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The effect of spin-orbit coupling on conduction electrons in metals Schmor, Paul Wesley 1973

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THE EFFECT OF SPIN-ORBIT COUPLING ON CONDUCTION ELECTRONS IN METALS by PAUL WESLEY SCHMOR B.Sc., ...McMas.ter U n i v e r s i t y , -196.5 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE ' DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We a c c e p t t h i s t h e s i s as conforming to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1973 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 fo r extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It 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 The University of B r i t i s h Columbia Vancouver 8, Canada Chairman: Professor Andrew V. Gold i i ABSTRACT Spin-orbit coupling for conduction electrons in metals i s studied by investigating g , an average of a spin-sp l i t t i n g factor for electronic states on extremal orbits of the Fermi surface. g is found from an examination of c the harmonic content of the de Haas-van Alphen oscillations in the magnetization, specifically from the dimensionless 2 ratio a = A^/ (A^A^i, where A^, A^, and A^ are the amplitudes of the f i r s t three harmonics. The method has been applied to lead, for which spin-orbit coupling effects are particu-l a r l y important, and a sensitive magnetoresistive-probe technique has been developed to detect the oscillations in disk-shaped samples. Values of g c have been found for two extremal orbits, and they d i f f e r significantly from the value g =2 for perfectly free electrons, c The pseudopotential interpolation scheme developed by Anderson, 0 1Sullivan, and Schirber to describe the band structure and Fermi surface of lead has been modified to include the effect of an applied magnetic f i e l d , thereby enabling the spin-splitting factors g(k) to be calculated for any point k in the Brill o u i n zone. The computed values of g(k) are found to depend markedly on the orientation of the f i e l d , and are compared with the experimental results. i i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF FIGURES v i LIST OF TABLES v i i i ACKNOWLEDGEMENTS i x Chapter I . INTRODUCTION 1 1 . 1 G e n e r a l C o n s i d e r a t i o n s 1 1 . 2 T h e s i s O u t l i n e 6 I I . g-FACTORS FROM THE DE HAAS-VAN ALPHEN EFFECT: THEORETICAL BACKGROUND 8 2 . 1 Zeeman S p l i t t i n g o f O r b i t a l Landau L e v e l s 8 2 i 2 Shoenberg E f f e c t : Magnetic I n t e r a c t i o n between Conduction E l e c t r o n s . . . . . 3 0 2 . 3 Other Systematic E r r o r s 4 3 (a) F i e l d Inhomogeneity 4 3 (b) Specimen Inhomogeneity due t o Mosaic S u b s t r u c t u r e , B e n d i n g , e t c 4 6 (c) Two Neighbouring F r e q u e n c i e s . . . 4 8 2 . 4 Summary 5 0 I I I . APPARATUS AND EXPERIMENTAL PROCEDURES . . . 5 2 3 . 1 G e n e r a l C o n s i d e r a t i o n s . . . . 5 2 Chapter i v Page 3.2 Magnet and C r y o s t a t . 55 3.3 Sample P r e p a r a t i o n 61 3.4 F i e l d M o d u l a t i o n 65 3.5 S i g n a l D e t e c t i o n C i r c u i t r y 72 3.6 Numerical A n a l y s i s 80 IV. g -VALUES FROM THE DE HAAS-VAN ALPHEN ^c ALPHEN EFFECT: EXPERIMENTAL RESULTS . . . . 83 4.1 I n t r o d u c t i o n . . . . . . . . . . . . . 83 4.2 g o s c i l l a t i o n s 86 (a) P r e l i m i n a r y R e s u l t s . 86 (b) F i e l d Inhomogeneity 89 (c) Specimen Inhomogeneity 92 (d) Neighbouring F r e q u e n c i e s 93 (e) g c - V a l u e s f o r the 3 O s c i l l a t i o n s 98 4.3 g c - v a l u e s f o r the y O s c i l l a t i o n s . . . 102 V. g(ic) VALUES FROM A BAND STRUCTURE MODEL 110 5.1 The H a m i l t o n i a n i n the Absence of a Magnetic F i e l d 110 5.2 The Magnetic F i e l d C o n t r i b u t i o n t o the H a m i l t o n i a n . . 116 5.3 g (Jc) a t P o i n t s o f High Symmetry i n the B r i l l o u i n Zone 122 5.4 g Ck) on the Extremal O r b i t s 133 V Chapter Page V I . CONCLUSION 139 6 . 1 G e n e r a l C o n c l u s i o n s and Comparison of Theory w i t h Experiment . . . . . . 139 6 . 2 Suggestions f o r F u r t h e r Study . . . . 141 APPENDIX I - AMPLITUDES AND PHASES FOR NON-QUADRATIC EXPANSIONS OF CROSS-SECTIONAL AREAS 143 REFERENCES 145 v i LIST OF FIGURES F i g u r e Page 2 . 1 THE ZEEMAN SPLITTING OF LANDAU LEVELS . . . . 1 2 2 . 2 THE RATIO OF THE THERMAL DAMPING FACTORS . . . . . 2 2 2 . 3 THE RATIO OF THE DINGLE COSINE TERMS . . . . 2 3 2 . 4 BRAGG REFLECTIONS AND THE RELATED MAXIMUM SPIN-SPLITTING 2 9 2 . 5 THE SHOENBERG EFFECT . 3 2 2 . 6 CORRECTION FACTORS FOR THE SHOENBERG EFFECT . 4 0 3 . 1 SIMPLIFIED SCHEMATIC OF THE EXPERIMENTAL ARRANGEMENT . . 5 3 3 . 2 CRYOSTAT TAILS . . . . . . 5 6 3 . 3 He 3 INSERT DEWAR 5 8 3 . 4 EXAMPLES OF X-RAY PHOTOGRAPHS 6 3 3 . 5 SAMPLE HOLDER 6 4 • 3 . 6 THE FREQUENCY DEPENDENCE OF THE MAGNETIC FIELD 6 9 3 . 7 THE MODULATION AMPLITUDE DEPENDENCE OF THE MAGNETIC FIELD . . . . . 7 0 3 . 8 BLOCK CIRCUIT DIAGRAM . . . . . . . . . . . . . 7 4 3 . 9 - WHEATSTONE BRIDGE ARRANGEMENT ........ 7 7 3 . 1 0 FIELD RESPONSE FOR THE MAGNETO-RESISTOR 7 8 4 . 1 A SKETCH OF THE FERMI SURFACE OF LEAD 8 4 v i i F i g u r e Page 4.2 PERIOD VARIATIONS IN LEAD 85 4.3 a 1 AS A FUNCTION OF THE DEMAGNETIZING FACTOR 88 4.4 FIELD PROFILE FOR HIGH FIELD POLE TIPS . . . 90 4.5 FIELD DEPENDENCE OF BEATING ENVELOPE FOR g OSCILLATIONS 96 4.6 VARIATION OF CROSS-SECTIONAL AREA FOR THE V ORBIT . . . . . . . . . . . . . 97 4.7 EXAMPLE OF £ OSCILLATIONS 99 4.8 FIELD DEPENDENCE OF BEATING ENVELOPE FOR Y OSCILLATIONS 104 4.9 VARIATION OF CROSS-SECTIONAL AREA FOR THE £ ORBIT 105 4.10 EXAMPLE OF y OSCILLATIONS 106 5.1 MODEL POTENTIAL . 113 5.2 PRIMITIVE BRILLOUIN ZONE . . . . . . . . . . 123 5.3 g(W) FOR FIELD ROTATION ABOUT z AXIS . . . . 126 5.4 g(W) FOR FIELD ROTATION ABOUT x AXIS . . . . 127 5.5 g(W) FOR FIELD ROTATION ABOUT y AXIS . . . . 128 5.6 EQUIVALENT CORNERS OF THE BRILLOUIN ZONE . . 130 5.7 g(X) FOR FIELD ROTATION ABOUT y AXIS . . . . 132 5.8 THE SHAPE OF THE V ORBIT 134 5.9 THE SHAPE OF THE ? ORBIT 136 v i i i LIST OF TABLES Ta b l e Page I . EXPERIMENTALLY OBSERVED g-VALUES IN • METALS 3 I I . FOURIER COEFFICIENTS FOR EQUATION (2.2.4) 35 I I I . AMPLITUDE COEFFICIENTS FOR FIELD MODULATION , . 67 IV. EXPERIMENTAL RATIOS FOR COS (w ) /COS (v^) . . 1 Q 9 i x ACKNOWLEDGEMENTS I t i s a p l e a s u r e t o thank P r o f e s s o r A.V. Gold f o r the many h e l p f u l s u g g e s t i o n s t h a t he o f f e r e d d u r i n g the course o f these i n v e s t i g a t i o n s . I would l i k e to extend thanks t o Dr. Dav i d Boyle f o r h i s h e l p i n d e s i g n i n g and i n s e t t i n g - u p the e x p e r i -mental apparatus and t o Dr. P e t e r Holtham f o r many-h e l p f u l d i s c u s s i o n s . I would a l s o l i k e to express my thanks to Dr. J.R. Anderson, who managed t o f i n d c o n s i d e r a b l e time i n a v e r y busy schedule t o e x p l a i n h i s band s t r u c t u r e programme, f o r a l l o w i n g me t o use t h i s programme and f o r g i v i n g me data p r i o r t o p u b l i c a t i o n . I would a l s o l i k e t o thank the P h y s i c s department a t the U n i v e r s i t y of Maryland f o r t h e i r generous h o s p i t a l i t y d u r i n g October 197 2. A b u r s a r y from the N a t i o n a l Research C o u n c i l , a H.R. M a c M i l l a n F a m i l y F e l l o w s h i p , and a U.B.C. graduate F e l l o w s h i p a r e g r a t e f u l l y acknowledged. F i n a l l y I would l i k e to thank my w i f e Irene f o r her c o n s t a n t p a t i e n c e and encouragement. CHAPTER I INTRODUCTION 1 . 1 G e n e r a l C o n s i d e r a t i o n s The e n e r g i e s of c o n d u c t i o n e l e c t r o n s i n the presence of a magnetic f i e l d a re q u a n t i z e d i n t o degenerate l e v e l s (Landau l e v e l s ) . The energy s e p a r a t i o n between these l e v e l s depends on the f i e l d and hence can be v a r i e d e x p e r i -m e n t a l l y . When e l e c t r o n s p i n i s c o n s i d e r e d these q u a n t i z e d l e v e l s are s p l i t ( r e f e r r e d to as e i t h e r s p i n or Zeeman s p l i t t i n g ) i n t o two s e t s ; one s e t f o r each of the two s p i n o r i e n t a t i o n s . i n a manner t h a t i s analagous t o the d e f i n i t i o n f o r the c o r r e s p o n d i n g atomic case, the s p l i t t i n g ^ E S p i n d e f i n e s a g-value f o r c o n d u c t i o n e l e c t r o n s , namely, 6E . 9 U B H where i s the Bohr magneton and H i s the magnetic f i e l d . * The de Haas-van Alphen ( d H v A ) e f f e c t — a n o s c i l l a t o r y v a r i a t i o n i n the magnetic s u s c e p t i b i l i t y — a r i s e s from energy l e v e l s c r o s s i n g the .Fermi energy as the f i e l d i s v a r i e d and c o n t a i n s i n f o r m a t i o n about the r e l a t i v e s p a c i n g between the two s e t s of l e v e l s . These o s c i l l a t i o n s a l s o c o n t a i n i n f o r m a t i o n about the p r o p e r t i e s of the Fermi * de Haas and van Alphen (1930). d i s c o v e r e d these o s c i l l a t i o n s i n bismuth c r y s t a l s . s u r f a c e (.the s u r f a c e determined by the l o c a t i o n o f the Ferm i energy i n momentum or k space), and i t i s o n l y a f t e r h a v i n g reached a f a i r l y thorough understanding of the Fermi s u r f a c e t h a t i t has been p o s s i b l e to study the Zeeman s p l i t t i n g o f energy l e v e l s i n metals ( i . e . , t h a t measure-ments c o u l d be made on the g-values of c o n d u c t i o n e l e c t r o n s ) . The dHvA experiments, however, are not the onl y means o f d e t e r m i n i n g g and i n T a b l e I a number of e x p e r i -* m e n t a l l y determined v a l u e s o f g i n metals are l i s t e d t o -gether w i t h the types o f experiments t h a t were used. A number o f the e n t r i e s i n t h i s T a b l e d i f f e r markedly from the f r e e e l e c t r o n g-value (g = 2\. Approximate c a l c u l a t i o n s of g from a p e r t u r b a t i o n theory (see Dupree and H o l l a n d (1967)1 show t h a t d e v i a t i o n s from the f r e e e l e c t r o n g-value depend d i r e c t l y on the s t r e n g t h of the s p i n - o r b i t i n t e r a c t i o n and i n v e r s e l y on the s p a c i n g between energy bands. Thus when two energy bands are v e r y c l o s e to one another the e l e c t r o n s i n these bands can have l a r g e g - v a l u e s . A l s o w i t h r e f e r e n c e to Tab l e I , i t should be noted t h a t the g-values f o r c e r t a i n metals depend on the types of experiments t h a t were used. There are two reasons f o r t h i s . F i r s t , the dHvA and r e l a t e d e f f e c t s (such as the o s c i l l a t i o n s i n temperature and i n r e s i s t a n c e ) measure a g-val u e f o r a v e r y s p e c i a l i z e d group o f e l e c t r o n s on the -F e r m i s u r f a c e . T h i s i s a l s o the case f o r the g i a n t quantum g- v a l u e s f o r Pb have been o m i t t e d as they have n o t been determined u n i q u e l y (see P h i l l i p s and Gold (1969)1. TABLE I EXPERIMENTALLY OBSERVED g-VALUES IN METALS Metal g Comments References . Ag 1.983 1*91 •*•*!. 95 CESR neck or b i t s dHvA _ _ _ _ _ _ _ Schultz et a l . (1967) Randies (1972) A l 1.997 CESR Schultz et a l . (1967) Au 2.26 2.0024 1.08 + 1.75 CESR CESR neck Dupree et a l . (1967 Monet et a l . (1971) Randies (1972) Be 2.0037 CESR Cousins and Dugree (1965) Bi 192. 27.5 225 16.7 1.3 62. 188. 27.5 H 11 binary CESR H || t r i g o n a l CESR H || binary, l i g h t electrons dHvA H j 1 t r i g o n a l , l i g h t electrons dHvA H || binary, l i g h t holes dHvA 70° o f f t r i g o n a l axis magneto-resistance H || binary magnetothermal H || t r i g o n a l GQO Everett (1962) II n McCombe and Seidel (1967) n II II n II II Eckstein and Kent (1965) Boyle et a l . (1960) Ilukor (1970) Cd 1.6 spin-- s p l i t t i n g zero dHvA Coleridge and Templeton (1971) Cs 2.005 (5) 2.0013 CESR CESR Walsh et a l . (1966) Schultz et a l . (1966) T a b l e I. Ccontinuedl M e t a l g Comments References Cu 2.033 2.08 •*• 2.11 2.07 -»• 2.05 1.87 •* 1.93 CESR - - - - - - - - - - - - - - -B e l l y dHvA Dogsbone dHvA Neck dHvA S c h u l t z Randies (1972) II II II n -Ga 0.94 or 30. or 32. GQO H along b a x i s - - - - - - - - S h a p i r a and Lax (1965) K 1.9997 2.77 •* 2.91 CESR _ _ _ _ dHvA _ _ _ _ _ _ Walsh e t a l . (1966) Randies (1972) L i 2.00223 CESR Vanderven (1968) Na 2.0015 CESR Feyer e t a l . (1955) P t 2.06 1.59 {** 2.00 s p i n - s p l i t t i n g z eros dHvA K e t t e r s o n and Windmill e r (1970) Zn 170. 0 ' S u l l i v a n and S c h i r b e r (1967) .the i n d i c a t e s the range o f v a l u e s measured f o r v a r i o u s f i e l d o r i e n t a t i o n s . it * the t h r e e g-values come from d i f f e r e n t extremal o r b i t s . o s c i l l a t i o n s (GQO, see S h a p i r a and Lax (1965)) which a r e found i n pure metals when the a t t e n u a t i o n c o e f f i c i e n t o f sound i s examined i n the presence of an a p p l i e d magnetic f i e l d . On the o t h e r hand, c o n d u c t i o n - e l e c t r o n - s p i n - r e s o n a n c e (CESR, see Walsh (1968)) experiments measure a g-value which has been averaged over the e n t i r e Fermi s u r f a c e . F u r -thermore, Janak (1969) and Kaplan and G l a s s e r (1969) argue t h a t an a d i a b a t i c experiment such as CESR measures a g-value which i s the f r e e e l e c t r o n g m o d i f i e d by s p i n - o r b i t i n t e r a c t i o n , whereas the dHvA experiments are i s o t h e r m a l and measure a g-value t h a t i s , i n a d d i t i o n , m o d i f i e d by many body e f f e c t s ( e l e c t r o n - e l e c t r o n and electron-phonon i n t e r a c t i o n s ) . Kaplan and G l a s s e r c l a i m t h a t f o r an i s o t r o p i c Fermi s u r f a c e o where g i s the i s o t h e r m a l g-value, <g> the a d i a b a t i c g-v a l u e , and B q a F e r m i - l i q u i d c o n s t a n t (see Platzman and Wolff (1967)) which i s approximately equal t o - 0.2 f o r po t a s s i u m (Randies (1972) ) . We d e c i d e d t o study the g-values of co n d u c t i o n e l e c t r o n s by means o f the dHvA e f f e c t and chose the metal l e a d f o r the f o l l o w i n g r e a s o n s . F i r s t , the Fermi s u r f a c e of l e a d has been i n v e s t i g a t e d i n c o n s i d e r a b l e d e t a i l and i s w e l l u n d e r s t o o d . I n a d d i t i o n , Anderson and Gold (1965) have found t h a t the e f f e c t s on the energy bands due t o s p i n -o r b i t i n t e r a c t i o n and due to the l a t t i c e p o t e n t i a l are comparable. S i n c e the s t r e n g t h of the s p i n - o r b i t i n t e r a c t i o n i s i m p o r t a n t i n d e t e r m i n i n g d e v i a t i o n s from the f r e e e l e c t r o n g - v a l u e , l e a d i s an e x c e l l e n t metal in--which to examine the e f f e c t s o f s p i n - o r b i t c o u p l i n g . F i n a l l y , t h e r e are a number of band s t r u c t u r e models (Anderson and Gold (1965), Anderson, O ' S u l l i v a n , and S c h i r b e r (1972). , and Van Dyke (.1973) ) and i f the e f f e c t o f a magnetic f i e l d i s i n c l u d e d i n these models i t i s p o s s i b l e t o c a l c u l a t e a g-value and hence t o make a comparison w i t h e x p e r i m e n t a l v a l u e s . 1.2 T h e s i s O u t l i n e The body of t h i s t h e s i s i s s u b d i v i d e d i n t o f i v e broad c a t e g o r i e s . I n the f i r s t s e c t i o n o f Chapter I I we develop a new approach f o r e x p e r i m e n t a l l y d e t e r m i n i n g the Zeeman s p l i t t i n g o f energy l e v e l s u s i n g the dHvA e f f e c t . T h i s approach i n v o l v e s measuring a d i m e n s i o n l e s s r a t i o and avo i d s many of the d i f f i c u l t i e s a s s o c i a t e d w i t h p r e v i o u s t e c h n i q u e s . The r e m a i n i n g s e c t i o n s o f Chapter I I d e a l w i t h the e f f e c t s of s y s t e m a t i c e r r o r s . Methods are g i v e n both t o a l l o w f o r and to minimize these e r r o r s . I n Chapter I I I the e x p e r i m e n t a l apparatus and procedures t h a t were used t o measure g-values i n l e a d a re o u t l i n e d . I n c l u d e d i n t h i s chapter i s a d e s c r i p t i o n o f a technique which uses magneto-resistors arranged in a Wheatstone bridge as sensors and involves double frequency modulation. In Chapter IV experimental g-values are given for two different groups of electrons (namely, for the electrons in the [100] v orbit and for the electrons in the [110] S orbit). The band-structure model of Anderson, O'Sullivan, and Schirber (1972), which has been used to calculate the Fermi surface of lead., i s outlined in the f i r s t section of Chapter V. This model does not include the effect of a magnetic f i e l d and in the second section of the chapter we add to the Hamiltonian a magnetic contribution and calculate the required modifications to the original model. The effect of the magnetic f i e l d s p l i t s the energy bands in proportion to the strength of the magnetic f i e l d (to lowest order) and allows us to compute a g-value for conduction electrons at any point Jc i n the B r i l l o u i n zone. In the f i n a l sections of this chapter, g-values which have been calculated from this modified model are- given for various symmetry points i n the B r i l l o u i n zone as well as for certain points relevant to the experimental results of Chapter IV. This i s the f i r s t known report of a pseudopotential model being used to compute g-values. Fin a l l y , i n Chapter VI we compare our computed and our experimental g-values, and offer suggestions for further study. CHAPTER I I g-FACTORS FROM THE DE HAAS-VAN ALPHEN EFFECT: THEORETICAL BACKGROUND 2.1 Zeeman S p l i t t i n g o f O r b i t a l Landau L e v e l s S i n c e the l i t e r a t u r e c o n t a i n s many e x c e l l e n t d e r i v a -t i o n s o f the dHvA e f f e c t ( c f . Gold (1968)), we s h a l l p r e s e n t here o n l y a s i m p l i f i e d and a b b r e v i a t e d o u t l i n e which a c c o u n t s f o r the main f e a t u r e s of the observed e f f e c t and s h a l l m e r e l y i n d i c a t e where a more complete theory i s n e c e s s a r y i n o r d e r to o b t a i n a s a t i s f a c t o r y d e s c r i p t i o n o f e x p e r i m e n t a l r e s u l t s . The motion o f a B l o c h e l e c t r o n i n the presence bf a magnetic f i e l d H i s q u a n t i z e d a c c o r d i n g to the s e m i - c l a s s i c a l Bohr-Sommerfeld q u a n t i z a t i o n r u l e , ^p»dr = (n+y) 2TTK . 2.1.1 -»• Here p i s the momentum c a n o n i c a l l y conjugate t o the p o s i t i o n r , n i s an i n t e g e r , and y i s a c o n s t a n t which equals 1/2 f o r a f r e e e l e c t r o n . I f we d e s c r i b e the motion o f t h i s B l o c h e l e c t r o n by the L o r e n t z f o r c e e q u a t i o n 9 |r- CK £) = -evx H , 2.1.2 where ft k i s the c r y s t a l momentum of the e l e c t r o n and where e i s the proton charge i n emu, i t i s p o s s i b l e t o show t h a t the r-space motion p r o j e c t e d on a pl a n e normal to H w i l l have the same shape as the ic-space o r b i t (except f o r a s c a l i n g f a c t o r eH/H and a r o t a t i o n o f TT/2) . Now i f p = ft ic - e A , 2.1.3 where c u r l A = H , i s s u b s t i t u t e d i n t o Eq. C2.1.1) the quantum c o n d i t i o n can be r e w r i t t e n Cwith the a i d of S t o k e s 1 theorem) as 10 H • r x d r 2 = (n+y) 2ir#/e 2 . 1 . 