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Orbital spin-splitting factors for conduction electrons in lead Ren, Yan-Ru 1985

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O R B I T A L S P I N - S P L I T T I N G F A C T O R S F O R C O N D U C T I O N E L E C T R O N S IN L E A D by J t E N YAN-RU B. Sc., Beijing University 1970 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E DEGREE OF DOCTOR OF PHILOSOPHY in T H E FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA April 1985 © Ren Yan-Ru, 1985 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of ) h y S i C £  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6 (.3/81) A B S T R A C T A detailed experimental study has been made of the spin-splitting factors gc for magnetic Landau levels associated with conduction electrons in extremal orbits on the Fermi surface of lead. This information has been derived from the waveform of the de Haas-van Alphen (dHvA) quantum oscillations in the magnetization of single-crystal lead spheres at temperatures of about 1.2 K and with applied magnetic fields in the range 50-75 kG. A commercial spectrum analyzer has been used to provide on-line values of the harmonic amplitudes in the dHvA waveform, and the values of gc have been extracted from the relative strengths of the harmonics. Serious systematic errors in ge can arise on account of waveform distortions caused by the small and subtle difference between the externally applied field H and the magnetizing field B acting on the conduction electrons. In 1981 Gold and Van Schyndel demonstrated that these 'magnetic-interaction* distortions could be suppressed to a large extent by using negative magnetic feedback to make the induction B within the sample be the same as H (or very nearly so). This thesis describes the first in-depth application of the magnetic-feedback technique to the systematic study of any metal. Particular attention has been paid to the effect of sample inhomogeneity, and Shoenberg's treatment of the magnetic interaction in a non-uniform sample has been generalized to include magnetic feedback. This theory accounts well for many features in the experimental data, especially those which remained a puzzle in the earlier work of Gold and Van Schyndel. Experimental gc values are given for the first time for most of the extremal orbits on the lead Fermi surface and for high-symmetry directions of the magnetic ii field. Indeed these are the most detailed data reported for any polyvalent metal. The gc factors for the different orbits and field directions are found to span the range from 0.56 to 1.147. These large net deviations from the free-electron value go = 2.0023 are consequences of the strong spin-orbit and electron-phonon inter-actions, and an attempt has been made to separate these two contributions to the y-shifts. iii T A B L E OF CONTENTS Page A B S T R A C T ii T A B L E O F C O N T E N T S iv LIST O F T A B L E S vii LIST OF F I G U R E S vi i i A C K N O W L E D G E M E N T S xi Chapter 1 I N T R O D U C T I O N 1 Chapter 2 SPIN-SPLITTING O F L A N D A U L E V E L S A N D T H E y - F A C T O R M E A S U R E M E N T 4 2.1 The de Haas-van Alphen Effect and the Spin-Splitting of Landau Levels 4 2.2 Methods for Measuring Orbital </c-Factors 9 2.2.1 Spin-Splitting Zero 11 2.2.2 Absolute Amplitude 12 2.2.3 Two-Harmonic Ratio 12 2.2.4 Three-Harmonic Ratio 13 2.3 Multiplicity of <jrc-Factor Determination 15 Chapter 3 S H O E N B E R G M A G N E T I C I N T E R A C T I O N A N D ITS S U P P R E S S I O N B Y M A G N E T I C F E E D B A C K 19 3.1 Shoenberg Magnetic Interaction 19 3.1.1 Distortion of Harmonic Content 20 3.1.2 Modulation of a Weak High-frequency Oscillation by a Strong Low-Frequency Oscillation 24 3.2 Suppression of Magnetic Interaction by Magnetic Feedback 27 iv Chapter 4 EXPERIMENTAL APPARATUS AND TECHNIQUE 29 4.1 General Description 29 4.2 Sample Preparation 30 4.3 The Main Magnetic Field and its Modulation 35 4.3.1 The Main Superconducting Magnet 35 4.3.2 The Superconducting Modulation Coil 37 4.4 Cryostat and Temperature Control 40 4.5 Signal Processing 42 4.5.1 Frequency Response of the Total Loop 42 4.5.2 Spectrum Analysis 48 4.5.3 The Ramp Shaper 52 Chapter 5 SAMPLE INHOMOGENEITY AND MAGNETIC FEEDBACK 58 5.1 Influence of Sample Inhomogeneity on Magnetic Interaction and Magnetic Feedback 58 5.2 Magnetic Feedback with One Fundamental Frequency and its Harmonics 65 5.3 Magnetic Feedback with Two Fundamental Frequencies 66 5.3.1 General Formulation 66 5.3.2 Experimental Evidence 75 Chapter 6 EXPERIMENTAL EVALUATION OF S FOR LEAD 86 6.1 The Fermi Surface of Lead 86 6.2 S for the [110] $ Orbit 90 6.2.1 The 7 Beat Pattern 90 6.2.2 Temperature Dependence 95 6.2.3 Field Dependence 98 6.2.4 S from Three-Harmonic Ratio 100 6.2.5 S from Two-Harmonic Ratio—a Cross Check 102 6.3 S for the [110] V Orbit 104 v 6.3.1 Preliminary Considerations 104 6.3.2 Temperature Dependence 106 6.3.3 S from Three-Harmonic and Two-Harmonic Ratios 111 6.4 S for the [111] rp and 9 Orbits 117 6.4.1 General Considerations 117 6.4.2 S for the [111] V Orbit from Two-Harmonic Ratio 119 6.4.3 S for the [111] 9 Orbit from Two-Harmonic Ratio 120 6.5 S for the [100] u Orbit 124 6.5.1 Extremal Orbits along the [100] Direction 124 6.5.2 The Beat Pattern 125 6.5.3 S from Two-Harmonic Ratio 129 6.6 Preliminary Results for the [100] rf> and £ Orbits 134 6.6.1 Effective Masses and Dingle Temperatures 134 6.6.2 Absolute Amplitude and S for the [100] £ Orbit 136 Chapter 7 C O N C L U S I O N S 140 7.1 Summary of Experimental Results 140 7.2 Theoretical Considerations and the Orbital g c-Factor Results 145 7.2.1 Spin-Orbit Interaction 145 7.2.2 Many-Body Interactions 146 7.3 Concluding Remarks and Suggestions for Further Study 151 B I B L I O G R A P H Y 154 A P P E N D I X A Magnetic Feedback with Two Fundamental Frequencies —Effect on the Higher Harmonics 159 A P P E N D I X B Absolute Amplitudes of the Fundamental dHvA Oscillations 162 vi LIST O F T A B L E S Table Page 1 Fourier Coefficients P, Q 22 2 Values of Reduction Factors p + and p~ 71 3 Experimental p + , p~ Values 82 4 D H v A Frequencies in Lead 89 5 Summary of Experimental S Values for [110] 7 Oscillation in Lead 103 6 Summary of Experimental 5" Values for [110] a Oscillation in Lead 116 7 Cyclotron Effective Masses in Lead 141 8 Dingle Temperatures for Pure Lead Samples 142 9 Summary of Possible Orbital </c-Factors of Lead 143 10 F i n a l Choice of Experimental Values for Orbital (^-Factors of Lead 150 11 Values of Reduction Factors p+ and p~ 160 12 Absolute Amplitudes of the Fundamental d H v A Oscillations 163 vii LIST O F F I G U R E S Figure Page 2.1 The Spin-Splitting of Landau Levels 8 2.2 The Spin-Split Magnetization at Absolute Zero 10 2.3 Diagram Illustrating Multiplicity of the Apparent Spin-Splitting 16 2.4 Multiplicity of S Values from d H v A Measurements 17 3.1 Fourier Spectrum of the dM/dH Waveform for a Lead Sphere with H||[110], as an Example for the Complexity of MI 24 4.1 Block Diagram of Apparatus 31 4.2 Sample Holder and Co i l Former 34 4.3 Field Inhomogeneity of the Main Superconducting Magnet 36 4.4 Inhomogeneity of the Modulation Field 39 4.5 General Schematic Cryogenic Assemply 41 4.6 Schematic Circuit Diagram of the Detecting System (I) 43 4.7 Schematic Circuit Diagram of the Detecting System (II) 44 4.8 Detection Arrangement 46 4.9 Total Loop Phase Shift without Compensating Circuit 47 4.10 Total Loop Frequency Response with Compensating Circuit 48 4.11 Apparent Spectrum Width and Amplitude Height 50 4.12 Sychronization of the Integrator and the Time Window 53 4.13 Modulation Field with and without the Ramp Shaper 56 4.14 Correct Amount of Quadratic Contribution Added to the Linear Ramp 57 4.15 T he Effect of the Ramp Shaper 57 viii 5.1 Sample Inhomogeneity and the Magnetization 59 5.2 Schematic Sketch of the Fourier Amplitudes as a Function of M F B Gain according to the P S M I Treatment 68 5.3 Schematic Sketch of the Fourier Amplitudes as a Function of M F B Gain according to the MIPS Treatment 75 5.4 Relative Phase of the [110] a Oscillation 79 5.5 Fourier Spectra of the a Oscillation and Sidebands 80 5.6 Fourier Amplitudes of the a Oscillation and Sidebands as a Function of M F B Gain 81 6.1 Primitive Brillouin Zone for an fee Lattice 87 6.2 The Empty-Lattice Model of the Lead Fermi Surface 88 6.3 Model of the T h i r d Zone Electron A r m of Lead 91 6.4 Beat Envelopes for the First Three Harmonics of the [110] 7 Oscillations . 94 6.5 Effective Mass Plots for the First Three Harmonics of the [110] 7 Oscillations 97 6.6 Dingle Plots for the First Two Harmonics of the [110] 7 Oscillations 99 6.7 Three-Harmonic Amplitude Ratio Plot for the [110] 7 Oscillations 100 6.8 Fourier Spectrum of the dM/dH Waveform with H | | [110] at a 7 Minimum 105 6.9 Effective Mass Plots for the First Three Harmonics of the [110]a Oscillation with and without MI Correction 107 6.10 Three«flarmonic Amplitude Ratio Plot for the [110] a Oscillation 112 6.11 Illustration of the Accesible Range of the Observable relative phase Values with MI 114 6.12 Dingle Plot for the Fundamental [110] a Oscillation 115 6.13 Fourier Spectra of the dM/dH Waveform with H || [111] 118 ix 6.14 Effective Mass Plots for the Fundamental and Second Harmonic of the-[111] a Oscillation 120 6.15 Dingle Plots for the Fundamental and the Second Harmonic of the [111] a Oscillation 121 6.16 Effective Mass Plots for the Fundamental and Second Harmonic of the [111] 8 Oscillation 122 6.17 Dingle Plots for the Fundamental and the Second Harmonic of the [111] 6 Oscillation 123 6.18 Beat Envelope for the [100] p Oscillation 126 6.19 Dingle Plot for the [100] p Oscillation after Beat Correction 127 6.20 Schematic Diagram of the Beat Envelopes for the [100] P Oscillation and its Second Harmonic 128 6.21 Relative Phase of the [100] p Oscillation 130 6.22 Fourier Spectra of the dM/dH Waveform with H || [100] 131 6.23 Effective Mass Plots for the First Two Harmonics of the [100] fi Oscillation 132 6.24 The M F B Gain Dependence for the Mixing Amplitude of the [100] a and 4/3 Oscillations 135 6.25 Effective Mass Plots for the [100] ir and a Oscillations 136 6.26 Dingle Plots for the [100] ir and a Oscillations 137 x ACKNOWLEDGEMENTS It is my sincere pleasure to thank Dr. A. V. Gold for his guidance throughout the course of this study, and for his helpful advice and many suggestions. Sincere thanks are due to the other members of my supervisory committee, Drs. B. Bergersen, J . F. Carolan and C. F. Schwerdtfeger, for their encouragement. I would like to thank Dr. A. J . Van Schyndel, then a fellow graduate student, for helpful and stimulating discussions, especially in the early stage of this work. I am grateful to Dr. P. M . Holtham for providing the results of his latest calculations of the curvature factors of the lead Fermi surface. I wish to express my gratitude to the Ministry of Education, the People's Republic of China, for financial support in the form of a scholarship, and to the Department of Physics, the University of British Columbia, for financial assistance. Finally it is a special pleasure to thank Wu Ying, my wife, for her understand-ing throughout the course of my graduate study. xi C H A P T E R 1 I N T R O D U C T I O N The energies of conduction electrons of a metal in the presence of a magnetic field are quantized into equally spaced degenerate levels known as Landau levels. When electron spin is considered, these quantized levels will split into two sets (referred to as spin or Zeeman splitting), one set for spin-up, one set for spin-down electrons. By analogy with the corresponding atomic case, the energy difference SEspiD defines a g value (Lande -^factor) for conduction electrons, where \IB is the Bohr magneton and H is the magnetic field. For free electrons, g = g0 = 2.0023, which for most practical purposes can be taken as 2, but in real metals g differs from its free-electron value on account of spin-orbit coupling and the many-body electron-electron and electron-phonon interactions. Indeed measurements of the ^-factor can yield valuable information about both the spin-orbit and the many-body effects. There are two main experimental techniques which are used to measure con-duction electron y-factors of metals. In the conduction-electron spin-resonance (CESR) experiment at microwave frequencies, the measured ^-factor is an average for electrons over the entire Fermi surface—the constant energy surface in momen-tum or k space which separates the occupied and empty states at the absolute zero 1 of temperature. The (/-factor thus obtained is not affected by the electron-electron interactions (see Yafet, 1963). On the other hand, studies of the magnetic quantum oscillations—known as the de Haas-van Alphen (dHvA) effect—allow determination of the g-factor for a well-defined group of electrons occupying a narrow band of states around an extremal section of the Fermi surface normal to the applied magnetic field. The (/-factor determined in this way is affected by both the electron-electron and electron-phonon interactions. Measurements at different field orientations for different extremal orbits may be used to study the anisotropy of the (/-factor over the Fermi surface in some detail. A comparison of (/-values from the CESR and dHvA methods provides a useful separation of the different interactions. Previous work on the dHvA measurement of the (/-factors has centred on the relatively simple alkali and noble metals, whose quasi-spherical Fermi surfaces exist only in the first Brillouin zone. For more complicated cases such as transition and polyvalent metals, only preliminary and fragmentary data are available for specific orbits in a limited number of metals. In the work presented here, an extensive study of the (/-values of conduction electrons in lead single crystals was carried out by means of dHvA effect. Lead was chosen for the following reasons. First, as a heavy polyvalent metal (with a high atomic number), the spin-orbit coupling is particularly important. Second, it is now well known that the electron-phonon interaction is very strong. Third, the Fermi surface has been investigated in considerable detail. In the dHvA effect, the (/-factor is obtained from the amplitude of the mag-netic quantum oscillations. Several different methods of extracting this informa-tion from the data are reviewed in Chapter 2. The discussion is illustrated with references to previous work on various metals. In analyzing the dHvA oscillations a serious complication arises from the small difference between the magnetic induction B and the applied field intensity H . 2 This subtle distinction gives rise to the Shoenberg magnetic interaction effect, and in Chapter 3 we discuss the origin of this effect and explain how it can have such a profound effect on the wave-shape analysis for the ^-factor. Fortunately this complication can be experimentally suppressed to a great extent by exploiting the technique of negative magnetic feedback. The experimental apparatus is described in Chapter 4. The apparatus used in this work was basically the same as that of Gold and Van Schyndel (1981; see also Van Schyndel 1980 for detail), but a great deal of effort went into refining various key factors for the apparatus so that reliable (/-factors could be obtained. In this thesis the emphasis is on the improvements achieved during this work. In studies of the dHvA effect there is a special need for single crystals of very high perfection. Despite taking great care with the crystal-growing process, as described in Chapter 4, it is impossible to prepare a sample without any defects, and sample inhomogeneity can limit the usefulness of the magnetic feedback technique. In Chapter 5, we discuss the influence of the sample inhomogeneity in detail, and also present the experimental evidence of the defects in the crystal with the aid of magnetic feedback. Due to the individual characteristics of the various orbits in lead, different pro-cedures are needed in order to obtain the required data for extracting the orbital ^-factors. In Chapter 6, we describe these procedures in detail. The g-factor ob-tained from dHvA amplitudes are not unique; for each extremal orbit normal to the applied field we can have a set of possible g values. To make the correct choices from these sets, the influence of the spin-orbit coupling and many-body interac-tions on the (/-factors must be taken into account. In the final chapter we discuss those effects and make our choice of the g-factors in keeping with these effects. We conclude the thesis with suggestions for further study. 3 C H A P T E R 2 S P I N - S P L I T T I N G OP L A N D A U L E V E L S A N D T H E g - F A C T O R M E A S U R E M E N T 2.1 The de Haas-van Alphen Effect and the Spin-Splitting of Landan Levels Under the influence of a magnetic field, the quasi-continuous energy spectrum of the conduction electrons in a metal is transformed into a set of highly degener-ate discrete Landau levels. The energy difference between two successive Landau levels is equal to %OJC, where uc = eH/m*. Here OJC is the cyclotron frequency, e the proton charge, H the magnetic field, and m* the cyclotron effective mass (to be defined below). This quantization into Landau levels results in many proper-ties of single crystals of metals showing an oscillating behaviour as a function of the reciprocal of the magnetic field 1/H . A specific and typical example is the magnetization. The oscillation of this quantity in a strong magnetic field at low temperatures, first observed by de Haas and van Alphen in 1930 in bismuth, is now known by the names of its discoverers. Since then, magnetization oscillations have been found in virtually every metal, and they have become one of the most power-ful means of revealing the electronic structure of metals. The literature contains many excellent reviews of the theoretical and experimental aspects of the dHvA effect, of which Shoenberg's newly published book ( Shoenberg 1984 ) is the most extensive and complete. 4 According to the theoretical work of Lifshitz and Kosevich (1955, referred to as LK), the oscillating part of magnetization* M associated with an extremal Fermi-surface cross-section of area A normal to the magnetic field H is described by the. expression r / F \ TTI M = J2Ar s i n [ 2 7 r r ( - ~ l) T-\. (2.1) r Here F is the dHvA frequency, 7 is the Onsager phase constant (Onsager 1952) which is 1/2 for free electrons, and the phase factor TT/4 is negative for a maximum cross-sectional area and positive for a minimum. Strictly speaking, equation (2.1) refers to the component of the vector M which is parallel to the applied field. The amplitude of the rth harmonic Ar, in an idealized situation for absolute zero of temperature, with perfectly sharp Landau levels, and without taking electron spin into consideration, is expressed as Ar = r-?D(H) (2.2) with D { H ) ~-4^c{-nc-) — U ) S K H B ) ( 2- 3 ) * In the literature, the oscillating magnetization is often denoted by Af. The tilde(~ ) will not be used in this thesis since we shall never be concerned with the steady part of the magnetization. 5 in which m* is the cyclotron effective mass defined as 2TT\dE/kHi m. and the curvature factor C = d2A dk\ is related to the geometry of the Fermi surface. Under practical conditions for observing the dHvA effect, several other factors must be included in the amplitude definition(2.2). 1. At finite temperatures, the Fermi surface becomes slightly diffuse, leading to a thermal damping of the oscillations. The result is to multiply each of the amplitudes AT in (2.2) by a reduction factor with _ 2ir*m*kBT _ T X - ehH " A / V ( 2 5 ) Here ks is the Boltzmann constant, A = 2n2kBm0/en, — 1.469 x 1 0 5 G K *, and y, = m* / m 0 is the ratio of the cyclotron effective mass to the mass of a free electron. 2. Another reduction factor, introduced by Dingle (1952), arises from the broadening of the Landau levels due to scattering. The amplitude Ar then becomes modified by multiplying by an exponential factor r r2ir2m*kBTD-] K r = eX?[ eHH J = exp[ - rA/ i^ ] . (2.6) 6 Tr> is called the Dingle temperature, which is related to the orbital average of the scattering rate 1/r by 3. The crucial modification to the amplitude as far as this work is concerned comes from the lifting of spin-degeneracy by the magnetic field. This results in a symmetrical spin (or Zeeman) splitting of each local k-state by the energy difference 8E(k, H), which defines a local Lande ^ -factor „ - 6E(k, H) , , g(k,H)= ' ] . (2.8 Since each dHvA oscillation is related to a cyclotron orbit on the Fermi surface, the relevant (/-factor in the dHvA effect must be the time-weighted average around the cyclotron orbit. This average was first suggested by Randies (1972) and shown semi-classically by Holtham (1973) to be given by where the points k are on the relevant extremal orbit, dkt is an orbital path element, and w±(k) is the component of the electron velocity in the plane of the orbit (i.e., normal to the magnetic field). The spin-splitting is illustrated in Figure 2.1. Because one full oscillation (i.e., a phase shift of 2w) corresponds to the passage of successive Landau levels across the Fermi level Ep, the spin-splitting 6E will correspond to a phase shift 2irS, where „ SE QCHBH 1 m* 1 , nujc nuc 2 m 0 2 (since huc = eH/m* and HB = eH/2mo). As a result, the contribution of each 7 1"l(JJ r-TieH 1 6E=g cyiBH i v Figure 2.1 The spin-splitting of Landau levels. spin subset of Landau levels to the amplitude of the oscillation is half of that in the absence of spin, with a phase shift of ±irrS for the rth harmonic. The total oscillatory magnetization then becomes M = X) j^sin[27rr(§ -1) =»= \ +rnS] + \A'*iii[2*r(j[ -1)T \ - r*s] I = J^i4 r cos rirS s in[2»rr(^ - 7) T ]^ • (2.11) r Thus the net effect of the spin-splitting is simply to reduce the amplitude of the rth dHvA harmonic by the factor cos rwS. In general, the Fermi surface is complicated and there are often more than one cross-section of extremal area, and correspondingly more than one F , A, m*, 8 C , gc, and TD for any given field direction. The total oscillating magnetization is then the sum of a number of contributions, each of which has the form of equation (2.1) but with different sets of parameters. Putting all the above modifications together, the general form of L K formula is o r b i t s r = E E o r b i t s r uFT exp(-rA/jgf) (rCH)? sinh(rA/i£) cos mS sm (2.12) with i / = ( l ) * ( - £ . ) * f c f l = 6.523 x l O ^ G ^ R " 1 . 2.2 Methods for Measuring Orbital ^-Factors In a few cases, the spin-splitting can be observed directly in the waveform of the magnetic oscillations. To illustrate the influence of the spin-splitting on the waveform, we consider the case at absolute zero (T = OK) and without broadening of the Landau levels (TD = OK) . Then equation (2.12) becomes, for a single orbit, M — ] j P r ~ * £ > ( # ) cos rvS s in[27rr(^ - ^ T J ] . Without considering the spin-splitting, the waveform is a cusp-like oscillation (see, for example, Gold 1968), with a sharp peak occuring when the uppermost Landau level suddenly becomes depleted as its energy exceeds the Fermi energy. As we have discussed in the previous section, the result of spin-splitting will be given by the sum of two such waveforms in which the contributions from the rth harmonic have been phase shifted by ±mS. The two cusp waveforms and their sum are shown in Figure 2.2. The splitting of peaks in the resultant waveform has been 9 10 observed directly in a few metals (e.g., bismuth; McCombe and Seidel, 1967), but it is a relatively rare occurence. The essential point for direct observation of spin-splitting is that the wave-form must show sharp cusps, i.e., be rich in harmonics, whereas in most instances the cusps have been rounded off by the combined result of thermal damping and broadening of the Landau levels. The reduction factor for the rth harmonic is like exp[—rX(i(T+TD)/H], so that when T increases or H decreases, the higher harmon-ics are more and more damped out, and direct splitting can no longer be observed. The effect of the spin-splitting is then reflected only in the harmonic amplitudes of the resultant oscillations. Here we shall describe a number of methods which make use of the dHvA amplitude to determine cos mS and hence the yc-factor of a particular cyclotron orbit. 