 + 1 — <*1) - P + ( z 2 + 1 — a2 ) + 2a+r+ — 2a_r_ +i[2a+p+ —2a_p_ + r+(x2 + 1 - a+) —r_x2 + 1 —ai)]}, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A.4) 72 where a± = ( u ± o/c)r, x = gZ, P± = J _ inr l ± j £ ± ^ ) ! 1 Ax 1 + (x —a±)2 r± = -^[tan_1(x + a±) + tan~l (x —a±)]. 2x (A.5) Integration of Eq. (A.2) is tedious but straight forward. The final forms are: <TXX = j^ { 2 a +p+ + 2a_p_ - 2 + r+(x2 + 1 - a2) + r_(x2 + 1 - a2_) +[(i - /)(e-^ - i)][x(u n - p a + ( u + r +) —0°—U - + /- ) ~ 0(9+ + 9+ + 9- + 9-) + (s+ + s+ + s- + O ^ 2] +i[a+ + a_ + p+(x2 + 1 —a+) + p~(x2 + 1 —a?_) —2a+r+ —2a_r_ +*'[(1 - /)(«“^ " 1)][*(^+ - 9 + + 9 - - 9 - ) - fa+(9+ + 9+) —0a-(9- + 9~) + 0(f+ + /+ + / - + /I ) + (t+ + CTxy = {a_ - a+ + p_(x2 + 1 - a 2_ ) — + t'_)x2]}, p+(x2 + 1 — a2 ) + 2a+r+ - 2a_r_ +[(1 ~ /) ( e " ^ “ !)][*(»- ~ 9 - ~ 9 + + 9+) + 0a+(9+ + 9+) -/3a-(g. + g'_) + /?(/_ + f'_ - f + - f'+) + (t_ + f_ - t + - t'+)x2} +i[2a+p+ —2a_p_ —r_x2 + 1 —a2_) + r+(x2 + 1 —a2)] +*■[/+(i - /)<=-% - 1]w /+ 1 '-) - /5 M /+ + /;> + 0a- ( f - + / I ) ~ 0(9 + + 0+ ~ 0- “ O + («+ + s+ - s_ - O x 2]}, (A.6) where l, t u± “ 2 l+ « ± i 1 + ~( ^ '+ q ± )'2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 v± = tan 1(a: + a ± )—tan x(a±), f± = I jS*+ (X- ~ (4 ~**~1)~ x1' x(x — 2 a ± ) . 2 ^ + ( x - ^ 1~ 2a±1,± + '1±(a±~ 3! _ 1 ) ---------2------11 S± “ [~ 2 a± U ± 1 r ~ s± = 2 j8> + ( l - 0a±)2 _ 1 -P(x - 0a±) u l * 2 Z?2 + (x —/3a±)2 0 / 2 2 -x ^ 111(1 + ^ + 0* (1 ~ 2 lv , , !/, P\\ ^ pp ( 2 (K 7\ ( ^ Purthennore, /±(x) = /± (—x), $±(x) = 3±(-x), s'^x) = s±(—x), and tf±(x) = t±(—x). The foregoing results are to be used in the integrand of Eq. (2.22), which must then be evaluated numerically. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 Appendix B: Polarization of the field inside an anisotropic metal Consider a metal in a high-frequency electromagnetic field. To learn how the wave is polarized we shall treat the normal skin effect for which Ohm’s law, J = aE, is valid and the conductivity is local. The relevant Maxwell equations are IdH „ Vx£ V x H = — J. (B.l) c We neglect the displacement current. Let us assume that the metal fills the z > 0 half-space, and that the wave is incident normal to the surface. For a wave propagating in the z-direction we shall seek a solution proportional to exp(iqz — iut). Eliminating the magnetic field H from Eq. (B.l), we can easily find: A'jr _ -V2£ + V (V - £) + ~ j = 0, (B.2) which reduces to d2 47riaj . a + ~ Ja~ ' (B.3) j z = o, (B.4) where a = x,y. The conductivity of a nearly-free electron system in the local approximation is, a8 = ne2r { 1 —iu)T —U)CT lOcT 1 —iuJT 0 0 (B.5) (l—iUT)2+(wcT)2 V o o l—tWT The conductivity of the Fermi-surface cylinder, as calculated in Sec. 2.3, is 771* (1 — IU>t ) 2 + (u /c T )5 ac = 7ine^Tc 771* / 1 (1 - iiJ T c ) 2 + (o>crc)2 1 — iu r c —U c Tc VCTC 1 —iuJTc ^ —tan 0(1 —iu>Tc) tan 6(u>crc) —tan 0(1 —iurc) \ —tan 0(u)ctc) tan20(1 —iojrc) j Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (B-6) 75 The total conductivity is a9+crc. The usual expression for the conductivity tensor is: ( <*xx G xy & xz\ 0yz &yy & yz \ & zx & zy &zz) a= (B.7) Because the number of electrons enclosed by the cylindrical Fermi surface is only a fraction, i/ = 4 x 10-4, of the total, the following inequalities prevail. 0~xzi &yzt & zx j &z y ^ x x i & yyi & zzi &xi11 Gyx~ (B.8) Using Ohm’s law to express Eqs. (B.3) and (B.4), we find a set of homogeneous equations: .2 V? 47ria; 4xiu> ^ *Txy£y ^ ^xz^z —0> 47tzu; . ^2 &xx)^x 47ria; _ , 2 47rzu; x „ 47ria; ^2 Gyx^x + (? ^ ayy)^y ^ Gyz^z ~ 0) &ZX^X “i" &z y £ y " f GZZ^Z = 0 . (B.9) We next eliminate £z in favor of £x and £y, using the third equation of (B.9). This allows us to express (B.9) with £x and £y only. , , (? 