4 Thus i n r e a l space t h e a r e a of an e l e c t r o n o r b i t p r o j e c t e d on a p l a n e normal t o the a p p l i e d f i e l d H i s q u a n t i z e d . F u r t h e r m o r e s i n c e t h i s p r o j e c t i o n of the r - s p a c e o r b i t has the same shape i n k-space, we o b t a i n the Onsager r u l e f o r the a l l o w e d a r e a s A n i n it-space, namely t h a t A n - 27reH (n+y)/K 2 . 1 . 5 The d i f f e r e n c e i n energy <5E orb between n e i g h b o u r i n g q u a n t i z e d o r b i t s i s 6E _ 3E o r b ~ 3A~ 6A = 27reH/(M §|) 2.1.6 Where u>c i s the c y c l o t r o n f r e q u e n c y . The o r b i t a l energy i s s a i d t o be q u a n t i z e d i n t o "Landau l e v e l s " s e p a r a t e d i n energy by Mw . 11 We now c o n s i d e r e l e c t r o n s p i n . In the presence of a magnetic f i e l d the energy d i f f e r e n c e SEgp-j^ f ° r e l e c t r o n s of d i f f e r i n g s p i n s i s 6E . = g y_H , • 2.1.7 s p i n ^c B where an a p p r o p r i a t e l y - a v e r a g e d o r b i t a l g-value (the a c t u a l a v e r a g i n g w i l l be d i s c u s s e d i n s e c t i o n (5.4)) and y B 2m ' i s the Bohr magneton. Thus t h e r e a r e i n f a c t two s e t s of Landau l e v e l s , each s e t having an e n e r g y - l e v e l s e p a r a t i o n }ioi = 2yT,Hm/m* . 2.1.8 * C B ' I f we a s c r i b e a phase d i f f e r e n c e o f 2TT t o energy l e v e l s h a v ing an energy s e p a r a t i o n of #to then the two s e t s of Landau l e v e l s d i f f e r i n phase by 27rg y H / ( K w ) = T r g m*/m . 2.1.9 The s p l i t t i n g o f the Landau l e v e l s by e l e c t r o n s p i n i s i l l u s t r a t e d i n F i g u r e (2.1). + H e r e m* i s the c y c l o t r o n (or o r b i t a l ) e f f e c t i v e mass.defined by m* = ()12/2TT) (3A/3E) . a 2 1 1 n+i n + i s n + i P + s • n - i "fitd. .n E + n - 1 n - i NO SPIN WITH SPIN F i g u r e ,2.1 The Zeeman s p l i t t i n g o f Landau l e v e l s . When s p i n i s c o n s i d e r e d each Landau l e v e l s p l i t s -i n t o two s e p a r a t e l e v e l s . 13 Now l e t us consider a thin s l i c e of thickness 6k H through the Fermi surface and normal to the applied f i e l d . At 0°K the only occupied orbits w i l l be those for which the cross-sectional areas A n are less than or equal to the cross-sectional area of the Fermi surface. The total energy 6u for a l l electrons in the s l i c e can be calculated and the magnetization for the s l i c e i s found to be 6M = - | H « u ' I^"5^ '^ff"n> ' a-1:1° assuming n » 1. As H i s increased the n t h Landau level w i l l pass through the Fermi level and suddenly become depopulated. There w i l l be a sudden change in the energy 5u and by Eq. (2.1.10) also a change in the magnetization 6M. This change corresponds to n in Eq. (2.1.10) being replaced by n-1 so that the magnetization jumps discontinuously by the amount 6 k H eMA A(6M) = — f • ' 2.1.11 4ir m* Thus the magnetization has a saw-tooth variation with respect to 1/H, and i s in fact periodic in 1/H with frequency F, where 14 F = — - . 2.1.12 27re I n o r d e r t o o b t a i n the t o t a l o s c i l l a t o r y m a g n e t i z a t i o n M , i t i s c o n v e n i e n t t o express the saw-tooth v a r i a t i o n i n Eq.. (2.1.10) as a F o u r i e r s e r i e s . Then i n t e g r a t i n g over k H we have t dk eHA . . r JrfA , M = —4 (-- Z - i - i i — s i n [ 2 u r — ^ ±] } 2.1.13 J k „ 4 T T J m* r r 2ire " S i n c e t h e argument o f the s i n e f u n c t i o n has been assumed t o be l a r g e (n>>l), c o n t r i b u t i o n s t o M are expected o n l y near s t a t i o n a r y v a l u e s of A^, t h a t i s near v a l u e s o f k H f o r which 3A. H A T a y l o r s e r i e s expansion about a s t a t i o n a r y p o i n t y i e l d s A_ = A I i a k j > ••• 2.1.14 f ex ^ 2 H where denotes the a r e a o f an extremal s e c t i o n through the F e r m i s u r f a c e . ^ I n t e g r a t i n g Eq. (.2.1.13) then g i v e s M = D(m Z s i n [27rr (| e x - y ) + j ] / 2.1.15 t We s h o u l d , of c o u r s e , make a s i m i l a r expansion about each e x t r e m a l s e c t i o n and when t h i s i s done the n e t e f f e c t i s s i m p l y t o sum Eq. (2.1.15) over a l l f r e q u e n c i e s . 15 where the upper s i g n i s a p p r o p r i a t e t o an area of maximum c r o s s - s e c t i o n and I n the event t h a t the c u r v a t u r e f a c t o r 'a' i s n e a r l y e q u a l t o zero we sh o u l d of course use h i g h e r - o r d e r terms i n the T a y l o r s e r i e s expansion (2.1.14) and i n s e c t i o n (4.2) we s h a l l i n f a c t f i n d i t necessary t o c o n s i d e r the e f f e c t s o f such h i g h e r o r d e r terms. A t temperatures other than a b s o l u t e zero the Fermi s u r f a c e w i l l n o t be p e r f e c t l y sharp and consequently Landau l e v e l s p a s s i n g through t h i s d i f f u s e energy s u r f a c e w i l l , become d e p l e t e d g r a d u a l l y . The L i f s h i t z - K o s e v i c h (1955) o r L-K f o r m a l i s m i n d i c a t e s t h a t f o r the r harmonic the amp l i t u d e D(H) i n Eq. (2.1.15) should be m u l t i p l i e d by the th e r m a l damping f a c t o r I r = X r / s i n h ( X r ) , 2.1.16 2 where X = 2ir rk„T/n.w r B ' c C r y s t a l l i n e d e f e c t s g i v i n g r i s e t o f i n i t e l i f e t i m e s f o r e l e c t r o n s t a t e s w i l l broaden the Landau l e v e l s . Assuming a L o r e n t z i a n broadening, D i n g l e (1952a) f i r s t suggested, and e x p e r i m e n t a l evidence has s i n c e confirmed, t h a t the ^where k_. i s Boltzmann's c o n s t a n t . a tt f o r the remainder of t h i s t h e s i s we s h a l l omit the s u b s c r i p t ex from the freque n c y determined from the extremal area. amplitude r e d u c t i o n can be accounted f o r by the e x p o n e n t i a l t h damping f a c t o r (again f o r the r harmonic) K r = e x p [ - X r T D / T ] 2.1.17 T D i s r e f e r r e d to as the D i n g l e temperature and i s r e l a t e d to the h a l f - w i d t h T of the L o r e n t z i a n broadening by s p i n . The spin-up and spin-down e l e c t r o n s w i l l i n f a c t g i v e r i s e t o two s e t s o f o s c i l l a t i o n s on account of the s p i n - s p l i t t i n g d i s c u s s e d on page. 11 . Each s e t has an a m p l i -tude D(H)/2 and the phase d i f f e r e n c e between the two o s c i l l a t i o n s i s g i v e n by Eq. (2.1.9). _The n e t e f f e c t of the s p i n - s p l i t t i n g i s simply t o reduce the amplitude of the r t f t harmonic by the f a c t o r (with no change i n the argument of the s i n e terms i n (2.1.15) T h i s r e d u c t i o n i n the amplitude was f i r s t p o i n t e d out by Sondheimer and Wi l s o n (1951)^and m o d i f i e d by D i n g l e (1952b) f o r t he case m* ^ m, a l b e i t f o r g c equal t o 2. Cohen and Blount (1960) appear t o be the f i r s t to use t h i s r e s u l t ^They c o n s i d e r the s p e c i a l case of f r e e e l e c t r o n s , i . e . , m* = m and g = 2) . T D = r/frk. B The amplitude i s a l s o a f f e c t e d by the e l e c t r o n i c 2.1.18 17 f o r the cases where g c d i f f e r s markedly from 2. To summarize, the dHvA e f f e c t can be c h a r a c t e r i z e d by the e q u a t i o n M = E A si n [ 2 i r r ( | - - y\+ 9] 2.1.19 j~ r n where A = D(H) -1 l r K r c o s ( T r r g c m * / 2 m ) r . ' and 6 i s u s u a l l y + IT/4 (see Eq. (2.1.15)). I n the p a s t , f o u r d i f f e r e n t methods have been used to o b t a i n v a l u e s o f g c from the dHvA e f f e c t . E v i d e n t l y the fundamental amplitude w i l l v a n i s h when g m*/m = 1, 3, 5, ••• e t c . . Thus i f _ a n u l l i s found f o r the fundamental amplitude, and i f the c y c l o t r o n mass m* can be e x t r a p o l a t e d smoothly a c r o s s such a n u l l p o i n t , then a v a l u e f o r g can be deduced. F o r example, K e t t e r s o n and Wi n d m i l l e r (197 0) have performed experiments i n which the dHvA s i g n a l i n P t has been examined f o r a l l f i e l d o r i e n t a t i o n s f o r n u l l s i n the fundamental term, and hence they have found some v a l u e s f o r g c i n a few i s o l a t e d d i r e c t i o n s . T h i s method i s i n g e n e r a l q u i t e u n s a t i s f a c t o r y s i n c e i t does not a l l o w g c to be measured f o r an a r b i t r a r y f i e l d d i r e c t i o n ; n u l l s i n the fundamental 18 amplitude a r e v e r y r a r e indeed, and may not occur a t a l l f o r c e r t a i n m e t a l s . A second approach has been t o measure the s t r e n g t h of the dHvA s i g n a l a b s o l u t e l y and then to compare t h i s v a l u e w i t h the v a l u e p r e d i c t e d by Eq. (2.1.19). The v a r i a b l e s H, T, T^, F, and m*/m can be e x p e r i m e n t a l l y determined and the c u r v a t u r e f a c t o r 'a' can be o b t a i n e d i f the shape of the Fermi s u r f a c e i s known. Thus the o n l y remaining unknown i s g c i n the c o s i n e term. T h i s method was used by P h i l l i p s and Gold (1969) t o es t i m a t e g c v a l u e s i n Pb. In a d d i t i o n to the d i f f i c u l t y of making a b s o l u t e measurements, t h i s approach g i v e s i n a c c u r a t e g v a l u e s because the shape of the Fermi s u r f a c e i s g e n e r a l l y not known to s u f f i c i e n t . a c c u r a c y to g i v e a r e l i a b l e c u r v a t u r e f a c t o r . I t should a l s o be noted t h a t the v a r i a b l e s H, T, T Q , and m*/m en t e r i n t o the amplitude through the r a p i d l y - v a r y i n g s i n h and e x p o n e n t i a l f u n c t i o n s and as a r e s u l t must be determined w i t h some degree of a c c u r a c y . R e c e n t l y Randies (1972) has deduced g v a l u e s f o r the noble metals by examining the r a t i o A^/A 2 as a f u n c t i o n o f o r i e n t a t i o n . While t h i s approach a v o i d s the n e c e s s i t y o f having t o make a b s o l u t e measurements o f the amplitudes, i t i s s t i l l n e c e s s a r y to o b t a i n p r e c i s e v a l u e s of H, T, T D , and m*/m i n order to e s t i m a t e g c r e l i a b l y . F i n a l l y i t i s p o s s i b l e to observe s p i n -s p l i t t i n g d i r e c t l y i n the dHvA waveform, p r o v i d e d t h a t the 19 h i g h e r harmonics tr = 2 , 3 , 4 , j ) i n Eq. (2.1.19) have amplitudes which are comparable t o t h a t o f the fundamental. Fo r X -*• 0, and T D ->- 0, the waveform g i v e n by Eq. (2.1.19) i s i n f a c t c u s p - l i k e , and when two c u s p - l i k e waveforms of equal amplitude a r e added to g e t h e r one expects i n g e n e r a l to f i n d two s e t s of d i s t i n c t peaks. The phase d i f f e r e n c e i s then g i v e n a t once from the s e p a r a t i o n of the peaks, and hence g c can be c a l c u l a t e d . U n f o r t u n a t e l y i t i s u s u a l l y not p o s s i b l e t o o b t a i n s u f f i c i e n t l y low v a l u e s o f X r and T Q t o a l l o w s p i n - s p l i t t i n g to be observed d i r e c t l y , a l t h o u g h the l i t e r a t u r e does c o n t a i n examples of s p i n - s p l i t t i n g ( c f . O ' S u l l i v a n and S c h i r b e r (1967)). S p i n - s p l i t t i n g was f i r s t observed d i r e c t l y (Boyle e t a l . (1960)) i n the waveform of the quantum o s c i l l a t i o n s i n the temperature of B i . In t h i s magnetothermal e f f e c t the thermal damping f a c t o r ( c f . Gold (1968)) i s found to be d l L = £ dX r = [ X r c o t h ( X r ) - l ] / s i n h ( X r ) which i n i t i a l l y i n c r e a s e s w i t h X^ and reaches a maximum a t X r = 1.6. Thus f o r X < 1.6, the h i g h e r harmonics (up to the r I w i l l be enhanced r e l a t i v e to t h e i r dHvA counterpart] thereby p e r m i t t i n g the d i r e c t o b s e r v a t i o n of s p i n - s p l i t t i n g . 20 I n t h i s t h e s i s we p r e s e n t a new approach f o r d e t e r m i n i n g g , i n which the v a r i o u s d i f f i c u l t i e s a s s o c -i a t e d w i t h the methods o u t l i n e d p r e v i o u s l y are circumvented, The amplitudes A^, A^, and A^ of the f i r s t t h r e e harmonics o f Eq. (2.1.19) are measured s i m u l t a n e o u s l y t o o b t a i n the r a t i o a = 4 A 1 A 3 3/2 (3/4) pCX^) y(w1) , 2.1.20 where p tx, ) 2 Z 2 1 1 ^ 3 1 9 1 + 3 tanh (X,) , 2.1.21 c o s 2 (w 2) cos (w 3)cos (w^) 2 2cos (w.^ ) - 1 2 2 cos (w1) [4cos (w^-3] 2.1.22 and w = rirg m*/2m - 2 . 1 . 2 3 r c i s the argument o f the D i n g l e c o s i n e f a c t o r (2.1.18). Thus knowledge o f a and X^ immediately g i v e s the v a l u e - o f the f u n c t i o n y (w..) , and hence g from equations (2.1.22) JL C and (2.1.23) . • S i n c e the r a t i o a i s a d i m e n s i o n l e s s q u a n t i t y , i t i not necessary t o make a b s o l u t e measurements o f the a m p l i -tudes, and thus t h i s method has the advantage of the Randies approach. Moreover a i s independent o f the D i n g l temperature s i n c e — — = l K 1 K 3 (see Eq. (2.1.17); T D should however be kept as low as p o s s i b l e t o ensure t h a t the t h i r d harmonic amplitude A^ i s w e l l above the n o i s e l e v e l . In a d d i t i o n a i s f a i r l y i n s e n s i t i v e to the thermal damping f a c t o r s 1^, I ^ r and I ^ as can be seen from F i g u r e (2.2) where the f u n c t i o n p(X^) i s p l o t t e d as a f u n c t i o n o f X^; p (X 1) v a r i e s m o n o t o n i c a l l y from 1 to 4/3. In f a c t an e r r o r o f 10% i n one o f the e x p e r i m e n t a l l y determined v a r i a b l e s T, H, or m*/m appearing i n X^ w i l l a f f e c t p (X^). and hence a by a t most 2%. F i n a l l y , s i n c e y(w^) i s a r a p i d l y v a r y i n g f u n c t i o n o f w-j_/ the argument w^  can be determined q u i t e a c c u r a t e l y . T h i s r a p i d v a r i a t i o n i s i l l u s t r a t e d i n F i g u r e (2.3), where ytw^) i s p l o t t e d l o g a r i t h m i c a l l y as a 2 f u n c t i o n o f cos (w^). 2 When u s i n g F i g u r e (2.3) to determine cos (w^) i t i s c l e a r t h a t t h r e e s o l u t i o n s are p o s s i b l e f o r | y | >_1 and two s o l u t i o n s f o r |y|<l . However a l l but one of these 0-1 -A 0-01 C O S (W,) Figure 2.3 The 2ratio of the Dingle cosine terms. This graph i s used to solve for cos (w,) in Eq. (2.1.22). to CO 24 s o l u t i o n s a re u n p h y s i c a l and i n o r d e r to s e l e c t the c o r r e c t v a l u e we need to examine the r a t i o A 2/A^. U s i n g Eq. (2.1.19) we can show t h a t A_ I 0 K_ -3/2 2-cos ( v O - 1 A. I, K, 1 > s 1 1 1 1 cos (w1) V 2*!* .-3/2 , 2>COB 2(W 1)-1 = 2 expl-X.TVT) ' 2.1.24 i 1 c x 1 ) cos(w 1) where we have used the f a c t t h a t K x = e x p C - X ^ / T ) Thus knowing the r a t i o A 2/A^ , X^, and T we can use Eq. (2.1.24) t o e s t i m a t e a D i n g l e temperature f o r each v a l u e f o r cos(w^). I t has been our experi e n c e t h a t o n l y one o f t h e p o s s i b l e s o l u t i o n s f o r cos(w^) g i v e s a r e a s o n a b l e e s t i m a t e f o r (we have found t h a t T Q f o r an i n c o r r e c t s o l u t i o n can be a r e l a t i v e l y l a r g e n e g a t i v e number). Another source of ambiguity i s i n h e r e n t i n a l l approaches u s i n g the dHvA e f f e c t to d e r i v e g c - v a l u e s , and r e s u l t s from the f a c t t h a t the argument f o r a p a r t i c u l a r v a l u e o f |cos(w-,)| i s m u l t i v a l u e d . I f w^w 1 25 i s a s o l u t i o n , then so i s w 1 = W1 + pit where P = 0, ± 1, ± 2, and hence we have g m*/m = 2*[w,/'IT + p] 2.1.25 F o r t u n a t e l y , an argument g i v e n by P i p p a r d (1969) a l l o w s us t o e l i m i n a t e most of t h i s m u l t i p l i c i t y . When c o n s t r u c t i n g the H a m i l t o n i a n used t o d e s c r i b e the motion o f an e l e c t r o n i n a p e r i o d i c l a t t i c e , i t i s p o s s i b l e to use an e f f e c t i v e p o t e n t i a l which can be c o n s i d e r e d to be the sum of two s e p a r a t e c o n t r i b u t i o n s , one p a r t coming from the e l e c t r o s t a t i c i n t e r a c t i o n and the o t h e r c o n t r i -b u t i o n coming from the s p i n - o r b i t i n t e r a c t i o n . The F o u r i e r components of these two i n t e r a c t i o n s a r e i n phase quadrature as a r e s u l t of the f a c t t h a t the e l e c t r o s t a t i c i n t e r a c t i o n i s determined by the p e r i o d i c l a t t i c e p o t e n t i a l V(r) whereas the s p i n - o r b i t c o u p l i n g i s determined by the g r a d i e n t of V ( r ) . S i nce V(r) can always be expressed as a F o u r i e r s e r i e s I V r c o s ( G * r ) , the c o r r e s p o n d i n g terms i n the F o u r i e r s e r i e s f o r the s p i n - o r b i t i n t e r a c t i o n w i l l ->-i n v o l v e o n l y s i n e f u n c t i o n s , i . e . , ±sin(G*r), where ± r e f e r s G to the two s p i n o r i e n t a t i o n s . The net r e s u l t i s an e f f e c t i v e p o t e n t i a l whose F o u r i e r c o e f f i c i e n t s have a s p a t i a l dependence l i k e cos ( G ' r i ^ ) , where the phase s h i f t ± $ G i s determined by the s t r e n g t h o f the s p i n - o r b i t i n t e r a c t i o n r e l a t i v e to the o r d i n a r y l a t t i c e i n t e r a c t i o n . The extreme phase s h i f t s ( i . e . , when the s p i n - o r b i t i n t e r a c t i o n i s dominant) are +TT/2 f o r an up-spin e l e c t r o n and -IT/2 f o r a down-spin e l e c t r o n . L e t us w r i t e the wave f u n c t i o n f o r an e l e c t r o n i n the absence of s p i n - o r b i t i n t e r a c t i o n and i n the'absence of a magnetic f i e l d i n the form y(k) = u k ( r ) e x p ( i k - r ) . Now when s p i n - o r b i t i n t e r a c t i o n i s i n c l u d e d i n the Hamilton-i a n , the r e s u l t s of the p r e c e d i n g paragraph i n d i c a t e t h a t the e f f e c t on the Ham i l t o n i a n i s e q u i v a l e n t to c o n s i d e r i n g the l i n e a r c o o r d i n a t e t r a n s f o r m a t i o n , -> ->- , 2 r ± 4>GG/G" , and the new wave f u n c t i o n s w i l l be of the form = u ' ( r e f f ) e x P C i k - r e f f ) . A f t e r a Bragg r e f l e c t i o n , k -*• k + G and the wave f u n c t i o n s -> ->• r -*• r rC = ef f 27 become V (£+G) = u £ ( ? e f f ) e x p [ i l k + G ) - r e f f ] . U s i n g t h e Bragg c o n d i t i o n k = - , we see t h a t s p m -o r b i t c o u p l i n g i n t r o d u c e s an e f f e c t i v e phase s h i f t between the wave f u n c t i o n s f o r up- and down-spin e l e c t r o n s which i s g i v e n by + 4>G/2 b e f o r e "Bragg r e f l e c t i o n and ± $Q/2 a f t e r r e f l e c t i o n . Thus the phases of the e l e c t r o n wave f u n c t i o n s f o r t h e two s p i n o r i e n t a t i o n s can d i f f e r by a t most 2c? = TT f o r each Bragg r e f l e c t i o n . G The wave f u n c t i o n o r a B l b c h e l e c t r o n i n the presence o f a magnetic f i e l d can be c o n s t r u c t e d from the z e r o - f i e l d f u n c t i o n s Y 1 ( k ) , and Chambers (1966) has g i v e n a p a r t i c u l a r l y c l e a r d e s c r i p t i o n o f t h i s c o n s t r u c t i o n i n which the phase ' s h i f t s d i s c u s s e d i n the l a s t paragraph w i l l be summed around an o r b i t . F o r an o r b i t i n v o l v i n g s Bragg r e f l e c t i o n s , t h e o r b i t a l f u n c t i o n s f o r the e l e c t r o n s of d i f f e r e n t s p i n w i l l t h e r e f o r e d i f f e r i n phase by 2st}>G up. to a maximum sir. Q u a n t i z a t i o n i n t o Landau s t a t e s r e s u l t s from the demand t h a t the o r b i t a l wave/ f u n c t i o n s be s i n g l e v a l u e d . S i n c e a phase change o f 2TT co r r e s p o n d s t o an energy change Kco c, t h e s p i n - s p l i t t i n g < S E s p i n f o r a n Y Landau l e v e l w i l l be i n p r o p o r t i o n t o the s e s p i n - i n d u c e d phase s h i f t s . Thus -2scb 6E . = 2y_H + Kto , s p i n B 2TT C where the term 2y_H i s the c o n t r i b u t i o n i n the absence o f a s p i n - o r b i t c o u p l i n g . There w i l l t h e r e f o r e be a maximum s p l i t t i n g g i v e n by (6E . ) • = 2y cH + % Kco , s p i n max B 2 f c s p i nand hence we o b t a i n an upper bound f o r the o r b i t a l g-f a c t o r (6E . ) = s p i n max = 2 + s { m / x a * ) , 2.1.26 ^c,max T T ' y BH ( s i n c e Hcoc = 2yBHm/m*) . Thus a l l but a few terms i n the i n f i n i t e s e t o f p o s s i b l e g -v a l u e s (Eq. (2.1.25)) can be c e l i m i n a t e d . In F i g u r e (2.4) we have shown a t r i a n g u l a r o r b i t , s i m i l a r to the £ o r b i t i n l e a d (see F i g u r e ( 5 . 