2.2.1 Spin-Splitting Z e r o This method is based on the fact that when 5 is a half integer, cos uS for the fundamental oscillation will be zero. If a null is found for the fundamental amplitude, and if m* is known, then the value of gc can be deduced. Zeroes may also be found for the second harmonic, but they will not occur at the same field orientations as the fundamental zeroes. The spin-splitting zeroes were first observed in Cu , Ag and A u by Joseph and Thorsen (1964), Joseph et al. (1966) and Halse (1969). They were also found in Sb (Windmiller 1966), Pd and Pt (Windmiller and Ketterson 1968; Windmiller, Ketterson and Hornfeldt 1970; Hornfeldt et ai. 1982; Gustafsson et al. 1983). This method is the most accurate and also the simplest. However, it is the least general method because the spin-splitting zeroes occur only rarely at some special orientations of the magnetic field in a few metals. It is not suitable for systematic study. 11 2.2.2 Absolute Amplitude The principle of this method is simple and straightforward, and involves mea-suring the absolute amplitude of the fundamental oscillation. Careful calibration is needed to find out the coupling between the detecting system and the sample, i.e., the total system gain. In order to extract |cos irS\ from the absolute ampli-tude, several quantities appearing in the L K formula (2.12) must be known. The dHvA frequency F , Dingle temperature Tp, and the cyclotron effective mass m* can all be determined experimentally, but the remaining quantity C, the curvature factor, can be calculated only if the shape of the Fermi surface is known; C cannot be determined experimentally. The absolute-amplitude method was first introduced by Knecht (1975). Since then, several calibration methods have been developed. However, the accuracy of this method is limited owing to the difficulties associated with the reliable calibra-tion of the detecting circuit and uncertainty in the sample volume, also with the lack of sufficiently detailed knowledge of the geometry of the Fermi surface to determine the curvature factor C. This method has been applied to the alkali and noble metals, whose Fermi surfaces are relatively simple (Knecht (1975) for K, Rb, Cs; Perz and Shoenberg (1976) for Na; Springford, Templeton-and Coleridge (1983) for Cs; Crabtree, Wind-miller and Ketterson (1977) for Au; Bibby and Shoenberg (1979) for Cu, Ag and Au). 2.2.3 Two-Harmonic Ratio Instead of measuring the amplitude absolutely, one can measure the harmonic amplitudes relative to the fundamental. As can be seen from the LK formula 12 (2.12), = r-$ exp[-(r - l)\nTD/H] sinh(A/i7y#) cos rnS 8inh{rXftT/H) cos irS ' (2.13) Therefore, the value of | cos rwS/ cos irS\ can be extracted from the harmonic ratio at given temperature and field, provided that To and y. are known. The most commonly used is the ratio of the second harmonic amplitude to that of the funda-mental, The great advantage of this method is that the difficulties of calibration are avoided, the sample volume need not be known, and the curvature factor is not involved in the ratio. The harmonic ratio method was first systematically applied by Randies (1972) to study <7C-factors for the 'neck' of the noble metal Fermi surfaces. Other studies of relatively simple metals using this method include: Crabtree, Windmiller and Ketterson (1975), Alles, Higgins and Lowndes (1975) on Au; Knecht (1975) on K, Rs, Cs; Springford, Templeton and Coleridge (1983) on Cs. This method has also been applied to certain transition metals, whose Fermi surfaces are complex and less precisely determined, so that C cannot be reliably estimated. Examples are the work of Cheng and Higgins (1979) on hole orbits in Rh; Startsev et al. (1984) on Ru and Os. In cases where the second harmonic is too weak to measure or badly distorted, one has to go back to the absolute amplitude method. 2.2.4 Three-Harmonic Ratio In this method, which was introduced by Gold and Schmor (1976), one ex-tracts gc by simultaneously measuring the (relative) amplitudes of the first three 13 A2 _ 1 exp(-\nTD/T) cos 2uS (2.13a) Ai 2\J2 cosh(XfiT/H) cos TTS harmonics at different temperatures, but about the same strength of the magnetic field. The only quantity other than ge involved is the cyclotron effective mass m*; neither the Dingle temperature Tr> nor the curvature factor C need be known. The dimentionless ratio a, is found from the LK formula (2.12) to be 1 a = ao [l + - tanh2 xj = i " 0 0 D + Itanh2 X] ' (2-14^ where a 2 l - 3 t a n 2 7 r 5 v ^ U - t a n 2 JTS) 2  Qoo = V \ o ^ . e • (2-15) The subscripts 0, oo refer to the limiting cases X —> 0 (zero temperature or infinite field) and X —> oo (infinite temperature or zero field). Once otoo is known, S can be found by solving equation (2.15). From (2.12) for r = 1, 2, 3, it also follows that £ — [ £ ) , - 3 G ) 3 . ™ where is independent of the bath temperature T. Therefore, the value |aroo| can be ob-tained as the slope of a straight line plot of |J4I/.A3| VS. (A1/A2)2 as the temperature is varied while the field is held constant. 14 This method is extremely convenient in cases where the Dingle temperature is difficult to measure. Unfortunately its application is limited because the ob-served third harmonic is often too weak. So far it has only been used for certain oscillations in Pb (Gold and Schmor 1976, Gold and Van Schyndel 1981). We have discussed various methods to determine the <jc-factor from the dHvA amplitudes. Each method has its own strengths and weaknesses. Since experi-mental errors appear differently in different techniques, using several methods si-multaneously in the same experiment will be a good cross-check on the reliability of the results. 2.3 Multiplicity of g c-Fac tor Determination Because the gc-factor appears in the argument of a cosine function in the dHvA amplitude, the gc values obtained from the quantum oscillations are necessarily multi-valued. This is shown by the diagram in Figure 2.3. Although in each case the energy difference between spin-up and spin-down states associated with one Landau level are totally different, the physically observable effect is the same. The degree of multiplicity also depends on the method used. For the prin-cipal interval of the argument of the function |cos T T £ | , i.e., for 0 < S < 1/2, the results are as follows (see Figure 2.4): 1. Spin-splitting zero: Only one possible value, i.e., S = 1/2, exists within the above interval. 2. Absolute amplitude of fundamental oscillation: In this case, we again have only one S within the principal interval. 15 b) c) d) F i g u r e 2.3 Diagram illustrating the multiplicity of the apparent spin-splitting. In a, b, c and d, real splittings are different, but the apparent splittings are all the same (after Shoenberg 1984). 3. Two-harmonic ratio: If we denote the experimental value of cos 2JT5/COS nS by c2, then after a simple trigonometric transformation we have What is found experimentally is the absolute value |c2|. If |c 2| > 1, the sign in (2.19) is impossible, since a cosine cannot exceed 1. The multiplicity is the same as in the absolute amplitude method. However, if |c 2| < 1, the multiplicity will double. (2.19) 16 0 05 0 05 0 4 05 6 s s s b) c) Figure 2.4 Multiplicity of S values from dHvA measurements. a) Absolute amplitude of fundamental oscillation (also showing spin-splitting zero when S = 1/2); b) Two-harmonic ratio; c) Three-harmonic ratio (log scale, after Gold and Schmor 1976). 4. Three-harmonic ratio: Here S comes from equation (2.15), interval. Otherwise there are two. If S refers to any one of the principal values in 0 < S < 1/2, then for all the above methods the basic trigonometric multiplicity gives ± n ± S as other possible solutions, where n = 1, 2, 3 ,— For simple metals such as lead, where the band structure can be derived from a weak pseudopotential together with the spin-orbit tan nS = ± 1 - V3< •a, (2.20) If l^ ool > \/3/2, there are as many as three possible S values within the principal 17 interaction, a physical argument given by Pippard (1969) gives an upper limit of possible values according to the inequality + ! (2.18) i.e., 9c < 2 + where s is the number of Bragg reflections undergone by an electron in one revolution around the cyclotron orbit. The multiplicity in the gc value determination may be further reduced with the aid of other information, for example, the absolute or 'relative' phase of the fundamental and higher harmonics. This will be discussed in Chapter 6 together with the evaluation of yc-factors for specific orbits. However, the ultimate choice relies on our knowledge about the energy band structure and the nature of many-body effects. 18 C H A P T E R 3 S H O E N B E R G M A G N E T I C I N T E R A C T I O N A N D ITS S U P P R E S S I O N B Y M A G N E T I C F E E D B A C K 3.1 Shoenberg Magnetic Interaction So far we have assumed that in the L K formula (2.12), the oscillating mag-netization is a function of magnetic field H. In experiments on the dHvA effect in the noble metals, Shoenberg (1962) found that the harmonic content of the os-cillations was much stronger than predicted by the L K formula. To explain this phenomenon, he suggested that the effective magnetizing field should be the total magnetic induction B rather than the applied field H alone, where B = H -f 4JT(1 - <5)M (3.1) for a second degree surface with H parallel to a principal axis. 6 is the demagnetiz-ing factor determined by the geometry of the sample (0 < 6 < l). This suggestion was later confirmed by Pippard (1963) on thermodynamic grounds. Thus the H'a in the formulae in the previous chapter should be replaced by B's. Under normal laboratory conditions, 4ir(l — S)M is very small compared to H, usually < 10 - 4 £T. However, since F/H is a large number typically of the order id of 103, replacing H by B amounts to changing the phase by 2 " r ( l - 1 ) - 2 " « —2nr F F_m lH + 4n(l-6)M H. F Air(\-6)M H H which can be quite comparable to the phase change for one cycle. The result is that the experimentally observed amplitude and phase may be severely distorted from the ideal LK form. This is called the Shoenberg magnetic-interaction (MI) effect. Since our experimentally-measured gc-factors rely on the amplitudes of the dHvA oscillations, the effect of MI must be understood, and methods to correct this distortion must be found. In what follows we discuss two cases which are important in practice. 3.1.1 Distortion of Harmonic Content First we discuss the case in which only one extremal orbit is involved. We rewrite equation (2.1) in a more convenient form by setting ( F \ 8ir2F - - 7 J and K=—r(\-6). (3.2) Since |4TT(1 - 6)M\ < H, (2.1) becomes M = J > s i ° [ ^ r ( g + = Y,A' s i n [r(z - k M ) T J ] • ( 3- 3) r This implicit equation of Af as a function of i can be solved to any desired accuracy by an iteration procedure developed by Phillips and Gold (1969), in which the nth 20 order approximation is given by n M ( n ) = £ Arsin[r(x - /cM<n-r>) T , (3.4) r=l with M= 0. Since the relative magnitudes of Ar are determined largely by the exponential damping factor exp[-r\fi(T+Tr))/H], we can classify the various terms in order 0(i) of small quantities according to the index i of the resulting negative exponentials. For example, both .A3 and A\Ai are of order of 0(3). With this assumption, the approximation Af(n) will be exact to the nth order. This has been carried out by Phillips and Gold (1969) to 4th order and later by Perz and Shoenberg (1976) to 8th order with the help of a computer program designed to perform algebraic manipulations. In both calculations, the results are displayed as a table of Fourier coefficients Pr and Qr, where n M{n) = 52 [P' 8 i n ( r i T j ) + Q r cos(rz T ] . (3.5) r—\ The results of Phillips and Gold (1969) to 4th order are listed in Table I. To obtain the amplitudes and phases of the resulting harmonics A'r and <p'r} we have A'r = {P* + <8)t (3.6) and Then (3.4) becomes M W = y ; ^ s i n ( r x T 7 + ^ ) . (3.8) 21 T a b l e 1 F o u r i e r C o e f f i c i e n t s P, Q (from Phillips and G o l d 1969) K5 Term 0(1) 0(2) 0(3) 0(4) Pi At KAIA2 K2A\ 2y/2 8~ K,AiA<2 KA\ KAJAS K?A\ 2 2 A P* M~2y/2 " 7 T 6 v ¥ 1 2 KA\ KAIA3 K3A* V 2 ± 7= ± 7=— T 7=-2>/2 y/2 6\/2 3KA\A2 A 3 - 2v/2 3KAIA2 ZK?A2 K?A\ _ 2KAJAZ _ KA% Sy/2 y/2 y/2 <*A\ 2KAI/ y/2 3y/2 y/2 For the first three harmonics, the result for the amplitudes is, to the lowest order, A[ =Ai+ 0(3) + • • (3.9a) .1 . r i KA2 I / / C A 2 \ 2 " I £ M . y/2 A2 A' = A ( 3KA1A2 [ 1 /3KA1A2\2r _ 1/ Ki4 2 \ j Earlier than Phillips and Gold, Shoenberg and Vuillemin (1966), and Shoen-berg (1968) developed another procedure with the assumption that in the MI, the dominant terms are those involving only the fundamental amplitude A\. Their result is Af = - E ( " l ) r ^ M r * A i ) « n [ r ( x T j)], (3.10) r=l where JR is the rth order Bessel function of the first kind. If we take the limit as A2 and A3 approach zero in (3.9) and take the leading term of the Bessel function expansion in (3.10) we obtain the same 'strong fundamental' results A\ = AX + 0(3) + • •• (3.11a) A'2 = -\KA\ + 0(4) + • • • (3.116) A'Z = IK2 A* + 0(5)+--. (3.11c) O 23 3.1.2 Modulation of a Weak High-Frequency Oscillation by a Strong Low-Frequency Oscillation Another noticable effect of MI appears in the case where two fundamental dHvA frequencies Fa and Fj> (F 0 > Fb) arising from different extremal orbits are present. As a result, not only the amplitude is distorted, but sidebands are also generated.There are combination tones with frequencies mFa ± nFf,, where m and n are positive integers. Several examples can be seen in Figure 3.1. Since we are concerned with only the two fundamental oscillations and their lowest order mixings with frequencies Fa + Ff, and Fa — Ft, respectively, we start r a |a+7 27 37 a-27 o a+27 20L 2CL-7 \2<X+7 za-27 a+37 ! \2a+27 ! \20i+37 100 200 300 400 DHVA FREQUENCY (MG) Figure 3.1 Fourier spectrum of the dM/dH waveform for a lead sphere with H || [110], as an example for the complexity of MI. Here a and 7 refer to fundamental frequencies, and a+ 7, 2a 4-37, etc. are sidebands generated by the MI. The details of this spectrum, and of how it was obtained will be discussed later. 24 with Shoenberg's 'strong fundamental' assumption. With the notation* o fFa,b \ _ * Za,b = 2«{-g--'1a,b)T-and _ 8 i r 2 ( l - g ) F f l , b « a , 6 = ^ , (3.12) a and 6 referring to two different extremal orbits, and with |4TT(1 — 6) (Ma+Mb)\H, we have, as in (3.3), M = Aa sin(xa - « a M ) + Ab sin(x6 - /cj,Af). (3.13) If we ignore terms causing higher harmonics of either fundamental frequency, which do not concern us at present, (3.13) becomes M = Aa s'm(xa - KaAb sin Xb) + Abs'm(xb - KbAa sin xa), (3.14) in which M's have been replaced by their first order approximations. With the aid of the formula +oo exp(-i>siny) = 52 Jn(/x)exp(-iny), (3.15) n=—oo equation (3.14) then becomes + 00 M = Aa 52 Jk(KaAb) sin(xa - kib) k= — oo + oo + Ab 52 MKbAa)Bin{xb - lxa). (3.16) /=-oo * Note the difference between the definition here (Tff/4 is absorbed into x) from that in the previous subsection. 25 Letting A: = I = 0 , we pick out the two fundamental frequency components which now have distorted amplitudes K = AaJ0{KaAb) (3.17a) and A'b = AbJ0(K.bAa). (3.176) On the other hand, setting fc = ± l , / = ± l gives us the two lowest-order sidebands having the sum and difference of the two frequencies FA±FB , and having amplitudes A'a+b = -AaJ^KaAl,) - AbJi(KbAa) (3.17c) and A'a_b = AaJi(KaAb) - AbJi(KbAa). (3.17a") If KbAa <C 1, the amplitude distortion of the low-frequency term can be ignored, since then Jo(KbAa) —• 1. If we consider just the leading term of the Bessel functions, these results become the same as the lowest order results in Phillips and Gold (1969) , which are K+b = -^Y^AAAB (3.18) Aa—b — 2 AaAb • The results obtained so far apply when the absolute amplitude of the dHvA oscillation is much smaller than the field spacing AH « H2/F, or KA « 1. In certain cases (e.g., very low temperature), the dHvA magnetization can approach the field spacing. When KA > 1, the magnetization is no longer uniform inside the sample, and Condon diamagnetic domains are formed (see Condon 1966, Condon and Walstedt 1968). 26 3.2 Suppression of M I by Magnetic Feedback In the experiments where information is to be obtained from harmonic content, the MI obviously adds a troublesome complication. The Phillips-Gold iteration procedure described in section 3.1 is a waveform analysis technique which has been developed to recover the intrinsic LK amplitudes and phases from the apparent ones (with MI) measured in the experiment. Application of this technique has been reviewed by Higgins and Lowndes (1980). If at all possible, a better method to deal with the MI complication is to minimize it experimentally. Since the MI induced harmonic content is determined by the product KAI oc (1 — 6)Ai/H2, one might consider reducing Ai by raising the temperature (lowering the field can also reduce A\ but will increase K at the same time). Unfortunately the L K second and higher harmonic amplitudes drop off even faster than their MI counterparts with increasing temperature, and the result is just the opposite to what is desired (Gold and Schmor 1976). But in cases where two fundamental frequencies are involved, the MI distortion on the amplitude of the high-frequency oscillation is determined by Jo(KaAb) (see equation 3.17a). Raising temperature will reduce the MI. Bibby and Shoenberg (1979) determined the <7c-factor of the 'belly' orbit in the noble metals by measuring the absolute amplitude at a temperature which was high enough for the MI distortion due to 'neck' oscillation to become insignificant. The usefulness of this method is limited because raising the temperature will also reduce the intrinsic L K amplitude of the high-frequency oscillation itself. Another approach would be to arrange the sample geometry in such a way that the demagnetizing factor 5 is as close to 1 as possible. Gold and Schmor (1976) used thin disks perpendicular to H so that 6 ~ 0.9. " This worked rather well in their experiment to determine gc for the @ oscillation in lead. The disadvantages of this method are that the signal detected is also reduced by a factor of (1 — 6) ~ 0.1 27 compared to that for a long cylinder for which 6 ~ 0, and that the thin disk (as thin as 0.5 mm) is so delicate that it is very likely to be damaged during handling and cooling. A more general experimental method, the magnetic feedback (MFB) tech-nique, which is being used in this work, was developed by Gold and Van Schyndel (1981). The principle is quite straightforward. In the field-modulation method (see, for example, Stark and Windmiller 1968; or Shoenberg 1984, section 3.4), a relatively weak time-dependent field h(t) is used to modulate the quasi-static back-ground field H0i so the magnetic induction B 'seen' by the conduction electron is B = Ho + 4TT(1 - 6)M + h{t). If we superimpose another weak magnetic field hj derived from measurement of M and let hj = — (M, where f is an experimentally adjustable feedback gain, then the equation for B can be made independent of M if h} = -cM = -4?r(l - 6)M. (3.19) Then the MI will be effectively compensated and within the sample we shall have B = Ho + h(t). The dHvA effect itself is used to establish the correct amount of feedback. The details of this technique will be discussed in chapter 4. The MFB method will work only for weak MI, i.e., under the condition KA < 1. In case of KA > 1, the sample will inevitably break into domains and the MFB method cannot return the sample to a single-domain state. Other limitations of this method will be discussed in chapter 5. 28 C H A P T E R 4 E X P E R I M E N T A L A P P A R A T U S A N D T E C H N I Q U E 4.1 General Description Before discussing the experimental apparatus and procedures in detail, we shall first give a simple description of our approach for observing the dHvA effect with the MFB technique. A spherical sample surrounded by a well-balanced pickup coil is placed in a liquid He4 dewar tail. The dewar tail is situated in a superconducting modulation coil which fits tightly into the bore of a high-field superconducting magnet. The former provides a time-dependent field hm(t) on top of the high steady background field (H ~ 70 kG) provided by the latter. The pickup coil is balanced to be insen-sitive to the uniform applied field. The above-mentioned time-dependent field is a uniform triangular variation, with typically 1 kG peak-to-peak amplitude and 0.5 to 1 Hz repetition rate. Thus the changing magnetization inside the sample will induce an e.m.f. proportional to dM/dt in the pickup coil. This signal is ampli-fied by a low-noise battery-operated preamplifier (PAR 113) of gain 2k to 5k, after which it is integrated during, say, an hm > 0 portion of the triangular modulation by an analog integrator to provide a signal proportional to Af which is then added to the triangular wave obtained from a function generator (HP 3312A). During the other portion (hm < 0) of the triangular wave, the integrator is reset to zero. This alternation between the integration and reset modes is controlled by an electronic 29 switch activated by a synchronizing signal from the function generator. The same power amplifier (Crown M600) and modulation coil provide both the triangular modulation hm and the feedback field hj = — f M . During the integration mode, the dM/dt waveform from the P A R 113 output is recorded by an H P 3582A digital spectrum analyzer, which provides amplitude and phase transforms of the data. The spectrum appears on a cathode-ray tube display, and it can be traced out on an X - Y recorder. A block diagram of the apparatus is given in Figure 4.1. The correct amount of feedback field can be achieved by adjusting a helipot between the integrator and the adder. The dHvA signal itself is used to provide criteria for the adjustment of the feedback gain f (to be discussed later). 4.2 Sample Preparation As we have pointed out in the introduction, high-quality single crystals are needed in the study of the dHvA effect, especially with the M F B technique; the influence of the sample inhomogeneity on the M F B technique is the sole topic of Chapter 5. The shape of the sample is also very important. The magnetization M inside the sample must be uniform if its contribution to B is to be suppressed by the uniform feedback field hj. Thus a sample having a second degree surface is required, with the applied field parallel to a principal axis. The simplest choice is a sphere; when preparing a sphere we need not to worry about the crystal orientation until the final stage of mounting the sample for an experiment. Phillips and Gold (1969) showed that the use of the Czochralski method of pulling the crystal from the melt was very successful in producing lead single crys-tals with low Dingle temperatures. Our technique was similar to theirs. Zone refined lead (69 grade) from Cominco Ltd , Trail, B . C . was electrically heated to melt in a vacuum of 1 0 - 6 to 1 0 - 7 Torr. A single-crystal seed was dipped into the melt, and the heat conduction through the seed was enough to keep all but 30 balanced pickup coil spherical I sample J » PREAMP. FILTER KROHN-PAR 113 HITE 3322 R POWER SUPPLY AND CONTROL SYSTEM INTEGRATOR L-^ vyv—I h f - £ M SPECTRUM ANALYZER HP 3582A X-Y RECORDER HP 7046A FUNCTION GENERATOR HP 33I2A RAMP SHAPER 'm ADDER POWER AMPLIFIER CROWN M600 Figure 4.1 Block diagram of apparatus. 31 the submerged portion solid. Then the seed was slowly pulled up from the melt. The temperature of the melt was controlled by adjusting the heater voltage in such a way that the crystal growing from the seed was first tapered to form a 'neck' having a diameter of 1 to 2 mm before pulling a crystal with a diameter of roughly 7 to 8 mm. After pulling a crystal cylinder about 4 cm long, it was separated from the melt by raising the heater voltage.