47rio; . . _ ~g~°xx)^x 47rio> . _ . , 47rzu; . _ ^ axy^v — 0) 47rio; . . „ + (« - = °. (B.10) where &O0 — &az&zf} (B.ll) Here a, 0 indicate x or y components only. This change of crQ/3 to a'a0 is the main contribution of the electrons in the tilted cylinder. Transverse conductivities are mixed with longitudinal conductivity on account of the longitudinal motion Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 of electrons in the cylinder. The determinant of (B.10) must vanish; andthis condition leads to the allowed propagation vectors: = Y ^rW xx + ff'n + >/(*£*-^ y P + ^V^v*]= y ^ r W z x + <r'vv- 'J(a'z x - ° ,n )2 + *°,xy°,vzl (B-12) Therefore the two electric-field modes are Ainoj(T'xy rCgjC2 - /k'Kiu)a'xx)a zy + A'Kiua'xv(Tzx Airicjcr'zyazz s, = Anuoo'^ _r «£I—~ Aziuja'^a^ + Arriucr^ajx Aniuja'^a^ (b .13) The amplitudes of the transmitted wave, £\a and £2o, can be obtained in terms of the amplitudes of the incident wave £Tby requiring the tangential field components to be continuous at the boundary. There are incident, reflected, and transmitted electric fields on the surface, z=0: E1 = (S'x + Sfye***-^, £* = (SZx + e f y + S^e-***-**, f? = £l0(x + aiy + piz)eullt~,ut + £2O(x + a2y + lhz)e%q2Z~%ut, (B.14) where T indicates the incident wave propagating along z withwave vector 50 = uj/c, ‘R’ indicates the reflected wave traveling along —z with wave vector —qa, and ‘T ’ indicates the transmitted wave. Equation (B.l) requires the microwave magnetic field to have x and y components only. H1 = (£j* + £/y)e<fl0*-<wt, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 = ^ . ( _ ai£ + y)ewi* - ^ + ^ 2 ( _ a2i + y)ewJ^ | (B.15) where Pi = g^c2 - 47Tta;q^z 47ria;a^y (g^c2 - 4'KiuoJxx)aty + $xiwo,xvozx 4iKVj}0,Xy0’zz (B.16) For the purposes of this appendix we treat potassium as a nearly-free electron gas characterized by the following parameters: effective mass, m* = 1.21m, electron density, n = 1.4 x 1022cm-3, Fermi radius, kp = 0.75 x 108cm-1, and electron scattering time, r = 2.0 x 10-10sec (which is appropriate at T = 2.5°K). The frequency of the applied microwave field is 23.9GHz. Accordingly, u j t = 30 is used for electrons on the spherical Fermi surface. On account of the small velocity for electrons in the Fermi surface cylinder urc = 150. (This value is required to fit the observed width of the cyclotron resonance peak in Fig. 2.1.) The inequalities of Eq. (B.8) are so extreme that ax and a 2 differ from i and —i by ~ 10~6. Specifically, ai « i, q2 (B.17) ~ —t. The ratio of the x or y components to the z component is about 100, so Calculation of £\0 and £2o is straightforward by using the continuity of the tan gential field at z=0. The final results are ; ---------------- -A a iE l - £ / ) (ai —a 2)(go + ?i) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 * • = (B19) The amplitude of the transmitted wave may be found by specifying the incident wave. For right-circular polarization, e ‘ = is ’. (B.20) On account of extreme inequality, (B.8), one mode dominates the other by a factor of at least 107 for all magnetic fields, i.e., €lo » €20 . (B.21) £ ' = - iS i, (B.22) For left-circular polarization, and £2o is much larger than £\0. The magnetic field at z=0 can be found from Eq. (B.15). H = 2aig1(a2gr ~ gy) _ 2a2g2(ai£7 - £j) (ax — a 2)(g0 + qi) ( — a 2)(?o + ) ’ 0:1 fl-,(0) = - W i - £ i ) (ai —a 2)(9o + 9i) From Eq. (B.17) and the fact that qi, q2 0) e T2i£j, ff„(0) » 2£j, 92 W i - £ ‘y) (ai —<*2)(?o + ?2) (B23) q0, the magnetic field at the surface is, /o r £[ = /o r S i = ±iS’. Therefore the total current defined by Eq. (2.40) is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (B.24) 79 The foregoing results are incorporated in the calculations of Sec. 2.3 at Eqs. (2.43) and (2.44). It must be appreciated that the Fermi-sphere electrons are treated non-locally in Sec. 2.3. The purpose of this appendix is to show that the microwave modes in the metal are essentially circularly polarized, (despite the broken axial symmetry caused by the tilt of the Fermi-surface cylinder) on account of the small value (4 x 10-4) of 77. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. VITA Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 VITA Mi-Ae Park was bom on February 3, 1964, in Seoul, Korea. She attended elementary and secondary school in Seoul, graduating from Jung-Eui High School in February, 1982. In March, 1982, she entered Ewha Womans University, where she received her Bachelor and Master of Science degree in physics in February, 1986 and February, 1988. In August, 1990, she entered Purdue University to study for a her Ph. D. Her doctoral research has been in the area of theoretical solid state physics. She was a Graduate Teaching Assistant from 1991 to 1996. In 1996 she was awarded the Edward Akeley Prize for Theoretical Physics. She is expected to complete her Ph. D. degree in August, 1996. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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