9 ) ) , super-imposed on a diagram of the plan e s of l a t t i c e p o t e n t i a l i n the absence o f s p i n - o r b i t i n t e r a c t i o n . In the presence o f s p i n - o r b i t i n t e r a c t i o n these p l a n e s a r e d i s p l a c e d and can move a t most to the p l a n e s r e p r e s e n t e d by the d o t t e d l i n e s . T h i s d i s p l a c e m e n t l e a d s t o the energy s h i f t s i n the Landau l e v e l s which have been shown f o r the case (m*/m) = 1. While the f o r e g o i n g suggests t h a t the e x t r a c t i o n o f 2 g c ~ v a l u e s from measurements of the r a t i o a = A 2 / (A^A^) ( a ) (b) Bragg reflections and the related maximum spin-splitting. The effect of spin-orbit interaction in (a)can s h i f t the planes of l a t t i c e potential from the positions represented by the solid lines as far as the positions represented by the dashed lines. This gives rise to the Landau level s p l i t t i n g shown in (b), for m = m* and s = 3. should be f a i r l y straight-forward, there are in fact experi-mental artifacts that may drastically affect the results unless these sources of systematic error are properly allowed for. In the next section we shall consider some of these extraneous effects and outline procedures to minimize them. 2 . 2 Shoenberg Effect: Magnetic Interaction between  Conduction Electrons As was f i r s t pointed out by Shoenberg (1962), the f i e l d experienced by the electrons should not be the applied f i e l d H but rather the total magnetic induction B = H + 4ir(l-6) M , 2.2.1 where 6 i s an appropriate demagnetizing factor (tensor) which depends on the sample geometry as well as the orientation of the f i e l d relative to the sample. The fact that the dHvA effect should be periodic in 1/B rather than 1/H has been j u s t i f i e d by Pippard (1963) and has been verif i e d many times by experiment (cf. Condon (1966)). Although M i s minute (< 1G) by comparison with H(~ 3 0kG), S t r i c t l y speaking the non-oscillatory component of the Landau diamagnetism (as well as the Pauli spin-paramagnetism) should be included in Eq. (2.2.1). However only the oscillatory part M contributes to the systematic error discussed in this section. 31 when B i s i n s e r t e d i n t o Eq. C2.1.19) one does i n f a c t f i n d an a p p r e c i a b l e e f f e c t because F/H i s such a l a r g e number, 3 t y p i c a l l y o f o r d e r 10 . T h i s can be seen by c o n s i d e r i n g the case where B , H, and hence M are p a r a l l e l and examining the e f f e c t on the argument of the s i n e terms i n Eq. (2.1.19). The argument becomes 2irr(|) . = 27rr(|) [1 + 4TT(1-6) (|) ] _ 1 4 27rr (|) [1-4TT(1-6) (|) ] , ' 2.2.2 so t h a t r e p l a c i n g H by B amounts t o changing the phase by -8Tr 2r(l-<$) (F/H) (M/H) , which can e a s i l y be of order 2ir. In F i g u r e (2.5) we have shown s c h e m a t i c a l l y one c y c l e o f the fundamental term i n the m a g n e t i z a t i o n M as a f u n c t i o n of B r a t h e r than 1 / B , s i n c e over j u s t one c y c l e a t s u f f i c i e n t l y l a r g e F / B the M vs B curve w i l l look v e r y s i m i l a r t o the M vs 1 / B c u r v e . A l s o shown i s the v a r i a t i o n o f the magnetiz-a t i o n as a f u n c t i o n of H = B-4TT (1-6) M. I t can be seen t h a t the p e r i o d i c i t y i s the same f o r both M vs B (or 1 / B ) and M vs H (or 1/H) c u r v e s , but the harmonic c o n t e n t does d i f f e r . L i k e w i s e the m a g n e t i c - i n t e r a c t i o n e f f e c t w i l l r e s u l t i n waveform d i s t o r t i o n s f o r each of the dHvA harmonics when M i s regarded as a f u n c t i o n H, the independent v a r i a b l e which i s measured i n an experiment. Thus the experimental r e s u l t s w i l l e x h i b i t a harmonic c o n t e n t which i s d i f f e r e n t from t h a t g i v e n by Eq. (2.1.19) w i t h H r e p l a c e d by B , \ \ \ M vs. H F i g u r e 2.5 The Shoenberg e f f e c t . The s o l i d curve i l l u s t r a t e s the s i n u s o i d a l dHvA e f f e c t - o f Eq. (2.2.3). The dashed curve i l l u s t r a t e s the e x p e r i m e n t a l l y observed dHvA e f f e c t i f <A1 = 0.8 (see Eq. (2.2.7)). OJ 3 3 i . e . , the amplitude A^ , of the observed r harmonic w i l l d i f f e r from A r g i v e n i n Eq. (2.1.19). T h i s i m p l i e s t h a t the observed r a t i o a 1 = A ^ / (A| A^) must be c o r r e c t e d f o r waveform d i s t o r t i o n b e f o r e g can be o b t a i n e d from c e q u a t i o n s (2.1.20) t o (2.1.23). The r e q u i r e d r a t i o a can be o b t a i n e d from the observed r a t i o a' i f the phases of the f i r s t t h r e e harmonics can be measured. The r e q u i r e d c o r r e c t i o n to a 1 u s i n g phase i n f o r m a t i o n f o l l o w s from a treatment of the magnetic-i n t e r a c t i o n problem presented by P h i l l i p s and Gold- (1969). The b a s i c Eq. (2.1.19) f o r M, but w i t h H r e p l a c e d by B, i s oo M = Z A s i n [ 2 n r ( ^ - y) + 6] . 2.2.3 I r a - • • T h i s e q u a t i o n can be f o r m a l l y r e w r i t t e n as a F o u r i e r s e r i e s of terms which a r e p e r i o d i c i n 1/H; 00 M = I {p» s i n [27rr(|- - y)+ 6] r = l r H + q £ cos [2irr ( | - y) + 6] } 00 2 A ' s i n [ 2 T T r ( | - y )+ 6 ' ] , 2 . 2 . 4 r=l r H r 34 where K = v P ' 2 + q; 2) 1 / 2 , 2.2.5 t h i s the measured amplitude o f the r harmonic and -1 q r 6 1 = 8 + t a n x (-4-) 2.2.6 r P r i s the measured phase. I f we w r i t e Eq. (2.2.2) as 2TrrF (|) = 2TrrF [| - ] , so t h a t K = 8 T T 2 ( 1 - 6 ) F / H 2 , 2.2.7 and c o n s i d e r M t o be s u f f i c i e n t l y weak so t h a t K A r < < 1 f o r a l l r , then the components p^. and q^ , of f o r r = 1, 2, 3 a r e g i v e n i n terms of the i d e a l L i f s h i t z - K o s e v i c h (L-K) amplitudes A i n T a b l e I I . These r e s u l t s have been d e r i v e d r f o r an a r b i t r a r y phase angle 6 u s i n g the i t e r a t i v e method of P h i l l i p s and Gold (.1969) , who c o n s i d e r e d the s p e c i a l case 6 = ~ir/4. The c o e f f i c i e n t s g i v e n i n Ta b l e I I are the l o w e s t - o r d e r ones i n the f o l l o w i n g sense. The a m p l i -tudes A' decrease w i t h r i n a manner which i s e s s e n t i a l l y T A B L E I I F O U R I E R C O E F F I C I E N T S F O R E Q U A T I O N ( 2 . 2 . 4 ) A X + • • 0 + A 2 2 cos 0 + KA. sin 6 + A 3 + | K 2 A 3 ( 2 cos 2 6 - 1 ) | s i n 9 + T k 2 A I S I N 6 C O S 9 + - < A ^ A 2 cos 6 + K = 8 T T 2 ( 1 - 5 ) ( | ) CO 36 exponential/ and the terms l i s t e d are a l l of the same exponential order r . The higher-order terms have a negative exponential dependence i n v o l v i n g (r+2) , (r+4), ••• etc. and may s a f e l y be neglected. I f we define a dimensionless parameter v such that K A 2 V - , 2.2.8 we can combine the Fourie r c o e f f i c i e n t s i n Table II to obtain che experimental amplitudes A^  i n terms of the i d e a l ones A^, the i d e a l r a t i o a, and the parameter v . The r e s u l t s are A£ = A 1 , 2.2.9 and AI = A 0 {1 - 2v cos 6+ v 2 } 1 / 2 A£ = A 3 {1 - 6 av cos 8 + (3av) 2 4"3av 2 (cos 2 6-sJn 2 3 ) 2.2.10 2 2 - (3av) 2 v cos 6- + •.}1/2 # 2.2.11 For v « l we have A^ = A r, i . e . the experimental amplitudes are the same as i n the L-K theory, while f o r V>>1 we f i n d 37 and ^ 2, ~~ ^ 1 / 2 • 2 • X 2 A 2 = A 2 V 2.2.13 3 a v 2 ^ "2.2.14 A 3 * A 3 ( 2 * • E q u a t i o n s (2.2.12) to (2.2.14), which p e r t a i n to what we s h a l l c a l l the S l i m i t , were f i r s t g i v e n by Shoenberg (1962) f o r 0 = + TT/4, and i n t h i s l i m i t the i n t r i n s i c L-K harmonic c o n t e n t i s n e g l i g i b l e compared t o the harmonic c o n t e n t t i n t r o d u c e d by the waveform d i s t o r t i o n . I t f o l l o w s from t h i s s e t of e quations t h a t a* = 2/3, i . e . i s independent of a i n the S l i m i t , and o b v i o u s l y no i n f o r m a t i o n about g c can be o b t a i n e d . I t i s t h e r e f o r e n e c e s s a r y t o perform the experiment under c o n d i t i o n s where v i s s m a l l , i . e . , as c l o s e as p o s s i b l e to the L-K l i m i t A ' = A ^ r r I f we i n t r o d u c e -1 <*r cj, = e« - e = tan A ( -4) , 2.2.15 T r r p' f o r the phase change due t o the magnetic i n t e r a c t i o n , then i t f o l l o w s from the e n t r i e s i n T a b l e I I t h a t + ~1 I t i s interesting to note that i n the S limit, A,=2K ( A 2 / A ^ ) , and as a result It i s possible to determine A , , absolutely, by simply measuring the ratio A 2 / A | and calculating K . 38 and t a n (2cb - <j> ) = — s i n 9 , 2.2.16 1 i - cos e (1 - v cos 8 ) s i n 6 t a n (3<J>, - <(>,) = . 2.2.17 l J 2_ •5—— - cosO + % cos 2 0 3av 2 E q u a t i o n s (2.2.16), (2 . 2 .17) , (2 . 2 . 9) , (2.2.10), and (2.2.11) can be combined to g i v e an e x p r e s s i o n f o r a/a 1 i n terms of the phase c o r r e c t i o n s <j> . For 0 = +TT/4 , as i s u s u a l l y the case, the e x p r e s s i o n reduces to a a' 1 + s i n 2 (2<J>1- <J>2) cos(3<j>^ - c^) + sin(3(j) - <\>^) [tan(2cj>^ - <j>2) + 1] 2.2.18 Thus we can o b t a i n the i d e a l r a t i o a from the measured r a t i o a 1 and the phase f a c t o r s (2cJ>^-c}>2) and O ^ - c f j ^ ) . The r e q u i r e d phase f a c t o r s a r e o f the form (r<j>^ -<J>r) f o r r = 2 or 3, and can be determined from the e q u a t i o n , r<f>.. - cb = r Z ' - Z 1 - (r-1) 0 , 2.2.19 Y l r r 1 r where Z£ = 2Trr (| - Y ) + e ; / 2.2.20 i s the argument o f the s i n e f u n c t i o n i n Eg. ( 2 . 2 . 4 ) . The arguments Z^ can be measured e x p e r i m e n t a l l y about the same v a l u e o f the a p p l i e d f i e l d H. However s i n c e (F/H - y) c a n c e l s out o f Eq. (2.2.19), the a c t u a l v a l u e s of F, H, and y need not be known when d e t e r m i n i n g the r e q u i r e d phase f a c t o r s (rcj>^ - cpr) . Contours of c o n s t a n t |a/a 1| — t h e c o r r e c t i o n f a c t o r f o r the magnetic i n t e r a c t i o n — c o m p u t e d from Eq. (2.2.18) ar e shown i n F i g u r e (2.6) as a f u n c t i o n of (2cb^-cb^) and (3cJ)^ —cf>^ ) « The ^ s i g n s i n the l a b e l s of the axes are ordered i n the same way as they appear i n Eq. (2.2.18); as has been our c o n v e n t i o n throughout, the upper and lower s i g n s r e f e r t o a maximum and minimum area of the c r o s s - s e c t i o n of the Fermi s u r f a c e , r e s p e c t i v e l y . The graph i s p e r i o d i c and may be extended by t r a n s l a t i o n s of i n t e g r a l m u l t i p l e s of TT a l o n g e i t h e r a x i s . The blank p o r t i o n s where no contours a r e shown r e p r e s e n t r e g i o n s which are i n a c c e s s a b l e as a consequence of the f a c t t h a t y(w^) i n Eq. (2.1.22) cannot take on v a l u e s between 0 and+1, which i m p l i e s by Eq. (2.1.20) t h a t t h e r e w i l l , be no v a l u e s f o r the r a t i o a i n the r e g i o n o 41 0 < a < ( | ) 3 / 2 -pCx^ , The form of the " s l a n t i n g " boundaries depends on the v a l u e o f X^; the d o t t e d curves r e f e r t o X^ = °°, w h i l e the dashed curves r e f e r t o X^ = 0. (The dashed curve f o r the r i g h t s i d e of the graph i s not shown but i s e s s e n t i a l l y g i v e n by the l i n e | a / a 1 | = 1.) The boundary formed by the a x i s of a b s c i s s a e r e s u l t s from a = 0. The p o i n t s l a b e l e d L-K and S r e f e r , r e s p e c t i v e l y , to no magnetic i n t e r a c t i o n and to the Shoenberg l i m i t (equations (2.2.12) to (2.2.14)). A l l contours pass through the S p o i n t , which i s a r e f l e c t i o n of the f a c t t h a t the r a t i o a ' i s independent of g a t t h i s p o i n t . We might c p o i n t out t h a t i t would a l s o be p o s s i b l e t o c o n s t r u c t con-t o u r s of c o n s t a n t g m*/m on the same s o r t of graph as c F i g u r e (2.6); i n f a c t the " s l a n t e d " boundaries of F i g u r e (2.6) a r e the c o n t o u r s g m*/m = 1. A l t h o u g h such a p l o t would enable g_m*/m to be found d i r e c t l y i n terms of phase i n f o r m a t i o n alone ( i . e . e n t i r e l y without amplitude measure-ments) , t h i s would be a v e r y t e d i o u s procedure s i n c e i t would be n e c e s s a r y to compute a f r e s h graph f o r each value o f X,. Furthermore, a l l contours o f c o n s t a n t g m*/m 1 3 c then pass through the L-K p o i n t and i n i t s neighbourhood the phase f a c t o r s would have t o be determined extremely a c c u r a t e l y i n order t o d i f f e r e n t i a t e between the c l o s e l y -packed c o n t o u r s . 42 These'phase f a c t o r s of Eq. (2.2.19) are not the o n l y means of d e t e r m i n i n g the s t r e n g t h of the magnetic i n t e r a c t i o n . S i n c e the v a r i a b l e v d e f i n e d i n Eq. (2.2.8) i s a f u n c t i o n o f temperature, e s s e n t i a l l y p r o p o r t i o n a l t o T f o r X^ > 2, the amplitudes A^ w i l l have a d i f f e r e n t temperature dependence than A r f o r r \ 1 ( P h i l l i p s and Gold (1969) ) . By o b s e r v i n g the' amplitude A^ , over a s u f f i c i e n t l y wide temperature range i t should be p o s s i b l e to c a l c u l a t e the magnitude of A . r P h i l l i p s and Gold (1969) found by examining the temperature dependence of v a r i o u s o s c i l l a t i o n s i n l e a d t h a t under t h e i r e x p e r i m e n t a l c o n d i t i o n s the harmonics of one term (3) i n d i c a t e d t h a t they were i n the S l i m i t whereas harmonics of o t h e r terms (a, 6) were adequately d e s c r i b e d assuming the magnetic i n t e r a c t i o n t o be n e g l i g i b l e . As w i l l be p o i n t e d out i n s e c t i o n (4.2), data taken v e r y e a r l y i n our e x p e r i m e n t a l programme u s i n g c y l i n d r i c a l l e a d samples suggested t h a t the magnetic i n t e r a c t i o n was masking th e L-K harmonics f o r the [100] 3 o s c i l l a t i o n s . In f a c t the r e s u l t s of P h i l l i p s and Gold suggest t h a t KA.^ would be 1.4, i . e . g r e a t e r than 1, a t the temperatures and f i e l d s t r e n g t h s a c t u a l l y used i n the experiments t o be d e s c r i b e d i n t h i s t h e s i s . As s t a t e d e a r l i e r , g v a l u e s should be determined 3 c as near the L-K l i m i t as p o s s i b l e . O b v i o u s l y the magnetic 43 interaction effect can be minimized by arranging the sample geometry in such a way that the demagnetizing factor 6 i s as close to unity as possible, that is for a sample in the form of a thin disk and with H normal to the plane of the disk. The thin-disk samples actually used in this investigation (section (3.3)) had 6 ^  0.9, so that the strength of the magnetic interaction (i.e. the parameter K A ^ ) was reduced to (1-6) ^  0.1 times that for a long cylinder (6 = 0, H parallel to the cylinder). The use of thin-disk samples to reduce the unwanted harmonic distortion arising from the magnetic interaction effect is not entirely without disadvantage. In any experiment designed to measure the volume average of the f i e l d inside the sample the magnetization w i l l always appear in the form 4TT(1 -6)M, S O that the signal as a whole w i l l be reduced by (i - 6) when compared to that for a long cylinder. This overall reduction did not prove to be a serious limitation in practice (Chapter IV). 2.3 Other Systematic Errors (a) Field Inhomogeneity An inhomogeneous f i e l d w i l l affect both the amplitude and the phase of the dHvA signal. In the absence of magnetic interaction the magnitude of this effect can. be formally c a l c u l a t e d i n the f o l l o w i n g manner. I f we c o n s i d e r a f i e l d p r o f i l e o f the form HCr) H 6 2.3.1 the volume average of Eq. (2.1.19) becomes M r s i n [2-rrr tc- - y)+ 6 ] dV/ dV 2 A r { G l r s i n [ 2 7 T r ( H " - ~ Y ) + 6 ] r 6 + G 2 r cos [ 2 i r r ( | - y) + 6 ] } , o 2.3.2 where G l r = cos[2Trr| n ( r ) d V / dV , 2.3.3 and '2r -I s i x i l 2 i r r |- n ( r ) ] d V / dV 2.3.4 The e f f e c t of f i e l d inhomogeneity i s thus t o m u l t i p l y the 2 2 1/2 a m p l i t u d e s A^ by the f a c t o r [G^ r + G 2 r ] ' , and to s h i f t -1 the phase by t a n ^ G 2 r ^ G l r ^ 45 Without assuming s p e c i f i c forms f o r n(r) i t i s d i f f i c u l t t o d i s c u s s the e f f e c t o f f i e l d inhomogeneity on the amplitude and phase i n g e n e r a l terms (Shoenberg (1969)). However, f o r 2i\r ( F / H Q ) n (r) << 1, the change i n phase w i l l be p r o p o r t i o n a l t o r , t o lowest o r d e r , and as a r e s u l t w i l l n o t a f f e c t the phase f a c t o r s (rcj ) ^ - c } > r) i n t r o d u c e d i n s e c t i o n (2.2). I f i n a d d i t i o n we r e s t r i c t o u r s e l v e s t o a s m a l l r e g i o n i n which H Q i s a " t u r n i n g p o i n t " about which we can approximate n(r) by D(?) = f ^ x 2 + g 2 y 2 + g 3 z 2 , 2.3.5 then we can c a r r y out the i n t e g r a t i o n s i n Eq. (2.3.3) and Eq. (2.3.4) f o r a r e c t a n g u l a r c e l l w i t h s i d e s X l ' X 2 ' X 3 * I t i s then found t h a t 2 2 ^ / 2 2 [G, + GZ ] = 1 - r p + ••• , 2.3.6 l r 2r ^ where P = 3T0 I 2*!