* After it had cooled down to room tem-perature, the cylindrical crystal was cut off around the 'neck' by the 'acid-cutting' technique. A beaker containing carbon tetrachloride to a depth of about 5 cm and a piece of cotton on the bottom was placed under the cylinder while it was still hanging vertically in the growing apparatus, and the beaker was then slowly raised until the- cylinder was immersed to the neck in the liquid. A drop of strong lead etchant** was then added to the CC1 4. This droplet floated on the CCI4 surface and surrounded the neck of the crystal. The etchant would cut through the neck in about one or two hours, and the cut portion then sank down to the protective cotton on the bottom of the beaker. The cylindrical crystal thus obtained was glued with silver print (GC Elec-tronics) on a piece of copper rod of about the same diameter as the crystal, and was then carefully mounted in a rotating chuck. A hollow copper circular cylinder with a diameter of 5.7 mm was used as a spark-cutting tool, which was also rotated during the erosion process.***The wall thickness of the hollow cylinder was kept * The heater voltage was set to be 3 1 . 3 0 V R M S to form the 'neck', and to be 3 0 . 4 0 V R M S for the required diameter ( 7 - 8 mm). The pulling rate was set to be 0.8 dial units, which was about 0.5 cm per hour, the lowest rate available for our puller. The heater voltage for separation from the melt was 3 3 . 2 0 V R M S -** The solution consists of 1/2 glacial acetic acid (CH3COOH), 3 / 8 distilled water, and 1/8 3 0 % H 2 0 2 by volume. *** Spark erosion machine: Agietron Model ABM, AG fur industrielle Elektronik, Losone-Locarno, Schweiz. Typical dial settings were: currents JT = 1/2 , JB = 0 , 32 below 0.05 mm. As the rotating tool was lowered with its axis tipped by about 5° from the perpendicular to the axis of the rotating crystal cylinder, a spherical sample resulted. The sample thus obtained usually formed a good sphere with a variance in diameter of about 0.5%, except for a ~ 0.5 mm high point or 'ear' where the sphere was last held to the crystal cylinder during the cutting process. This 'ear' was etched down by dabbing it with a pointed cotton swab which carried a drop of strong etchant. In order to obtain adequate Laue back-reflection photographs, it was necessary to immerse the sphere in the strong etchant for 30-45 minutes to remove the pitted surface generated by the spark erosion. The diameter of the final sphere was about 4.5 mm. Then it was carefully mounted on a goniometer with modeling plasticine, and X-ray photographs were taken for orientation. The oriented sample was then mounted on the sample holder. The sample holder shown in Figure 4.2 consisted of a pedestal and a protective sheath, both made of nylon. For better access of liquid helium, the top of the pedestal on which the sample sphere would sit was cut into four fingers and a hole was drilled along the axis of the pedestal. Several windows were also cut into the top and wall of the sheath for the same purpose. The lower half of the hole through the pedestal was threaded so that the pedestal could be held by a screw. The screw holding the pedestal upside down was then mounted in the chuck of a milling machine, and the pedestal was lowered to touch the oriented sample on the goniometer, which was fixed on the horizontal table of the machine. With a drop of G.E. adhesive varnish on one of the fingers, the sample was glued on the pedestal. After 24 hours for glue Jp = 0 dial units; impulse duration t = 5/JS; duty factor T = 19 dial units; servo GAP voltage (^ ) 5 dial units; servo amplification (0) 4 dial units; sensitivity (ty) 2.5 dial units; current limit t = 3 dial units; S-box selector C = 1.5 ~ 3.3 nF. 33 protective sheath sample sphere central hole coil former balanced pickup coil TOP VIEW OF THE P E D E S T A L sample pedestal 1 cm finger central hole scale Figure 4.2 Sample holder and coil former. 34 drying, the sample was separated from the goniometer by dissolving the plasticine in trichloroethylene, and then the pedestal holding the sample was carefully removed from the chuck. A small wad of loose cotton was placed over the sample, then the protective sheath was placed on top of them and glued at its edge to the pedestal. Another X-ray photograph of the sample was taken through the top hole on the sheath to confirm that the above mounting process had not changed its orientation. Then the sample holder was inserted into the bottom of the pickup coil former which also functioned as the housing for the sample. A threaded nylon disk held the sample holder rigidly in place. The error of the sample orientation relative to the axis of the pickup coil was believed to be less than 1°. 4.3 The Main Magnetic Field and its Modulation 4.3.1 The Main Superconducting Magnet The main applied magnetic field was provided by a superconducting magnet built by American Magnetics Inc. (originally A . M . I . # 10-066). The magnet was rated at 80 k G with an inhomogeneity of less in 1 part in 10 5 over a 1 cm diameter sphere at its centre. The field-to-current ratio was quoted as 1229 G / A by the manufacturer. Unfortunately a high resistance developed at one of the internal joints after this work was started, and a new magnet with same specifications was wound by American Magnetics Inc. (A.M.I . # 2237). The quoted field-to-current ratio for the new magnet was 1263 G / A . The field inhomogeneity is one of the major concerns about the main mag-net. Any inhomogeneity makes the dHvA phase lirrF/H vary over the sample, thus causing a reduction of the amplitude and errors in the observed phase (Paton and Slavin 1973, Hornfeldt, Ketterson and Windmiller 1973). A n inhomogeneity 35 plot for the magnet A.M.I. # 2237 was obtained by using the dHvA effect. The field strength was set to correspond to a zero-crossing of the [110] a-oscillation in Pb and a D.C. current was provided in the modulation coil. This D.C. current was adjusted to maintain the same zero-crossing position as the spherical sample was moved along the axis of the magnet. Since the field-to-current ratio of the modulation coil was known (see next subsection), the axial change of the main field could be found. A 7 mm axial region over which the inhomogeneity was less than I O - 5 at about 65 kG was found to be centred at 8 mm above the geometric centre of the magnet, or 11.4 cm down from the uppermost surface of the top plate of the magnet (Figure 4.3). io to i o X S i x -10 I 1 1 1 1 r — TOP 3.0 4- 4-BOTTOM — 2.0 1.0 DISTANCE (cm) 0.0 -1.0 Figure 4.3 Field inhomogeneity of the main superconducting magnet A.M.I. # 2237. Distance along the axis is given relative to the geometric centre of the mag-net (positive for upwards and negative for downwards). The circle indicates the approximate size and position of the spherical sample.-36 4.3.2 The Superconducting Modulation Coil The modulation coil was designed with the following considerations in mind: (1) it should be capable of producing a modulation field of about 1 k G peak-to-peak, when energized by a current whose maximum value does not exceed about 5 A; (2) since the modulation field should be about 1/50 of the static magnetic field, the tolerance on the inhomogeneity can be 50 times the inhomogeneity of the main magnet, i.e., 5 x 10 - 4 over a 1 cm-diameter sphere; (3) maximum rigidity should be achieved in order to prevent any mechanical vibration in the strong magnetic field. The coil was designed to be wound on a 11.43 cm (4^ ") long coil former with 1.905 cm ( |") outer diameter. It was decided to use four layers over the whole length and two double-layer compensating coils, one at each end. The coil was wound with a Nb 5 2Ti 48 alloy superconducting wire,* and 563 turns of.this wire could be accommodated in each layer. To determine the number of turns for each compensating coil, the field inho-mogeneity must be calculated as a function of the number of turns. The magnetic field inside a cylindrically symmetric solenoid can be expanded around its geometric centre in a power series involving Legendre polynomials (Garrett 1951). In prac-tice, the on-axis field at a point whose distance from the centre is z can be expanded as *.(.) = fT.[l + ft(±)' + E,(j-)' + * ( ± ) ' + . . . ] , (4.1) * T48B type S superconducting wire; Supercon. Inc., 9 Erie Drive, Natick, Mas-sachusetts, 01760. The nominal diameters quoted by the manufacturer were: NbTi core 0.005", copper cladding 0.0065", heavy (double) Formvar insulation 0.008". The critical-current rating was given to be 13 A at the field of 80 k G for short sample tests. 37 where Ho is the axial magnetic field at the centre, E2 , E4, E6... are the 'error coefficients' which depend on the coil geometry, and ai is the inner radius of the solenoid. The detailed treatment of this problem can be found in the book by Montgomery (1980). With the aid of the numerical table in that book for 'outside notch' coils, the on-axis field was calculated to the 6th order terms in the expansion, and the number of turns for each of the compensating layers was determined to be 69 for the field inhomogeneity to be less than 5 x 10~4 in a ±1.5 cm axial region. The coil former was made from a 1.905 cm (|") thin-wall tube of 'non-magnetic' stainless steel (type 321), onto which two brass end pieces were soldered. Low temperature epoxy* was applied as the coil was being wound, so as to improve both mechanical rigidity and electrical insulation. Additional insulation between the layers was provided by ultra-thin (~0.002 cm, Zig-zag brand) cigarette paper. The brass end pieces of the coil fitted snugly into the bore of the main magnet, and a thin layer of grease (Apiezon) was used as a low temperature 'glue'. The field inhomogeneity of the completed solenoid was tested experimentally at room temperature. A 20 V peak-to-peak sinusoidal signal was connected to the solenoid. A matched pair of pickup coils** were used as probes to detect the e.m.f. induced by the alternating magnetic field. With only one pickup coil, the gross variation of the magnetic field along the axis of the solenoid could be mapped out, but differential measurements of much greater accuracy could be made by connecting the coils in series-opposition. The field difference AH _ Hz-H0 H Ho * Stycast LN 78058 and Catalyst 11; Dielectric Materials Division, Emerson & Cuming, Inc. Canton, Massachusetts. The epoxy was mixed with 18-20 parts of Catalyst 11 to 100 parts of resin (by weight). ** The coils were wound to be 66 turns with a length of 2 mm and a inner diameter of 3 mm with # 40 AWG copper wire. 38 J 1 I I I I I I I I 3 2 ! 0 - 1 - 2 - 3 DISTANCE (cm) Figure 4.4 Inhomogeneity of the modulation field. Distance along the axis was measured from the geometric centre of the coil, positive upwards and negative downwards. The circle indicates the approximate size and position of the sample. thus obtained is shown in Figure 4.4, and it agrees well with the design criteria. The field-to-current ratio 7 for the superconducting modulation coil was cal-ibrated by the dHvA effect with the conventional weak modulation method. A D.C. current t 0 was added to the sinusoidal current t m sin u>t to provide a magnetic field ho + hm sin ut = 7(t 0 4- t ' m sin ut). The [110] a-oscillation in Pb was detected by a lock-in detector and displayed on a chart recorder. A particular zero-crossing of the waveform was used as a sensitive indicator for this calibration. When t*o was decreased, the main field H had to be increased in order that the recorder pen should not move. Knowing the change in 3d the main field H, the field-to-current ratio of the superconducting modulation coil was found to be 7 = 245.3 G / A . 4.4 Cryostat and Temperature Control Housing the main magnet is a stainless-steel liquid helium dewar with the usual liquid-nitrogen jacket (Oxford Instrument Inc.). Inside this outer dewar is the sample dewar with its tail extending into the main magnet and the modula-tion coil. The inner (sample) dewar was designed by H . Bless and made by the Physics Machine Shop at U B C . By pumping the helium inside the sample dewar, a temperature as low as ~1.2 K can be achieved. Both the outer and the inner dewars contain liquid helium level detectors.* After precooling the main magnet by direct contact with liquid nitrogen, about 16 litres of liquid helium are needed to fill both dewars, and the fill lasts about 5 hours. Figure 4.5 shows a schematic drawing of the cryogenic apparatus. To determine the temperature of the sample, the vapour pressure over the liquid helium bath was measured. The pressure was read on standard mercury and oil (butyl-phthalate) manometers, and then was converted to the bath temperatures using the revised 1958 He 4 Temperature Scale (Brickwedde et al. 1960). A 3.2 mm (|") diameter stainless-steel tube for sensing the vapour pressure had its open end low in the cryostat, just above the liquid level. A l l the experimental data were taken below the A-point (T\ = 2.178 K ) , and then there was no need to correct for the hydrostatic-head effect (Phillips and Gold 1969). With the above mentioned methods and precautions, the accuracy of temperature measurement was better than 0.01 K . To regulate the helium vapour pressure and thus the temperature, a controller of Walker-type (Walker 1959) was adopted. A short section of the pumping line * Model 110; American Magnetics Inc. 40 to manometer inner dewar outer dewar sample holder and pickup coi level detectors liquid helium (4.2K) iquid helium (1.2-42K) helium reservoir of outer dewar modulation coil main magnet Figure 4.5 General schematic cryogenic assembly (not exactly to scale). 41 was replaced by a piece of thin rubber tubing, which acted as a valve controlled by the reference pressure in a closed volume of gas surrounding it. The reference pressure could be set by connecting this container either to a vacuum line or to an external supply of helium gas. This device maintains pressure for at least one half hour without any observable change of the manometer reading. 4.5 Signal Processing A general description of the detection circuitry has been presented in sec-tion 4.1, together with a block diagram (Figure 4.1). We now discuss in more detail some of the features which are of crucial importance for the successful perfor-mance of the experiment. Most of the relevant details are presented in Figures 4.6 and 4.7. 4.5.1 Frequency Response of the Total Loop Because the applied field is ramped in a linear fashion, oscillations with dif-ferent dHvA frequencies F show up as different frequencies / in the time domain. The frequency response of the circuit will therefore affect the accuracy of the experi-ment. This is especially true for the MFB technique because complete suppression of MI can occur only when hj and —4TT(1 — S)M are exactly in phase. The pickup coil, designed and wound by A. J. Van Schyndel, consists of two counter-wound coaxial coils wound on top of each other. It was balanced at room temperature to be insensitive to a uniform field, and the final configuration had 9500 and 6016| turns of # 46 AWG copper wire for the inner and outer coils respectively. A lead from the centre tap of these coils was made available at the top of the cryostat to fine-tune the balance as the coils were cooled. The balancing could be done to an accuracy of about 1 in 105 (Van Schyndel 1980). Good compensation of the pickup coil is necessary to prevent overloading the PAR 113 preamplifier, and to 42 anced coil ba pickup balancing circuit SPECTRUM ANALYZER HP 3 5 8 2 A trigger channel signal channel PREAMP FILTER PAR 113 KH 3 3 2 2 R 0 10K A / W 10K triangular wave to ramp shaper A sync. FUNCTION GENERATOR HP 33 I2A dM gain dt adjust integrator invert or M gain adjust Figure 4.6 Schematic circuit diagram of the detecting system (I). For details of the integrator reset circuit see drawing for MFB integrator E94. identical phase compensating circuits triangular wave ^ from function generator squarer inverter adder Ramp Shaper power amplifier modulation coil 5Q non-inductive resistor Figure 4.7 Schematic circuit diagram of the detecting system ( I I ) . For details of the ramp shaper see drawing SD-G130-1. prevent a systematic amplitude error in the spectrum analysis. Use of spherical samples ensures a uniform magnetizing field inside the crystal, while outside the sphere we have the inhomogeneous field from a point dipole (Figure 4.8). It is this dipole field which links with the compensated pickup coil. To make sure that the field of the modulation coil faithfully follows the sum of the triangular waveform from the function generator and the M waveform from the integrator, a tap was taken between the 5 ft non-inductance monitor resistor* and the modulation coil (see Figure 4.7). This tap was used to provide a feedback voltage for the operational amplifier (adder) feeding the power amplifier. Using this arrangement, the system could deal with signals of up to 1 kHz frequency, above which the system began to oscillate. Because of this oscillation problem, the upper cutoff frequency of the PAR 113 preamplifier was never set higher than 1 kHz, which in turn limited the frequency response of the total loop. It was found that the noise level could be reduced by inserting a Krohn-Hite 3322R filter, with the same bandwidth (DC-1 kHz) but twice the attenuation rate beyond the cutoff frequency, between the output of the PAR 113 and the input to the integrator ('dMjdt gain adjust' point in Figure 4.6). An experimental procedure was devised for testing the frequency response of the total circuitry at liquid-helium temperature and in an applied field of ~60 kG. In our method, a small coil was used as a 'dummy' sample.** A 1 kft resistor was connected in series with the coil in order to make the voltage across the pair track the current at frequencies of DC-1 kHz. In the first part of the test, the 'dummy' was energized by the reference signal from a phase-sensitive lock-in detector (PAR 124) * DALE NH-100. The resistance was calibrated to be 4.9822 O, and the induc-tance was measured to be 0.5 J I H . ** The dimensions of the coil itself were i.d. 0.254 cm, o.d. 0.296 cm, with 200 turns of # 46 AWG copper wire. The resistance and inductance of the coil were 29 ft and 95 pH respectively at room temperature. 4 5 Figure 4.8 Detection arrangement. The spherical sample with radius a is situated within a carefully balanced pickup coil. The compensated coil is insensitive to changes in the uniform external field H + hm + hj, but responds to changes in the nonuniform part of the induction outside the sphere (field of a point dipole at the centre of the sphere, after Gold and Van Schyndel 1981.) 46 to simulate the dHvA signal, and the phase of the signal from the output of the voltage follower (i.e., '—cM to (U)' point in Figure 4.6) was compared with the phase of the original energizing (reference) signal by the lock-in detector. In the second part, the reference signal was fed to the input of the adder (i.e., '—cM from (I)' point in Figure 4.7) and the phase of the Am signal induced in the small coil was compared by the same PAR 124 with the phase of the reference signal. The total loop phase shift should be the sum of the two, and is shown in Figure 4.9. Part of the phase shift came from setting the bandwidth of the PAR 113 preamplifier to be DC-1 kHz, a fairly low high-frequency roll-off point. The other part must be due to effects which are rather difficult to tie down specifically, such as the capacitance between the stainless-steel dewar tail and the modulation coil, and the coupling between the modulation coil and the main superconducting magnet. "i—i i i I 11 1 1—i • i i i i i i 1 1 — i — i i I i 0°<> x V) ID CO < X Q_ -80° -J — I I I I 1111 I I I I I 1111 1000 FREQUENCY (Hz) Figure 4.9 Total loop phase shift without compensating circuit, as the sum of two parts discussed in the text, measured at liquid helium temperature, in the field of 60 kG, with the PAR 113 bandwidth DC-1 kHz. 47 The total-loop frequency response was improved by inserting two identical RC circuits as 'phase compensators' at the two adder inputs (Figure 4.7). The final arrangement showed a phase shift less than ±1°, and a normalized gain variation less than 3% over the range 1 Hz< / < 200 Hz (figure 4.10). The optimum values of C and R to be used in the phase-compensation circuit were found by experiment to be 0.396 pF and 629 ft, respectively. 1—i I I I I 11 1 1—I I I i I 11 1 1—I i\ I II I FREQUENCY (Hz) Figure 4.10 Total loop frequency response with the phase-compensation circuits (solid line—phase shift; dashed line—normalized gain). Note the expanded scale for phase shift when comparing with Figure 4.9. 4.5.2 Spectrum Analysis As discussed in section 4.1, a triangular wave was used to provide a current ramp in the modulation coil, and only the dM/dt signal from the rising portion of the triangular wave was integrated; the integrator was reset to zero during the 48 falling portion. The integration and reset modes were switched automatically by a sync signal from the function generator providing the triangular waveform. The dM/dt signal fed into the integrator (i.e., the signal from the output of the Krohn-Hite filter, see Figure 4.6) was recorded in the same rising portion by a 256-channel Hewlett-Packard 3582A spectrum analyzer, which provided the amplitude and the phase information. The HP 3582A spectrum analyzer is essentially a microcomputer specially programmed to perform fast Fourier transforms (FFT), and it is equiped with a cathode-ray tube (CRT) display. The input signal is recorded during a finite time T (the window width) and is converted into a set of 1024 discrete points with spacing At = T/N(N = 1024). This data set is then transformed into a set of TY = 1024 frequency points with spacing Af = 1/T in the frequency range [—F, F] with F — NAf/2 = N/2T. Both positive and negative frequency points are calculated but the negative points provide no new information. The upper half of the positive frequency range [F/2,F] is then discarded as being potentially aliased, and finally only N/4 = 256 frequency points within the range [0,F/2\ are displayed on the CRT. For a general description of FFT, see, for example, Higgins (1976). As a result of the finite time window T, the peaks in the spectrum will be broadened and sidelobes will appear. The broadened spectrum and the sidelobes are described by the usual diffraction function «•>[»(/ - fo)T] tU-htr (4'2) with the first zeroes at / = f0 ± 1/T for the central peak, giving a peak-to-zero width 1/T. Since the spacing in the frequency domain is Af = 1/T in a FFT, there are theoretically no more than two sampling points per peak in the amplitude spectrum. The apparent height of the peak (thus the accuracy of the measured amplitude) depends very much on how close the actual frequency fo comes to one 49 of the sampling points / * = k/T (with k — 0,1,2 . . . ,T/4). Figure 4.11 shows two extreme cases in which one is 'on channel' (/o = k/T), the other is 'off-channel' (/o = (k + \)/T). The apparent amplitudes are 1.00 and 0.64 times the real amplitude, respectively. Because the time frequency of the dM/dt signal depends on both the amplitude and frequency of the modulation field (see below), care was always taken during the experiment to make the frequency in question (and hence its harmonics) lie at the centre of the peak by fine adjustment of either the modulation amplitude or frequency. CO >-cr < cr m < UJ o Z3 \~ Z o < 2 O N - C H A N N E L S A M P L E S •A O F F C H A N N E L S A M P L E S sin(wT) /CJT F R E Q U E N C Y ( A R B I T R A R Y U N I T S ) Figure 4.11 Apparent spectrum width and amplitude height (from Higgins 1976). The F F T of a sine wave within a finite window T samples 8m(irfT)/irfT at intervals of 1 /T. When frequency / 0 of the sine wave is commensurate with 1 /T, the samples all lie at zeroes except for one at the top of the central lobe. On the other hand, when /b lies midway between commensurate points (/0 = (k + %)/T), the discrete spectrum samples all of the sidelobe peaks, and there are two points of reduced amplitude on the central lobe. 50 The small number of points per peak is basically insufficient to determine the amplitudes reliably, and the large sidelobes can hide spectral components of lesser strength. This difficulty can be circumvented by use of a time-window weighting function which increases the width of the peak (and hence the number of points per peak) and decreases the height of the sidelobes, all at the cost of losing fre-quency resolution. Three different windows are available for the HP 3582A. The uniform window is the result of using no weighting. It has the worst amplitude error (—36% off-channel) but the best frequency resolution (with a peak-to-zero width equal to the frequency spacing A / = 1/T). In contrast, the Bat-top window is optimized for minimum amplitude error (—1% off-channel) but the worst fre-quency resolution (peak-to-zero width 5A/). Lastly, the Hanning window offers a compromise between the flat-top and the uniform windows. Its worst (off-channel) amplitude error is -16% with a peak-to-zero width of 2A/ for the central peak. In most of our experiments, the Hanning window was used with the above mentioned precaution of always keeping the frequency of interest at the centre of the spectral peak. The flat-top window was used only in cases where the frequencies were quite separated so that resolution was not a serious problem. The appropriate strength of the modulation field was determined by the re-quirement of the spectrum analyzer. When the Hanning window is used, at least four or five cycles of the lowest frequency component are required in order to achieve a reliable spectrum. The lowest dHvA frequency in our experiment was F = 18 M G (fundamental of 7-oscillation), so that the field spacing at a field of about 60 or 70 kG was AH « IP/F « 200 G, and thus a ramp of 1 kG peak-to-peak was needed. Use of a linear field ramp transforms a dHvA oscillation of frequency F in the 51 1/H domain to an oscillation of frequency / in the time domain with , _ d ( F \ ^ 2Fh t f~Tt\H0 + hm(t)) ~ ~BJ/MOD' ( 4 3 ) where Ho is the steady background field, h the peak-to-peak modulation ramp amplitude, and / m o d the repetition rate for the modulation ramp. Circuit char-acteristics set a limit of / to be less than 200 Hz (Figure 4.10). The highest dHvA frequency involved in our experiment is F = 480 M G (the third harmonic of a-oscillation), and we have chosen / m o d = 0-5 Hz at a field of 60 kG to give / = 130 Hz, comfortably within the 200 Hz bandwidth. Because the dM/dt sig-nal was recorded only in the rising portion of the modulation ramp, the time window on the analyzer should be a little less than l / 2 / m o d = 1 sec, triggered at the be-ginning of the rising ramp. The synchronization was achieved by using the same trigger signal which controlled the switch for the integration and reset modes. This is shown in Figure 4.12. The phase spectrum was also provided by the HP 3582A. For a given waveform sin(27r/f + <p), its phase constant <p is defined by comparing with a reference sinusoid with the same frequency / , crossing zero with a positive slope at some arbitrary time t = 0 (or any convenient t — m/f with m = 1,2,3...). Associated with this fundamental reference are harmonics at frequencies rf crossing zero with a positive slope at the same time t = 0. The HP 3582A measures phase by filtering off the frequency in question, and comparing it with the appropriate hypothetical reference sinusoid. 