- ] 2 [ 31 4 + 32 X 2 + 33 X 3 ] * 2 ' 3 * 7 o Thus the amplitudes remain u n a f f e c t e d to second o r d e r (the l e a d i n g c o r r e c t i o n term f o r r = 1, i s r o u g h l y 1/20 of the square o f the argument of the s i n u s o i d a l f u n c t i o n s of Eq. (.2.3.3) and Eq. (2.3.4) when e v a l u a t e d a t a co r n e r of the c e l l ) . Furthermore, a l t h o u g h f o r A^, A 2 , and A^ the i n d i v i d u a l f r a c t i o n a l e r r o r s a r e -p, -4p, and -9p, 2 r e s p e c t i v e l y , the f r a c t i o n a l e r r o r i n the r a t i o a = (A^A^) i s j u s t + 2p. (b) Specimen Inhomogeneity due to Mosaic S u b s t r u c t u r e , Bending, e t c . I f a d i s k sample i s bent (or e x h i b i t s mosaic sub-s t r u c t u r e ) b o t h the phase and the amplitude of the dHvA s i g n a l w i l l v a r y over the sample as a r e s u l t of the f a c t t h a t H w i l l make s l i g h t l y d i f f e r e n t a ngles t o the c r y s t a l axes a t each p o i n t r i n the sample. Thus the a p p l i e d f i e l d w i l l be s e l e c t i n g out s l i g h t l y d i f f e r e n t c r o s s - s e c t i o n a l a r eas and the r e s u l t a n t dHvA s i g n a l w i l l be a volume average of the m a g n e t i z a t i o n d i s t r i b u t i o n M ( r ) . T h i s e r r o r can be t r e a t e d f o r m a l l y i n a manner i d e n t i c a l t o the f i e l d inhomogeneity problem d i s c u s s e d i n s e c t i o n (2.3a) i f Eq. (2.3.1) i s changed t o P = P [1 + n ' ( ? ) ] , 2.3.8 to account f o r the v a r i a t i o n i n frequency r e s u l t i n g from the v a r i a t i o n i n c r o s s - s e c t i o n a l a r e a s . For t h i s e r r o r the frequency inhomogeneity n'(r) w i l l depend on how r a p i d l y F changes w i t h f i e l d d i r e c t i o n and on the e x t e n t of the mosaic spread ( c f . Shoenberg (1969)). The r e s u l t i n g 47 phase and amplitude c o r r e c t i o n s w i l l be minimized i f we exam-i n e o s c i l l a t i o n s a t symmetry d i r e c t i o n s f o r which the d e r i v a t i v e of frequency w i t h r e s p e c t t o the f i e l d d i r e c t i o n i s z e r o . (This i s indeed the case f o r the o s c i l l a t i o n s examined i n Chapter IV.) 48 Ic). Two Neighbouring F r e q u e n c i e s I f two f r e q u e n c i e s l i e c l o s e to one another, the presence o f one term can g i v e r i s e to amplitude and phase a b e r r o r s i n the o t h e r . C o n s i d e r f r e q u e n c i e s F and F having phases 6 a and 0^, r e s p e c t i v e l y . Then cl b M R = A A s i n [ 2 7 r r C | - Y> + S a ) + A £ s i n [ 2 T T r ( | - Y) + 6 b ] = [ ( A A 2 - A £ 2 ) + 4 A A A B c o s 2 A r ] 1 / 2 s i n [ 2 T r r £ ^ - y) + ^ j r _ + a r ] , 2.3.9 where and A A - A B tan to.) = C f f ) tan (A ) , 2.3.10 R A A + A B R r r A r = 2 7 T r C ? ~ i l ~ ) + ^~ir~ * 2.3.11 I f the data a n a l y s i s i s unable t o d i f f e r e n t i a t e between the two f r e q u e n c i e s then an averaged frequency w i l l be measured and the phase r e l a t i o n s i n Eq. (2.2.19) become r<j>{ -<{>; = r Z | - - (r-1) ( ^ J 9 ^ ) - ro± + o r , 2.3.12 f o r r = 1 and 2. While these phase f a c t o r s s t i l l measure the s t r e n g t h of the magnetic i n t e r a c t i o n , the r e s u l t s o f Eqs. (2.2.9) through (2.2.19) are now i n v a l i d s i n c e i s n e i t h e r c o n s t a n t nor l i n e a r i n the harmonic number However the m o d i f i c a t i o n t o Eqs. (2.2.10) and (2.2.16) s t r a i g h t forward and merely amounts to r e p l a c i n g 0 by 0 re where ef f 6 = ^ — + 2a, - a_ . 2.3 e r r z 1 2 The replacement e q u a t i o n f o r Eq. (2.2.17) i s "3 t a n (3<J>1- (j>3) = - ^-r , where q 3 , . ,0 a+0 b . . ,. a v 2 = - 3 v a s i n ( — = — + a. + a 0 - a_) + A 3 - — — v 2 ~ x « 2 - 3 / 2 { s i n ( 0 a + 0 b + 3 0 ^ 0 3 ) 0 a+0 b + 4 c o s 0 e f f s i n ( — 2 — + 0 ^ + 0 ^ - o^)} , and = 1 + |<xv 2cos(0 a+0 b+ 3 a 1 - a 3 ) - 3 a v s i n t ^ | ^ + a i + a 2 . 3 The e f f e c t on the r a t i o a 1 can be c a l c u l a t e d from Eq. (2.3.9) i f A a/A b and A are known. r The r a t i o (A a - A b) / (A a + A b) needed t o determine a r as w e l l as the r a t i o A a/A b, can be c a l c u l a t e d by examining the amplitudes of the beat minima and the beat maxima. A study of the beat s t r u c t u r e a l s o p e r m i t s the phase A 1 of the beat envelope f o r the fundamental o s c i l l a t i o n t o be cl b p l o t t e d as a f u n c t i o n o f the a p p l i e d f i e l d , and ( 0 - 0 ) i s then found by e x t r a p o l a t i n g t h i s p l o t to i n f i n i t e f i e l d . T h i s i n f o r m a t i o n a l l o w s us t o c a l c u l a t e fia_flb A = rA - ( r - l ) 0 0 , '2.3.14 1C ^  A. • £ and hence i t i s p o s s i b l e to s o l v e f o r from Eq. (2.3.10). The c o r r e c t i o n f a c t o r needed f o r the r a t i o a can then be found a t once from the r e s u l t a n t amplitude i n Eq. (2.3.9). 2.4 Summary S i n c e t h i s chapter has been concerned w i t h a v a r i e t y of d e t a i l s i n a l o g i c a l but not n e c e s s a r i l y e x p e r i m e n t a l l y useable o r d e r , we s h a l l b r i e f l y summarize our approach f o r o b t a i n i n g g c - v a l u e s . F i r s t the amplitudes Aj^, A^ and A^ are measured and the phase f a c t o r s (2cj)^-(|)^) and (3cfj^—cj)^) a r e c a l c u l a t e d from the measured phase i n f o r m a t i o n . C o r r e c t i o n s , i f n e c e s s a r y , are then made to the r a t i o a 1 and t o the phase f a c t o r s to a l l o w f o r s y s t e m a t i c phase s h i f t s and amplitude e r r o r s due t o f i e l d inhomogeneties, e t c . . Eq. (2.2.18) (Fi g u r e (2.6)) i s used t o c o n v e r t the measured r a t i o a' i n t o the i d e a l r a t i o a. and then p (X^) are determined from knowledge of m*/m, T, and H. Eq. (2.1.20) i s then used t o s o l v e f o r ytw^) , and by means o f Eq. (2.1.22) (Fi g u r e (2.3)1 we o b t a i n cos (w,). F i n a l l y g i s found from Eq. (2.1.25), u s i n g Eq. (2.1.26) to s e l e c t the s o l u t i o n s which a r e p h y s i c a l l y r e a l i s t i c . CHAPTER I I I APPARATUS AND EXPERIMENTAL PROCEDURES 3.1 G e n e r a l C o n s i d e r a t i o n s B efore d i s c u s s i n g the ex p e r i m e n t a l apparatus and procedures i n d e t a i l , we s h a l l i n i t i a l l y g i v e a s i m p l i f i e d d e s c r i p t i o n o f our approach f o r o b s e r v i n g the dHvA e f f e c t i n t h i n , s i n g l e - c r y s t a l d i s k s of l e a d , w i t h r e f e r e n c e t o the schematic drawing i n F i g u r e (3.1). A s m a l l m a g n e t o - r e s i s t i v e probe was a t t a c h e d to one s i d e of a d i s k , and both the 3 sample and the sensor were p l a c e d i n the t a i l of a He c r y o s t a t and c o o l e d t o temperatures as low as 0.35°K. The t a i l of the c r y o s t a t was s i t u a t e d between the p o l e - f a c e s o f an electromagnet, which c o u l d be r o t a t e d about the v e r t i c a l a x i s of the c r y o s t a t u n t i l the a p p l i e d magnetic f i e l d was a l i g n e d normal to the f a c e s of the d i s k sample. T h i s a p p l i e d f i e l d was the sum o f a l a r g e q u a s i - s t a t i c component (H < 35 kG) from the electromagnet i t s e l f and a s m a l l e r o s c i l l a t o r y component (h = h Q s i n (to^t) w i t h h Q < 100 G) d e r i v e d from a p a i r of modulation c o i l s wound around the p o l e - t i p s . As a r e s u l t of the c o n t i n u i t y ->-of B normal t o the f a c e s of the d i s k , the m a g n e t o - r e s i s t o r which was l o c a t e d a t the s u r f a c e of the d i s k , sensed the 53 WHEATSTONE BRIDGE V cc M(t) LIQUID He" PHASE SENSITIVE AMPLIFIER MAGNETIC TAPE RECORDER V ct M(H) LIQUID He CRYOSTAT BISMUTH MAGNETO-RESISTOR H + h0sin(w,t) DISK SAMPLE Figure 3.1 Simplified schematic of the experimental arrangement v 54 o s c i l l a t o r y m a g n e t i z a t i o n M (as w e l l as the a p p l i e d f i e l d ) . The r e s i s t a n c e of the m a g n e t o - r e s i s t o r thus c o n t a i n e d a F o u r i e r component which was p r o p o r t i o n a l t o M(t) (where we have w r i t t e n M.(t) t o s t r e s s the f a c t t h a t the a p p l i e d f i e l d was time dependent i n t h i s experiment), and a v o l t a g e p r o p o r t i o n a l to t h i s component was o b t a i n e d through the j u d i c i o u s use of a second m a g n e t o - r e s i s t o r i n a Wheatstone b r i d g e . A p h a s e - s e n s i t i v e a m p l i f i e r was used to demodulate t h i s v o l t a g e and gave as output, a v o l t a g e p r o p o r t i o n a l t o the o s c i l l a t o r y m a g n e t i z a t i o n M(H) a t the f i e l d H. The H component of the a p p l i e d f i e l d was s l o w l y v a r i e d over a s p e c i f i e d range AH, i n order to observe e x p l i c i t l y the o s c i l l a t o r y f i e l d dependence M(H). The v o l t a g e o u t p u t p r o p o r t i o n a l to M(H) was d i g i t a l l y r e c o r d e d onto magnetic tape as a f u n c t i o n of H and subsequently used i n the n u m e r i c a l a n a l y s i s programme t h a t was employed t o e x t r a c t the amplitudes A* and the phase r e l a t i o n s (rZ' - Z') r x r d i s c u s s e d i n Chapter I I . The g c v a l u e s were then o b t a i n e d by means of the r e s u l t s which we have a l r e a d y d e r i v e d i n Chapter I I . In t h i s s e c t i o n we have o n l y o u t l i n e d the b a s i c components t h a t were e s s e n t i a l i n our experiment and i n the r e m a i n i n g s e c t i o n s o f t h i s Chapter the i n t e r e s t e d r e a d e r w i l l f i n d these i n d i v i d u a l components d e s c r i b e d i n much g r e a t e r d e t a i l . 55 3.2 Magnet and C r y o s t a t In order t o o b t a i n the l a r g e s t p o s s i b l e dHvA s i g n a l , i t can be seen from Eq. (2.1.19) t h a t the two v a r i a b l e s X r (which can be e x p e r i m e n t a l l y c o n t r o l l e d through the f i e l d H and the temperature T) and the D i n g l e temperature T^ should be as c l o s e to zero as p o s s i b l e . In our experiments the a p p l i e d f i e l d H was produced by means of a 15" V a r i a n f i e l d - r e g u l a t e d magnet which allowed us to o b t a i n a maximum f i e l d o f about 35 kG when u s i n g s p e c i a l p o l e t i p s having a 5/8" a i r gap. Given t h i s maximum f i e l d , X r c o u l d be f u r t h e r d e c r e a s e d f o r a p a r t i c u l a r dHvA frequency by lowering the temperature, and i n order t o o b t a i n a s u f f i c i e n t l y 3 s m a l l v a l u e f o r X r we had t o s e t up a He system. T h i s system a l l o w e d us to o b t a i n s i g n a l s t r e n g t h s comparable t o 4 systems u s i n g a superconducting magnet a t He temperatures (our v a l u e of X r a t 0.4°K was approximately equal t o t h a t f o r a 100 kG magnet a t 1.2°K). 4 A He dewar (model MD3) al o n g w i t h a s e t 1 of narrow 3 t a i l s and He i n s e r t dewar were purchased from the Oxford Instrument Company. The dewar t a i l s a re shown i n F i g u r e (3.2). Out of the many p o s s i b l e c r y o g e n i c manufacturers, we c o u l d f i n d o n l y one company who f e l t they c o u l d meet the c l o s e t o l e r a n c e s demanded by the dimensions of the lower f o u r i n c h e s of the t a i l assembly. As mentioned i n the p r e c e d i n g paragraph t h e r e was o n l y a 5/8" a i r gap between 56 (b) (c) p (d) F i g u r e 3.2 C r y o s t a t t a i l s . The lower 4" of the t a i l s have O.D.s o f ; (a) 9/16", (b) 1/2", (c) 7/16" and <d) 3/8" p l u s a 5/16" t a i l i n s i d e . The.upper p o r t i o n s have O.D.s of (a) 1" and (b) 3/4". 57 the magnet p o l e - f a c e s , and i t was f e l t t h a t a t a i l w i t h a 9/16" o u t s i d e diameter would be the maximum standard w i d t h t h a t c o u l d be used and s t i l l p ermit magnet r o t a t i o n about t h i s t a i l . There a r e fo u r t a i l s t h a t must f i t i n s i d e t h i s t a i l ; each one having an o u t s i d e diameter 1/16" l e s s than i t s neighbour. S i n c e the w a l l t h i c k n e s s o f these t a i l s was 0.008", t h e r e was o n l y a 0.015" c l e a r a n c e between each t a i l . In order t o prevent thermal s h o r t s and to keep these tubes c o n c e n t r i c , e a c h ' t a i l had a t h i n n y l o n c o r d s p i r a l l i n g 3 down i t s l e n g t h . The innermost two t a i l s belong t o the He i n s e r t dewar (see F i g u r e (3.3)) and were i n s e r t e d i n t o 4 4 the He t a i l which was a t t a c h e d t o the l i q u i d He r e s e r v o i r and s e a l e d from an o u t e r vacuum by means of an indium 0 - r i n g 3 s e a l . The two i n n e r t a i l s on the He i n s e r t dewar separated 3 4 r e g i o n s of l i q u i d - H e temperatures and l i q u i d - H e temperatures and were r i g i d l y mounted tog e t h e r a t the top and a t the bottom i n a manner such t h a t n i t r o g e n gas a t NTP was s e a l e d 4 i n the space between the two t a i l s . When l i q u i d He was p r e s e n t the n i t r o g e n gas f r o z e out to l e a v e a vacuum. The 4 next o u t e r t a i l t o the He t a i l was connected t o the l i q u i d n i t r o g e n r e s e r v o i r . F i n a l l y the outermost t a i l had j u s t a 9/16" o u t s i d e diameter f o r the lower f o u r i n c h e s (the upper p a r t had a 1" O.D.). The two spaces between the 4 outer t h r e e t a i l s of the He dewar were i n t e r c o n n e c t e d and were evacuated. A l l t a i l s were made from type 321 s t a i n l e s s s t e e l except f o r the upper p o r t i o n o f the t a i l i n c o n t a c t C O 59 w i t h the l i q u i d n i t r o g e n bath. A l l but the lower 4" of t h i s t a i l was made of copper and the e n t i r e t a i l was g o l d p l a t e d t o ensure t h a t the e n t i r e t a i l was maintained a t 77°K. (The i n t e r e s t e d reader may o b t a i n more d e t a i l e d i n f o r m a t i o n about our system i n the Oxford Instrument Co. L t d . b l u p r i n t D1935.) 3 He gas was condensed i n a " s i n g l e - s h o t " procedure a t the b e g i n n i n g o f each e xperimental run. There was 3 s u f f i c i e n t gas t o y i e l d about 8.5 ml of l i q u i d He which would g e n e r a l l y l a s t f o r about seven hours a t 0.4°K. Access 3 to the He chamber was a c h i e v e d through a Rose's metal s o l d e r j o i n t a t t h e top o f the 'pot' which had t o be r e s o l d e r e d each time a new sample was used. The temperature was determined by m o n i t o r i n g the r e s i s t a n c e of a l / 1 0 t h watt A l l e n - B r a d l e y r e s i s t o r w i t h a nominal room temperature r e s i s t a n c e of 33 fl. T h i s r e s i s t o r had been c a l i b r a t e d as a f u n c t i o n of temperature by comparing i t s r e s i s t a n c e w i t h t h a t o f a c a l i b r a t e d , 100 ft, Speer r e s i s t o r s u p p l i e d by Dr. A r t Burgess of the U.B.C. low temperature P h y s i c s l a b o r a t o r y . In Chapter IV we have g i v e n r a t h e r l a r g e e r r o r bounds on the measured temperature, t o a l l o w f o r the f a c t t h a t the our r e s i s t o r was temperature c y c l e d many times and hence may have changed r e s i s t a n c e s l i g h t l y . However, t h i s need not concern us s i n c e as has a l r e a d y been s t a t e d i n Chapter I I the temperature need not be known a c c u r a t e l y i n our approach f o r d e t e r m i n i n g g - v a l u e s . 3 c 60 As was s t a t e d e a r l i e r , we used a 15" V a r i a n e l e c t r o m a g n e t , which a l l o w e d us t o o b t a i n a maximum f i e l d o f about 35kG. The magnet used a H a l l probe as a f i e l d s e n s o r , and as i t was ne c e s s a r y t o move t h i s probe e a c h time the dewar was i n s e r t e d between the p o l e f a c e s Cas a s a f e t y p r e c a u t i o n ! , we r e p e a t e d l y checked the r e a d i n g o f the f i e l d d i a l of the magnet by u s i n g a Rawson-Lush r o t a t i n g c o i l gaussmeter w i t h a 0.1% a c c u r a c y . The magnet was mounted on a s p e c i a l base which a l l o w e d us to r o t a t e t h e magnetic f i e l d about the dewar t a i l , and hence about t h e sample i n the i n n e r t a i l . ''A Helmholtz p a i r o f m o d u l a t i o n c o i l s was d e s i g n e d and p l a c e d over the p o l e t i p s so t h a t we c o u l d use the f i e l d m o d u l a t i o n t e c h n i q u e d e s c r i b e d i n s e c t i o n (3.4). I n the. c o u r s e o f our measurements we found t h a t the a p p l i e d f i e l d , was v a r y i n g p e r i o d i c a l l y i n time and t h a t t h i s v a r i a t i o n c o u l d -be c o r r e l a t e d t o d r i f t s i n the room temperature, (the a i r c o n d i t i o n e r came on a t r e g u l a r i n t e r v a l s ) . In f a c t as the room temperature changed by 5 ° t h e f i e l d would change by 500 mG. T h i s f i e l d change was due t o the f a c t t h a t the H a l l probe was temperature dependent. However we found t h a t the f i e l d s t a b i l i t y c o u l d be c o n t r o l l e d t o under 10 mG p r o v i d e d the f o l l o w i n g p r e c a u t i o n s were taken. The room temperature had t o be m a i n t a i n e d c o n s t a n t to w i t h i n 1.5°C. The magnet was covered to prevent a i r c u r r e n t s from f l o w i n g by and hence changing the temperature of the H a l l probe. The magnet had t o be l e f t on a t l e a s t 24 hours p r i o r t o t a k i n g d a t a i n o r d e r f o r the system to r e a c h an e q u i l i b r i u m . A t h e r m i s t o r p l a c e d near the H a l l probe proved t o be an . adequate sensor f o r e s t i m a t i n g the f i e l d d r i f t s due t o temperature v a r i a t i o n s . 3.3 Sample P r e p a r a t i o n In o r d e r t o ensure s m a l l v a l u e s f o r the D i n g l e temp-e r a t u r e , the l e a d c r y s t a l s were grown by techniques s i m i l a r t o t h o s e d e s c r i b e d by P h i l l i p s and Gold (1969). We used a C z o c h r a l s k i type c r y s t a l p u l l e r ( s i m i l a r t o the one used by P h i l l i p s * and Gold) which was shock mounted i n order to a v o i d c r y s t a l d e f e c t s due to mechanical v i b r a t i o n s . The temperature of the melt was c o n t r o l l e d i n a manner such t h a t the c r y s t a l growing on the end o f a seed was i n i t i a l l y t a p e r e d t o form a neck having a diameter o f 1 t o 2 mm b e f o r e growing a c r y s t a l 1 t o 2 cm l o n g . These samples were n o r m a l l y about 5 mm i n diameter. A f t e r a c r y s t a l of d e s i r e d l e n g t h had been grown, the temperature o f the m e l t was a g a i n i n c r e a s e d and the c r y s t a l diameter t a p e r e d o f f u n t i l the c r y s t a l was no longer i n c o n t a c t w i t h 62 the m e l t . The c r y s t a l was then c u t from the seed u s i n g the a c i d c u t t i n g t e c h n i q u e s of P h i l l i p s and Gold. The c r y s t a l was o r i e n t e d a l o n g a symmetry a x i s ( g e n e r a l l y v e r y near the a x i s a l o n g which i t was grown) u s i n g b a c k - r e f l e c t i o n Laue X-ray p a t t e r n s . The c r y s t a l was then p l a c e d i n a spark c u t t e r and d i s k samples were v e r y s l o w l y c u t . ( I t was found t h a t u s i n g a r a z o r b l a d e as the c u t t i n g t o o l , r a t h e r than copper w i r e , y i e l d e d super-i o r samples.) A f t e r each c u t was completed a s m a l l p i e c e o f paper coated w i t h S i l v e r P r i n t was i n s e r t e d i n t o the c u t opening and a l l o w e d to dry b e f o r e b e g i n n i n g the next c u t , i n order to p r e v e n t bending damage. The d i s k s were t y p i c a l l y 5 mm i n diameter and 0.4 t o 0.8 mm t h i c k . •These d i s k s were then p o l i s h e d w i t h a l e a d e t c h a n t which ..automatically rounded o f f the sharp edges. A t t h i s stage the d i s k samples were c a r e f u l l y X-rayed i n order t o determine t h e i r q u a l i t y . In g e n e r a l e x c e l l e n t s i g n a l s were observed from c r y s t a l s f o r which the X-ray s p o t s from t h r e e exposures on the same n e g a t i v e taken near two edges and the c e n t r e of the sample d i d not d i f f e r i n d i f f u s e n e s s from a n e g a t i v e of s i m i l a r exposure time taken a t o n l y one spot on the f a c e . An a c c e p t a b l e photograph as w e l l as an example of a r e j e c t are shown i n F i g u r e (3.4). The sample was then put i n the sample h o l d e r shown i n F i g u r e (3.5). The sample was p l a c e d a g a i n s t the f l a t Figure 3.4 Examples of X-ray photographs (.a) i s a t r i p l e exposure (30 minutes) photograph of an acceptable [110] disk; (b) i s a single exposure (15 minutes) of an unacceptable disk. 64 T O P S P A C E R S SAMPLE CAVITY . . C A P ft WOOL^ \ DISK SAMPLE RAY WINDOW FRONT SIDE Figure 3 . 