4.5.3 The Ramp Shaper The fact that Af is a periodic function of 1/H rather than H leads to the result that the time-domain frequency / of the dM/dt (or Af) signal will not be 52 500Gr FIELD MODULATION 0 -500G SYNC SIGNAL INTEGRATOR INTEGRATE RESET INTEGRATEl RESET 1 TIME OPEN CLOSED 1 OPEN | CLOSED WINDOW Figure 4.12 Synchronization of the integrator and the time window. 53 constant if the sweep of the modulation field is truly linear with time. Expanding (4.3) to the second order gives j _ 2 F / t / m od / j _ 4 / l / m o d A _ , „ ( ! _ « & - , ) , (4.4) with fo = 2F/i/mod/HQ referring to the frequency at the mid-point of the ramp. The instantaneous frequency at the top will be less than / 0 by the amount f0h/Ho, and likewise the instantaneous frequency at the bottom of the ramp will be greater than fo by the same amount. Thus the instantaneous frequency changes by a total of 2foh/Ho over the duration of the ramp. In our usual experimental conditions with h — 1 kG and H0 — 60 ~ 70 kG, this amounts ~ 3% of / 0 . Both theory and numerical calculation show that this varying instantaneous frequency will broaden the spectrum peak and reducing the apparent amplitude. To eliminate this effect, we ideally want the phase of the dHvA oscillation to be linear with time, i.e., 2 * F « r . 2 * F l. P N = -2TT/O< + — - (4.5) H0 + h{t) ' v Ho where t = 0 is chosen to be the zero-crossing of a rising ramp, i.e., h(t) = 0 with t = 0, and the negative sign is used because the phase decreases when h(t) increases. Then (4.5) becomes If we expand the right side to second order, and replace fo by 2F / i / m o d/-ffoi we finally have MO = + {2hf™At)\ (4.7) From the above argument, we see that the instantaneous time-frequency can be held constant to second order if we add to the linear modulation ramp a field which 54 varies as the square of the linear ramp field and is inversely proportional to the main steady field (Figure 4.13). This was achieved by inserting a 'ramp shaper' between the triangular wave function generator and the adder (Figure 4.7). Since the modulation field is rather strong, undesirable coupling between the modulation coil and the main magnet is expected. The modulation field induces an e.m.f. in the main superconducting magnet, which causes a fluctuation of the main field. Unfortunately the current regulator in the energizing circuit of the main magnet cannot suppress this interaction completely. In normal conditions, the amount of the fluctuation is < ± 5 G, as inferred from the voltage ripple across the monitoring resistor in the energizing circuit. Because the fluctuation occurs at the frequency of the modulation field, we treat it as a second-order perturbation in the modulation field, and the ramp shaper is just the correct instrument to suppress it. - The main parts of the ramp shaper are an IC package to perform the squaring operation and an adder. Figure 4.7 contains the schematic diagram of its circuitry, which is self-evident. The correct amount of the quadratic contribution can be found experimentally for a particular value of the main field H0 by adjusting the ramp-shaper setting until the spectrum analyzer gives the narrowest and highest spectral peaks. Figure 4.14 shows an example of this procedure. The quality of the spectrum was indeed improved with the ramp shaper cor-rection. This can be clearly seen in Figure 4.15. 55 Figure 4.13 The modulation field with and without the ramp shaper: a) linear sweep without the ramp shaper, b) quadratic field which is being added to the linear sweep, c) the resultant modulation field with the ramp shaper. 56 Figure 4.14 Finding the correct amount of the quadratic contribution added to the linear ramp. The amplitude (normalized to the value with shaper off) reaches a maximum when the correct sign and amount of quadratic correction is used. The data shown here refer to the [100] /? oscillation in Pb in an applied field of 56.38 kG. Of I I a) b) c) Figure 4.15 The effect of the ramp shaper. Spectra show [ 1 1 0 ] 7,a oscillations and their MI sidebands in Pb at H = 67.95 kG and T = 1.2 K: a) without ramp shaper; b) optimum setting of ramp shaper, with which the peaks are highest and the spectral lines are narrowest; c) with incorrect sign of the quadratic term, spectrum is badly distorted. 57 C H A P T E R 5 S A M P L E I N H O M O G E N E I T Y A N D M A G N E T I C F E E D B A C K 5.1 Influence of Sample Inhomogeneity on Magnetic Interaction and Magnetic Feedback Sample inhomogeneity plays an important part in the MI. In chapter 3, we have introduced the MI effect and have discussed its suppression by the MFB tech-nique assuming implicitly that the sample is homogeneous. However, in practice it is impossible to prepare a sample without any defects, and the MI problem then becomes more complicated. The difficulty comes from the fact that the lo-cal magnetization M# is no longer uniform. Then in principle the MI cannot be compensated for every location in the sample by feeding back a uniform magnetic field derived from the average magnetization M over the sample.* Thus it is cru-cial to discuss in this chapter to what extent the MFB technique can be applied to an inhomogeneous sample. Can the MFB technique be used with inhomogeneous samples to help us to obtain reliable <7c-factors? To study this problem, we have to first set up a model for the local mag-netization MR as a function of the variable dHvA frequency FR throughout the sample. The next step will be to find the average magnetization M, which is the detectable quantity in the experiment. This problem is simple when the MI is negligible. For a sample (see Figure 5.1), we assume that locally, over a region JR, * M should not be confused with the steady magnetization. 58 Figure 5.1 Schematic diagram showing an ellipsoidal sample of average magneti-zation M in an applied field H . In different regions R, the local magnetization M/z is different (after Shoenberg 1976). which is large compared with the cyclotron orbit size, but small enough so that the local magnetization MR is uniform, the magnetization has the L K form with dHvA frequency F R and amplitude AR, in which the Dingle damping factor is determined only by impurity scattering. We next assume that AR has the same value Ao from region to region, and that it is the phase which varies throughout the sample due to a small spread of the dHvA frequency F R . This spread can be caused by a spread of the crystal orientation (mosaic structure), or by the varying strains in the sample (due to defects such as dislocations). Although we cannot derive the distribution function of the dHvA phase from first principles, the phase is assumed to be sym-metrically distributed about its average value <p, and we take the distribution to be Lorentzian for computational convenience. For simplicity, we write the local 59 dHvA magnetization as MR = AQ sin(^ + A<p), (5.1) and the normalized distribution function as 1 D ( A i p ) = f (5.2) ^ l + ( r f ) where p is the average phase, Ap the departure from the average phase Ip, and r^ > the half width of the distribution function. Obviously Tp, A<p and T^ , are re-lated to the average dHvA frequency F, the deviation AF and the half width IV, respectively, by Jp = 2nF/H Atp = 2ir AF/H and Tv = 2trTF/H. The average magnetization is then obtained from /+ 0 O A0D(A<p) sin(^ + A<p)d(A(p) -oo = A0e~r,fi s'mp = As'mlp (5.3) with i4 = Aoe~r,p, thus resulting in a Dingle-like exponential reduction factor. We can therefore define a Dingle temperature for this phase smearing by r" = V ' (5-4) 60 so that exp 2nTF" H . = exp (5.5) and then A = AQK. This is exactly of the same form as (2.6), where the Dingle reduction is due to ordinary electron scattering. Thus in the absence of any significant MI, the two kinds of Dingle reduction are not distinguishable. Usually in pure metals at low temperatures, where scattering by impurities is very weak, the observed Dingle re-duction is predominantly due to phase smearing arising from sample inhomogeneity. When the MI is important, however, the calculation becomes much more com-plicated, and two different treatments have been developed successively by Shoen-berg (1968, 1976). In the first treatment, the MI is assumed to be related to the average of magnetization. Thus the phase smearing is done first and then the effect of MI is calculated as if the sample were homogeneous with a magnetization corresponding to the average value M . We call this the 'phase smearing before MP (PSMI) treatment. Shoenberg calls it the 'old' treatment because it had been developed earlier than the other treatment. On the contrary, in the other treatment, the MI is assumed to be a local phenomenon. Thus the effect of the MI is calculated first for each local region, and only after this is the phase smearing carried out over the whole sample. We call this procedure the 'MI before phase smearing' (MIPS) treatment. Shoenberg calls it the 'new' treatment because it was developed later. The essential difference of the two treatments is the order in which the calcu-lations are carried out, namely, 61 PSMI: phase smearing before magnetic interaction (Shoenberg's 'old' treatment), MIPS: magnetic interaction before phase smearing (Shoenberg's 'new' treatment). In the following we will discuss the M F B problem in an inhomogeneous sample for the two different treatments. 1. The PSMI treatment.-According to the PSMI treatment, the phase-smearing calculation is carried out first, so that the magnetization appearing in the M I calculation is the average magnetization M rather than the local value MR. In the presence of M F B , the feedback field hj, which is proportional to the volume average of the magnetization M , is subtracted from the externally applied field so that, for a sample with a second-order surface and with the applied field H parallel to one of its principal axes, the magnetic induction inside the sample becomes e = l means no feedback (feedback gain £ = 0), and e = 0 refers to the optimum feedback = 4ir(l - 6)). Comparison of (5.6) with (3.1) leads to the conclusion that to obtain the measured quantities (resultant amplitude, phase) in the presence of M F B we need B = H + 4TT(1 - 6)M - cM = H + 4ir{\ - 6)eM (5.6) where the 'feedback factor' e is defined by 4TT(1 - 6)' (5.7) 62 only replace M by Af, and replace 4JT(1 — 8) by 4;r(l — 8)e in all the formulae in section 3.1 for no feedback. 2.The MIPS treatment. In this treatment, the MI is calculated first, thus the magnetization involved is the local value MR rather than the average value M. The difficulty is that the local value of induction BR is determined not only by the local magnetization MR in the same region, but it is also influenced by the magnetization elsewhere in the sample. The determination of M as a function of the variable dHvA frequency FR throughout the sample leads to a complicated implicit equation for which an explicit solution has yet to be found. Shoenberg (1976) suggested a simplified model which worked well at least qualitatively. We shall follow his procedure. We assume that outside the small region with local magnetization M J J , the magnetization can be treated as having a uniform value M , i.e., the average magne-tization over the whole sample. If the region R is also an ellipsoid with a principal axis parallel to H, both M and W/LR are parallel to the applied field H. Let the demagnetizing factors for the region and for the whole sample be 6R and 8, respec-tively. Then inside the small region, BR is given by Further simplification can be made by assuming that the shape of the small region is the same as that of the whole sample, so that we may set SR = 8. Finally we have the local magnetic induction BR = H + 4TT(1 - 6R)MR + 4TT(6R - 8)M. (5.8) BR = H + 4TT(1 - 8)MR . (5.9) 63 In the presence of MFB, the local induction within the sample becomes BR = H + 4ir(l - 6)MR - cM = H + 4TT(1 - 6)eM + 4JT(1 - 6)(MR - M). (5.10) The local magnetization is then M * = E E 8 i n[ 2 ? r r(fe _ 7 ) T 9 • ( 5 1 1 ) o r b i t s r With the same symbols as denned in (3.2), i.e., x = 2 i r ( - - 7 ) and « = _ ( ! - « ) , (3.2) equation (5.11) can be rewritten as MR= E E S I N {rlx ~ EKM ~ K ( M R - A?)l T ^  + rAtp} (5.12) o r b i t s r provided that \eK,M +K(MR — M)\ <S 1. When the phase smearing is finally carried out we obtain Ar,oD(A<p) sin | r [ i - e/cM-K{M R - M)] =F - + rA<p\d(A<p). o r D i t s r - ° ° (5.13) We can see that the complication in the MIPS treatment arises from the extra term 4JT(1 — 8)(MR — M) in the magnetic induction BR. Two interesting cases of practical importance will be discussed in the following sections for both P S M I and MIPS treatments, and the theory will be compared with the experimental data. 64 5.2 Magnetic Feedback with One Fundamental Frequency and Its Harmonics In this section we discuss a single extremal orbit on the Fermi surface so that only one fundamental dHvA frequency and its higher harmonics are involved. As discussed in the previous section, to derive the formulae with MFB by using PSMI treatment, we simply replace Af in the formulae for no feedback by Af, and 4TT(1 — 6) by 4JT(1 — 6)e. Thus in the presence of MFB, K should be replaced by eK throughout Table 1, and in particular the apparent amplitudes of the first three harmonics become A\ = Ay + 0(3) + • • • AI A r. 1 ^KAI i / € / C i 4 ? \ 2 i 3 f _ 3 ^ 4 , 1 t ^ M ^ T _ i ( + (-8A7±) } + Turning now to the MIPS treatment, no such simple substitutions can be found which allow us to relate the formulae in the presence of MFB to those where feedback is absent. We still use the Phillips-Gold iteration procedure. Starting from equation (5.12),the nth order approximation can be given by MRn)=f^Ar,0sm{r[x-eKM(N-R)-K(M^ (5.15) r=l 4 with Afj^ = 0. The phase smearing is then given by A f ( n ) = D{A<p)MRn)d(A<p). (5.16) J — OO (5.14a) (5.146) (5.14c) 65 The mathematics of the iteration is even lengthier than that for a homogeneous sample, treated in section 3.1.1. However, to the lowest order it is found that the results for the observed amplitudes in (5.14) for the PSMI treatement can be applied equally well to the MIPS treatment. It is only for the higher order terms in equations (5.13), i.e., for 0(3), 0(4), 0(5).. .etc., that the difference between the PSMI and MIPS treatments show up. According to MIPS, these higher order terms cannot be suppressed completely by MFB. Our experiment has shown no significant difference between the PSMI and the MIPS treatments. This is an indication that the influence of these higher order terms is negligible for lead under our experimental conditions. Thus we conclude that in the case of only one extremal orbit on the Fermi surface, MI can be effectively suppressed by using the MFB technique, even if the sample is inhomogeneous. 5.3 Magnetic Feedback with Two Fundamental Frequencies Now we will discuss the situation in which two dHvA frequencies from differ-ent extremal orbits are involved. It is in this situation that the effects of sample inhomogeneity show up very directly and that the two treatments give significantly different results. 5.3.1 General Formulation Previous discussions have shown that formulae for the PSMI treatment can be easily derived by replacing Ka,b by eKa,b, and M by M . Thus equations in section 3.1.2 for no feedback become, in the presence of MFB, + 00 M = Aa E Jk(e*aAb) sin(za - kxb) k= — oo +oo + Ab E MeKb^a) sin(xb - lxa). (5-17) l=-oo 66 In particular, the apparent amplitudes of the two fundamental components are A'a = A aJ 0(e/c oA&) (5.18a) and A'b = AbJo{£KbAa). (5.186) The two lowest order sidebands with frequencies Fa ± Fb have amplitudes As we have pointed out in section 3.1, if a refers to a weak high-frequency oscillation, and 6 to a strong low-frequency oscillation, then if e/c&Aa <S 1, J0(eKt,Aa) —• 1, and we have A'b = Ab. From (5.18) one sees that when e = 0, i.e., f = 4JT(1 — 6), A'a = Aa, A'b = At,, and Aa+b = A'a_b = 0. Thus according to the P S M I treatment, M I can be com-pletely suppressed when the M F B gain is optimized. In particular, all sidebands of frequencies mFa ± riFb should then be absent. T he feedback gain dependence of apparent amplitudes A'a, A'b, A'a+b and A'a_b is indicated schematically in Fig-ure 5.2. For the MIPS treatment, we use the same notation as in (3.12), with subscipts d and 6 again denoting two frequencies from different extremal orbits. T h e local magnetization is (compare with equations (3.14) and (5.12)) and A'a+b = -AaJi(eKaAb) — AbJi(eKbAa) A'a_b = AaJi(tKaAb) - AbJ\(e.KbAa). (5.18d) (5.18c) MR = Aa0sin|xa + A<pa - KaMR + Ka{l - e)M + Abo sin xb + A<pb - KbMR + *ct(l - e)M\ (5.19) 67 0 Q5 1 FEEDBACK GAIN fAjr(1-6) 1.5 F i g u r e 5.2 Schematic sketch showing, the apparent amplitudes A'a, A'b, A'a+b and A'a_b as a function of feedback gain according to the P S M I treatment. M F B should suppress MI to zero when f/4fr(l — 6) = 1. As before, we ignore terms causing higher harmonics, and replace MR and A f by their first-order approximations. Then (5.19) becomes (5.20) 68 With the aid of the expansion (3.15), we finally have +00 +00 MR = Aao 52 \52 ^fc(«aA6o)^/((l - e ) * ^ ) s i n J (x a + A<pa) - k(xb + A<pb) + / X f c ] fc=—00 /=—00 +00 +00 + Ab0 52 52 Jk{KbAa0)Ji([[l-€)KbAa^8m^{xb + A<pb)-k(xa + A<pa) + lxa k=—00 /=—00 (5.21) We are now ready to examine the effect of phase smearing, i . e . , average over the whole sample. In the general form, this average involves phases A<pa - kA<pb or A<pb - kA<pa, and the results will clearly depend on how the phases A<pa, A<pb are correlated with each other. In practice, only A; = 0 and k = ± 1 terms are important, and we will discuss them respectively. The relevant integrals for k = 0 are straightforward, namely, /+00 D(A(pa) sin(x + A<pa)d(A<pa) = Ka sin x -00 and •+00 /0  D(A<pb) sin(x + A<pb)d(A<pb) = Kb sin x; -00 the integration simply gives rise to Dingle reduction factors Ka or Kb as defined in(5.5). O n the other hand, for k = ± 1 , i . e . , for A<pa ± A<pb, the correlation between Aipa and A<pb is very important. For example, if A<pa and A(pb are completely 69 uncorrelated, we must consider the integral /+ oo r + oo j D(A<pa)D(A<pb) sin(x + A<pa ± A<pb)d(A<pa)d(A<pb) = KaKb sin x. -oo J —oo Whereas if A<pa and A<pb are completely correlated with A(pb = ZA<pa, (5.22) where £ is a positive or negative constant, we then have only one integral f+°° D{A<pa) sin [x + (1 ± 0Apjd(A<p a ) = Kll±(\ sin x. J —oo L From (5.22) for a Lorentzian distribution, we have r6 = |£|r a, (5.22a) thus Kb = KW . (5.226) But the values of Kal±^ in terms of Ka and Kb depend both on the sign of £ and on whether £ is smaller or greater than 1, as indicated in Table 2. In general, we can define two correlation factors p + and p~ with respect to A<pa + A<pb and A<pa — A<pb, so that the reduction factors can always be expressed as p+KaKb and p~KaKb respectively.* Thus in the completely uncorrelated case, p + = p~ = 1, whereas for perfect correlation, p + and p~ are given in Table 2. In between lies a grey area of partial correlation for which p + and p ~ can be other values provided Our p + and p correspond to p and q, respectively, in Shoenberg (1984). 70 Table 2 Values of Reduction Factors p + and p~ when A<pt = £A<pa (Shoenberg 1976, 1984) * P+ t\<Pa ~ A^fc o<e<i KaKb 1 jjp(KaKi,) 1 1 •jp(KaKb) 1 KI - 1 < t<o 1 KaKb 1 Z < - \ 1 KaKb 1 * The quantities p + and p~ correspond to p and q, respectively, of Shoenberg (1984). A slightly different definition for p and q appears in Shoenberg (1976). Ka,Kb < 1. that both p + KaKb and p KaKb must be less than 1. Actual values of p + and p~ can be estimated by experiment, as will be discussed later. Now we consider the components of magnetization with frequencies Fa, Fb, Fa + Fb and Fa - Fb respectively. We assume Fa » Fb and Aa < Ab, so that the amplitude of Mb is essentially not affected by either MI or MFB. Therefore we have A'b = Ab. (5.23) Only the components with frequencies Fa, Fa+Fb and Fa—Fb need to be considered. 