5 Sample holder. inner face of the 6 mm diameter hole i n the sample holder. The magneto-resistance probe (to be described i n more d e t a i l l a t e r ) was placed on top of the sample, with the f i e l d s e n s i t i v e area near the centre of the disk. A small wad of loose cotton wool was placed over the magneto-resistor. F i n a l l y a cap was screwed i n t o the hole, u n t i l the sample was gently h e l d i n place. The sample was then X-rayed i n s i t u i n order to check the alignment, i . e . , that the req u i r e d c r y s t a l a x i s ( p a r a l l e l to the axis of the,disk) would l i e i n the h o r i z o n t a l plane when the sample holder was i n s e r t e d i n t o the dewar. The f i n a l alignment was carried, out by r o t a t i n g the electromagnet about the v e r t i c a l a x is u n t i l the d i r e c t i o n of the applied f i e l d coincided with the sample a x i s , as judged by the symmetry of the dHvA r e s u l t s - Using t h i s procedure i t i s estimated that the f i e l d could be set to w i t h i n 3 ° of a p a r t i c u l a r a x i s . 3 . 4 F i e l d Modulation In these experiments we used the f i e l d modulation technique, i n which there was a small a.c. modulation f i e l d superimposed on top of the much larger d.c. (or rather so slowly v a r y i n g that i t can be considered to be s t a t i c ) a p p l i e d f i e l d . Since the dHvA e f f e c t i s non-linear with respect to the ap p l i e d f i e l d , the dHvA s i g n a l contains harmonics of the modulation frequency f ^ . Using a phase s e n s i t i v e a m p l i f i e r , we examined the dHvA s i g n a l i n f o r m a t i o n c a r r i e d by the n t h harmonic ( i . e . , a t the f r e q u e n c y n f 1 = . I f we r e p l a c e H by H + n" i n Eq. (2.2.4) , where h = h s i n (to., t) , o l i t i s p o s s i b l e to show t h a t the d e t e c t e d amplitude i s p r o p o r t i o n a l t o E A 1 J (A ) s i n ( n c o n t + nr/2) , r n r 1 ' n and t h a t the i n t r i n s i c phase of the dHvA e f f e c t i s s h i f t e d by nir/2. Here = 2irrFh /H 2 r o and J (A ) i s the n ^ o r d e r B e s s e l f u n c t i o n , n r V a l u e s of A_,h F/H 2 ( = A ^ / 2 i r ) , and J „ ( A ) which J o J n r maximize J n U 3 ) are g i v e n i n T a b l e I I I f o r v a r i o u s n. ( I t i s i n t e r e s t i n g t o note t h a t f o r a number o f cases (n>4) the maximum dHvA s i g n a l a m plitude response o c c u r s when the f d e l d modulation amplitude h Q i s l a r g e r than the f i e l d s p a c i n g AH o f a s i n g l e dHvA p e r i o d . ) A number o f t e c h n i q u e s have been attempted i n measuring A|, A ^ r and A^. I n i t i a l l y each amplitude was determined s e p a r a t e l y . A l a r g e v a l u e of n(n = 8, 9, or 10) was chosen and the f i r s t extremum of J n ^ r ) w a s used t o examine A 1 f o r r = 1, 2 and 3. T h i s was an e x c e l l e n t r ' method f o r d i s c r i m i n a t i n g a g a i n s t Unwanted lower dHvA TABLE I I I AMPLITUDE COEFFICIENTS FOR FIELD MODULATION n * v X 3/2TT J n U 3 ) w W 1 1.841 0.2930 0.5819 0.4130 0.2910 2 3.054 0.4861 0.4865 0.3608 0.1187 3 4.201 0.6686 0.4344 0.2727 0.05050 4 5.318 0.8463 0.3997 0.2113 0.02194 .5 6.416 1.021 0.3741 0.1659 0.009611 6 7.501 1.194 0.3541 0.1310 0.004225 7 8 .578 1.365 0.3379 0.1042 0.001868 8 9.647 1.535 0.3244 0.0832 0.000828 9 10.711 1.705 0.3128 0.0665 0.000367 10 11.771 1.873 0.3027 . 0.0534 0.000163 \~ i s the argument a p p r o p r i a t e t o the f i r s t extremum of J ( x ) . f r e q u e n c i e s , but p r e s e n t e d two d i f f i c u l t i e s . F o r the B o s c i l l a t i o n s i n l e a d the temperature had t o be kept c o n s t a n t t o 1 m i l l i d e g r e e K e l v i n d u r i n g the time the data was being r e c o r d e d i n order to o b t a i n a 1% e s t i m a t e f o r a 1 . Secondly, the dc magnetic f i e l d was dependent on the modulation f i e l d l e v e l h as w e l l as the modulation o t frequency f . In F i g u r e (3.6) we show the f i e l d s h i f t f o r h Q = 37.6 G as a f u n c t i o n o f modulation frequency. In F i g u r e (3.7) we show the f i e l d s h i f t a t f ^ = 23 Hz, as a f u n c t i o n of the modulation f i e l d l e v e l . Thus when-a data s e t , f o r say the fundamental, was completed and the modulation amplitude was changed i n order to maximize the response of the second harmonic amplitude A ^ j t h e f i e l d H would change as w e l l . Unless t h i s f i e l d s h i f t was known i t was i m p o s s i b l e t o make phase comparisons between the fundamental and the second harmonic. S i n c e i t would have been n e c e s s a r y to determine t h i s s h i f t by a s e p a r a t e experiment f o r each run, we d e c i d e d i n s t e a d t o use a lower v a l u e of n ( u s u a l l y n = 4 ) , t h a t a l l o w e d us to r e c o r d a l l three harmonics s i m u l t a n e o u s l y . We t h e r e f o r e used a p r o -cedure i n which the response of the t h i r d harmonic amplitude A^ was maximized (X^ was chosen to g i v e the f i r s t extremum v a l u e f o r Jj^ CA )^) r and n was chosen such t h a t a l l t h r e e harmonic amplitudes were r o u g h l y o f equal magnitude. The frequency f ^ was chosen s u f f i c i e n t l y low t o ensure complete s i g n a l p e n e t r a t i o n of the sample t A d i s c u s s i o n of these e f f e c t s can be found i n the V a r i a n b u l l e t i n 25-F-013-86. H = 34-3 kG hp=38G MODULATION FREQUENCY (Hz) F i g u r e 3.6 The f r e q u e n c y dependence of the magnetic f i e l d . T h i s i l l u s t r a t e s the f i e l d s h i f t a t 34.3 kG w i t h h = 38 G over a frequency range from 0 t o 100 Hz. 0 50 100 h o ( G ) F i g u r e 3.7 The m o d u l a t i o n amplitude dependence of the magnetic f i e l d . T h i s i l l u s t r a t e s the f i e l d s h i f t a t 34.3 kG f o r f]_=23 H 2 as a f u n c t i o n of m o d u l a t i o n amplitude h . o (determined by a s e p a r a t e experiment). The frequency = 20 Hz was avoided as the f i e l d r e g u l a t i o n c i r c u i t r y o f the V a r i a n magnet was u n s t a b l e a t t h i s frequency. We modulated the f i e l d h a t f r e q u e n c i e s (f-^) r a n g i n g from 23 to 35 Hz and g e n e r a l l y examined i n f o r m a t i o n c a r r i e d by the f o u r t h harmonic ( 4 f ^ ) . The magnet was r o t a t e d so t h a t the a p p l i e d f i e l d H was a l o n g a d e s i r e d symmetry a x i s ( i . e . , normal t o the f a c e of the d i s k ) . T h i s d i r e c t i o n c o u l d be determined q u i t e a c c u r a t e l y 0.1°) f o r the h o r i z o n t a l plane by examining the dHvA s i g n a l as the f i e l d d i r e c t i o n was r o t a t e d w i t h r e s p e c t t o the c r y s t a l axes. We c o u l d not r o t a t e the f i e l d about the v e r t i c a l p lane and as a r e s u l t c o u l d have been as much as 3° away from being p e r p e n d i c u l a r to the d i s k . The phase of the r e f e r e n c e s i g n a l and the amplitude (h Q) of the f i e l d modulation were both v a r i e d t o p r o v i d e a maximum output response f o r the t h i r d harmonic of the dHvA frequency of i n t e r e s t . The r e f e r e n c e phase was a d j u s t e d u n t i l a v a l u e g i v i n g minimum response was found and then changed by 90° from t h i s s e t t i n g i n o r d e r to g e t the maximum response. The amplitude of the modulation f i e l d was s l o w l y i n c r e a s e d from zero and a curve f o r a B e s s e l f u n c t i o n of o r d e r n was t r a c e d out. The v a l u e of h a t o the f i r s t zero c r o s s i n g was r e c o r d e d and used to determine^ the h f o r the b e s t response ( i . e . , the 1 s t extremum of the B e s s e l f u n c t i o n ) . The modulation f i e l d was monitored by a d i g i t a l v o l t m e t e r connected t o a s m a l l pickup c o i l l o c a t e d between the modulation c o i l s . The dHvA s i g n a l from each new sample was examined as a f u n c t i o n of modulation frequency ( f j ) , to determine an optimum modulation frequency. A t each frequency the r e f e r e n c e phase and modulation amplitude were o p t i m i z e d f o r maximum s i g n a l response. The a p p l i e d f i e l d H was v a r i e d so t h a t a few c y c l e s of adHvA o s c i l l a t i o n were t r a c e d out and the amplitude f o r the o s c i l l a t i o n was r e c o r d e d . I t 3 was found f o r a l l samples t h a t the He bath temperature began to r i s e b e f o r e the e v e n t u a l decrease i n amplitude r e s u l t i n g from s k i n depth was found i n the dHvA amplitude f o r t he fundamental, A^. S i n c e the maximum modulation frequency a t which an extremum o f the B e s s e l f u n c t i o n c o u l d be examined was about 400 Hz, we deci d e d t h a t an e x p e r i m e n t a l d e t e c t i o n frequency l e s s than 2 00 Hz should a v o i d problems due t o s k i n depth e f f e c t s . 3 . 5 v S i g n a l D e t e c t i o n C i r c u i t r y In a f i e l d modulation experiment £he dHvA s i g n a l i s c o n v e n t i o n a l l y measured by wrapping a c o i l around the sample and then the output v o l t a g e o f t h i s c o i l i s a m p l i f i e d u n t i l i t can be r e c o r d e d on s t r i p - c h a r t r e c o r d e r . With our 3 d i s k samples we d i d not have s u f f i c i e n t space i n the He 73 system t o i n c l u d e a p i c k - u p c o i l and as a r e s u l t had t o use a c o m p l e t e l y d i f f e r e n t approach. As our system was unique we s h a l l d e s c r i b e i t i n some d e t a i l . A b l o c k c i r c u i t diagram f o r our experiment i s shown t i n F i g u r e (3.8). Two low d i s t o r t i o n o s c i l l a t o r s o perated a t f r e q u e n c i e s f-^ and f^' f]_ w a s f e d i n t o an audio a m p l i f i e r t t which drove the f i e l d modulation c o i l s . A s q u a r i n g module b u i l t by Dr. D. B o y l e , c o u l d be i n s e r t e d between o s c i l l a t o r #1 and the a m p l i f i e r . When i n use the modulation f i e l d h o 2 2 was k e p t p r o p o r t i o n a l t o H , t h a t i s h Q/H = a c o n s t a n t , and the argument X^ of the B e s s e l f u n c t i o n J n ( ^ 3 ) was kept c o n s t a n t as the f i e l d H was swept over a s p e c i f i e d range. The frequency output of o s c i l l a t o r #1 was a l s o f e d i n t o a f r e q u e n c y d o u b l e r which p r o v i d e d the f r e q u e n c i e s n f ^ , where n = 2, 4 or 8. The i n p u t s i g n a l V . = V.. sin(co..t) i n 1 1 i n t o the d o u b l e r was squared to g i v e an output Vout = V l s i n 2 ^ l * ' - ~~2 {1 - c o s t e a ^ t ) } , which was f i l t e r e d to e l i m i n a t e the d.c. component l e a v i n g o n l y ^ O p t i m a t i o n model RCD-1 and model RCD-10. t t The harmonic c o n t e n t from the c o i l s was l e s s than 0.5% of the fundamental. 74 FREQ. DOUBLER rif. MODULATION COILS nf ,±f 2 B R I D 6 E LOW NOISE DIFF. V PRE-AM P. m f r V Sig. In STRIP CHART RECORDER PHASE SENSITIVE AMPLIFIER (PAR 124) Out, Ref. Sig. nfj+ f 2 FILTER v MAG. TAPE RECORDER F i g u r e 3.8 B l o c k c i r c u i t , d i a g r a m . T h i s f l o w diagram shows • t h e arrangement of the b a s i c e l e c t r i c a l com-:c- ponents i n our experiment v ' 75 V Q U t = -CV^/2)cos (2o)1t) This frequency 2f^ could in turn be doubled to give 4f^ by feeding the signal into a second doubler. Thus by cascading, these squaring modules we were able to obtain any multiple 2 pf^ (where p i s an integer) of the input frequency f^. (The basic c i r c u i t can be found in Figure 14 of the specifications and Applications Information form for the Motorola monolithic four-quadrant multiplier MC1595 . Figure 15 of this same data sheet contains the c i r c u i t diagram of the frequency mixer referred to next.) The output frequency nf^ was fed into a frequency mixer together with the frequency f 2 from oscillator #2. The mixer multiplied V-^sin (nu^t) by V 2sin(co 2t) giving vo u t = V1 V 2 sinCnw1t) sin(w 2t) V l V2 = —2— {cos[nco1~ u>2)t]- cos [ (nu^+co^ t] } . Thus the output of the mixer contained the two frequencies n f l ± f 2 ' w h ^ c h were passed through a Kronhite f i l t e r in order to select either nf^ + f 2 (most commonly used) or n f l - f 2 -The output frequency f 2 from o s c i l l a t o r #2 was also fed into an attenuator before going into the Wheatstone. bridge which i s shown schematically in Figure ( 3 . 9 ) . Two matched magneto-resistors (type MRA-11 purchased from American Aerospace Controls Inc. of Farmingdale, N.Y.), Although the use of magneto-resistors to observe the dHvA effect i s not unique (see Thomas and Turner (1968) and Condon (1966)) our technique does give greater sensitivity. A B l a b e l e d and R ^ i n F i g u r e t3.9), were p l a c e d i n the sample h o l d e r i n a manner such t h a t they both e q u a l l y sampled the e x t e r n a l l y a p p l i e d f i e l d s (both a.c. and d . c ) . However, o n l y one of the r e s i s t o r s was l o c a t e d on the s i d e o f the d i s k sample; the other was on the o p p o s i t e s i d e of the h o l d e r away from the sample. Both o f the probes 3 ( r e s i s t o r s ) were l o c a t e d i n the He. bath. A t y p i c a l f i e l d response f o r the m a g n e t o - r e s i s t o r s i s shown i n F i g u r e (3.10). The remaining two r e s i s t o r s o f the Wheatstone b r i d g e were 4 metal f i l m r e s i s t o r s which were l o c a t e d i n the He bath. A low r e s i s t a n c e t e n - t u r n potentiometer was p l a c e d i n the c i r c u i t o u t s i d e the c r y o s t a t i n o r d e r to p e r m i t a f i n e b a l a n c i n g adjustment. The r e s i s t a n c e o f the metal f i l m C D r e s i s t o r s R and R was approximately equal t o t h a t of the m a g n e t o - r e s i s t o r s a t maximum f i e l d and a t temperatures l e s s than 4.2°K each r e s i s t o r i n the Wheatstone b r i d g e had a r e s i s t a n c e of order s e v e r a l Kft, w h i l e the b a l a n c i n g r e s i s t o r was about 100ft, and hence we had minimized Johnson n o i s e s i n c e the b u l k of the r e s i s t a n c e was c o o l e d t o w e l l below room temperature. When the Wheatstone b r i d g e was balanced the output v o l t a g e V Q u t was r e l a t e d to the i n p u t v o l t a g e V ^ n by the equa t i o n vout - v m '""ai^1 'JXTTC1 " i 1 - " " -Jt\_ • • R B l - ' 77 B A L R • ~ i v i n R R Bi R R B BI DISK SAMPLE out ROOM TEMP. 1 He TEMP, HeQ TEMP. H+ hosin(wlt) F i g u r e 3 . 9 Wheatstone b r i d g e arrangement. 7 8 0 10 20 30 H (kG) F i g u r e 3.10 F i e l d r e s p o n s e f o r the m a g n e t o - r e s i s t o r . The r e s i s t a n c e of the m a g n e t o - r e s i s t o r (shown i n o u t l i n e i n the i n s e r t ) i s l i n e a r w i t h r e s p e c t to the f i e l d p r o v i d e d H > 15kG. T h i s c a l i b r a t i o n c u r v e was o b t a i n e d a t 4.2°K. where Rg^ i s the r e s i s t a n c e of the m a g n e t o - r e s i s t o r sampling the dHvA s i g n a l , R i s the r e s i s t a n c e of one of the metal f i l m r e s i s t o r s Ccf. F i g u r e (3.9)), and (dR^./dH) i s the o 1 A s l o p e of the curve f o r the r e s i s t a n c e of R n. as a f u n c t i o n o f the f i e l d H, which f o r H g r e a t e r than 15 kG was e s s e n t i a l l y a c o n s t a n t . As s t a t e d p r e v i o u s l y , i n a f i e l d modulation experiment the m a g n e t i z a t i o n c o n t a i n s a F o u r i e r component of the form sin(ntot + nir/2) . S i n c e i n our experiment the i n p u t v o l t a g e V. i n t o the Wheatstone b r i d g e c o n t a i n e d a term of the xn ^ form s i n ( c o 2 t ) , the output v o l t a g e v " o u t c o n t a i n e d s i n u s o i d a l terms i n v o l v i n g the mixed f r e q u e n c i e s n f ^ ± f 2« The output v o l t a g e from the b r i d g e was f e d i n t o - t a low-noise (battery-operated) - p r e a m p l i f i e r which was o p e r a t e d i n a d i f f e r e n t i a l mode. ( I t was n e c e s s a r y to use a d i f f e r e n t i a l a m p l i f i e r s i n c e an e l e c t r i c a l ground had been d e f i n e d by the i n p u t v o l t a g e o s c i l l a t o r ) . The p r e a m p l i f i e r was connected i n t o a p h a s e - s e n s i t i v e d e t e c t o r . (We used a P r i n c e t o n A p p l i e d Research model 124 as w e l l a s a n H R - 8 ) . The o u t p u t s i g n a l which was p r o p o r t i o n a l to M(H) was d i s p l a y e d g r a p h i c a l l y by means of a s t r i p - c h a r t r e c o r d e r as w e l l as d i g i t i z e d and put onto magnetic tape f o r subsequent n u m e r i c a l a n a l y s i s . In a d d i t i o n a v o l t a g e p r o p o r t i o n a l t o the f i e l d was d i g i t i z e d and a l s o put onto + P r i n c e t o n A p p l i e d Research Model 113, which f o r our experiment had a n o i s e f i g u r e o f 5dB. 80 the tape. The d i g i t i a l r e c o r d i n g system was d e s i g n e d and b u i l t by E l e c t r o m a g n e t i c Research C o r p o r a t i o n of C o l l e g e Park, Maryland. 3.6 N u m e r i c a l A n a l y s i s A l i m i t e d number of c y c l e s (from 2 t o 6 c y c l e s of the fundamental) of the o s c i l l a t i o n under study were d i g i t i z e d a t e q u a l increments i n time to g i v e about 250 d a t a p o i n t s f o r a p a r t i c u l a r d a t a s e t . The. f i e l d dependence of the a m p l i t u d e s A^. was n e g l i g i b l e over so few c y c l e s , and hence c o u l d be i g n o r e d . A l i n e a r l e a s t squares s u b r o u t i n e was used t o f i n d the c o n s t a n t s 'B Q, B R , and C r i n the e q u a t i o n M F ± T = B q + Z { B r s i n ( 2 i r r '!')+ C r c o s (27rr ~ ) } , f o r r = 1, 2, and 3, which g i v e the b e s t l e a s t - s q u a r e s f i t t o the e x p e r i m e n t a l d a t a assuming the f r e q u e n c y to be F 1 . F 1 was then incremented and the c o n s t a n t s B , B , and C o' r r were r e d e t e r m i n e d u n t i l we found an o v e r a l l minimum i n the sum o f the squares of the d i f f e r e n c e between the a c t u a l and the f i t t e d p o i n t s . Then the phase f a c t o r s o f Eq. (2.2.19) were determined u s i n g C C = ( r j t a n " 1 ^ ) - t a n " 1 (^) - ( r - l ) 6 , 81 and the amplitudes were determined u s i n g A' = [B 2 + C 2 ] 1 / 2 . r 1 r r T h i s approach handled the problem of n o i s e i n a s t r a i g h t -forward manner and a l s o e x p l i c i t l y imposed the c o n d i t i o n t h a t the i n p u t data should be s i n u s o i d a l as a f u n c t i o n o f 1/H. The number of f i t t i n g parameters c o u l d be e a s i l y i n c r e a s e d t o account f o r any a d d i t i o n a l f r e q u e n c i e s t h a t were p r e s e n t . I n i t i a l l y , we used the more common approach t o dHvA a n a l y s i s , and F o u r i e r a n a l y z e d the data c o m p u t a t i o n a l l y by means of a f a s t - F o u r i e r - t r a n s f o r m s u b r o u t i n e (UBC Four2). However, i n a f a s t - F o u r i e r t r a n s f o r m t h e r e are very few p o i n t s ( . t h e o r e t i c a l l y no more than two) per peak i n the "power spectrum" t h a t i s c a l c u l a t e d i n an i n t e r m e d i a t e step b e f o r e d e t e r m i n i n g the amplitudes A^,. A l t h o u g h such a sm a l l number of p o i n t s per peak was i n s u f f i c i e n t to determine the amplitudes A^, r e l i a b l y , t h i s d i f f i c u l t y was circumvented by use of a "window" ( c f . R i c h a r d s (1967)), which i n c r e a s e d the w i d t h of the peak (and hence the number of p o i n t s per peak) and a l s o decreased the s c a t t e r r e s u l t i n g from the n o i s e i n the d a t a . However, once a "window" had been used i t was d i f f i c u l t to extract, the phase i n f o r m a t i o n needed to f i n d the c o r r e c t i o n f a c t o r |a/a 1| . T h i s approach a l s o r e q u i r e d t h a t many c y c l e s (.^  100) of a p a r t i c u l a r o s c i l l a t i o n be examined i n order to o b t a i n a r e a s o n a b l e e s t i m a t e of the amplitudes A^,. Data taken e a r l y i n t h i s e x p e r i m e n t a l programme, b e f o r e we a p p r e c i a t e d the s i g n i f i c a n c e o f the phase information,were a n a l y z e d u s i n g the f a s t - F o u r i e r -t r a n s f o r m t e c h n i q u e . CHAPTER IV g -VALUES FROM THE DE HAAS-VAN ALPHEN EFFECT: c EXPERIMENTAL RESULTS 4.1 I n t r o d u c t i o n We have e x t r a c t e d g c ~ v a l u e s from e x p e r i m e n t a l dHvA data u s i n g the approach d e s c r i b e d i n Chapter I I and have determined g c ~ v a l u e s f o r two d i f f e r e n t o s c i l l a t i o n s i n l e a d . The e l e c t r o n o r b i t s which g i v e r i s e to these (and other) dHvA o s c i l l a t i o n s can be seen i n F i g u r e (4.1), where Anderson and Gold (1965) have shown the extremal o r b i t s f o r the h o l e s u r f a c e of the second zone as w e l l as f o r the e l e c t r o n s u r f a c e of the t h i r d zone. In F i g u r e (4.2) we have shown the angular v a r i a t i o n o f t h e i r c o r r e s p o n d i n g (experimental and computed)dHvA p e r i o d s (P = 1/F). In order to minimize s y s t e m a t i c e r r o r s r e s u l t i n g from specimen inhomogeneity we wanted t o have (%F/dB)- 0, as has been p o i n t e d out i n s e c t i o n (2.3b). When we added the requirement t h a t the o s c i l l a t i o n s be r e l a t i v e l y s t r o n g we were l e f t w i t h o n l y the B o s c i l l a t i o n s a t [100] and the y o s c i l l a t i o n s a t [110]. —1 F i g u r e 4.1 A s k e t c h o f the Fermi s u r f a c e of l e a d . The h o l e s u r f a c e i n the second zone i s shown on the l e f t - h a n d s i d e . c e n t r a l [111] extremal o r b i t ; non-extremal o r b i t . The e l e c t r o n s u r f a c e i n the t h i r d zone i s shown on the r i g h t hand s i d e . The o r b i t s K and t are non-extremal w i t h r e s p e c t t o a r e a , and the broken curves d e p i c t the open o r b i t s p and u. ( a f t e r Anderson and Gold (1965)). C O 85 0° 20° 40° 60° 80° 100° F i g u r e 4.2 P e r i o d v a r i a t i o n s i n l e a d . The s o l i d l i n e s come \ from a band s t r u c t u r e model f i t t e d to e i g h t symmetry o r b i t s (from Anderson and Gold (1965)). The f i e l d d i r e c t i o n s are i n the (110) plane and the p e r i o d s are p l o t t e d on a l o g a r i t h m i c s c a l e . 86 We examined f i r s t the 8 oscillations. These o s c i l l a -tions arise from electrons in the orbit labeled v in Figure (4.1). This orbit encircles the junction of the four arms about the symmetry point W (see Figure (5.2) where we have labeled the symmetry points on a drawing of the Brillouin zone and also Figure (5.8) where the calculated cross-sectional area i s shown) and has a minimum area in the (100) plane. Our results for the 8 oscillations are given in section (4.2). We examined next the y oscillations. These o s c i l l a -tions arise from the orbit labeled C. This orbit encircles an arm cross-section about the symmetry points K and U (see Figure (5.2) and Figure (5.9)) and has a minimum in the (110) plane. Our results for the y oscillations are given in section (4.3) . 4.2 8 Oscillations (a) Preliminary Results Our f i r s t dHvA amplitude measurements were obtained from lead samples in the shape of cylinders, which had a length to diameter ratio of approximately 4:1 (and hence a demagnetizing factor 6 - 0 . 0 7 ) . We wanted to examine 2 the ratio a' = A_ l/(A|A'ias a function of temperature and c o n s e q u e n t l y we determined A ^ . a t a number of temperatures. The temperature dependence of the amplitudes A ^ and A ^ suggested t h a t we were measuring i n t e r a c t i o n harmonics and t h a t our r e s u l t s c o u l d not be used t o determine g . In 3 c f a c t the r e s u l t s o f P h i l l i p s and Gold (1969) i m p l i e d a v a l u e o f 1.4 f o r K A ^ f o r the 3 o s c i l l a t i o n s a t 0.4°K, and as a r e s u l t the approximations made i n s e c t i o n ( 2 . 2 ) , t o a l l o w f o r i n t e r a c t i o n r e s u l t s were i n v a l i d . In o r d e r t o determine g c i t was necessary t o decrease K A ^ and as has been p r e v i o u s l y noted we d e c i d e d t o accomplish t h i s by changing the sample geometry, i . e . , we d e c i d e d t o use d i s k samples w i t h the a p p l i e d f i e l d H normal t o the f a c e s of t h e s e d i s k s . In o r d e r t o determine an a p p r o p r i a t e geometry f o r these d i s k s , the c o n s t a n t s a p p r o p r i a t e f o r the 3 o s c i l l a t i o n s a t [100], were used t o compute the ( i n t e r a c t i o n ) amplitudes A' a t T = 0.5°K and H = 34 kG, as a f u n c t i o n of b o t h r the demagnetizing f a c t o r 6 and the D i n g l e - c o s i n e f a c t o r cos (Trg EI*/(2m)) . The r a t i o a 1 was determined and i s shown i n F i g u r e (4.3) as a f u n c t i o n o f the D i n g l e - c o s i n e f a c t o r f o r f o u r v a l u e s o f 6. We concluded from t h i s c a l c u l a t i o n t h a t i n o r d e r t o determine cos (w^) r e l i a b l y we would have t o use samples f o r which the demagnetizing f a c t o r 6 was t h e o r d e r o f 0.9. Osborne (1945) has c a l c u l a t e d and g i v e s c h a r t s f o r the demagnetizing f a c t o r of a g e n e r a l e l l i p s o i d and these c h a r t s were used t o show t h a t i n o r d e r F i g u r e 4 . 3 a ' a s a f u n c t i o n o f t h e d e m a g n e t i z i n g f a c t o r . T h e s e a m p l i t u d e r a t i o s h a v e b e e n c a l c u l a t e d f o r t h e B o s c i l l a t i o n s a t 3 4 k G a n d 0 . 4 ° K . 89 t o reduce KA^ by approximately a f a c t o r of 10 i t would be ne c e s s a r y t o use d i s k samples f o r which the r a t i o of t h i c k n e s s 3 to diameter was 1/10. S i n c e the t a i l of the He i n s e r t dewar had an i n s i d e diameter of o n l y 7.5 mm, i t was n e c e s s a r y t o use samples whose diameter c o u l d not exceed t h i s dimension and hence d i s k s of width l e s s than 0.7 mm had t o be c u t . The procedure f o l l o w e d i n making these d i s k s has a l r e a d y been o u t l i n e d i n s e c t i o n (3.3). (b) F i e l d Inhomogeneity The 3 o s c i l l a t i o n s were used t o measure the f i e l d i n -homogeneity a t 35 kG. T h i s was done by measuring the change i n phase of the fundamental o s c i l l a t i o n as the dewar (and hence the sample) was moved both h o r i z o n t a l l y and v e r t i c a l l y w i t h r e s p e c t to the magnet. S i n c e the dimensions of' the m a g n e t o - r e s i s t i v e probe normal t o the f i e l d were 1/16" by 1/16" we were i n e f f e c t measuring a change i n the average f i e l d i n the volume of the sample bounded by the two s i d e s of the d i s k and t h i s 1/16" square. The f i e l d p r o f i l e about a t u r n i n g p o i n t i s shown i n F i g u r e (4.4) f o r a v e r t i c a l -dewar di s p l a c e m e n t . The f i e l d p r o f i l e f o r a h o r i z o n t a l -dewar displacement p a r a l l e l t o the p o l e f a c e s of the magnet, about t h i s t u r n i n g p o i n t , was v e r y s i m i l a r . The dewar c o u l d not be moved i n a d i r e c t i o n normal to the p o l e f a c e s and hence no f i e l d p r o f i l e was o b t a i n e d a l o n g the d i r e c t i o n o f a p p l i e d f i e l d . However, the inhomogeneity a l o n g H was 90 Figure 4 . 4 F i e l d p r o f i l e for the high f i e l d pole t i p s . The f i e l d p r o f i l e about a turning point was ineasured using the dHvA e f f e c t as a f i e l d sensor. 91 expected to be n e g l i g i b l e and s i n c e the volume of the sample g i v i n g r i s e t o the measured dHvA s i g n a l had a dimension al o n g the f i e l d 1/3 t h a t of the dimensions normal t o the f i e l d , we have s e t t h i s component of n(r) e q u a l to zero i n our c a l c u l a t i o n s . With r e f e r e n c e to the v a r i a b l e s d e f i n e d i n s e c t i o n (2.3a), we found t h a t 3 2 - 3 X - -2.8 x 10~ 5 mm"2 and s e t B 3 « 0 . Then w i t h the a i d of F r e s n e l - I n t e g r a l T a b l e s , G i n Eq. (2.3.3) , and G2 r i n Eq. (2.3.4) , were c a l c u l a t e d f o r the 3 o s c i l l a t i o n s and hence the c o r r e c t i o n s to the r a t i o a 1 and t o the phase f a c t o r s of Eq. (2.2.19) were determined, making use of the f a c t t h a t the m a g n e t o - r e s i s t o r was c e n t e r e d about a t u r n i n g p o i n t . The r e s u l t s showed t h a t the r a t i o a 1 should be m u l t i p l i e d by 0.997 and t h a t the c o r r e c t i o n t o the phase f a c t o r s was l e s s than O . O O I T T . These c o r r e c t i o n s were n e g l i g i b l e when compared t o the a c t u a l a c c u r a c y t o which these q u a n t i t i e s c o u l d be measured. 92 (c) Specimen Inhomogeneity We d i d not measure the specimen inhomogeneity w i t h a g r e a t d e a l of a c c u r a c y , but were a b l e to conclude from the X-ray n e g a t i v e s t h a t i f the c r y s t a l s were bent, the normal t o the f a c e of the c r y s t a l s v a r i e d by l e s s than 0.5° between the c e n t e r and the c i r c u m f e r e n c e of the d i s k b e f o r e c o o l i n g . A l s o , any spread of the X-ray spots due t o mosaic s t r u c t u r e was l e s s than 0.5°. In o r d e r t o f i n d an upper l i m i t on the c o r r e c t i o n s t h a t would be r e q u i r e d i f the c r y s t a l s were indeed bent 0.5°, we c a l c u l a t e d f i r s t , the second d e r i v a t i v e of the frequency w i t h r e s p e c t to an angle 0 (the angle the f i e l d H was away from the symmetry d i r e c t i o n [100]), from the p e r i o d v a r i a t i o n s shown i n F i g u r e (4.2) and found t h a t F - F Q ( 1 + 0.00046 2) We then had t o assume a p a r t i c u l a r f u n c t i o n a l dependence on p o s i t i o n i n o r d e r t o d e s c r i b e how the angle 8 may have been a f f e c t e d by t h i s p o s s i b l e bending. We chose t o use the approximation o = / 0.0004 93 and c a l c u l a t e d g' t o be 8 x l O - 5 f o r 6 •» 9 = 0.5°. G, and x max xr G„ were c a l c u l a t e d and we found t h a t the r a t i o a 1 would 2r be a t most 2% too l a r g e and t h a t the maximum e r r o r i n the pha f a c t o r s would be l e s s than O . O I T T . (d) N e i g h b o u r i n g F r e q u e n c i e s The dHvA s i g n a l a t [100] was co m p l i c a t e d by the presence o f f o u r b a s i c o s c i l l a t i o n s a r i s i n g from f o u r d i f f e r e n t c r o s s - s e c t i o n a l areas of the Fermi s u r f a c e . In a d d i t i o n t o t h e f o u r s e t s ( i n c l u d i n g the harmonics) of o s c i l l a t i o n s t h e r e were a l s o the sum and d i f f e r e n c e f r e -q u e n c i e s between these s e t s due t o magnetic i n t e r a c t i o n . (Gold (1969) has shown how magnetic i n t e r a c t i o n can l e a d t o frequen c y m i x i n g ) . F o r t u n a t e l y the amplitudes o f these terms were weaker than the o s c i l l a t i o n s from the v o r b i t . In a d d i t i o n t o these f r e q u e n c i e s , a n a l y s i s o f the $-o s c i l l a t i o n s i s c o m p l i c a t e d by the f a c t t h a t the envelope of t h e s e o s c i l l a t i o n s i s not a monotonic f u n c t i o n o f f i e l d , b u t e x h i b i t s l o n g beats w i t h approximately 600 c y c l e s per beat . I t i s b e l i e v e d t h a t the b e a t i n g p a t t e r n i s due to the presence o f two d i f f e r e n t dHvA o s c i l l a t i o n s a r i s i n g from two s l i g h t l y d i f f e r e n t extremal areas f o r the v o r b i t . Presumably one area i s a maximum and the o t h e r a minimum, and we s h a l l d i s c u s s the evidence f o r t h i s i n t e r p r e t a t i o n s h o r t l y . I t t u r n s o ut t h a t the two o r b i t s 94 must be q u i t e c l o s e i n it-space and hence should have e s s e n t i a l l y the same p r o p e r t i e s , and t h i s i s inde e d the ca s e f o r the e f f e c t i v e masses which are e s s e n t i a l l y the same f o r b o t h s e t s o f e l e c t r o n s . I n o r d e r t o a n a l y z e the data i t was n e c e s s a r y t o assume t h a t the e l e c t r o n s i n the two o r b i t s a l s o had the same g - v a l u e s , i . e . , we assumed the r a t i o A a / A b was x' r independent o f r . T h i s was ne c e s s a r y s i n c e we c o u l d o n l y measure the amplitude r a t i o o f beat minima t o b e a t maxima f o r the fundamental terms. T h i s r a t i o was found t o be | ( A A - A * J ) / ( A J + A £ ) I = 0 . 6 3 ± 0 . 0 2 . We were o n l y a b l e t o study the beat s t r u c t u r e o v e r a v e r y l i m i t e d f i e l d range. The f i e l d inhomogeneity became p r o g r e s s i v e l y worse as the f i e l d was d e c r e a s e d from 3 5 kG and a l t h o u g h t h i s d i f f i c u l t y c o u l d have been accounted f o r by d e t e r m i n i n g the f i e l d p r o f i l e a t each f i e l d p o i n t , the l e n g t h o f time i n v o l v e d i n making these measurements made t t h i s approach i n f e a s i b l e . I n s t e a d we removed the h i g h -f i e l d p o l e - t i p s and s t u d i e d the b e a t i n g p a t t e r n i n an extremely homogeneous f i e l d ; however we now c o u l d not exceed 2 5 kG. The E x t r a p o l a t i o n needed i n o r d e r t o "'"The h i g h - f i e l d p o l e t i p s had been s p e c i f i c a l l y shimmed t o g i v e the b e s t f i e l d p r o f i l e a t 3 5 kG. determine (6 - 8 )/2 would have been f a i r l y i n a c c u r a t e from our r e s u l t s alone and f o r t h i s reason we have used some unpu b l i s h e d data g i v e n t o us by Dr. J . Lee and Dr. J.R. Anderson o f the U n i v e r s i t y of Maryland. They have s t u d i e d the b e a t i n g p a t t e r n from 20 to 40 kG and t h e i r r e s u l t s have been used t o draw the dashed l i n e of F i g u r e (4.5) . The i n t e r c e p t a t (1/H) = 0 i s (- 0.25 ± 0.01)TT. Our d i s k sample c o u l d p o s s i b l y have been s l i g h t l y m i s a l i g n e d (^  3°) and as a r e s u l t we expected a change i n the s l o p e ( i . e . , the r e l a t i v e f r e q u e n c i e s ) but not the i n t e r c e p t ( i l e . , the types o f extrema). Thus we used the same i n t e r c e p t ( s h i f t e d f by TT) t o g e t h e r w i t h our r e s u l t s t o draw the s o l i d l i n e of F i g u r e (4.5). In s e c t i o n (4.2e) we have g i v e n our phase and amplitude data and have made use of t h i s l i n e t o c a l c u l a t e the c o r r e c t i o n s which we have o u t l i n e d i n s e c t i o n (2 . 3 c ) . We conclude by n o t i n g t h a t the l a r g e r and the s m a l l e r cl JD frequency (F and F ) must have come from maximum and minimum c r o s s - s e c t i o n a l a r e a s , r e s p e c t i v e l y , as a f u n c t i o n , o f k H . Since we know t h a t the area must e v e n t u a l l y i n c r e a s e as a f u n c t i o n of k , the o n l y p o s s i b l e shape must be s i m i l a r t o t h a t sketched i n F i g u r e (4.6). Van Dyke (1973) has r e c e n t l y computed the Fermi s u r f a c e of l e a d from a pseudo-p o t e n t i a l model which he c l a i m s t o be s u p e r i o r t o oth e r e x i s t i n g models and f i n d s no evidence f o r a second o r b i t . T h i s prompted us to see i f an a l t e r n a t i v e e x p l a n a t i o n was % h e p o i n t s were s h i f t e d by TT S O t h a t the two l i n e s c o u l d be d i s t i n g u i s h e d from one another. ray 96 <f 2-CL > C LU CJ) c CD o J C CL A ( l/H = 0) = (-0-25*0-01) TT =(ea- eb)/2 0 l /H (kG"') 0-05 F i g u r e 4 . 5 F i e l d dependence of the b e a t i n g envelope f o r B o s c i l l a t i o n s . The dashed l i n e comes from p o i n t s o b t a i n e d by Lee and Anderson ( p r i v a t e communi-c a t i o n ) . The s o l i d l i n e uses t h e i r i n t e r c e p t ( s h i f t e d by TT) t o g e t h e r w i t h our da t a p o i n t s . F i g u r e 4 . 6 V a r i a t i o n o f c r o s s - s e c t i o n a l a r e a f o r the v o r b i t . The i n t e r c e p t of the l i n e i n F i g u r e 4.5 •  i m p l i e s t h a t the c r o s s - s e c t i o n a l a r e a normal to the a p p l i e d f i e l d must be s i m i l a r * to the shape sketched h e r e . : 98 p o s s i b l e . The c u r v a t u r e f a c t o r f o r the 3 o s c i l l a t i o n s i s n e a r l y zero and i n view of t h i s we re-examined the T a y l o r s e r i e s expansion of Eq. (2.1.14). A number of r e a d i l y i n t e g r a b l e e x t e n s i o n s are g i v e n i n Appendix I . While some of these p o s s i b i l i t i e s p r e d i c t a b e a t i n g p a t t e r n , none of the s o l u t i o n s p r e d i c t the c o r r e c t phase as w e l l as a p a t t e r n of b e a t s which i s p e r i o d i c i n 1/H. (e) g^-Values f o r the 3 O s c i l l a t i o n s The r e s u l t s r e p o r t e d here were o b t a i n e d d u r i n g two s e p a r a t e e x p e r i m e n t a l runs u s i n g d i s k sample IXr the o n l y sample f o r which both the f i e l d p r o f i l e and the b e a t i n g s t r u c t u r e were s t u d i e d . The modulation f r e q u e n c i e s f o r the f i e l d h and the Wheatstone b r i d g e were 35 Hz and 115 Hz, r e s p e c t i v e l y , and the f o u r t h time harmonic (4f^ = 140 Hz) was d e t e c t e d a t 255 Hz ( 4 f ^ + f 2 ) . The o s c i l l a t i o n s were s t u d i e d over a f i e l d range of 45 G c e n t e r e d a t 35.2 kG. The output of a s t r i p - c h a r t r e c o r d e r f o r a t y p i c a l d a t a s e t i s shown i n F i g u r e (4.7). The average amplitudes A^ . f o r s i x data s e t s about a t u r n i n g p o i n t of the f i e l d H and a t T = (0.40 ± 0.05)°K were computed to be ( i n a r b i t r a r y u n i t s ) 9 9 12 4 - 1 0 6—• H(6) F i g u r e 4 . 7 Example o f B o s c i l l a t i o n s . These d a t a were o b t a i n e d at a f i e l d o f 35.2 kG and a t a temperature o f 0 . 4°K. The sweep r a t e was 5 G per minute and the - . time c o n s t a n t was 3 sec w i t h a Q o f 20. The maximum v e r t i c a l e x c u r s i o n c o r r e s p o n d s t o a v o l t a g e change o f 5 x 10*"8 v o l t s a t the i n p u t o f the p r e a m p l i f i e r . 100 A | = (293 ± 12)/J 4(X 1) = (1.34 ± 0.05)104 , &2 = C287 ± 15l/J 4(X 2) = (1.36 ± 0.07)103 , A 3 = (140 + 20)/J 4(X 3) = (3.5 ± 0.5U0 2 . . * The average phase differences rZj^ - Z^ . for r = 2 and r = 3 were found to-be and 2 Z 1 " Z 2 = ( 1 ' 1 2 ± ° - 0 2 ) 7 T ' 3ZJ = Z 3 = (1.16 ± 0.06)TT, . From Figure (4.5) we find A± = (-0.25 ± 0.01) TT , and then using the ratio of the beat maxima to beat minima given i n section (4.2d), Eq. (2.3.10) can be solved for o*r and this gives rise to the phase corrections 2a 1 - a 2 = (0.651 ± 0.007)TT , and 3c 1 - a 3 = (0.27 ± 0.02)TT , +The quoted errors are the standard deviations for the six data sets. The s t a t i s t i c a l errors for a given data set are a factor of 20 less. 101 The c o r r e c t i o n f a c t o r |a / a'| can now be determined. The o n l y p h y s i c a l s o l u t i o n has A > A and ||, | = 1-48 ± 0.07 , and hence a = - 0.36 ± 0.12 From Eqs. (2.1.20) to (2.1.22) we f i n d Icostw-jjl =0.810 ± 0.001 , Case I or I c o s t w ^ l = 0.55 ± 0.03 . Case I I U s i n g Eq. (2.1.24) Case I can be e l i m i n a t e d and we can s o l v e f o r g . U s i n g m*/m = 1.2 ( P h i l l i p s and Gold (1969)) and Eq. (2.1.26) w i t h s = 8, we f i n d t en p o s s i b l e g c ~ v a l u e s , namely g = 0.54, 1.12, 2.21, 2.79, 3.88, 4.46, 5.54, 6.12, 7.21, or 7.79, w i t h a p o s s i b l e e r r o r ±0.02. The g-value c l o s e s t t o the f r e e e l e c t r o n g-value has been underscored. P h i l l i p s and Gold (1969) have c a l c u l a t e d the v a l u e of cos(wj) r e q u i r e d t o g i v e exact agreement between t h e i r e x p e r i m e n t a l and t h e i r c a l c u l a t e d amplitudes f o r the 8 o s c i l l a t i o n s . They f i n d cos (w n) =0.3 , whereas we find 102 cos(wx) = 0.55 ± 0.03 . If we assume that their absolute amplitude measurements had no systematic errors, the most probable cause for the discrep-ancy would be the value of the curvature factor 'a' (see Eq. (2.1.14)). Our results suggest that they should have used a = 0.21 instead of 0.07. 