71 (1) Fa. Letting k = I, the first term of equation (5.21) will give +00 MRa = Aa0 52 Jk(KaAb0)Jk ((1 - e)KaAb^ sin(i a + &<pa - kA<pb). (5.24) fc=—00 After separate consideration of the phase smearing of the first three terms (k = 0 , ± 1 ) in the series (5.24), we have the apparent amplitude of the a oscil-lation in the presence of MFB as A'a = ^ [ M ^ ) J o ( 4 £ * ) + (P + + P l i T | J , ( ^ ) J , ( 7 ^ y ) ] , where Aa = AaoKa. Since the arguments of all the Bessel functions are assumed to be small (weak (5.25) MI) we expand (5.25) to the second order terms, and have . (5.26) We now find, for given p + , p , i) A'a reaches its maximum when the MFB gain is set to be <r = & = 2 4TT(1 - 6) (5.27) rather than 4JT(1 - 6) for the PSMI treatment. max is no longer equal to Aa. In general, Aa ~ (K) Aa max [*r>-(^)1- (5.28) 72 (2) Sidebands Fa + Fb and Fa - Fb. In the general formula (5.21), terms with / = k + 1 in the series will give rise to the sideband with frequency Fa + Fb, whereas terms with / = k — 1 the Fa - Fb sideband. Thus we have + 0O MR,a±b = Aa0 E Jk{KaAbo)Jk±i({l - e)KaAb^ 8in[(xa ± xb) + A<pa - kA<pb] fc=—oo + 0O + Ab0 E Jk(KbAao)Jk±i({l - e)KbAa^sm[{xa±xb) + A(pb-kA<pa . k=—oo (5.29) If we take only the dominant terms, i.e., terms with JQ(X) and J ± i ( x ) , into account, after phase smearing over the sample, we finally have, for the apparent amplitude of the Fa + Fb sideband, (5.30) and for the Fa — Fb sideband, A'a_b = Aa[p~ KbMKaAboUo^^—^j) ~ M*a A^) J X ( ^ " " ^ j ) ] - Ab[p~KaJi(KbAao)Jo(4^ag)) ~ M^o)^ ( J^^j)] • (5.31) 73 The second order expansion of the Bessel function gives, for the sideband amplitudes, < + l = - ^ ( ^ + « . ) [ p + - i ^ r 7 y ] , (5.32) ^ = ^ *-*>[»--i^rij]- <5-33' Therefore, to reduce those amplitudes to zero, different MFB gain settings are needed for the two sidebands, namely $+ = 4TT(1 - 6)p+ (to make A'a+b = 0), (5.34a) = 4TT(1 - 6)p~ (to make A'a_b = 0). (5.346) We note that the feedback gain ft at which A'a reaches its maximum is (see equation 5.27) just the average of t + and i.e., 9, = - * ) - £ t ± f c . (5.35) The feedback gain dependences of the apparent amplitudes A'a, A'b, A'a+b and A'a_b are shown schematically in Figure 5.3. From the above discussion, it is found that if the sample is inhomogeneous, and if there are two dHvA frequencies Fa and Fb arising from different extremal orbits, the PSMI and MIPS treatments lead to different results (compare Figures 5.2 and 5.3). According to the former, the effect of MI can be completely eliminated if the MFB gain is optimized by setting the gain t = 4TT(1 — 6); while according to the latter, the effect of MI can only be partially suppressed with MFB technique, and different MFB gain settings are needed to minimize various sidebands mFa + nFb respectively. Yet another gain setting is required to maximize the apparent 74 0 p- p + FEEDBACK GAIN {/Wl-6) Figure 5.3 Schematic sketch of the apparent amplitudes A'a1 A'b, A'a+b and A'a_b as a function of feedback gain according to the MIPS treatment (assuming p+ > p~). The zeroes of the two sidebands now occur at different MFB gains, and the maximum of the amplitude A'a occurs midway between those gain settings. For the MIPS treatment, the maximum of A'a is not simply Aa. It is assumed that the amplitude A'b of the strong low-frequency oscillation is not affected by MI and is thus independent of MFB gain. amplitude A'a, and in general (A'a)mBLX ^ Aa. Experiment is needed to judge which treatment is closer to the facts. 5.3.2 Experimental Evidence Before this study was started, two experiments were reported whose results tended to favour the MIPS treatment. Both experiments were concerned with the MI associated with two extremal orbits, but neither exploited MFB. The first of 75 these experiments was carried out by Bibby (1976) on an Au sphere where the 'belly' oscillation was both frequency- and amplitude-modulated by the 'neck' oscillation. The experimental results were interpreted as being closer to the prediction of the MIPS treatment rather than the PSMI treatment (Shoenberg 1976). Secondly, experiments were carried out by Aoki and Ogawa (1978, 1979) on the MI between the two sets of oscillations in Pb, and in order to appreciate what Aoki and Ogawa have done it is necessary to introduce some of the experimental features at this point. When a single crystal of Pb is oriented with the applied field along [110], it exhibits strong 7 oscillations consisting of two closely-spaced dHvA frequencies Fnl = 17.64 MG and Fl2 = 18.07 MG. These two components have approximately equal amplitudes, and the result is a beat pattern with about 42.5 cycles in one beat. There are also weaker a oscillations at a frequency of 159.12 M G (Anderson, Lee and Stone 1975).* The MI is expected to distort the a oscillation seriously. When no MFB field is applied, the apparent amplitude of the a oscillation, according to the PSMI treatment, should be given by A'a = AaJ0{KaAi), (5.18a) while for the MIPS treatment A'a = AaJo(taA7o) • (5.25) The arguments of the Bessel functions in these two formulae are different, one involving the amplitude A1 after phase-smearing, the other involving the amplitude A 7o before phase-smearing. Since the two components of the 7 oscillations have approximately equal amplitudes, the beats resulting from their superposition have * For a more detailed description of the Pb Fermi surface , see section 6.1. 76 very narrow waists where the resultant amplitude A1 (or A^o) almost vanishes. As the applied field is varied, the MI effect on the high-frequency cr oscillation can thus be made to range from almost zero at the 7 beat waists all the way to a strong MI at the 7 beat maxima. In their experiments, Aoki and Ogawa measured the absolute values of A'a and A'^ — Alt and worked out the A'a vs. A^ dependence during a 7 beat cycle. Comparing with the predictions of the two different treatments, they also found that experimental results were much closer to the predictions of the MIPS treatment. In their first report on the use of MFB, Gold and Van Schyndel (1981) studied the feedback gain dependence of both A'a and the principal sidebands A'a±lt and they compared their results with the predictions of the older PSMI treatment. They found that while for A'a the agreement was good, but for the sidebands only 'the general trends seem correct'. They noted that the sidebands A'a±^ could never be made to vanish completely and they suggested that sample inhomogeneity might be the reason. They also found (see Van Schyndel 1980) that the minima for the two sidebands occurred at different gain settings, but this was not discussed at all. The major difference between the two treatments is that for PSMI there ex-ists a unique MFB gain setting f = 4ir(l — 6) at which all the MI effects will be eliminated, and thus the minimization of the sidebands gives a natural criterion for setting the optimum MFB gain. On the other hand, for the MIPS treatment this is no longer true. At a beat minimum of the 7 oscillation, there exists virtually only the a oscillation, and the MFB gain required to eliminate MI between the a oscillation and its second harmonic is ft=47r(l-$); (3.19) 77 by contrast when the 7 oscillations are strong (near a beat maximum), to minimize the two sidebands a + 7 and a — 7 requires different MFB gains, namely, f+ = p +4jr(l - 6) and = p_4*r(l - 6). (5.34) If we know the three gain settings ft, f+, and we can in fact determine p + and p~ from p + = and p - = f_/ft. (5.36) In this work the correct ft was found experimentally by measuring the phases of the a oscillation and its second harmonic. The L K theory requires that (see equation 2.12 ) the quantity rtpx - <pr ( the 'relative phase') satisfies r<p, - <pr = r[2*(^ - 7) T - [ 2 i r r ( £ - 7) =F ^  = T ( r - l ) J , (5.37) so that when r = 2, we have 2<pi — <p2 = Figure 5.4 shows the 'relative phase' obtained from phase spectra given by the HP 3582A as the MFB gain was varied, and the LK result 2<pi —<p2=-'K]4 was found to correspond to a gain setting of 4.95 dial units.* To study the effect of MFB on the MI between the a and 7 oscillations, we set the intensity of the applied field in such a way that the 7 oscillations are close to a maximum of their beat pattern, i.e., to give the strongest MI. The oscillations were Fourier-transformed and the amplitude spectra were studied at various MFB gain settings. Figure 5.5 shows the Fourier spectra of the a oscillation and several of its MI sidebands of frequency Fa±nF1 at different MFB gain settings, and in Figure 5.6 * A more detailed discussion of the relative phase measurements will be given later in section 6.3. 7 8 FEEDBACK GAIN (did units) Figure 5.4 Relative phase 2<pi — <p2 of the a oscillation with the applied field H parallel to [110] direction of sample 1, measured as a function of MFB gain. The gain values are given in dial units, which are proportional to c. The measurements were made at the field H = 58.14 kG, which corresponded to a beat minimum of the 7 oscillation. This gave ft = 4.95 dial units when 2(pi — <p2 = —45°. we present the dependence of the measured amplitudes A'a, A'a+^ and J44_7 on the MFB gain. The results are in accord with our conclusions for the MIPS treatment. The minima of the sideband amplitudes A'a+1 and A'a_^ appeared at f+ = 7.15 and = 5.85 dial units respectively. Three samples have been studied with the above-described method and the correlation factors p + and p~ have been estimated from MFB gain ratios according to equations (5.36). These p+ and p~ values are presented in Table 3. 7 9 a Of a-y oe+y 0 1 7 a-y a+y d) e) f ) Figure 5.5 Fourier spectra of the a oscillation and its MI sidebands, at a beat maximum of the 7 oscillation (H = 67.95 kG, H | | [110]), showing the dependence on the MFB gain: a) without MFB; b)-f) with increasing MFB. c) A'a_^ at minimum, d) A'a at maximum, e) A'a+^ at minimum. The frequency range shown corresponds toF = 90 M G to F = 260 MG. 80 1 I 1 1 1 T FEEDBACK GAIN (dial units) Figure 5.6 Fourier amplitudes as a function of MFB gain, measured for sample 1 at H = 60.48 kG and T = 1.19 K. MFB gains are in dial units. Amplitudes are normalized with A 7 = 1. Curves are the least-squares fit to eqations (5.25), (5.30) and (5.31). Experimental parameters used in the fit are Ka = 2.26 G _ 1 , « 7 = 0.257G"1; Ka = 0.54, JT7 = 0.79; and p+ = 1.44, p~ = 1.18. Adjustable fitting parameters Aao and A 7o are found to be 0.181 G and 0.534 G respectively. It is difficult to deduce the exact reason for the phase smearing in our crystals without direct study of the microscopic sub-structures of the samples. However, the p + , p~ values in Table 3 give us a clue. The fact that both p + , p~ are larger than 1 implies a certain degree of correlation between phases &<pa and Ay>7 for the a and 7 oscillations, though the correlation is not complete. Because p~ ta 1 and p + > p - , we may consider equation (5.22) A p 7 = £A<pa 81 Table 3 Experimental p + , p values Deduced from MFB studies of the MI between the [110] a and 7 oscillations in Pb P + = f+/fi P sample 1 1-44 1.18 sample 3 1.30 103 sample 5 1.35 1.06 to be approximately valid with f < 0 (see Table 2). A negative £ value implies that the phase A<pa and A<p^ are oppositely affected in the a and 7 oscillations. In the [110] direction, the dHvA frequency Fa has its maximum value while is at its minimum (Anderson and Gold 1965). If the sample has a mosaic structure with tilts between adjacent grains, such an angular spread will then cause a negative correlation between the two oscillations. To complete our comparison between the experiment and the theory, the apparent amplitudes A'a, A'a+1 and A'a_^ were fitted to the equations (5.25), (5.30) and (5.31); the curves in Figure 5.6 were obtained by this method. In the fitting procedure, experimentally determined values were used for the six parameters Kat /c7, Ka, K^, p + and p~, namely, 8TT2( 1 — 6}F /ca,7 = " 7 (with 6 = 1/3 for a sphere) Kan = expj— Xfi g^'7^] (with measured Dingle temperatures) and p+ = , p" = f_/f!. The other two remaining quantities Aa0 and Al0, the ideal L K amplitudes in the 82 absence of any phase smearing, are evidently not directly accessible and they were therefore treated as adjustable fitting parameters. The best fit for the data from sample 1 was found with Aa0 = 0.181 G , jijo = 0.534 G (which refer to the experimental conditions temperature T — 1.19 K and H = 60.48 kG). The corresponding amplitudes after phase smearing were Aa = KaAa0 = 0.098 G , AlQ = JT7 Al0 = 0.534 G . Figure 5.6 shows good overall agreement between the MIPS theory and experiment, especially if one bears in mind that only two parameters were adjusted in the fitting process. The MIPS treatment of this chapter predicts zero sidebands at appropriate, though different, MFB gains. While the sidebands never vanish completely, we have been able to reduce the residual sideband amplitudes to typically 4 % of their values without MFB. This represents a significant improvement over the earlier work of Gold and Van Schyndel (1981), where the corresponding figure for the residual sidebands was never less than about 20 %. * * * So far we have verified that the MIPS treatment gives a better description of the effects of MFB in an inhomogeneous sample than does the PSMI treatment. Now we can answer the question raised at the beginning of this chapter: Can the MFB technique be used with inhomogeneous sample to help us to obtain reliable t7c-factors? Our conclusions may be summarized as follows: 8 3 (1) When the dHvA oscillations arise from one extremal orbit on the Fermi surface, the influence of sample inhomogeneity is a higher order effect as far as the MFB is concerned. In most cases these higher order effects can be ignored, and the MI can be effectively eliminated with the MFB technique at a unique feedback gain setting. The optimum MFB gain can be determined, for example, by consideration of the relative harmonic phases r<pi —tpr, as in Figure 5.4. Once the optimum MFB gain setting is achieved, the LK form of the harmonic amplitudes, which we need for the <7c-factor measurements, is restored. (2) When the dHvA oscillations arise from two or more extremal orbits, the influence of the sample inhomogeneity is highly dependent on the correlation be-tween the phases of these oscillations, and the MFB technique in general cannot eliminate the MI completely at a single feedback gain setting. For example, to minimize different sidebands requires different MFB gains, thus the minimization of a sideband is not a precise criterion for the feedback gain adjustment. Though we can set MFB gain to be the average of the two gains f+ and f_, at which the Fa + Fb and Fa — Fb sidebands are minimized respectively, and at which average gain the apparent amplitude A'a is indeed maximized, yet (A'a)max is not the L K amplitude Aa (see equations (5.27), (5.28) and(5.34)). Fortunately, in most cases in our experiment, this was not a serious problem. For example, in the [110] di-rection the 7 oscillation can be set to be at a beat minimum so that virtually only the a oscillation exists. There are also cases where the difference between ( j4 0 ) m a x and Aa is small, so that ( i4 a ) m a x can be used as a good approximation for Aa. These will be discussed in detail in the next chapter along with the experimental evaluation of the (fe-factors. 84 To make a long story short, we conclude that the MFB technique is still a powerful means to deal with the MI problem, and it is still helpful in our ^-factor measurement even if the sample is inhomogeneous. 85 C H A P T E R 6 E X P E R I M E N T A L E V A L U A T I O N O F S F O R L E A D 6.1 The Fermi Surface of Lead Before presenting our experimental study of the orbital spin-splitting factors S = a brief discussion of the Fermi surface of lead is appropriate. Since the early work of Gold in 1958 (Gold 1958), extensive study has been made of the lead Fermi surface by means of the dHvA effect (Anderson and Gold 1965, Phillips and Gold 1969, Anderson, O'Sullivan and Schirber 1972, Anderson, Lee and Stone 1975, Ogawa and Aoki 1978, Ogawa, Aoki and Nakatani 1979, Joss 1981) as well as Azbel-Kaner cyclotron resonance (Khaikin and Mina 1962, Mina and Khaikin 1963, see also Agababyan, Mina and Pogosyan 1968; Young 1962, Krasnopolin and Khaikin 1973, Onuki, Suematsu and Tanuma 1977). Other ex-perimental studies include magneto-resistance (Alekseevskii and Gaidukov 1961, Caroline 1969), ultrasonic attenuation (Mackintosh 1963, Rayne 1963, Ivowi and Mackinnon 1976) and Shubnikov-de Haas effect (Tobin, Sellmyer and-Averbach 1969). Several semi-empirical band structure models have been given based on the experimental data (Anderson and Gold 1965, Anderson, O'Sullivan and Schir-ber 1972, Van Dyke 1973, Joss 1981), and also ab initio relativistic band struc-ture calculations using Augmented-plane-wave (APW) method (Loucks 1965) and 86 Figure 6.1 Primitive Brillouin zone for the fee lattice. The coordinates for points of high symmetry are: T = (0,0,0), W = (1/2,1,0), X = (0,1,0), U = (1/4,1,1/4), K = (3/4,3/4,0) and L = (1/2,1/2,1/2) in units of 2ir/a, where a is the lattice constant. Korringa-Kohn-Rostoker (KKR) method (Sommers, Juras and Segall 1972, Neto and Ferreira 1976, Looney and Dreesen 1979). Both the experimental results and the theoretical calculations confirm in detail the correctness of a nearly-free-electron Fermi surface based on four conduc-tion electrons per atom. In what follows we discuss the Fermi surface in relation to the Brillouin zone for the face-centred cubic (fee) lattice (Figure 6.1). The empty-lattice model—ideally free electrons subject only to Bragg reflections—predicts that the first Brillouin zone is full and that the Fermi surface consists of a large closed hole surface in the second zone and a multiply-connected electron surface in the third zone, as shown in Figure 6.2. Figure 6.2 also indicates the various extremal orbits which contribute to the dHvA effect. The empty-lattice model also predicts 87 b) Figure 6.2 The empty-lattice model of the lead Fermi surface (from Anderson and Gold 1965). Surface a is centred on the point T of the Brillouin zone, while surface b may be thought of a 'thickening' of the zone edges KWU (see Figure 6.1). The non-central [111] V2 orbits do not exist due to the rather large rounding by the potential. The orbits K and r are nonextremal with respect to area, and the broken curves depict the open orbits p and /J. small electron pockets in the fourth zone, centred at the zone corner W (see Fig-ure 6.1), but no evidence for these pockets has ever been found with the dHvA effect. Unexpected oscillations of low frequency (~4 MG) have been reported in Shubnikov-de Haas experiments (Tobin, Sellmyer and Averbach 1969) and in sound attenuation (Ivowi and Mackinnon 1976), and it has been suggested that these low-frequency oscillations might arise from electrons in the fourth zone. However, band-structure calculations fitted to the experimental data as well as first-principles calculations show the fourth zone to be empty. The only exception is found in the calculation of Joss (1981), which does give some electrons in the fourth zone, but the calculated extremal cross sections of Joss' fourth-zone pockets are between two 88 to three times too small to account for the low-frequency oscillations found by Ivowi and Mackinnon (1976). Their origin, if they are indeed genuine, remains a mystery. In Table 4, we present dHvA frequencies obtained in high symmetry directions by Anderson, O'Sullivan and Schirber (1972), and by Joss (1981) more recently. These results are in nearly perfect agreement. Table 4 D H v A Frequencies in Lead oscillation orientation zone orbit F (MG) Anderson et al. (1972) Joss (1981) a [100] 2 204.4 ± 0.4 204.4 ± 0 . 1 a [110] 2 159.11 ± 0 . 0 5 159.17 ± 0 . 1 0 a [111] 2 155.8 ± 0 . 4 154.72 ± 0.10 7 [100] 3 i 25.38 1 — 7c [110] 3 18.072 ± 0.004 2 18.06 ± 0.02 7nc [110] 3 17.64 ± 0.06 2 17.64 ± 0.02 7 [111] 3 22.37 ± 0.05 — P [100] 3 V 51.25 ± 0 . 0 1 51.26 ± 0.02 ir [100] 3 i 36.04 ± 0.20 35.64 ± 0.02 6 [111] 3 9 109.5 ± 0.3 109.21 ± 0.10 1 From Anderson and Gold (1965). 2 From Anderson, Lee and Stone (1975). 89 After this brief introduction to the Fermi surface of lead, we now turn to the main experimental matter of this thesis and give an orbit-by-orbit account of our determination of each spin-splitting factor S = \\t.gc (for the purposes of the present chapter it turns out to be more convenient to discuss the results in terms of S rather than gc). Just which approach is best suited for a particular orbit depends on,many factors such as the relative strengths of the d H v A harmonics, interference from oscillations due to other orbits, beating patterns,etc., etc. In short, no universal procedure can be applied to all cases; each orbit is unique. T h i s means that the matter of this chapter is necessarily rather detailed and technical as we proceed from one orbit to the next. The final results for the individual orbits will be drawn together in Chapter 7. 6.2 S for the [110] c Orbit 6.2.1 The 7 Beat Pattern When lead single crystals are oriented with the applied field H along [110], the oscillations with lowest d H v A frequency are the 7 oscillations, which consist of two closely-spaced frequencies arising from a central orbit f c centred at the symmetry point K (or U) and two non-central orbits f n c between K (or U) and W around a third zone electron arm (Figure 6.3). The two components have approximately equal amplitudes and the result is a beat pattern with about 42.5 cycles in one beat period. Obviously the harmonic amplitudes, and hence the two-harmonic ratio Ar/A\ or the three-harmonic ratio a = A\fA\Az will depend on where in the 42.5-cycle beat pattern the data are recorded. 90 Figure 6.3 Model of the third zone electron arm of lead (schematic). The vertical axis is parallel to [110]. In general we can represent the magnetization due to two nearly equal dHvA frequencies Fa and Fb (here Flc and Flnc) by (6.16) and the resultant magnetization will be (6.1a) Af = £ 4 s i n » ^ - 7 U J Af — Af ° + Af*, where 6F — Fa - Fb, 91 F = (Fa + Fb)/2 with Fa > Fb. If we let the beat modulation factor be _ Aa-Ab  V ~ Aa + Ab' i.e., the ratio of the amplitude at beat waists to that at beat maxima, then elemen-tary trigonometric manipulations lead to the resultant magnetization M = + A>)f[(1 + , • ) + (1 - , * ) - » ( ^ ) ] * r x sin 2 * T ( ^ - 7) + <pr (6.2) with the resultant phase furSF u <pr = tan 1 "»(-»—«)J <6-3' and the beat amplitude v/2 . (2urbF\\k A*"* = (A- + Ab)^-[(1 + v2) + (1 - rf) * H ^ ) J • (6.4) Without MI , i.e., B = H, equation (6.4) shows that the beat envelope is also a periodic function of 1/Hy with the repetition rate r6F for the rth harmonic. As a result, the beat length for the rth harmonic should be r times shorter than that for the fundamental. But when M I is significant, the apparent amplitude for a higher harmonic will contain part of the amplitude of the fundamental (see Table 1). Thus the beat envelope for the higher harmonic may in fact follow that of the fundamental (the fundamental amplitude is essentially not affected by MI). Gold and Van Schyndel (1981) studied this effect extensively and found that the 92 shape of the beat envelope for the second harmonic is very sensitive to the MFB gain and can be used as a criterion for setting the optimum MFB gain. They have proved (see details in Van Schyndel 1980) that near optimum feedback, the ratio of two successive beat lengths (i.e., A(1/£T) between two apparent amplitude minima) is linear in the MFB gain; it should be 1 at optimum feedback. As discussed in section 4.5.2, a linear ramp of 1 kG peak-to-peak was used to modulate the 60—70 kG static applied field so that about 5 cycles of the fun-damental 7 oscillations (F = 18 MG) could be recorded. For better resolution, the modulation frequency was chosen to be 1 Hz, and the first three harmonics of the 7 oscillations appeared at 10, 20 and 30 Hz respectively in the time domain. To find out the optimum MFB gain, the amplitudes of 27 (second harmonic of 7 oscillations) were recorded at constant temperature for at least two periods of the beat pattern. Three minima at the fields Hi, H2 and Hz gave two successive beat lengths Pa = \/H2 — 1 /Hi and Pb = I/H3 — 1/H2. Optimum feedback was achieved by adjusting the MFB gain settings until Pa = Pb. At this gain setting, the beat frequencies showed a ratio of 1:2:3 for the first three harmonics (Figure 6.4). The beat envelopes can look quite different when the MFB gain is not optimized (or zero), and illustrations are given in Gold and Van Schyndel (1981). In the three-harmonic ratio method for measuring S, the amplitudes of the first three harmonics are to be measured simultaneously at different temperatures with the applied magnetic field held constant. For better accuracy of amplitude measurement, when beat patterns exist, this field must be carefully chosen so that over one sweep of the modulation ramp, the beat amplitudes j4)?eat do not change too much. It appears that there is such a favoured field within each beat cycle where the beat envelope of the third harmonic is a maximum, and the beat envelopes of both the fundamental and the second harmonic are close to their maximum values. These fields, which were called 'magic fields' by Van Schyndel, are indicated in 93 6 0 0 1 /H (MG - 1 ) Figure 6.4 Beat envelopes for the first three harmonics of the [110] 7 oscillations from sample 3, at T =1.17 K with optimum feedback gain. The beat frequencies scale as 1:2:3. Vertical broken lines indicate 'magic fields' (see text) at which the beat envelopes of all three harmonics are close to their maximum values (maxima for r = 3). 