4.3 g -Values for the y Oscillations  Jc __ •—— For the y oscillations the corrections needed to allow for f i e l d and specimen inhomogeneity are much smaller than for the 3 oscillations. There are two reasons for this. F i r s t , the argument of the sinusoidal terms in Eqs. (2.3.3) Y 3 /-, and (2.3.4) is reduced by a factor of three, since F -F /3. ->-In addition, n 1(r) in Eq. (2.3.8) i s also a factor of two smaller. Hence, as was the case for the 3 oscillations, we do not make any corrections for these effects . As can be seen from Figure (4.21, the frequency spectrum at [110] is much simpler than the spectrum at [100] . In fact, only the frequency co l i e s within our range of interest and this frequency has never been experi-mentally observed. There are, however, two y frequencies of nearly equal amplitude which give rise to a signal having-42.5 cycles per beat. This beating pattern was studied as 103 a function of magnetic f i e l d . The results are shown in Figure (.4.8). The intercept at tl/H) = 0 i s (0.40±0.20) TT. This implies that, as a function of k„, the cross-sectional ti • area must be similar to that for the 8 oscillations, and this i s in fact the shape calculated in the empty-lattice approximation. In Figure (4.9) we have sketched the kjj-dependence for-three, cases; the empty-lattice approxi-mation, 1 the calculated shape using the band model of Anderson and Gold (1965), and what we believe to be the correct shape. Our data analysis could separate out the two fre-quencies and as a result the actual analysis i s simpler than that for the 8 oscillations. We studied the y oscillations over a f i e l d range of 450 G centered at 34.2xkG. The modulation frequencies for the f i e l d h and the Wheatstone bridge were 35 Hz and 80 Hz, respectively, and the fourth time harmonic was detected at 220 Hz. The output from a strip-chart recorder i s shown in Figure (4.10). The average amplitudes A^ for the higher frequency (i.e., k„ = 0) at T = CO.40 ± 0.05) were computed to be, again in arbitrary units, This curve was computed by Dr. Peter Holtham. 1 0 4 005 Figure 4 . 8 Field dependence of beating envelope for y o s c i l l a t i o n s . 0 01 k H(2TT /a ) F i g u r e 4.9 V a r i a t i o n , of c r o s s - s e c t i o n a l area f o r the £ o r b i t . The s o l i d l i n e g i v e s the c r o s s - s e c t i o n a l a r e a normal to the a p p l i e d f i e l d as computed by Anderson and G o l d (1965). The dashed l i n e shows the e m p t y - l a t t i c e curve. The d o t t e d c u r v e i n d i c a t e s the form expected from F i g u r e 4.8. Each curve has been n o r m a l i z e d a t k„ = 0 and the v e r t i c a l a x i s r e p r e s e n t s a 17% change i n a r e a . 106 100 6 H ( k G ) Figure 4.10 Example o f y o s c i l l a t i o n s . These d a t a were o b t a i n e d at a f i e l d o f 33.4 kG and a t a temperature of 0.4°K. The sweep r a t e was 20-G per minute and t h e time c o n s t a n t was 3 sec w i t h a Q of 100. The -maximum v e r t i c a l e x c u r s i o n corresponds to a v o l t a g e change o f 7 x 10~7 v o l t s a t the i n p u t of t h e p r e a m p l i f i e r . 107 Aj^ = (80±4)/J 4(X 1) = (3.8±0.2)103 , A^ = (105±4)/J 4 (X 2 ) = (5.0±0.2)102 A 3 = (69±4)/J 4 (X 3 ) = (1.7±0.1)102 . + The average phase differences were computed to be 2Zj[ - 2^ = (0.85 + 0.05) TT and '" 3Z£ - Z 3 = (0.70±0.10)TT . The rather large errors in the phases result from d i f f i c u l t y i n finding an overall best f i t (see section (3.6)). Now, using 9 = - 0.25TT, we find from Figure (2.6) that . |£r| = 0.4 ± 0.2 . Using m*ftn = 0.56 (from P h i l l i p s and Gold (1969) together with Eqs. (2.1.20) to (2.1.22) we find o r |cos(w1)|= 0.58 ± 0.05 , Case I |cos(w1)| = 0.79 ± 0.02 . Case II Again, these errors are the standard deviations. 108 Then, using Eq. (2.1.24) and a similar equation for A3/A^ we can eliminate Case II. Then using Eq. (2.1.26) with s = 3 to terminate the i n f i n i t e series of Eq. (2.1.25) we fin d four possible values for g , namely g„ = 1.09, 2.48, 4.66, or 6.05 , with a possible error ± 0.07. The g c~value closest to the free electron g-value has been underscored. P h i l l i p s (1967) has measured the amplitudes A^ . of the Y o s c i l l a t i o n s , absolutely, for the f i r s t four harmonics and gives both the experimental and the calculated ratios, A^/A^ and Ar/A^, respectively. His data were taken at much higher f i e l d s (71 to 119 kG) than we had at our disposal and from temperature dependence studies he was able to conclude that at the lower values of T/H, he was indeed measuring a L-K result. By examining his ratios i t i s possible to deduce values for cos (wr) /cos (w^ ) . We have l i s t e d these values i n Table IV together with our ratios, for the cosine values appropriate to both Case I and Case I I . This comparison with P h i l l i p s results reinforces our conclusion that Case I i s the physically meaningful solution. I t i s of interest to note that P h i l l i p s could not . estimate a g-value for the y oscillations because his c calculated amplitude A, was smaller than the measured 109 TABLE IV EXPERIMENTAL RATIOS FOR COS(w^)/COS(w^) Co s i n e P h i l l i p s 1 R a t i o R e s u l t s P r e s e n t Work Case I Case I I cos(w~) z cos(w 1) 0.37 0.5 ± 0.2 0.32 ± 0.09 cos(w,) > cos(w 1) ' 1.9 1.7 ± 0.2 0.5 ± 0.1 cos(w.) cos(w^) 1.1 I 1.3 ± 0.4 1.4 ± 0.2 109a amplitude A^. T h i s i l l u s t r a t e s the p o i n t we have p r e v i o u s l y s t a t e d i n Chapter I I , namely t h a t i t i s d i f f i c u l t t o o b t a i n r e l i a b l e r e s u l t s when measuring amplitudes a b s o l u t e l y . 2 However, i f our r a t i o a = A^/ ^ 3 ^ 3 ) ^ s u s e d t n e g c ~ v a l u e s can be c a l c u l a t e d and indeed those c a l c u l a t e d from P h i l l i p s 1 amplitudes are i n agreement w i t h our own. CHAPTER V g(£) VALUES FROM A BAND STRUCTURE MODEL 5.1 The H a m i l t o n i a n i n the Absence of a Magnetic F i e l d Anderson and Gold (1965) found t h a t they c o u l d d e s c r i b e the Fermi s u r f a c e of l e a d q u i t e a c c u r a t e l y i f they i n c l u d e d the e f f e c t of s p i n - o r b i t i n t e r a c t i o n i n an i n t e r -p o l a t i o n scheme, which used plane waves o r t h o g o n a l i z e d t o a tomic-core f u n c t i o n s ( r e f e r r e d to as OPWs).. T h e i r scheme i n v o l v e d a H a m i l t o n i a n which was made up of two p a r t s , a band Ha m i l t o n i a n H, p l u s a s p i n - o r b i t H a m i l t o n i a n H , b c * so . and i n c l u d e d f o u r a d j u s t a b l e parameters which were determined by o p t i m i z i n g the l e a s t - s q u a r e s f i t between the c r o s s -s e c t i o n a l a r e a s c a l c u l a t e d from t h i s model and the a c t u a l c r o s s - s e c t i o n a l areas as determined from e i g h t e x p e r i m e n t a l f r e q u e n c i e s . They used f o u r OPWs combined w i t h the two s p i n o r s |+> and [->, to o b t a i n e i g h t b a s i s v e c t o r s (product wave f u n c t i o n s ) , but because of Kramers' degeneracy, o n l y f o u r energy e i g e n v a l u e s E n ( ^ ) were found a t each p o i n t it. In t h i s work we removed t h i s degeneracy by i n c l u d i n g i n the H a m i l t o n i a n a.magnetic f i e l d c o n t r i b u t i o n H and ^ mag then determined g (k) from the r e s u l t a n t energy e i g e n v a l u e s E*(k) and E~ (k) . ( i n t h i s Chapter n i s the band index.) R e c e n t l y Anderson, O ' S u l l i v a n , and S c h i r b e r (1972) found t h a t the model o f Anderson and Gold d i d not c o r r e c t l y a c c o u n t f o r the observed changes i n the Fermi s u r f a c e as a f u n c t i o n o f e x t e r n a l p r e s s u r e and t h a t i n order t o a d e q u a t e l y d e s c r i b e t h e i r h i g h - p r e s s u r e r e s u l t s they needed t o use a n o n l o c a l - p s e u d o p o t e n t i a l model g i v e n by Animalu (1966). F o r t h i s reason we d e c i d e d t o c a l c u l a t e g (k) v a l u e s i n l e a d u s i n g Animalu's model and i t i s t h i s model which we w i l l now b r i e f l y o u t l i n e . Animalu used the model p o t e n t i a l o f Heine and Abarenkov (1964) , i n which the atomic p o t e n t i a l i s r e p l a c e d by a s u p e r p o s i t i o n of square w e l l s i n s i d e a core r e g i o n (r < R ) around an atom. The depths A 0 of these square w e l l s depend on the angular momentum s t a t e t, and were + 3 determined from the atomic spectrum of Pb and then a d j u s t e d s l i g h t l y by Anderson e t a l . i n order t o d e s c r i b e the observed F e r m i s u r f a c e of l e a d more a c c u r a t e l y . Outside t h i s c o r e r e g i o n (r > R^) the p o t e n t i a l i s s e t e q u a l to - Z / r , where Z = +4 f o r l e a d . The core p o t e n t i a l i n t h i s scheme i s g i v e n i n g e n e r a l by VHA " - E V * ' 5' 1- 1 where the a r e the p r o j e c t i o n o p e r a t o r s t h a t s e l e c t the component of the wave f u n c t i o n w i t h a n g u l a r momentum I. I f we now use the approximation o f Heine and Abarenkov (1964) and s e t A^ = A 2 f o r a l l I > 2, and use the f a c t t h a t EP^ = 1, then the p o t e n t i a l reduces t o VHA = " A2 " <VA2>P0 ~ ( A 1 " A 2 ) P l ( r < V | (r > R m) . 5.1.2 T h i s p o t e n t i a l as a p p l i e d to Pb by Anderson e t a l . i s shown i n F i g u r e (5.1). A nimalu and Heine (1965) have shown t h a t i t i s n e c e s s a r y t o add to the p o t e n t i a l V the terms V , V , and V q i n o r d e r t o take i n t o account c o r r e l a t i o n , exchange, and o r t h o g o n a l i z a t i o n , r e s p e c t i v e l y . The r e s u l t i n g t o t a l p o t e n t i a l g i v e s r i s e t o i n d i v i d u a l m a t r i x elements ( l a t t i c e -kk' p o t e n t i a l form f a c t o r s ) i n the H a m i l t o n i a n which a r e o f t h e form v k k - < k ' ] v H A | k > + v c c + v e x + v p b e<£ - P) where k and k 1 d e s c r i b e the i n i t i a l and f i n a l s t a t e s , and where i t has been assumed t h a t V , V and V a r e indepen-c c ' ex o * dent of k. A l t h o u g h Anderson e t a l . e s t i m a t e the s i z e of t h e s e added p o t e n t i a l terras, the accuracy of the e s t i m a t e i s not a l l t h a t important as any e r r o r s can be absorbed i n t o the p o t e n t i a l w e l l depths A^ i n the a c t u a l e(k-k') i s the H a r t r e e d i e l e c t r i c f u n c t i o n . 113 F i g u r e 5.1 Model p o t e n t i a l . T h i s i l l u s t r a t e s the model p o t e n t i a l o f Heine and Abarenkov as a p p l i e d t o l e a d by Anderson e t a l . (1972). 114 f i t t i n g p rocedure. In o r d e r to d e s c r i b e the e f f e c t of s p i n - o r b i t i n t e r a c t i o n , Animalu (1966) added.a s p i n - o r b i t c o u p l i n g p o t e n t i a l V" s o which i s g i v e n by V = ZX„s*t P„ r < R so 1 1 m = 0 r > R , 5.1.4 m ' where s i s the s p i n o p e r a t o r , ~l i s the angular momentum ope r a t o r and the c o e f f i c i e n t s X^ measure the s t r e n g t h of the s p i n - o r b i t i n t e r a c t i o n f o r the v a r i o u s angular momenta Z . By analogy w i t h the approximation used f o r the w e l l depths A^, Animalu s e t X^ = ^ f ° r a l l I > 2, which r e s u l t s i n V s o = ( X 1 ~ V P l + X2 r < R m = 0 r > R , 5.1.5 m Anderson e t a l . s i m p l i f i e d the problem s t i l l f u r t h e r by s e t t i n g X^ = 0. Thus i n t h i s model the s p i n - o r b i t i n t e r a c t i o n i s d e r i v e d w h o l l y from the i o n - c o r e r e g i o n , kk1 and o n l y from the p s t a t e s I = 1. The form f a c t o r s V J c so d e r i v e d from the s p i n - o r b i t p o t e n t i a l Eq. (5.1.5) are shown by Animalu t o be g i v e n by 115 V kk' so = iX,G kxk' 1 1 k T •<V s v> 5.1.6 where v and v' are the s p i n o r s |+> or |-> f o r the i n i t i a l and f i n a l s t a t e s . i s determined by performing the r a d i a l i n t e g r a t i o n 12TT f Rm . Ck'r)j- 1(kr)r , 6dr , 5.1.7 where Si i s the volume o f the u n i t c e l l and i s the s p h e r i c a l B e s s e l f u n c t i o n o f o r d e r 1. We can now e x p l i c i t l y w r i t e out the H a m i l t o n i a n m a t r i x f o r e i g h t b a s i s v e c t o r s o b t a i n e d by combining f o u r OPWs w i t h the s p i n o r s |+> and |-> . The H a m i l t o n i a n i s b so 5.1.8 where r T 1 1 v ? ; 2 + h 1 2 b so v 1 3 + h 1 3 so so v j 4 + h b _ h s o T2 2 23 23 b so 24 h s o v 2 3 - h 2 3 b so T 3 3 v 3 4 + h b - h 1 4 . so v 2 4 - h 2 4 b so v 3 4 - h 3 4 b so T44 so 24 so 34 so 5.1.9 and -q -q .12 so 13 so 14 'so 12 fs0 -q .23 so 24 [so 13 ^ o 23 'so -q .34 so .14 so 24 so 34 so \ J 116 5.1.10 The s p i n - o r b i t components V k.k. so have been broken i n t o the two q u a n t i t i e s h1-^ and q1-', which r e s u l t from p a r a l l e l and ^ so ^so a n t i p a r a l l e l s p i n o r s , r e s p e c t i v e l y , and the d i a g o n a l elements ,11 a r e the k i n e t i c energy terms which are g i v e n by ,11 £1 k2 2m i 5.1.11 5.2 The Magnetic F i e l d C o n t r i b u t i o n t o the H a m i l t o n i a n In o r d e r t o i n v e s t i g a t e the e f f e c t o f a magnetic f i e l d on the energy l e v e l s , the u s u a l magnetic c o n t r i b u t i o n "mag = ^B S'& + 2*> 5.2.1 was added to the H a m i l t o n i a n ; here y n i s the Bohr magneton a ehV(2m). Each of the f o u r energy e i g e n v a l u e s E^ (it) f o r a 117 g i v e n band n, were i n g e n e r a l s p l i t i n t o the two e i g e n v a l u e s E (k) and E ( k ) , and the e f f e c t i v e g-value f o r the s t a t e k n n was then found from E + ( k ) - E" (k) g (k) = L i m ^ g { - 2 * } . 5.2.2 The c a l c u l a t i o n o f the form f a c t o r s V k k ' = <k'v'\H |kv> , 5.2.3 mag 1 mag' a r i s i n g from ^mag was s i m p l i f i e d by making use o f two assumptions which have a l r e a d y been used when s e t t i n g up W , namely t h a t s p i n - o r b i t i n t e r a c t i o n was c o n s i d e r e d o n l y f o r p s t a t e s and even f o r these p s t a t e s the s p i n - o r b i t i n t e r a c t i o n was taken t o be zero o u t s i d e the core r e g i o n . Atomic g-values were then s u b s t i t u t e d i n t o the e x p r e s s i o n f o r the form f a c t o r s i n the c o r e r e g i o n ; t h i s i s analagous t o the procedure used to determine the s q u a r e - w e l l depths A^. The magnetic form f a c t o r s can then be w r i t t e n out Vmg = < k ' V ' l y B 5 ' C^ + 2s> l k v > = y 1 DH<k lv 1 |g TH«j|kv> (r<R , 1=1) a j m UgRXk'v*|H»(1 + 2s)|kv> ( r > R m ' l = 1 ) u BH<k fv'|H-(1 + 2s)|kv> Call r , tyl) 5.2.4 118 where H i s the u n i t v e c t o r a l o n g H ( d e f i n e d to be the z a x i s ) and g j i s the Lande atomic g-value which i s equal t o 2/3 f o r J = 1/2 and i s equal t o 4/3 f o r J = 3/2. Sub-s t i t u t i n g these g - v a l u e s i n t o Eq. (5.2.4) and a f t e r r e a r r a n g i n g the terms the form f a c t o r s can be r e w r i t t e n i n the form Vmag = ^ B H < k ' V ' J F * s ' k V > ( r < Rm' J = 1 / 2 ) +y BH<k'v' IjH«t|kv> (r<R m, J=3/2) -y BH<k'v 1 ||H«£|kv> Cr>R m, £=1) +y BH<k'v'|H« ( t + 2s) |kv> ( a l l I, a l l r) . 5.2.5 In order t o c a l c u l a t e the v a r i o u s m a t r i x elements i n Eq. (5.2.5), the no r m a l i z e d p l a n e wave |k> was expressed i n s p h e r i c a l c o o r d i n a t e s as |k> = 1 ,v e x p ( i k r c o s 8) SI*-'* = '-J72- 2iJl(2£ + 1) j £ ( k r ) P £ ( c o s 8 ) , 5.2.6 where P^(cos 6) i s a Legendre p o l y n o m i a l . The a n g u l a r -momentum o p e r a t o r i s g i v e n i n s p h e r i c a l c o o r d i n a t e s by 119 1 = - i r x t G f j + J |_) . s.2.7 S i n c e it and s, but n o t J a r e the n a t u r a l quantum o p e r a t o r s i n Eq. (5.2.5), two more con v e n i e n t o p e r a t o r s were i n t r o d u c e d where _ &+l+2s'£ 2£+l f o r J = I + 1/2 . 5.2.8 These new o p e r a t o r s s e l e c t out the d e s i r e d angular-momentum components J = 1/2 and J = 3/2. We need not g i v e the i n d i v i d u a l s t e p s i n v o l v e d i n c a r r y i n g out the i n t e g r a t i o n s i n Eq. (5.2.5) s i n c e they can be found a t v a r i o u s p l a c e s i n Animalu's (1966) p^per on the f o r m a l i s m when tfmag = 0 . The magnetic c o n t r i -b u t i o n t o the form f a c t o r s was f i n a l l y found to be 120 rkk* mag g y B H G 1 U ( k k ^ - k yk x)6vv' - 8(k-k')<v•|s z|v> + 4<v'|-Vx8x- k z k y S y + < k x k x + k y k y > S z I V > y y + 4i<v' s„s (k k 1 - k k')- + s s (k k • -k k') 1 z x y z z y z y z x x z + s s (k k« - k k 1) v>} z z x y y x 1 + 2 y B H 6 k k , < v ' | s j v > 5.2.9 where i s given in Eq. (5.1.5) , k^, k^, and k z are the x , y, and z components, respectively of the unit vector k, and s x, s y r a n ^ s z a r e *-he Pauli spin matrices. In the presence of a magnetic f i e l d the total Hamiltonian i s f i n a l l y found to be / w , + ti. ti = ti. + ti + ti b so mag 1 ' "3 tit + ti] 2 4 H2 + H 4 where and ti^ have already been given in Eqs. (5.1.9) and (5.1.10), and where with and H 4 = y B H N N N 3 D _ 3-D _ 3 D _ 2 9 " i 1 42 i N 2 N 1 4 2 N f <2 N 2 3 2 N 2 4  W 2 W 2 N 3 2 33 Ni N 3 4  w 2 N 4 1  W 2 »« 22 N3 33 W 3 <2 44 N W 3 9 ^ 2- (k 9 l K i z k. + fr. jz a • k.)G 3 i<V k i ' * 9 D X k. - k As. . k. ) xy ix (kj2) H * j y - £ e jx> Gl <V k: (-k. k JZ . + k. ix jx k i z > G l ( k i ' , k # (k. 9 A\ AS. k. + k iy . k. ) jy " • \ 122 5.3 g(k) at Points of High Symmetry in the Bril l o u i n Zone The form factors VV , V , and V in Eqs. (5.1.3), b ' so ' mag ^ ' (5.1.6), and (5.2.9) are of course independent of the number of OPWs used in the calculation, whereas the corresponding Hamiltonian matrices have been e x p l i c i t l y written assuming a basis set of four OPWs. This number has been chosen for the following reason. Most of the third-zone Fermi surface of lead (the sheet to which our experimental observations refer) l i e s close to the corner W of the Br i l l o u i n zone; this surface can be thought of as the zone lines "fattened up". Consequently, i f the Fermi surface and band structure near the Fermi level are to be calculated with f a i r r e l i a b i l i t y i t i s necessary to use a model that predicts the energy splittings at the corner W reasonably accurately. Since at least four OPWs (eight product wave functions) are required to resolve the empty-lattice degeneracies at W, Anderson et a l . have assumed that this i s a sufficient number of OPWs for any point % in the B r i l l o u i n zone. However, i t should -be noted that i t i s possible to show from symmetry arguments that a set of four OPWs i s not an appropriate set for most of the other major symmetry points. Thus at the point Y in Figure (5.2), for. example, the smallest symmetrized basis sets consist of 1, 9, ••* , OPWs but not 4. Nevertheless, since the four-OPW model of Anderson et a l . 