94 Figure 6.4 by Hi and H2, and the former [Hi = 69.45 kG) was chosen as the field at which the three-harmonic measurement was performed. At a 'magic field', the amplitudes of the fundamental and the second harmonic were not exactly at their beat maxima, so that small amplitude corrections were needed to recover the resultant amplitudes Ar = A" + Abr (r = 1,2) from the experimental values A^EAT (r = 1,2). These corrections were made from knowledge of rj and smftwrSF/H) (see equation 6.4). It should be a good approximation to take n = 0 because the beat minima were almost zero. From Figure 6.4, the 'magic fields' are seen to be displaced (in 1/H) by 0.17 beat cycle to the left of the maxima in the fundamental beat envelope, which gave the argument of the sine function to be 0.16TT radians. Thus the correction factor was found for the fundamental to be Ai/A\eat = 1.16, and by the same method, the correction factor for the second harmonic was found to be A2/A\E&T = 1.02. After the above amplitude correction, what we have is A R = A A + A B , where both A " and ABR have the L K form in equation (2.12) since the MFB gain has been optimized. Because A% and A B arise from extremal orbits with nearly equal area on the same sheet of the Fermi surface, it is reasonable to assume that their sum A R also has the L K form, with m*, TD and gc understood to be appropriately weighted averages of the relevant quantities for the two orbits. All the experimental quantities measured for the [110] 7 oscillations should be understood in this way. 6.2.2 Tempera ture Dependence At a constant field, the temperature dependence of the dHvA amplitude A R is given by the L K formula (2.12) to be 95 with T X = A^—, (2.5) where A = 146.9 k G / K and fi = m*/mo. In the high temperature limit (rX > 3), the hyperbolic sine function can be replaced with its exponential approximation Since X = XpT/H, a plot of ln(Ar/T) vs. T/H should yield a straight line of slope —rA/z, i.e., the slope for the rth harmonic should be r times that of the fundamental. This is the conventional method of measuring the cyclotron effective mass from the dHvA effect, and the \n(Ar/T) vs. T/H plot is usually called an effective-mass plot. The approximate form of the hyperbolic sine usually holds, but in the case of fundamental 7 oscillations at [110], the value of X at 1.2 K and a field of 70 k G is about 1.4, causing an error of about 6% in the exponential approximation. Since sinh rX « -e 1 2 so that Ar oc rXe (6.6) sinh rX - 2 r X we should have, from (6.5) (6.7) A n iteration scheme can be developed with the nth order value fi^n) for the mass parameter obtained from l n j ^ l - e x p ^ r A / x ^ - 1 ) ^ ) ] } = - r A ^ I + const (6.7a) 96 with fj.(°) —» oo. Convergence is achieved when /*(n) = ySn ^ within the desired accuracy. With the previously determined optimum MFB gain and at the 'magic field' (H = 69.45 kG), the amplitudes of the first three harmonics were measured simul-taneously at various temperatures, and after beat correction for the amplitudes, the values of In(j4r/!T) vs. T/H were plotted (Figure 6.5). The slopes were de-termined by linear regression, and they were found to be in the ratio 1:1.95:3.11, which is very close to the theoretical ratio 1:2:3. These results are further confir-mation that the choice of the MFB gain setting was correctly made. The weighted Figure 6.5 Effective mass plots for the first three harmonics of the [ 1 1 0 ] 7 oscilla-tions from sample 3 at H = 69.45 kG with optimum MFB gain. The slope ratio is 1:1.95:3.11. The uncertainties quoted for the effective mass values are a combi-nation of the statistical error in the determination of the slopes and the estimated uncertainty in setting the appropriate MFB gain. 97 average of the [110] 7 effective-mass values was found to be /i = 0.555 ± 0.003. 6.2.3 Field Dependence At a constant temperature, the field dependence of the dHvA amplitude Ar is given by the LK formula (2.12) to be exp(-rAi i^) ArocH-* ^ i U (6.8) sinhf rA/ijj] with the quantities defined earlier. Thus a plot of ln[Ar>/H sinh(rA/iT/J?)] vs. l/H yields a straight line of slope —rXfiTo, so that the slope of the rth harmonic should be r times that of the fundamental. Knowing the effective mass, the Dingle tem-perature can be derived from the slope of this 'Dingle' plot. In most cases the exponential approximation of the hyperbolic sine function holds and (6.8) can then be simplified as Ar oc f r* e x p [ - r A / i ^ ± ^ ] , (6.9) and a Dingle plot of ]n(Ary/H) vs. l/H will yield a straight line of slope — rXfi(T + TD). In our experiment the dM/dt signal, instead of Af, was recorded and Fourier-transformed, so that the amplitudes in the spectrum were proportional to fAr, with / = 2Fhfmoa/H2 being the time-frequency of the dHvA oscillation (see equation 4.3). Better accuracy was achieved by using this method because the dM/dt signal was stronger, and because it enhanced the higher harmonics. The time-frequency / was held constant by re-adjusting h (the amplitude of the modulation ramp) for each of the field settings H in our field-dependent measurements. As a result, the relations (6.8) and (6.9) were still satisfied. 98 The beat pattern of the amplitudes does not change the relations (6.8) or (6.9), as long as the points used on the graph are at corresponding positions in the beat pattern. The obvious choice is to use the field and amplitude at the maxima of the beat envelope. Figure 6.6 shows a Dingle plot of ]n(Ary/H) vs. 1/H for the first two harmonics of the 7 oscillation at constant temperature with the optimum feedback found earlier, and the slope ratio is found to be 1:2.05. The weighted average of the Dingle temperature was found to be TD = 0.29 ± 0.04 K. The third harmonic could be measured reliably only over a very limited field range, and the Dingle temperature obtained from Az was the least reliable, namely 0.42 ± 0 . 1 8 K. S 8 0> N >s I— o i _ lo i _ o c 1 1 1 ( — ^ o r ( T D =0.27±0 . 05K) — < S *T(T D =0.31±0 . 07K) — 1 I I I — 18 22 1/H (MG"1) Figure 6.6 Dingle plots for the first two harmonics of the [110] 7 oscillations from sample 3 at T = 1.17 K, with optimum MFB. The slope ratio is 1:2.05. The plot for the third harmonic is not shown, as explained in the text. 99 6.2.4 S from Three-Harmonic Ratio According to the discussion in section 2.2.4, the three-harmonic ratio |aoo| is the slope of a straight line plot of |j4i/i43| vs. (ili/^a) 2, = a~[(£)2-Ki)o] (2.16) with the amplitudes Ar (r = 1,2,3) measured at various temperatures with the applied field held constant. The temperature-dependent data in section 6.2.2 ob-tained at the 'magic field' with optimum MFB were used for this plot (Figure 6.7), and linear regression yielded a slope |Qroo| = 0.556 ±0.035 and an intercept —11.5 ± 2.9. The quoted uncertainties include the statistical error of the linear regression and also the error in the amplitude measurements. 4 0 0 T T T 2 0 0 -< 0 2 0 0 4 0 0 6 0 0 VA 2 Figure 6.7 Three-harmonic amplitude ratio plot for the [110] 7 oscillations from sample 3 at i f = 69.45 kG with optimum MFB. Linear regression gives the slope Ictool = 0.556 ± 0.035 and the intercept -11.5 ± 2.9. The quoted errors include the statistical error of the linear regression and the uncertainty in the amplitude measurements. 100 The quantity S = \pgc can be obtained from aioo according to equation (2.20) tan nS = ± (2.20) The inner square root in (2.20) yields an imaginary result for = +0.556, so we must conclude that ttoo — —0.556. This gives us two possible solutions of S in the basic interval 0 < S < 1/2, Si = 0.343 ± 0.003 and S2 = 0.192 ± 0.009. To decide between the two S values, we use the intercept in Figure 6.7 to obtain (Ai/A2)o as can be seen in equation (2.16). From equation (2.17), i.e., ( £ ) „ - * * « * ( * £ ) Tn\ COS ICS cos 2ITS ' (2.17) it is easy to derive an expression for the Dingle temperature corresponding to 5, H , TD = —\n (Ai/A2)ocos 2irS~ 2V/2COS nS (6.10) When the experimental quantities are substituted into (6.10), our two solutions for S correspond to the following Dingle temperatures: TD = 1.12 ± 0.15 K for Sx = 0.343 ± 0.003 and TD = 0.28 ± 0.15 K for S2 = 0.192 ± 0.009. The relatively large error in TJJ arises from the large uncertainty in the intercept of Figure 6.7. From the more direct field-dependent measurement, we have 101 TD = 0.29 ± 0.04 K, which allows us to select 5 = 0.192 ± 0.009 as the proper principal value. 6.2.5 S from Two-Harmonic Ratio—a Cross Check As discussed in section 2.2.3, the ratio c r = cos nrS/ cos irS can be evaluated from the ratio of the amplitude Ar of the rth harmonic to that of the fundamental Ai, as long as the effective mass y and the Dingle temperature TD are known. Specifically, for the second and third harmonic, we have, from (2.13), cos 2?r5 „ r- , / \ T\ /, TD\A2 ^ - i ^ ^ ^ M ^ - i ) ^ ( 6 1 1 ) and COS3JTS r- &mh{3\yT H) / TD\ A3 c3 = - = \/3 . * m ' y e x p ( 2 A j i - £ ) - ^ . (6.12) COSTT5 smh(\yT/H) K\ * HI Ax K ' In our calculation, Ci and c2 were deduced from each temperature setting in the effective mass data (section 6.2.2), with the effective mass p. = 0.555 ± 0.003 and the Dingle temperature TD = 0.29 ± 0.04 K. The averaged value of c2 and c3 were found to be \c21 = 0.484 ± 0.028 and |c31 = 0.431 ± 0.052 . The errors quoted for c2 and c3 come chiefly from the uncertainty of the Dingle temperature TD, and from the uncertainty of the amplitudes A2 and A3. The S values can now be inferred by inversion of the measured values c2 and c3. Simple trigonometric transformation gives us, from the definition of c2 and c3, cos 2irS = ^c 2 c2 ± \Jc% + 8 (2.19) 102 and cos TTS = ± ^ 0 3 + 3 . (6 .13) This procedure gave S = 0 .184 ± 0 . 0 0 5 from |c2| = 0 .484 ± 0 .028 and S = 0 .204 ± 0 .005 from |c3| = 0 .431 ± 0 . 0 5 2 , respectively. Two other possible S values, namely 0 .297 from c2 and 0 . 123 from c 3, were rejected as being unphysical; they are not consistent with each other. The different results from various methods are summarized in Table 5 . The weighted average of the S value obtained from the three methods is found to be 0 . 1 9 4 ± 0 . 0 0 6 , and is in good agreement with the preliminary result S = 0 .197 given by Gold and Van Schyndel ( 1981) . T a b l e S Summary of Experimental S Values for [ 1 1 0 ] 7 Oscillations in Lead method w I«oo| 5 ( 0 < S < 1 /2) 0 . 4 8 4 ± 0 . 0 2 8 — 0 . 1 8 4 ± 0 . 0 0 5 A3/A1 0 . 4 3 1 ± 0 . 0 5 2 — 0 . 2 0 4 ± 0 . 0 0 5 A2/A\Az — 0 .556 ± 0 .035 0 . 1 9 2 ± 0 . 0 0 9 weighted average — — 0 . 1 9 4 ± 0 . 0 0 6 103 6.3 S for the [110] ip Orbit 6.3.1 Preliminary Considerations Now we turn to the a oscillation arising from the second-zone hole orbit tp for H || [110]. From the experimental point of view, it is much more difficult to measure the a oscillation and its higher harmonics due to the following reasons. Firstly, the dHvA frequency for the [110] a oscillation is about nine times that for the [110] 7 oscillations, requiring a good frequency response in the total cir-cuitry over a wider frequency range. Secondly, the effective mass is about twice that of the 7 oscillation. Since the damping factor for the dHvA amplitude Ar is approximately exp[-rA//(!T + TD)/H], it is expected that the a amplitudes (espe-cially for the higher harmonics) drop off much faster then the 7 amplitudes with increasing temperature or decreasing field. Thus the field and temperature ranges over which the required signal can be detected are much more limited. Thirdly, the influence of MI between a and 7 oscillations cannot be completely suppressed with the MFB technique because of sample inhomogeneity (the two orbit-case as discussed in section 5.3). As a result, the a amplitudes for both fundamental and higher harmonics are expected to be distorted to some extent. The first problem was easily solved by lowering the modulation frequency to 0.5 Hz. As discussed in section 4.5.2, the frequency / in the time domain for a dHvA frequency F is given by r • 2 F h , J = - ^ 2 " / m o d , (4.3) with parameters defined before. For the third harmonic of the [110] a oscillation, F3a = 480 MG, and with h = 1 kG, / m o d = 0.5 Hz, H0 = 70 kG, equation (4.3) 104 gives a time frequency / ~ 100 Hz, comfortably within the 200 Hz bandwidth limited by the frequency response of our detecting system (Figure 4.10). Because of the beat pattern of the [110] 7 oscillations, the applied field can be chosen to correspond to a beat minimum. This will clearly minimize the influ-ence of the MI between a and 7 oscillations. The highest field at which a 7 beat minimum could be found (within the available range of the main superconducting magnet) was found to be H = 73.39 kG. A typical Fourier spectrum of the oscil-lating magnetization at that field is shown in Figure 6.8. The amplitude of the a zy -y >3y 200 ZOL 301 - A . 400 F (MG) 4 a 600 Figure 6.8 Fourier spectrum of the dM/dH waveform for sample 3 with H J| [110], at H = 73.39 kG and at T = 1.19 K with optimum MFB. This field setting cor-responds to a minimum in the 7 oscillation beats. Notice that the fundamental 7 peak is small compared with the peak height at a beat maximum in Figure 3.1. 105 fundamental a peak was then quite insensitive to the MFB gain setting (variation less than 2%), which indicates that the a peak is now essentially unaffected by MI with the weak 7 oscillations. Having the 7 oscillation set at a beat minimum does not, of course, have any bearing on the self-interaction between the fundamental a oscillation and its own harmonics, which can be suppressed by MFB. The relative phase between the a fundamental and its second harmonic were measured as a function of MFB gain, and the optimum MFB gain was achieved when 2<pi — <p2 = —JT/4 ('—' sign since the orbit is one of maximum area).* This procedure has already been illustrated in Figure 5.4. 6.3.2 Temperature Dependence As we did for the 7 oscillations, temperature-dependent measurements were performed for the first three harmonics for the a oscillation at the above-mentioned field (H = 73.39 kG) with optimum MFB. Then the graphs of In(i4r/T) vs. T/H were plotted and their slopes were determined by linear regression. The slope ratio thus obtained was 1:1.87:2.67 instead of the expected 1:2:3 (Figure 6.9). One possible reason for this discrepancy might be that the MFB used was, in fact, not perfectly optimized, so that a small amount of MI still existed between the a fundamental and its higher harmonics. To investigate this possibility, the influence of MI on the slope ratio was studied and the slope ratio itself was used as an alternate criterion for setting the correct MFB gain. As shown in section 5.2, the MFB gain dependence of the apparent amplitude * For more detailed discussion about the relative phase see section 6.3.3. 106 Figure 6.9 Effective mass plots for the first three harmonics of the [110]a oscillation from sample 3 at H = 73.39 kG, which corresponds to a 7 beat minimum, and with optimum feedback. Solid lines indicate experimental results with slope ratio 1:1.87:2.67. Broken lines represent results after MI correction for the second and third harmonic amplitudes with slope ratio 1:2:3 (see text). 107 for the second harmonic is given by t, . r KeA2 1 / KeA2 \ 2 i i where e = 1 — f/4TT(1 — 6) is the 'feedback factor' with f representing the M F B gain, e can also be expressed as e = l - f , (6.14) with ft = 4fr(l — 6) being the optimum M F B gain. The temperature dependence of the L K harmonic amplitudes should be well expressed by the exponential approximations Ai=piTe~x and A2=p2Te~2X1 (6.15) where p i , p2 are factors which are independent of T, and X = XpTfH. If we sub-stitute (6.15) into (5.14b) and introduce another temperature-independent quantity we obtain A'2 = p2Texp(-2A/z|) [l - uT + \(i>T)2} * . (6.17) v should be zero if the feedback were optimized (e = 0), and we now treat u as a correction factor. For v ^  0, it is a plot of l n { ^ [ l - ^ + i ( r v r ) 9 ] " * } vs. 1 (6.18) which should yield a straight line of slope — 2\p. Taking p from the temperature dependence of the fundamental (which should not be affected by a possible error in the M F B gain setting), we find the value v which is required to obtain the correct 108 slope. This procedure is then repeated for another MFB gain setting. If the two fitted values corresponding to nearly optimum MFB gains ft and f2 are v\ and t/2, respectively, then "2 ^2 fo — ft where ft is the optimum MFB gain. Following this process, a revised MFB gain fo was selected at which the slope ratio was expected to be exactly 1:2. Unfortu-nately the slope ratio was still 1:1.86, i.e., revising the MFB gain setting gave no improvement at all. Another possible reason might be the remaining higher order terms in the Phillips and Gold expansion of the MI amplitudes A'r (r = 1,2,3) which in principle cannot be completely suppressed by MFB in the presence of sample inhomogeneity (see equations (5.13) in section 5.2). But the following argument will also rule this out. The order of the first term for the MI amplitude A'r is O(r), determined by the exponential damping factor exp[—r\/i(T + TD)/H], and the succeeding term is of the order of 0(r -f 2), giving a ratio of 0(r+2) r T + TD O(r) e x p [ - 2 A / x - ^ ] . (6.20) For approximately the same T, TD and H, this ratio is larger for a smaller effective mass. Because fia w 2filt we should expect that the higher order terms in Ar are more important for 7 oscillations than for a oscillation. Since we have not found any importance of the higher order terms for 7 oscillations, it is most unlikely that they will play an important rule for the a oscillation. The most probable reason for the unexpected effective-mass ratio for the a harmonics might be the MI between them and the low-amplitude 7 oscillation which remains at the 7-beat minimum. This 7 oscillation is much weaker than the fundamental a oscillation, and the MI between them can be ignored, as has been 109 verified experimentally (section 6.3.1). However, the second and third harmonics of the a oscillation are weak and quite comparable with the remaining 7 oscillation, so that the MI of these harmonics with 7 should be taken into consideration. This sort of MI cannot be completely eliminated by the MFB. When the MI between the a fundamental and its higher harmonics is well compensated, the apparent amplitude A'ra for the rth harmonic, due to the MI with 7, can be expressed as A' -A + ( p t + p 7 ) K l M r K M ( ^ ) } (6.21) with MFB gain $ (see Appendix A). Expanding the Bessel functions and retaining only the leading terms, we have ^ ^ { . - ^ [ ( ^ - ^ ^ - ( ^ n } . ^ ) where both in (6.21) and (6.22) p+ and p~ are the two correlation factors relating to rA(pa + A<Pi and rA<pa — A<p~,. Equation (6.22) may be written simply as A'ra = Ara[l - (r<rrT)2exp(-2 A * i 7 ^ ) ] (6.23) where ar is independent of temperature. The L K amplitude Ara can be recovered for each temperature from (6.23) if <rr is known. We use p. = 1.126 ± 0.004 from the slope of the plot of ln(Ai/T) vs. TIH for the fundamental a oscillation, i.e., assuming <j\ = 0. <rr for a harmonic was found by forcing the slope of a plot of b{ T [ l - ( ^ r T ) a e x p ( - 2 A,,,-)] } vs. - (6.24) to be —rA/x. Such plots are shown as broken lines in Figure 6.9, and give a2 = 2.75 and 03 = 8.40. The corrected harmonic amplitudes Ara were then obtained from no the measured ones A'ra and the trr values, using equation (6.23). It is these Ara which were used to evaluate the S factor for the [110] a oscillation. The above correction did not take into account the possible MI with the second and third harmonics of the 7 oscillations, and in principle, we did not know for sure how strong this MI was. The final judgement of this correction depends on the consistency of the various S values evaluated from these corrected amplitudes by different methods. 6.3.3 S from Three-Harmonic and Two-Harmonic Ratios The amplitudes Ar (r = 1,2,3) in the temperature-dependent measurement (after the MI correction described in the last section) were used to calculate |vl 1 /A31 and (A1/A2)2, and the results are shown in the three-harmonic plot of Figure 6.10. Linear regression gives a slope of | a r o o | = 0.705 ± 0.059, and a r o o was shown to be negative for the same reason as for the 7 oscillations. Two possible S values within the basic interval 0 < S < 1/2 were found from (2.20) to be Sx = 0.325 ± 0.004 and S2 = 0.189 ± 0.012. The intercept in Figure 6.10 was found by linear regression to be —3.36±9.86. The intercept for the 7 oscillation was much better denned (—11.5 =fc 2.9; section 6.2.4) and we were able to make use of that information to choose between two possible S values. Unfortunately this cannot be done with any degree of confidence for the a oscillation. This ambiguity between the two S values can, however, be resolved by the relative phase measurements (Alles, Higgins and Lowndes 1975, Crabtree, Wind-miller and Ketterson 1975; see also Higgins and Lowndes 1980). This procedure can be illustrated as follows. i l l 400 200 <|< 0 0 200 AOO Figure 6.10 Three-harmonic amplitude ratio plot for the [110] a oscillation, from sample 3 at H = 73.39 kG with optimum MFB. Amplitude values for the sec-ond and third harmonics have been corrected. Linear regression gives a slope \aoo\ = 0.705 ± 0.059, and an intercept -3.36 ± 9.86. Without MI correction for the amplitudes, the result would be |aoo| = 0.72 ± 0.06. The difference in the two slopes is less than the experimental error. The Phillips and Gold expansion for the MI may be written, to the second order in negative exponentials, as M = Ci|cos jr5|sin[iT ^ + (1 +i0^] + C 2|cos 27rS|sin[2zT ^ + (1 + te)^] - ^[Ct cos irS}2 sin [ 2 ( i ^ ) ] , (6.25) with x and K denned before. The spin-independent term Cr in the L K amplitude Ar has been separated from the spin-splitting cosine, i.e., | j i r | = C r|cos ncS\, (6.26) 112 and the intrinsic minus sign and the sign of cos rirS have both been absorbed in the phase factor (1 + jV)f. where . _ ( +1, for cos nrS > 0 3 r ~ { -1, for cos mS < 0. The first two terms in (6.25) are the fundamental and the second harmonic in the L K form, while the last term is an additional contribution to the second harmonic caused by MI. If the LK term is dominant (^- <tC 1), the relative phase will have the LK value (pure LK) 2<P! -<P2 = T^~ (1 +;2) J • (6.27a) 4 I On the contrary, if the MI term is dominant {^A^ ^ *)» 2<p\ -<p2 = -n (6.276) (pure MI). The actual observed relative phase falls between the pure LK and pure MI limiting values, and will move towards the L K limit with increasing MFB gain cf as illustrated in Figure 6.11 for all the four combinations of j2 and the sign of JT/4. For a given orbit whose sign for T^"/4 is known, the two possibilities in Figure 6.11 are mutually exclusive, and the value of 2<pt — <p2 can be used to determine the sign of cos 2TS.* For the [110] a oscillations arising from the ip orbit with maximum cross sectional area, the measured relative phase with MI falls in the interval * In practice, due to the sign of the net gain of the amplifiers, or the connections of the pickup coil, we may have a possible u ambiguity in both pure LK and pure MI relative phase 2ipi — f>2. For a given orbit, those two accessible regions remain mutually exclusive with the addition of a factor u. Thus there is no ambiguity in the sign of cos 2nS. 113 Orbit a maximum (-•?•) a) COS(2TTS)K) (j2=1) b) cos(27rS)*0 (j2=-1) LK Ml V Ml \ LK Orbit a minimum (+*^ -) a) COS(2TTS)-0 (J2=1) b) COS(2TTS)-0 (j2=-1) Ml Ml f LK Figure 6.11 Illustration of the accessible range of the observed relative phase 2y?i — <P2 values with MI (after Higgins and Lowndes 1980). For a given orbit with known sign of n/4, the two possible ranges are mutually exclusive and depend on the sign of the spin-splitting cos 2JTIS. The measured relative phase values also vary in opposite direction (shown by arrows on the arc) with increasing negative MFB gain. 114 - ir<(2y?i - y?2)<-fl"/ 4( 8 e e Figure 5.4). Upon comparison with Figure 6.11, this is seen to be the case for cos 2nS < 0. Because S i = 0.325 ± 0.004 yields a negative value of cos 2TXS while S2 = 0.189 ± 0.012 a positive one, the correct choice should be Si. In order to make two harmonic ratio calculations, we need to know the Dingle temperature. It is found to be Tp — 0.26 ± 0.02 K from the slope of the Dingle plot for the a fundamental shown in Figure 6.12. The MFB with which the field-dependent measurement was made was determined by minimizing the sidebands, and the fact that a good straight line was obtained indicates that the MI between 7 and fundamental a oscillations was effectively suppressed. Had it been otherwise, the Dingle plot would have exhibited a periodic ripple, a reflection of the beat 15 20 1/H (MG H) F i g u r e 6.12 Dingle plot for the fundamental [110] a oscillations from sample 3 at T = 1.14 K, with optimum MFB. The slope gives TD = 0.26 ±0.02 K. 115 pattern in the 7 oscillation.* With TD found, as done for the 7 oscillations, = 0.26 ± 0.02 K and fi = 1.126 ± 0.004, we |c 2| = 1.416 ±0 .077 S = 0.356 ± 0.004 and |c 3| = 2.53 ± 0 . 2 7 0.389 ± 0.039 Because both |c 2| and |c 3| are larger than 1, there are no other possible solutions for S in the basic interval. The results from the three methods are summarized in Table 6, and they are in good agreement with each other. The weighted average of S was found to be S = 0.354 ± 0.004. method Table 6 Summary of Experimental S Values for [110] a Oscillation in Lead S{0 < S < 1/2) A2/Ax Az/A, Al/A> A* weighted average 1.416±0.077 2.53±0.27 0.705 ± 0.059 0.356±0.004 0.389±0.039 0.352±0.004 0.354±0.004 * Dingle plots for the second and third harmonics of the a oscillation did, in fact, show periodic ripples, and they could not be used to determine TD reliably. 116 6.4 5 for the [111] rp and 9 Orbits 6.4.1 General Considerations There are three groups of oscillations when lead single crystals are oriented with the applied field H || [111]. The oscillations with lowest frequency are the 7 ones from the f orbit around the electron arm in the third zone. In general, there are three branches for the 7 oscillations, but lattice symmetry requires that they cross one another to give just a single frequency in the [111] direction. Experimen-tally, small deviations in the orientation from [111] are almost unavoidable, so that the 7 oscillations always exhibit a rather complicated beat pattern. They are also less interesting because [111] is not the axial direction of an electron arm. Because of these reasons, S was not measured for the 7 oscillations at [111]. The second oscillation, as in the [110] direction, is the a oscillation from the central second-zone ip orbit. The empty-lattice model actually predicts two additional noncentral orbits xp2 (see Figure 6.2) with maximum cross sections, but they are 'sandpapered away' by the potential. Experiments show just a single frequency, no beats have ever been found. The third group, which does not exist within the vicinity of [110], is the 6 oscillation from the hexagonal shaped 9 orbit centred on L in the third zone (see Figure 6.2). This 9 orbit is one of minimum area and gives rise to a single frequency. The signals for the [111] a and 8 oscillations were much weaker than those for the [110] or oscillation under roughly the same experimental conditions. In particular, no third harmonic signals could be detected, so that the three-harmonic ratio method was not applicable. In the absence of M F B , the [111] spectrum turned out to be very complicated, as shown in Figure 6.13a. Fortunately it was found long ago (Phillips and Gold 117 F i g u r e 6.13 The Fourier spectra of the dM/dH waveform from sample 11 with H || [111], at T = 1.18 K. a) at H = 72.07 kG, without MFB; b) at H — 72.07 kG, with optimum MFB; c) at H = 71.58 kG, corresponding to a minimum in the 7 beat pattern, with optimum M F B . 1969) that the MI between the fundamental and the second harmonic was not significant for both a and 6 oscillations. Then the major consideration for the MFB was to get rid of the MI between the 7 oscillations and the a and 6 ones. This was achieved by first of all choosing optimum MFB gain to be the average of the MFB gain settings with which the sidebands or ± 7 and 6 ± 7 were minimized (Figure 6.13b) and then adjusting the applied field to give a minimum in the beat pattern for the 7 oscillations (Figure 6.13c). As discussed in section 5.3, this would not guarantee in general that the LK forms of Aa and As were recovered completely, but further experiments performed at this MFB gain proved that the measured values for the amplitudes of both a and 6 oscillations were at least rather good approximations to the LK amplitudes. 6.4.2 S for the [111] rb Orbit from Two-Harmonic Ratio In sections 6.2 and 6.3 we have described in great detail the procedure leading to the evaluation of S values for a specific orbit. Here we simply present the experimental results. Temperature dependent measurements were made for the first two harmonics of the [111] a oscillation with the above-mentioned MFB gain setting and the effective mass plots are shown in Figure 6.14. Linear regression gave a slope ratio 1:1.99, in excellent agreement with the L K prediction of 1:2. The weighted average of the effective mass from the two slopes was found to be p. = 1.113 ± 0.005. Field-dependent measurements were also performed for the fundamental and the second harmonic, and the Dingle plots (Figure 6.15) gave a slope ratio 1:2.02, again in excellent agreement with the LK prediction. The weighted average of the Dingle temperature was found to be TD = 0.58 ± 0.03 K. Two |c 2| = I cos 2JT5/COS ITS\ values were calculated from the temperature-dependent and field-dependent data separately. This was used as a cross check 119 Figure 6.14 Effective mass plots for the fundamental and the second harmonic of the [111] a oscillation from sample 11, at H = 68.80 k G with optimum M F B . The slope ratio is found to be 1:1.99. for the correctness of the | c 2 | evaluation. The average value of | c 2 | for all the temperature settings was found to be 7.91 ± 0.32; while the average of | c 2 | for all the field settings was 7.39 ±0 .34 . The weighted average of the two was 7.67 ±0 .27 , and the S value obtained from it was 0.460±0.004. No other S values were possible within the basic interval because | c 2 | > 1. 6.4.3 S for the [111] 9 Orbit from Two-Harmonic Ratio The experiments on the 6 oscillation were more difficult than those on the a oscillation, because of the relative weakness of the second harmonic of 6 under the 120 1 / H ( M G ' 1 ) Figure 6.15 Dingle plots for the fundamental and the second harmonic of [111] a oscillation from sample 11, at T = 1.16 K with optimum MFB gain. The slope ratio is found to be 1:2.02. same experimental conditions. This made the experimental results less reliable for the [111] 6 oscillation. Effective mass plots for the first two harmonics of the [111] 6 oscillation are shown in Figure 6.16, and their slope ratio was found to be 1:1.92 from linear regression. This result showed a certain amount of MI between the remaining 7 and the 26 oscillations, as in the case of [110] 2a oscillations. But no correction was attempted because the main error in the evaluation of |c 2| came from the uncertainty in the Dingle temperature TD, which is much larger (9%). The weighted average for the effective mass was p. = 1.196 db 0.005. 121 Figure 6.16 Effective mass plots for the fundamental and the second harmonic of the [111] 5 oscillation from sample 11, at H = 71.58 kG, with optimum MFB. The slope ratio is found to be 1:1.92 by linear regression. 122 The Dingle temperature TD could only be deduced from the slope of the Dingle plot for the fundamental (Figure 6.17), and it was found to be TD — 0.22 ± 0.02 K. The Dingle plot for the second harmonic was not a straight line, but more or less followed the complicated beat pattern of the [111] 7 oscillations, which is definite evidence for some residual MI between 7 and 26 oscillations. The spin-splitting cosine ratio |c 2| was taken as the average value for all the temperature settings in the temperature-dependent measurement. It was found to be |c 2| = 1.426 ± 0.070, and the relevant S value is S = 0.357 ± 0.003. Figure 6.17 Dingle plot for the fundamental [111] 6 oscillations from sample 11 at T = 1.16 K with optimum MFB. The Dingle temperature is found to be TD = 0.22 ± 0.02 K. A Dingle plot for the second harmonic is not available as explained in the text. 123 6.5 S for the [100] u Orbit 6.5.1 Extremal Orbits along the [100] Direction The oscillations with the applied field H along [100] are the most complicated among the three main symmetry directions. They arise from four extremal orbits on the Fermi surface. The oscillations with the lowest dHvA frequency are the 7 oscillations arising from the ( orbit around the third zone electron arm centred at the symmetry point K. Like [111], [100] is also a crossing point for 7 branches due to lattice symmetry, and for the same reason as for the [111] direction, a measurement of S was not attempted. The second set, the n oscillation, arises from the square-shaped £ orbit centred on X and formed by four electron arms (Figure 6.2). It has a minimum cross sectional area. This oscillation is very weak and no higher order harmonics can be detected. We will discuss this in section 6.6. The oscillation with the highest dHvA frequency is the ct oscillation from the large \b orbit on the second-zone hole surface with centre at the Brillouin zone centre T. As for the ir oscillation, the [100] a oscillation is also very weak and will be discussed later together with the TT oscillation. The oscillation with the strongest amplitude is the oscillation from the u orbit. This is an orbit around the junction of four electron arms, and is centred at the point W. It is for this orbit that the most detailed discussion of the S measurement will be presented. One of the difficulties with the [100] direction is the mixing of the various oscillations. The dHvA frequency for the /? oscillation is approximately twice that 124 for 7, and the a frequency is close to four times the /? frequency. As a result, the following pairs of terms will be essentially indistinguishable as far as frequency is concerned: 27 (50.8 MG) and P (51.3 MG), ZP (154 MG) Ap (205 MG) SP (256 MG) and a-/?(153MG), and ct (204 MG), and ct + p (256 MG). The reliability of the amplitude for the fundamental P oscillation will not be a problem, because the p amplitude is much stronger than that for the 27. However, there exists the possibility of misinterpreting the data ascribed to the three other pairs. The three-harmonic ratio method was not attempted because of the possible confusion of 3/? with a — p. 6.5.2 T h e B e a t P a t t e r n Like the [110] 7 oscillations discussed in section 6.2.1, the [100] p oscillation also exhibits abeat pattern with about 600 cycles per beat. These extremely long beats are quite genuine, and have been observed in several laboratories using differ-ent experimental techniques (for example, Phillips and Gold 1969, Anderson, Lee and Stone 1975, Gold and Schmor 1976) and they are believed to indicate the exis-tence of a pair of noncentral extremal orbits v' on either side of the central orbit v. A calculation by Phillips (1967) based on the band-structure model of Anderson and Gold (1965) results in a beat pattern with about 1200 cycles per beat, but later calculations (Van Dyke 1973; Gold, Holtham and Van Schyndel^  to be published) gave only a simple minimum at W. Other possible explanations are under study. Figure 6.18 shows a plot of Ln[AH~% smh^XpT/H)\ vs. 1/H, which is domi-nated by the beat envelope. After beat correction for the amplitude (i.e., to find out Aa + Ab from A b e a t , see equation 6.4), the Dingle plot should be a straight 125 20 25 1/H (MG - 1) 30 35 Figure 6.18 Beat envelope for the [100] /? oscillation from sample 18, at T = 1.15 K without MFB. The magnitude of the modulation factor \n\ was found to be 0.280 ± 0.005, and the frequency difference 6F to be 87.7 ± 0.1 kG. line. This was used as a least-squares fitting criterion for the determination of |n| and 6F, which were the adjustable parameters in the fitting procedure. We finally found |n| = 0.280 ± 0.005 and 6F = 87.7 ± 0.1 kG, which gave 6F/F = 584 cycles per beat, in good agreement with the earlier findings (570 cycles by Anderson, Lee and Stone 1975, 565 cycles by Gold and Schmor 1976). But our value for |»7| is slightly smaller than the results reported earlier (see Gold and Schmor 1976). The Dingle plot after beat correction with the above |»7| and 6F values turned out to be an excellent straight line (Figure 6.19). The field range over which reliable amplitudes for the second harmonic could be measured was very small. This field range is shown in the schematic diagram of the beat envelopes in Figure 6.20. Hence it was impossible for us to determine the optimum MFB gain from the successive beat lengths of the second harmonic as 126 Figure 6.19 Dingle plot for the [100] /? oscillation after suppression of the long beats, using the same data as in Figure 6.18. The Dingle temperature is found to be TD = 0.11 ± 0.01 K from the slope of the straight line. 127 /' \ / \ / \ |FUNDAMENTAL _ / ' \ / 1 v 2 ND HARMONIC \ / / \ ' v 1 1 1 1 I I I I _ >^ 20 ft 30 1/H (MG-1) F i g n r e 6.20 Schematic diagram of the beat envelopes for the [100] (3 oscillation and its second harmonic with |n| = 0.28 and 8F = 87.7 kG. Solid curves indicate the regions in which reliable amplitudes can be measured; they are limited on the left by the maximum available field from the superconducting magnet. T h e three 'magic field' points (indicated by $ on the 1/H axis) lie outside the region for reliable second-order data. Vertical line shows the field setting at which the temperature-dependent measurements for the first two harmonics were made. we had done for the much shorter [110] 7 beats (section 6.2.1). There were also no 'magic field' points inside the field region for appreciable second harmonic. We chose the field setting at a maximum of the beat envelope for the second harmonic as the appropriate field for the temperature-dependent measurements. T h i s field was calculated to be H = 53.97 kG. In our experiment, the field used was H = 53.29 kG, quite close to the calculated value. The beat correction factors for the fundamental and the second-harmonic amplitudes were 2.40 and 1.01 respectively. 128 Because of the beat, the resultant phase <pr is no longer and is field dependent even if the MI does not exist. This has important effects when we try the relative phase was calculated to be 2<pi — <p2 = ±0.427JT = ±77° at the field H = 53.29 kG. The assignment of the sign is uncertain, because we cannot determine the sign of n only from the beat pattern. 6.5.3 S f r o m T w o - H a r m o n i c R a t i o The MI of the /? oscillation was found to be very strong. Without MFB, the observed amplitude of the second harmonic was predominantly due to the MI. Thus the reliability of the S measurements depend strongly on the choice of the MFB gain. The optimum MFB gain was determined from the relative phase measure-ments with the calculated value 2<pi — <p2 = ±77° as a criterion. As a result of the possible ±TT ambiguity in the determination of the phase, two possible MFB gain settings were found (see Figure 6.21). They were ft = 2.32 dial units for 2<pi — <p2 — 1°3° and $2 = 2.48 dial units for 2(pi — <p2 = 77°. Fourier spectra with no MFB (f = 0) and with optimum MFB (ft> = 2.48 dial units) are shown in Figure 6.22. The actual spectra for the two possible optimum settings ft and £ 2 are virtually identical, and the reasons for selecting & hinge on temperature-dependence data, as will now be discussed. The temperature-dependent measurements were performed for the /3 oscil-lation and its second harmonic with both possible MFB gain settings, and the to determine the optimum MFG gain later from the relative phase information. According to equation (6.3) (6.3) 129 F i g u r e 6.21 Relative phase of the [100] /? oscillation from sample 18 as a function of M F B gain. The measurements were taken at H = 53.29 k G and T = 1.15 K. Two possible optimum gain settings were found to be 2.32 and 2.48 dial units, corresponding to two possible relative phase values of 103° and 77° respectively. T h e second gain setting ( f 2 = 2.48 dial units) was later found to be the correct choice. effective-mass plots were made. With the first gain setting (2.32 dial units corre-sponding to 103°), the slope ratio was found to be 1:1.77; while with the second one (2.48 dial units corresponding to 77°), the slope ratio of 1:1.97 was much closer to the ideal ratio 1:2 (see Figure 6.23). The value £ 2 = 2.48 dial units was therefore taken to be the correct one, and the amplitudes measured with this M F B gain setting (af-ter beat correction) were used in the evaluation of the spin-splitting cosine ratio |c2|. The weighted average of the effective mass was found to be /x = 1.233 ± 0.005. 130 p p W<*-P) A ; m«+p) i 1 1 «y-" n f 20 30(Ot-Pj ! (*{4P) yuyM— 200 F(MG) 400 o) 200 b) 400 F i g u r e 6.22 Fourier spectra of the dM/dH waveform from sample 18 with H || [100], at H = 53.29 kG and T = 1.15 K: a) without MFB; b) with optimum MFB (f = 2.48 dial units). The higher harmonics are predominantly determined by the MI when MFB is absent. The weak ir oscillation cannot be seen in the spectra. J 1 1 1 I I L H ^MG' Figure 6.23 Effective mass plots for the first two harmonics of [100] /? oscillation from sample 18 at H = 53.29 kG, with optimum M F B ($ = 2.48 dial units). The slope ratio is found to be 1:1.97 by linear regression. With the other M F B gain setting (£ = 2.32 dial units), shown as a broken line for the second harmonic, the slope ratio is 1:1.77. The Dingle temperature required in calculating | c 2 | was found to be TD = 0.11 ± 0.01 K from the slope of the Dingle plot for the /? fundamental (see Fig-ure 6.19). This was the lowest of all the Dingle temperatures measured in this work. The spin-splitting cosine ratio | c 2 | was taken as the average for all the tem-perature settings in the temperature-dependent measurement. It was found to be 132 0.561 ± 0.035. Two possible S values within the basic interval 0 < S < | were solved from |c2|, according to equation (2.19). They are Si = 0.170 ± 0.007 with cos 2*rS = 0.482 ± 0.035 and S2 = 0.303 ± 0.003 with cos 2uS = -0.327 ± 0.015. To choose between these two 5* values, again we use the relative phase to determine the sign of cos 2irS. Despite the fact that the resultant L K phase <pr is no longer because of the intrinsic beat pattern, the discussion in section 6.3.3 still holds. The only change needed is to replace T«"/4 in both equation (6.27a) and in Figure 6.11 by the calculated values, i.e., —77° and +77°, respectively, for H = 53.29 kG. Whether the value of 2<pi — <p2 is positive or negative, cos 2ixS < 0 corresponds to the situation in which the angle between pure MI and pure LK is larger than TT/2, while cos 2nS > 0 corresponds to the situation in which that angle is less than n/2. We already verified that f = 2.48 dial units is the correct choice of optimum MFB gain setting, giving an angle between pure MI and pure LK larger than JT/2 (see Figures 6.21 and 6.11). We conclude that S 2 = 0.303 ± 0.003 is the correct choice since it gives the required negative value of cos 2TXS. It is noticed that the S value (within the basic interval) obtained in this work (S = 0.303 ± 0.003) is about 30% higher than the earlier result S = 0.236 ± 0.002 measured by the three-harmonic ratio method (Gold and Schmor 1976). Since the frequencies of the 3/3 harmonic and the a — f3 sideband are virtully identical, it may be that the oscillation believed by Gold and Schmor to be 3/? actually consisted of an admixture of the 3/7 and the a — (3 terms. Because of this confusion we have decided to draw our conclusions only from the 0 and 2/3 harmonics. 133 6.6 Preliminary Results for the [100] ib and £ orbits In the [100] direction, as mentioned in section 6.5.1, both the ir oscillation arising from the £ orbit and the a oscillation from the ip orbit are weak. Only their fundamental components are observable under our experimental conditions. The two-harmonic and three-harmonic methods for measuring S are therefore not applicable. Here we first present our experimental results for their effective masses and Dingle temperatures, and then we shall obtain a value of S for the £ orbit from measurement of the absolute amplitude of the TT oscillation. 6.6.1 Effective masses and Dingle Temperatures The dHvA frequency of the [100] a oscillation is very close to four times that of the [100] (3 oscillation. They were indistinguishable in our experiment. Both the self-MI between the /3 harmonics and the mutual-MI between (3 and a oscilla-tions were strong. Without MFB, the observed amplitude at this frequency was predominantly the MI component of 4/? (Figure 6.22a). With increasing negative MFB gain, this amplitude first dropped off until a minimum was reached, and then behaved as the mutual-MI case described in section 5.3 (Figure 6.24). Thus the observed amplitude was believed to be the L K amplitude of the a oscillation when it was maximized with MFB gain f = 3.44 dial units. With this MFB gain setting, both the temperature- and field-dependent measurements were made. The effec-tive mass plot (Figure 6.25) and the Dingle plot (Figure 6.26) were straight lines. The effective mass was found to be y. = 1.53 ± 0.03, with the quoted uncertainty from both the standard error in the slope and from the possible mixing with the remaining 4(3 amplitude. The Dingle temperature was found to be 0.68 ± 0.05 K, the highest in this work. At approximately the same field and temperature, the TT amplitude was buried in the extremely strong /? amplitude in the Fourier spectrum (see Figure 6.22). To 134 T 1 r 2 U FEEDBACK GAIN (dial units) Figure 6.24 The MFB gain dependence for the mixing amplitude of the [100] a and 4fi oscillations from sample 18, with H = 62.72 kG and T = 1.18 K. With MFB gain setting £ = 3.55 dial units, this amplitude is believed to be the amplitude of fundamental a oscillation. get a better frequency resolution, the modulation frequency was increased so that the time-domain frequencies for the corresponding oscillations were better separated in their Fourier spectrum. At the same time, we had to cut the main field down to about 50 kG in order to avoid the extremely strong fi signal which would make the the PAR 113 preamplifier saturate. The fundamental JT amplitude was quite insensitive to the MFB gain (about ~ 5% variation with 0 < c < 5). Therefore the same MFB gain setting (c = 3.55 dial units) was used for the JT measurements. The effective mass and Dingle plots are shown in Figures 6.25 and 6.26 respectively. The effective mass was found to be H = 0.945 ± 0.005, and the Dingle temperature TD = 0.20 ± 0.02 K. 135 7 I 1 1 I L 20 30 LQ - I (JL) H \MG / Figure 6.25 Effective mass plots for the [100] IT and a oscillations from sam-ple 18. The former was measured at the field H = 51.40 kG, and the latter at H = 62.75 kG. 6.6.2 Absolute Amplitude and S for the [100] £ Orbit Since higher harmonics are not detectable, the only method applicable for measuring the orbital spin-splitting factor for the TT oscillation is the absolute am-plitude method. The MFB technique provides an easy way to measure the absolute magnetiza-tion. Earlier measurements of the absolute amplitude involve careful calibration of the sample geometry, the coupling constant between the sample and the detecting 136 Figure 6.26 Dingle plots for the [100] w and a oscillations from sample 18, at r = 1.16K. coil, as well as the net gain of the amplification system (see, for example, Knecht 1975). With MFB, the feedback field is related to the magnetization by either hf = - f t M = -4?r(l - 6)M (in case of self-MI) (3.19) or hf = -cQM = - P + + P 4?r(l - 6)M (in case of mutual-MI) (5.35) 137 when the MFB gain is optimized. Thus the absolute magnetization Af can be measured via the feedback field hj. Since the field-to-current ratio of the modula-tion coil is known (7 = 245.3 G/A), the only other quantity to be measured is the feedback current if in the modulation coil. However, it is difficult to obtain an accurate measurement of this weak currentif (~1 mA), because if is superimposed on a strong linear ramp of modulation current i m (~5 A). Instead, we measured the input voltage (v,- oc — f Af) feeding one of the two arms of the adder (i.e., '—£Af from (I)' point in Figure 4.7). With the knowledge of the gain for the last stage of amplification ('adder+power amplifier' configuration ; gain calibrated to be 2.18 ± 0.02)} and the 4.9822 ft resistance of the monitor resistor in series with the modulation coil, the feedback current if could be found. This procedure gave an amplitude of the feedback field kf = 0.213 G, with the applied field H = 51.40 kG, and the temperature T = 1.18 ± 0.01 K. Since the optimum MFB gain setting was found from maximizing the funda-mental a amplitude, equation (5.35) should be the appropriate relation between hf and Af. The factor (p + + p~)/2 was found experimentally from the ratio of the two different MFB gain settings, P++P~ (0 3.55 ~ 2 ~ " = 23l8 = L 4 3 ' and the demagnetizing factor was 6 = 1/3 for a spherical sample, so that the absolute amplitude for the fundamental JT oscillation was calculated to be 0.018 ± 0.001 G, with the quoted error arising from the uncertainties in both the feedback field measurement and the setting of the MFB gain. The curvature factor C was calculated to be C — \d2A/dkjj\ = 4.90 from a new pseudopotential band-structure model of Gold, Holtham and Van Schyndel (to be published) fitted to the latest dHvA frequency data of Joss (1981). With experimental parameters H = 51.40 kG, T = 1.18 ± 0.01 K, /i = 0.945 ± 0.005, 138 TD = 0.20 ± 0.02 K and with the dHvA frequency F = 35.64 ± 0.02 MG given by Joss (1981), we finally have , from the L K formula (2.12), corresponding to S = 0.263 ± 0.026 within the basic interval. With this S value, | cos 2JT5| for the second harmonic is found to be 0.082. This near-vanishing of cos 2nS offers a natural explanation for the unobservably-small second harmonic of the TT oscillation (~ spin-splitting zero for the second harmonic 2n). The curvature factors for the other orbits were not yet available when this the-sis was being written. The absolute amplitudes deduced from experimental para-meters for the rest of the orbits are listed in Appendix B. A{CH)i vFT = 0.677 ±0 .061 , 139 C H A P T E R 7 C O N C L U S I O N S 7.1 Summary of Experimental Results In Chapter 6, we have discussed in considerable detail the experimental mea-surements of the quantities required for evaluating the orbital spin-splitting factor S for the various cyclotron orbits in lead. In this section we shall draw these experimental results together. 