123 F i g u r e 5.2 P r i m i t i v e B r i l l o u i n Zone. The c o o r d i n a t e s f o r p o i n t s of h i g h symmetry a r e r = (o. ,o,o), w = ( i / 2 f o f i ) , x = ( o , i , o ) , K = (1/4,1/4,1) , U = (3/4,3/4,0) , and L = (1/2,1/2,1/2) i n u n i t s of (2u/a), where a i s t h e l a t t i c e c o n s t a n t . 124 d e s c r i b e s the o v e r a l l Fermi s u r f a c e r e a s o n a b l y a c c u r a t e l y , a t both normal and h i g h p r e s s u r e s , we have d e c i d e d to c a l c u l a t e g-values u s i n g t h e i r s e t of f o u r OPWs. The f o u r b a s i c OPWs a p p r o p r i a t e t o the zone corner W a t (1/2, 0, 1) (2Tr/a) have the k-components, k 1 = (1/2, 0 , l ) , k 2 = (1/2, 0, 1) , k 3 = (1/2, I, 0) , and k 4 = (1/2, 1, 0) , 5.2.10 i n u n i t s o f (2Tr/a) . Each of these plane waves was combined w i t h the s p i n o r s |+> amd |-> t o form e i g h t p r o d u c t wave f u n c t i o n s , and then used t o compute the i n d i v i d u a l m a t r i x elements i n Eq. (5.1.2), Eq. (5.1.4) and Eq. (5.2.9). As f a r as Eqs. (5.1.2) and (5.1.4) a r e concerned, we have used the w e l l - d e p t h s and s p i n - o r b i t parameter which Anderson e t a l , found t o g i v e the b e s t f i t t o the Fermi s u r f a c e , namely A Q = 3.93 r y , Aj^ = 4.20 r y , = 1.60 r y , and A. = 0.60 r y 125 The f i n a l H a m i l t o n i a n i n Eq. (5.2.9) was then d i a g o n a l i z e d by computer i n order to f i n d the e i g e n v a l u e s E n ( k ) and 9 n ( k ) was determined by s u b s t i t u t i n g these e i g e n v a l u e s i n t o Eq. (5.2.2) . In order t o t e s t the r e l i a b i l i t y of the c a l c u l a t i o n , g-values were f i r s t computed f o r the zone corner (1/2, 0, 1) (2ir/a) and examined as a f u n c t i o n o f f i e l d d i r e c t i o n . The f o u r g n(W) v a l u e s , where n l a b e l s the bands i n order of i n c r e a s i n g energy, a r e shown i n F i g u r e (5.3) as a f u n c t i o n o f a c r y s t a l r o t a t i o n about the z a x i s , i . e . , about the f i e l d d i r e c t i o n H. The g-values are expected to be indepen-dent of a r o t a t i o n about the z a x i s , as i s indeed found to be the case. The c r y s t a l was then r o t a t e d about the x a x i s (which i s e q u i v a l e n t to r o t a t i n g the f i e l d i n the yz p l a n e ) , and the g n(W) v a l u e s are shown as a f u n c t i o n of the angle of r o t a t i o n i n F i g u r e (5.4). A g a i n t h e r e i s no s i g n i f i c a n t dependence on the d i r e c t i o n o f the f i e l d . F i n a l l y the c r y s t a l was r o t a t e d about the y a x i s ( e q u i v a l e n t to r o t a t i n g the f i e l d i n the xz plane) and the c a l c u l a t e d g n(W) v a l u e s a r e shown i n F i g u r e (5.5), a g a i n as a f u n c t i o n of the angle of r o t a t i o n . For f i e l d r o t a t i o n s i n t h i s p l a n e , o n l y the v a l u e g^(W) remains c o n s t a n t , w h i l e g^(W) and g^(W) now depend v e r y s t r o n g l y upon the f i e l d d i r e c t i o n . In f a c t f o r a g e n e r a l f i e l d o r i e n t a t i o n g^(W) w i l l always equal 2, s i n c e the lowest (n=l) l e v e l a t W i s a pure s - s t a t e which i s 2-g,(W) g4(W) U ) - J < > I I-g9(W) 90' 180° ANGLE OF ROTATION F i g u r e 5.3 g (W) f o r f i e l d r o t a t i o n about z a x i s . The s u b s c r i p t s l a b e l the energy K bands i n o r d e r b e g i n n i n g a t the lowest band. 2 0 • * -1 1 | | / 0° 90° 180° ANGLE OF ROTATION Figure 5 . 4 g(W) for f i e l d rotation, about x axis. I . 1 2 9 u n a f f e c t e d by s p i n - o r b i t c o u p l i n g . On the o t h e r hand the l e v e l s n = 2, 3 and 4 a r e p - l i k e a t W and g i v e g-values which are extremely s e n s i t i v e t o the c h o i c e of f i e l d d i r e c t i o n ( i . e . , the a x i s of s p i n q u a n t i z a t i o n ) . In the f o r e g o i n g , our e x p l i c i t c a l c u l a t i o n s were f o r the p a r t i c u l a r p o i n t W = (1/2, 0, 1) (2Tr/a) . E q u i v a l e n t l y we may c o n s i d e r the f i e l d d i r e c t i o n f i x e d a t an a r b i t a r y o r i e n t a t i o n r e l a t i v e t o the c r y s t a l axes and ask how g n(W) depends on which of the 24 p o i n t s W i s b e i n g c o n s i d e r e d . With r e s p e c t to F i g u r e (5.6) the c o r n e r s W have been d i v i d e d ± ± ± i n t o s i x groups W&/ W^ , and W c,each of which c o n s i s t of f o u r c o r n e r s r e l a t e d t o one another by t r a n s l a t i o n s of r e c i p r o c a l l a t t i c e v e c t o r s . The f o u r c o r n e r s w i t h i n a group must a l l have the same g-values, which w i l l i n g e n e r a l be d i f f e r e n t from those i n the other f i v e groups (with the e x c e p t i o n of g^(W), which i s e q u a l to 2 ) . The dependence of the f a c t o r s g n(W) f o r the p - s t a t e s on the f i e l d o r i e n t a t i o n i s p r e c i s e l y analagous to the dependence on f i e l d o r i e n t a t i o n which i s found f o r the t r a n s i t i o n - m e t a l ferromagnets N i and Fe when s p i n - o r b i t c o u p l i n g i s c o n s i d e r e d (Hodges, Stone, Gold (1967), Gold (1968)). In these metals i t i s the d - s t a t e s which are a f f e c t e d , and because the s a t u r a t i o n m a g n e t i z a t i o n (which a r i s e s from a l a r g e s p l i t t i n g (y 1 eV) between the u p - s p i n and down-spin energies) i s " b u i l t i n " and cannot be e x t r a p o l a t e d t o zero, the a c t u a l band s t r u c t u r e E (k") and shape of the c F i g u r e 5 . 6 E q u i v a l e n t c o r n e r s o f the B r i l l o u i n zone. With the f i e l d i n a a r b i t r a r y d i r e c t i o n o n l y t h e W p o i n t s r e l a t e d t o one another by t r a n s l a -t i o n s o f r e c i p r o c a l l a t t i c e v e c t o r s are - e q u i v a l e n t 131 Fermi s u r f a c e depend on the m a g n e t i z a t i o n d i r e c t i o n . In l e a d , on the othe r hand, o n l y the g - f a c t o r s depend on the f i e l d o r i e n t a t i o n , whereas the band s t r u c t u r e E (it) does n o t — a t l e a s t i n the modest f i e l d s which the e x p e r i m e n t a l i s t has a t h i s d i s p o s a l . As s t a t e d p r e v i o u s l y , symmetry r e q u i r e s t h a t a t l e a s t f o u r OPWs be used a t W but a d i f f e r e n t number of OPWs i s r e a l l y r e q u i r e d f o r each o f the ot h e r major symmetry p o i n t s . The model o f Anderson e t a l . had been a d j u s t e d t o f i t the Fermi s u r f a c e of l e a d u s i n g o n l y f o u r OPWs for* a l l ' k and by our use of t h i s model as a b a s i s f o r our c a l c u l a t i o n s we have r e s t r i c t e d o u r s e l v e s to t h i s incomplete s e t of b a s i s v e c t o r s . I n order t o see whether t h i s incompleteness i s s e r i o u s a t p o i n t s away from W, we have computed the 9 n(X) v a l u e s f o r the p o i n t X = (0, 1, 0) (2ir/a) (see F i g u r e (5.2)) as a f u n c t i o n of a c r y s t a l r o t a t i o n about the y a x i s ( i . e . , e q u i v a l e n t t o a f i e l d r o t a t i o n i n the xz p l a n e ) . The 9 n(X) v a l u e s a r e shown i n F i g u r e (5.7) as a f u n c t i o n o f the angle o f r o t a t i o n . Symmetry o b v i o u s l y r e q u i r e s t h a t the 9 n(X) v a l u e s a t o ° , 90°, and 180° be i d e n t i c a l whereas t h i s i s c l e a r l y n o t the case. T h i s i n c o r r e c t angular dependence i s s u r e l y a consequence of the f a c t t h a t an incomplete s e t o f b a s i s v e c t o r s was b e i n g used t o c a l c u l a t e g n ( X ) . Symmetrized s e t s a t X c o n s i s t o f 2, 6, ••• OPWs, but not 4. Figure 5.7 g(X) for f i e l d rotation about y axis. An unsymmetrized set of basis vectors was used to calculate these g-values. H CO to 133 5.4 g (3c) on the Extremal O r b i t s F i n a l l y we computed g (k) v a l u e s f o r the two dHvA o r b i t s a p p r o p r i a t e to our experimental g r e s u l t s o f c C h a p t e r IV, i . e . , the v o r b i t f o r the f i e l d p a r a l l e l t o [001] and t h e r, o r b i t w i t h the f i e l d p a r a l l e l t o [101] , both o f which l i e on the t h i r d - z o n e s u r f a c e which i s d e r i v e d from the band n = 3; see F i g u r e (4.1). The v o r b i t f o r H p a r a l l e l to i n F i g u r e (5.6) i s c e n t e r e d about the p o i n t s W^ . In F i g u r e (5.8) we have shown the shape o f t h i s o r b i t as computed from the band s t r u c t u r e programme of Anderson e t a l . and we have c a l c u l a t e d the g-values f o r the two p o i n t s A and B shown i n the f i g u r e to be g 3 ( k A ) = 1.72 and g 3 ( k B ) = 1.62; (the p o i n t s A and B a r e , r e s p e c t i v e l y , the c l o s e s t and most ± -*• d i s t a n t p o i n t s from the o r b i t c e n t r e W^). Thus g 3 ( k ) appears to be s l i g h t l y k-dependent. However, i n view o f our r e s u l t s f o r the symmetry p o i n t X, we must conclude t h a t t h i s k-dependence c o u l d p o s s i b l y r e s u l t from the f a c t t h a t an i n a p p r o p r i a t e s e t of b a s i s v e c t o r s was used. In f a c t p o i n t B i s c l o s e r by 28% t o the n e a r e s t symmetry p o i n t K (not i n the p l a n e of the o r b i t ) than i t i s to the o r b i t c e n t r e W, 134 4 — 0-411 (2 r r /a ) — p F i g u r e 5.8 The shape o f the v o r b i t . P o i n t A has c o o r d i n a t e s (0.832 7 0.168, 0.5) (2ir/a) . P o i n t B has c o o r d i n a t e s CO.589, 0, 0.5) C2ir/ai. 135 and f o r K the symmetrized s e t s c o n s i s t o f 3, 5, ••• OPWs whereas we have used f o u r OPWs throughout. We t u r n now to the C o r b i t which e n c i r c l e s the p o i n t s K and U of F i g u r e C5.2). In F i g u r e (5.9) we have shown the shape o f t h i s o r b i t and the g-values found f o r the p o i n t s C, D, and E were g 3 ( k c ) = 1.92 , g 3(*D) = I-* 2 " . ' and g 3 ( k E ) = 1.91 . Thus g 3 ( k ) appears t o v a r y v e r y l i t t l e around the £ o r b i t . However, a g a i n we must be somewhat c a u t i o u s about the i n t e r p r e t a t i o n o f these v a l u e s , s i n c e f o r t h i s o r b i t i t would have been more a p p r o p r i a t e t o use a s e t o f b a s i s v e c t o r s symmetrized f o r K, r a t h e r than the s e t f o r W. I f a symmetrized s e t o f b a s i s v e c t o r s c o u l d have been used and i f t h e r e s t i l l were a k-dependence f o r the g ( 5 ) v a l u e s around an o r b i t , then i t would be ne c e s s a r y t o c a l c u l a t e the a p p r o p r i a t e l y - a v e r a g e d g-value, g c , i n or d e r t o make a comparison w i t h the exp e r i m e n t a l i n f o r m a t i o n . Holtham (-1973) has examined t h i s a v e r a g i n g problem and has found t h a t the proper average i s the time-weighted average 136 c « — O330(2TT/a) — • [010] F i g u r e 5.9 The shape o f the C o r b i t . P o i n t s C , D, and E have c o o r d i n a t e s (0.543, 0, 0 . 543) , . (0.820, -0.127, 0.820), and (0.845, 0, 0.845) i n u n i t s o f (2-rr/a) , r e s p e c t i v e l y . 137 gCk)dkt 9 ~ v± (k) g c = ' 5.4.1 dk, v ± (k) t where (k) i s the component o f the v e l o c i t y normal to the magnetic f i e l d , k f c i s the component of k t a n g e n t i a l t o the o r b i t and the l i n e i n t e g r a l s a r e e v a l u a t e d around the dHvA o r b i t . The parameters k t , v ( k ) , and g (k) can a l l be c a l c u l a t e d from the p s e u d o p o t e n t i a l model, but d o i n g so does n o t seem worthwhile f o r the p r e s e n t model w i t h i t s s y m m e t r i z a t i o n d i f f i c u l t i e s . We conclude t h i s c h apter by n o t i n g a p o s s i b l e consequence o f our c a l c u l a t i o n s to some r e c e n t computations of Randies (1972). Randies has c o n s t r u c t e d an a n a l y t i c a l r e p r e s e n t a t i o n f o r g Ck) i n copper which p e r m i t t e d him to i n v e r t the i n t e g r a l e q u a t i o n (5.4.1) and hence determine g(k) from the g - v a l u e s which he had measured u s i n g the dHvA e f f e c t f o r a v a r i e t y of f i e l d d i r e c t i o n s H. H i s p a r a m e t e r i z e d scheme i s based on the assumption t h a t g (ic) i s a s c a l a r q u a n t i t y which depends o n l y on the v e c t o r k and n o t on the f i e l d d i r e c t i o n H. In view o f our r e s u l t s f o r the zone c o r n e r s W (where a p r o p e r l y symmetrized s e t was used), we b e l i e v e t h a t Randies' i n v e r s i o n procedure i s i n v a l i d s i n c e i n r e a l i t y we know t h a t g(k, H) Lim ' { -H •+ 0 y B V H E ( k , H) CHAPTER VI CONCLUSION 6.1 G e n e r a l C o n c l u s i o n s and Comparison of Theory w i t h  Experiment The s i g n i f i c a n t r e s u l t s of t h i s t h e s i s can be c l a s s e d i n t o the f o l l o w i n g f o u r d i v i s i o n s : ( i ) A new approach has been developed f o r e x t r a c t i n g g - v a l u e s from dHvA data by measuring the t h r e e harmonic amplitudes A^, A^, and A^ and then by examining the dimen-2 s i o n l e s s r a t i o a' = A^ / (AjA^1) (see Eq. (2.1.20)). ( i i i T h i s approach ( i ) has been used t o determine g -P v a l u e s f o r two e x t r e m a l areas of the Fermi s u r f a c e i n l e a d u s i n g a s e n s i t i v e m a g n e t o r e s i s t i v e - p r o b e t e c h n i q u e to measure the dHvA amplitudes i n t h i n d i s k samples. ( i i i ) A p s e u d o p o t e n t i a l model has been augmented (see Eq. (5.2.9)) t o i n c l u d e the e f f e c t o f a magnetic f i e l d and g(k) v a l u e s have been computed from the r e s u l t i n g energy band s p l i t t i n g . ( i v ) The computed g (ic) v a l u e s were found t o depend on f i e l d d i r e c t i o n and 'as a r e s u l t we b e l i e v e the i n v e r s i o n scheme (to c a l c u l a t e g (Ic) from e x p e r i m e n t a l g - v a l u e s ) of Randies (1972) i s i n v a l i d . We have a l r e a d y e x p r e s s e d c a u t i o n about the r e l i a b i l i t y of our computed g (lc) v a l u e s . N e v e r t h e l e s s , i t i s of i n t e r e s t 140 t o compare these v a l u e s w i t h our e x p e r i m e n t a l v a l u e s . I f we assume the g - v a l u e s c l o s e s t t o the f r e e e l e c t r o n v a l u e s 3 c are the c o r r e c t ones, then we f i n d f o r the g o s c i l l a t i o n s g c , e x p 1 * 3 1 g c , c o m p u t e d ' and f o r the y o s c i l l a t i o n s g = 1.29g . , ^c,exp Recomputed As we have noted i n Chapter I, the dHvA g -value i s m o d i f i e d c by many body e f f e c t s . However, many body e f f e c t s have not been i n c l u d e d i n the computed g c ~ v a l u e s . I f we assume a formula s i m i l a r t o Eq. (1.1.1) i s a p p l i c a b l e t o l e a d , we f i n d B = - 0.23 ± 0.05 o The c o e f f i c i e n t B has been both c a l c u l a t e d and measured i n o potassium (see Randies (1972)) and i t has been found t h a t - 0.30 < B < - 0.21 — o — Thus i t appears t h a t a comparison between e x p e r i m e n t a l and computed qQ-values may y i e l d i n f o r m a t i o n about the magnitude of the Landau c o e f f i c i e n t B . I t would be of c o n s i d e r a b l e o 141 i n t e r e s t i f B q can i n f a c t be e s t i m a t e d by t h i s procedure as t h e c a l c u l a t i o n of B q from f i r s t p r i n c i p l e s i s not an easy problem. 6 . 2 S u g g e s t i o n s f o r F u r t h e r Study I t would be o f i n t e r e s t t o map out g - v a l u e s over c the e n t i r e F e r m i s u r f a c e i n o r d e r t o determine j u s t how a n i s o t r o p i c t h e g - v a l u e s i n l e a d a r e . T h i s s h o u l d not be d i f f i c u l t u s i n g our approach f o r g - v a l u e e x t r a c t i o n . c However, many o f our e r r o r s and d i f f i c u l t i e s a r o s e 'from the f a c t t h a t t h e experiment was not performed on the most s u i t a b l e a p p a r a t u s . I t would have been to our advantage t o d e c r e a s e the s t r e n g t h of the magnetic i n t e r -a c t i o n by g o i n g t o h i g h e r f i e l d s r a t h e r than using, d i s k samples and t r y i n g to. cope w i t h the r e d u c t i o n i n s i g n a l s t r e n g t h . F o r t h i s r e a s on we suggest t h a t any f u r t h e r study s h o u l d i n v o l v e the use o f a s u p e r c o n d u c t i n g magnet. I t would a l s o be of i n t e r e s t t o determine whether ^ c , exp ~ ^ ' ^ c , computed ' h o l d s f o r a l l o r b i t s i n l e a d . I n o r d e r t o c a l c u l a t e the r e q u i r e d g _ - v a l u e s f o r t h i s comparison, i t w i l l be n e c e s s a r y t o p r o p e r l y symmetrize the band s t r u c t u r e model t h a t we have o u t l i n e d i n C h a p t e r V. There a r e s e v e r a l approaches t h a t can be u s e d . One approach i s t o weight the b a s i s v e c t o r s with a suitably chosen function that gives the properly symmetrized set at each symmetry point. An alternative approach i s to use a sufficiently large set (in principl an i n f i n i t e set but in practice about 60) basis vectors that symmetrization no longer i s a factor. +Ehrenreich and Hodges (1968) and Smith (1971). +See Van Dyke (.1973) . APPENDIX I AMPLITUDES AND PHASES FOR NONQUADRATIC EXPANSIONS OF CROSS-SECTIONAL AREAS There a r e a number o f a l t e r n a t i v e a p proximations f o r Eg. (2.1.14) f o r which Eq. (2.1.13) can be i n t e g r a t e d . C o n s i d e r A f - A e x 4 I a k H k H i b = 0 k H > b Then i n t e g r a t i n g Eq. (2.1.13) we f i n d and M r a ( C 2 ( z ) + S 2 ( z ) ) 1 / 2 s i n (2TTr | ± e r ) ' , 6 r = t a n 1 (|) , where C and S a r e the c o s i n e and s i n e F r e s n e l i n t e g r a l s , r e s p e c t i v e l y , and where and - v - e x 2ne F o r z » , we f i n d 144 and ,„2 . e 2 N l / 2 _^  0 - l / 2 r i . - 1 / 2 -1 • , 2 T K , (C + S ) ' - > 2 ' [1 + TT ' z s m ( z " j ) 1 / a ^ * ^ „~l/2 -1 • * 2 TT v 6 -r + TT ' Z S i n ( Z + -r) r 4 4 Hence both the amplitude and the phase have an o s c i l l a t o r y component. However, the envelope p a t t e r n of the amplitude i s not p e r i o d i c i n (1/H)for the s m a l l v a l u e s of z 2 o f Sec.4.2(d) Next, l e t us c o n s i d e r A f = A e x ± ^4 + 1A b kH> Now i n t e g r a t i n g Eq. (2.1.13) once a g a i n , we f i n d M r a J _ 1 / 4 ( z ' ) s i n ( 2 - r r r | ±z' ± J) ± J + 1 / 4 t z ' ) cos.C2Trr| ± z • + | l ) where z , 3 r f l a2 4bH While J _ i / 4 C Z ' ^ a n ( ^ J + l / 4 ^ 2 ' ^ a r e k o t h o s c i l l a t o r y , a computer study o f the a c t u a l "amplitude d i d not r e v e a l a b e a t i n g p a t t e r n . In the l i m i t a -*• 0, we f i n d , the phase tends t o TT ± g- . Hence, m g e n e r a l the phase 6 of a dHvA o s c i l l a t i o n need not be TT/4 and f o r t h i s reason we have attempted to develop the formulae i n t h i s t h e s i s f o r a g e n e r a l 6 . REFERENCES Anderson,J.R., Gold, A.V., Phys. Rev. 139, A1459 (1965). Anderson,J.R., O ' S u l l i v a n , W.J., S c h i r b e r , J.E., Phys. Rev. B 5, 4683 (1972). Animalu, A.O.E., Heine, V., P h i l . Mag. 12_, 1249 (1965). Animalu, A.O.E., P h i l . Mag. 13_, 53 (1966). B o y l e , W.S., Hsu, F.S.L., K u n z l e r , J.E., Phys. Rev. L e t t e r s 4_, 278 (1960) . 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