1. Effective masses. In Table 7, we present a summary of our experimental results and compare them with previously reported data from both d H v A and the cyclotron-resonance experiments. There is good overall agreement with the work of other investigators. 2. Dingle temperatures. The Dingle temperatures for our samples are shown in Table 8, where they can be compared with the results of Phillips and Gold (1969) for long-rod samples. Our results fall in between their smallest and largest values, but closer to the latter. Unlike Phillips and Gold, we have never found essentially zero Dingle temperatures for any of our spherical samples. This is probably because our single-crystal rods as pulled from the melt were more than ten times the diameter of the thin rods pulled by Phillips and Gold. The larger size of our crystal increased the possibility 140 T a b l e 7 Cyclotron Effective Masses in Lead oscillation, orientation orbit effective this work mass p. = m*/m0 d H v A 1 cyclotron resonance' a [100] i> 1.53 ±0.03 1.51 ± 0.03 1.58 a [110] i> 1.126 ±0.004 1.10 ±0.01 1.12 or [111] i> 1.113 ±0.005 1.12 ±0.01 1.14 1 [no] ?3 0.555 ± 0.003 0.56 ± 0.01 0.56 P [100] V 1.233 ±0.005 1.22 ±0.01 1.23 ir [100] 0.945 ± 0.005 0.89 ± 0.02 0.93 6 [111] 9 1.196 ±0.005 1.19 ±0.01 1.20 1 Phillips and Gold (1969) 2 Khaikin and M i n a (1962), M i n a and Khaikin (1963), as read from graphs (uncertainties not given). 3 Quoted results refer to an average for the £ c central orbit and the f n c noncen-tral orbit. Separate values for the central orbit and £ n c noncentral orbit are given to be 0.539 ±0.006 and 0.571 ±0.007 respectively by d H v A effect (Ogawa and Aoki 1978). of microcrystalline substructure. Furthermore, our samples were subjected to an extra degree of handling when they were spark-machined into spheres. 3. The spin-splitting factors S and ge = 25//i. S and the related values of gc = 2S/p are summarized in Table 9. The ef-fective masses fi used to convert S to gc are our own results from Table 7. When 141 T a b l e 8 Dingle Temperatures for Pure Lead Samples oscillation, -orientation j orbit Dingle this work* temperature TD (K) Phillips and Gold (1969) smallest largest cr [100] 0.68 ±0.05 (18) -0.09 ± 0.07 0.87 ±0.14 a [110] 0.26 ± 0.02 (3) -0.01 ±0.07 0.26 ± 0.20 a [111] 0.58 ±0.03 (11) -0.01 ±0.09 0.84 ±0.14 7 [HO] 0.29 ± 0.04 (3) — 0.30 ±0.10 p [100] u 0.11 ±0.01 (18) 0.08 ± 0.05 0.41 ± 0.05 TT [100] i 0.20 ± 0.02 (18) -0.01 ± 0.07 0.98 ± 0.06 6 [111] e 0.22 ±0.02 (11) -0.02 ± 0.06 0.68 ±0.10 * The Dingle temperatures listed here are those being used to extract S values in our experiments. Sample numbers for our spheres are given in parentheses. discussing individual oscillations in Chapter 6 we have simply presented the prin-cipal S values in 0 < S < |, whereas information extracted from trigonometric functions is necessarily multivalued, as discussed in section 2.3. The full range of possible values n ± S for all integers n becomes truncated according to Pippard's inequality (section 2.3) 5 < M + | , (2.18) with s denoting the number of Bragg reflections undergone by an electron in one revolution around the cyclotron orbit. T h is truncation still leaves us with a choice 142 Table 9 Summary of Possible Orbital yc-Factors of Lead oscillation, Bragg upper limit orbit reflection of S <7C = 2S//1 orientation number (s) (^max) a [100] V- 8 5.53 — a [110] 6 4.13 0.354 > 0.628 ± 0.007 0.646 1.147 ±0 .008 1.354 2.41 ± 0 . 0 1 1.646 • ±0 .004 2.92 ± 0.01 2.354 4.18 ± 0 . 0 2 2.646 4.67 ± 0.02 3.354 5.96 ± 0.02 3.646' 6.48 ± 0.03 a [111] 6 4.11 0.460 x 0.827 ± 0.008 0.540 0.970 ± 0.008 1.460 2.64 ± 0.01 1.540 2.77 ± 0.01 •±0.004 2.460 4.42 ± 0.02 2.540 4.57 ± 0.02 3.460 6.22 ± 0.03 3.540-' 6.36 ± 0.03 1 [no] 3 2.06 0.194 x 0.70 ± 0.02 0.806 2.91 ± 0.03 • ±0 .006 1.194 4.30 ± 0.03 1.806' 6.51 ± 0 . 0 4 143 Table 0 (continued) oscillation, orientation Bragg upper limit orbit reflection of 5 S < 5n number (5) (5 m a x) gc = 2S/n p [100] v 5.23 0.303 \ 0.697 1.303 1.697 2.303 2.697 3.303 3.697 4.303 4.697'' > ± 0.003 0.491 ± 0.005 1.131 ±0.007 2.11 ±0.01 2.75 ± 0.01 3.74 ± 0.02 4.38 ± 0.02 5.36 ± 0.02 6.00 ± 0.03 6.98 ± 0.03 7.62 ± 0.03 ir [100] £ 2.95 0.263 s 0.737 1.263 1.737 2.263 2.737' > ± 0.026 0.56 ± 0.05 1.56 ± 0.05 2.67 ± 0.06 3.68 ± 0.06 4.79 ± 0.06 5.79 ± 0.06 s [111] e 4.20 0.357^ 0.643 1.357 1.643 2.357 2.643 3.357 3.643/ > ± 0.003 0.597 ± 0.006 1.075 ±0.007 2.27 ± 0.01 2.75 ± 0.01 3.94 ± 0.02 4.42 ± 0.02 5.61 ± 0.02 6.09 ± 0.03 144 of about 4 ~ 10 possible gc values for each orbit, and in the following section we appeal to theoretical considerations for help in making the final selection. 7.2 Theoretical Considerations and the Orbital a c-Factor Results In order to decide which of the possible gc values listed in Table 9 are the correct ones, recourse must be made to our theoretical understanding of the spin-splitting factors for conduction electrons. T h e main mechanisms responsible for the departure of the conduction electron g values from the free-electron value go = 2 are the spin-orbit and many-body interactions. In the following we will discuss them in turn. 7.2.1 Spin-Orbit Interaction The theoretical considerations of the spin-splitting in metals on the basis of band-structure theory have been reviewed by Yafet (1963), and later developments of this theory can be found, for example, in de Graaf and Overhauser (1969, 1970), Moore (1975a, b), etc. Here we shall not attempt to give detailed discussions of the rather complicated theory involved, but shall only give a semi-qualitative discussion for the magnitude of shift from g0. A useful guide for setting bounds on the y-shift due to spin-orbit interaction is given by Dupree and Holland (1967). Their calculation is based on the tight-binding approximation and on first-order perturbation theory. They show that the maximum o-shift can be expressed as \SgUx = { 2 l 4 l * ) 6 E , (7.1) where 6g is the departure of the conduction electron g factor from the free-electron value <7o, A the maximum spin-splitting for the atomic states contributing to the 145 wave function for a state at the Fermi surface, / the corresponding azimuthal quan-tum number, and SE the band gap to the nearest other band with appropriate symmetry. This criterion was applied to the alkali and noble metals by several authors (Dupree and Holland 1967, Randies 1972). This upper bound does not, however, give any way to sum up the contributions from an orbit of mixed symme-try, or even to predict the sign of the shift. Furthermore, the Dupree and Holland calculation has nothing to do with the average around the cyclotron orbit. Nev-ertheless, equation (7.1) gives a range into which the gc-factor should fall, more or less, as far as the spin-orbit interaction is concerned. The strength of the spin-orbit interaction depends roughly on Z4f where Z is the atomic number of the metal. As a heavy metal (Z = 82), the spin-orbit interaction in lead is expected to be strong. The upper limit of the g-shift due to this mechanism can be estimated according to equation (7.1). In our estimation, the spin-orbit interaction is considered only for p states, i.e., 1=1. The quantity A is taken to be A = 1.35 eV from Table 2 in Yafet (1963). The energy differ-ence SE used is that between the third-zone bands near the zone-corner W. It is estimated to be 3.3 eV from the band-structure calculation of Looney and Dreesen (1979). Upon substituting these values into equation (7.1), |5</|max is found to be 0.55. Although this is just a rough estimate of the g-shift resulting from only the spin-orbit interaction, it will clearly allow us to rule out most of the entries in Table 9. However we must first consider the many-body effects before the final selection can be made. 7.2.2 Many-Body Interactions Since the L K theory is based on the single-particle assumption, many efforts have been made to take many-body interactions into account (see, for ex-ample, Luttinger 1960, Fowler and Prange 1965, Engelsberg and Simpson 1970). It has been proved that the LK formula retains its validity in the presence of 146 both electron-electron and electron-phonon interactions, in so far as the experimen-tal conditions are not too extreme, but the parameters entering into the formula have to be appropriately renormalized. This renormalization can be expressed within the framework of the Landau theory of Fermi liquids and for an isotropic metal the interactions are determined by the coefficients AQ,A\ and Bo, By ,... of the Legendre expansion of the spin-symmetric and spin-antisymmetric parts, respectively, of the Landau scattering function. Here again we shall not attempt to present the basic theory, but only summarize the key results of the theory which can be used as an aid in the choice of the correct gc values. The general background of the many-body theory involved is given, for example, by Hedin and Lundqvist (1969) and by Wilkins (1980). For the gc-factor, Kaplan and Glasser (1969) suggest that the renormalization is simply * " TTB0 <7-2> where gs depends on spin-orbit coupling, but is not affected by the many-body effects. The coefficient B0 is determined by both electron-electron and electron-phonon effects and can be separated in the form l + B0 = (l + B*E)(l + BSp). (7.3) The cyclotron effective mass is renormalized as m*e=ml(l + A$p), (7.4) where mj is the 'bare mass', which contains contributions from the band-structure and electron-electron interactions (Heine, Nozieres and Wilkins 1966). Again the 147 bare mass can be expressed as m; = m c % ( l + A f * ) , (7.5) where m*r is the 'crystalline mass' obtained from band-structure calculations and should in principle not depend on electron-electron interaction (Prange and Sachs 1967). Since band-structure calculations usually include screening effects, equation (7.5) must be considered as a slight oversimplification. Usually AfE is rather small, so that m£ and m*r are nearly equal. With the aid of equations (7.2)~(7.5), the product gcm* can be expressed as 9 c m < - ^ ( i + « ( i + r ) " ( } Engelsberg and Simpson (1970) have proved that this product gcm* is enhanced by the same amount as the electronic (Pauli) spin susceptibility x» depends only on the electron-electron interaction but not on the electron-phonon effect (Prange and Sachs 1967). This is only possible if the electron-phonon factors in the numerator and the denominator of equation (7.6) cancel, i.e., if 1 + Ajjp = 1 + B§P = l + \ , (7.7) where A is usually called the electron-phonon enhancement parameter. Therefore we finally have, from equat ions (7.2), (7.3) and (7.7), The electron-phonon interaction in lead is known to be strong because of the the high transition temperature Tc of superconductivity and the high electronic specific heat 7 ej. The effective-mass enhancement parameter 1 + A can be found 148 from the ratio of the experimental value of the effective mass fiexp to the calculated value p,hc from an energy band model if we assume that = m*r. The p.hc values were calculated from the band-structure model of Gold, Holtham and Van Schyndel (to be|published), and the 1 + A values are listed in Table 10. These values are close to 2, and the values in the second zone are in general smaller than those in the third zone. Of all the Fermi-surface orbits listed in Table 10, the two with the highest enhancement parameters are £ and f; these two orbits also happen to be the two with the lowest cyclotron masses (Table 10) and the smallest orbital areas (see dHvA frequencies in Table 4). According to equation (7.8), the spin-splitting yc-factors should be reduced by the same factor of 1 + A. As a result, the gc-factors for the [110] c and [100] f orbits are expected to be lower than the others, if the spin-orbit coupling strength is about the same. The electron-electron renormalization for the gtc-factors in lead is thought to be small compared with the effect of the spin-orbit and electron-phonon interactions. If we assume that BEE w 0, the ac-factor should fall between 00 + |<$fif|max , go — \6g\max 9max — r — — : and <7min = 1 + A 1 + A for each orbit. We therefore choose the experimental value within this interval as the physically meaningful value of the orbital yc-factor for each orbit, as shown in Table 10. It is found from Table 10 that there are two possible gc values, namely <7d = 0.827 and gc% = 0.970, between gmax and <7min for the [111] rp orbit (a oscillation). The corresponding S values for them are Si = 0.460 and S 2 = 0.540 respectively. Since Si gives cos uS > 0, while 5 2 gives cos irS < 0, the fi-nal choice depends on the sign of cos trS. Experimental determination of the sign 149 Table 10 Final Choice of Experimental Values for Orbital Factors of Lead* oscillation, orientation orbit electron-phonon fie*p fihc enhancement 1 + A 9min 9m ax 9lxp '9. a [100] 1.53 0.750 2.04 ± 0.04 0.71 1.25 — — a [110] 1.126 0.556 2.025 ± 0.007 0.71 1.26 1.147 ±0 .008 I 0.321 ± 0.008 a [111] 1.113 0.546 2.038 ± 0.009 0.71 1.25 0.827 ± 0.008 or 0.970 ± 0.008 -0.32 ± 0.01 or-0.023 ± 0.009 7 [HO] 0.555 0.246 2.26 ± 0.01 0.64 1.13 0.70 ± 0.02 -0.42 ± 0.03 P [100] V 1.233 0.607 2.031 ±0.008 • 0.71 1.26 1.131 ± 0 . 0 0 7 +0.295 ± 0.007 TT [100] i 0.945 0.362 2.61 ± 0.01 0.56 0.98 0.56 ± 0.05 -0.54 ± 0.09 6 [111] e 1.196 0.549 2.18 ±0 .01 0.67 1.17 1.075 ± 0 . 0 0 7 +0.342 ± 0.007 * References for / i e x p and / i b c are explained in the text. The experimental errors for nexp are listed in Table 7. |5o|m a x is taken to be 0.55, as explained in the text. 1+A = M « P / ^ , gmia = g°-jyi—, w = g 0 +lt glm a x» a D d = (l+A)a-P-ff 0, with g0 = 2.0023. of cos nS requires measuring the infinite-field phase of the dHvA oscillation (see Coleridge and Templeton 1972, 1980). Further work is required to resolve this ambiguity. The fir-shift 6g, due to spin-orbit coupling can be found from the experimental value gr*xp by These 8gt values are presented in the last column of Table 10. 7.3 Concluding Remarks and Suggestions for Further Study In this thesis we have presented in detail a study of orbital spin-splitting factors of conduction electrons for various extremal orbits at the Fermi surface of lead by means of measuring the dHvA amplitudes. As a method for the reduction of Shoenberg's M I effect, the M F B technique has been employed in our investigation. Shoenberg's 'new treatment' (Shoenberg 1976, MIPS treatment as we call it) has been generalized to the case of M F B . It has been found that the new model gives a better understanding and accounts well for many of the observed features in the experiments. The MIPS treatment predicts that although the M F B can no longer suppress the M I completely in case of two or more oscillations from different orbits, it is still a powerful means to reduce the M I effect as long as the imperfection of the sample is not too severe. It would be highly desirable to relate the correlation factors p+ and p~ quan-titatively to the microscopic structure of the inhomogeneous sample, i.e., to the mean tilt between adjacent grains in a mosaic structure, or to the density of lattice dislocations. This question is open for further study. 151 The values of the spin-splitting 0 c-factors for lead are given for almost all the extremal orbits at the Fermi surface when the applied field is directed along the three major symmetry orientations. So far this is the most complete data for the <7c-factors in polyvalent metals. Experimental values are compared with our theoretical considerations on the y-shift for the conduction electrons from its free-electron value go = 2. The gc values for most of the orbits are close to 1, except for the [110] ( and [100] £ orbits, where gc has dropped to 0.70 and 0.56, respectively, because of the strong electron-phonon interaction. Extracting $»c-factors from the absolute amplitude would be worthwhile to do, as soon as the more reliable curvature factors from the new band-structure model (Gold, Holtham and Van Schyndel) are available. T h i s would be the only means of measuring the <7C-factor for the [100] a oscillation under conventional experimental conditions; it would also provide further confirmation for the ^-factors for the other orbits. In order to have a better physical understanding of the y c-factors, it is re-quired to separate the major effects responsible for the g shifts, namely, spin-orbit, electron-phonon and electron-electron interactions from one another. Theoretical calculations for the ^-factors based on the band-structure theory should be per-formed, so that the contribution from spin-orbit coupling can be estimated. The contribution from the electron-phonon interaction can be easily ascertained from the ratio of the experimental value of the cyclotron effective mass to that calculated from band-structure theory, as we have already done in the previous section. 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Mag., 7, 2065. 158 A P P E N D I X A Magnetic Feedback with Two Fundamental Frequencies —Effect on the Higher Harmonics In section 5.3, we have studied the M F B with two fundamental frequencies using the MIPS treatment. We have found that the effect of the M F B is highly dependent on the correlations between the phase deviations of the two frequencies. However, we have limited our discussions only to the M F B dependence of the fundamental amplitude A'a and two lowest order sidebands A'a±b. Here we will expand our discussions to the influence of M F B on harmonic amplitudes A'ra under the condition that the MI between the fundamental and its own harmonics can be ignored or has been effectively suppressed. We use the same notation as in section 5.3. With the above assumption, the M I between the rth harmonic of the o oscillation (with frequency Fra = rFa) and the fundamental of the b oscillation (with frequency Ft,) can be treated as if they were both fundamental. As a result, we can still use equation (5.24) as an expression for the component with frequency rFa in the local magnetization except for that Aao should be replaced by Ara0t xa by rxa, A<pa by rA<pa and *ca by r/c a. Then we have 159 in which only the terms k = 0,1, — 1 are significant. The next step will be the phase-smearing average over the whole sample. With k = 0, we simply have /+oo sin(rxa + rA(pa)D(rA<pa)d(rA<pa) = Jf£sin rxa; -oo while with k = ± 1 , the average involves phases rA<pa ± A<pbl and the results will depend on the correlation between two phases rA<pa and Aipb. Calculations show that the discussions on the correlations in section 5.3 still hold, and we can still use the p+, p~ factors to indicate correlations. Note that now p+ and p~ might be different for different r's. In case of completely lack of correlation, we have the same result p+ = p~ = 1; while in case of complete correlation, we have p+ and p~ as shown in Table 11. In case of partial correlation, no explicit solutions of p+ and p~ have yet been found mathematically. Table 11 Values of Reduction Factors p+ and p~ when A<pb = £A(pa rA<pa + &<Pb Pr rA(pa-A<pb p r 0 < £ < r KraKb 1 •jq\KlKb) 1 Kl 1 -j^(KraKb) a 1 KIr - r < £ < 0 ; •Jp{KaKb) 1 Kl K^Kb 1 i<-r - k J r { K K b ) 1 KlKb 1 160 The observed amplitude of the rth harmonic of a oscillation with MFB can be expressed, similar to equation (5.25), A'ra = Ara[MrKaAbo)Jo(^^) + (vt + p r J ^ x ^ ^ y ^ ^ f ^ j ) ] , (A.2) with Ara = AraoKa- Retaining only the leading terms of the Bessel functions involved, we have the approximation 4 . - * . { . - [ ( ^ - * ± £ ) ' + ^ - ( * ± i £ ) ' ] j . (A.3) With f = 4?r(l — 6), the MI between the harmonics of oscillation a can be effectively compensated, but according to (A.3), the MI due to 6 oscillation can still distort the amplitude j4 r o, A'ra = Ara{l- \r2KlA2b [l - (p+ +p-) + Kh~2]} . (AA) This effect could be the reason for the fact that no matter what the MFB gain was set, the slope ratio of the effective mass plots for the first three harmonics of the [110] a oscillation was still away from 1:2:3 (see section 6.3.2). 161 A P P E N D I X B Absolute Amplitudes of the Fundamental d H v A Oscillations In this appendix, we present our experimental results of the absolute ampli-tudes for the fundamental dHvA oscillations in lead measured from the feedback field (see section 6.6.2 for experimental details). Due to difficulties in the direct measurement of the feedback current in the modulation coil, as discussed in sec-tion 6.6.2, the actual physical quantities we measured are the input voltage of the feedback integrator (i.e., output voltage of the Krohn-Hite filter, see Figure 4.6). Since the total amplification gain of the feedback integrator depends on dM/dt and M gain settings as well as the time-frequency of the signal, it was calibrated indi-vidually for each of the dHvA oscillations at the same gain and frequency settings as the particular dHvA oscillation whose amplitude was being measured. The quoted error includes both the uncertainties in the voltage measurement and in setting the appropriate feedback gain. 162 Table 12 Absolute Amplitudes of the Fundamental d H v A Oscillations* oscillation, orientation orbit H (kG) T ( K ) TD (K) sample A(G) a [100] 62.75 1.16 0.68 ± 0.05 18 (2.89 ±0.17) x IO" 3 a [110] 73.39 1.16 0.26 ± 0.02 3 0.178 ±0.007 a [111] 68.80 1.16 0.58 ± 0.03 11 0.049 ± 0.004 7 [HO] ? 69.45 1.17 0.29 ± 0.04 3 0.122 ±0.005 0.122 ±0.005 fi [100] V 53.29 1.15 0.11 ±0.01 18 0.113 ±0.004 0.064 ± 0.004 JT [100] i 51.40 1.18 0.20 ± 0.02 18 0.018 ±0.001 6 [111] e 71.58 1.16 0.22 ± 0.02 11 0.077 ± 0.006 * Because orbits f and v occur in inversion pairs, the quoted amplitudes have been halved. The oscillations from these orbits show two-frequency beat patterns, and the two entries for each orbit are the separate amplitudes A A and A B of the two beating components (equation 6.1). 


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