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Temperature dependence of the exchange splitting in ferromagnetic metals Lonzarich, Gilbert G. 1973

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TEMPERATURE DEPENDENCE OF THE EXCHANGE SPLITTING IN FERROMAGNETIC METALS by GILBERT G. LONZARICH M.S., U n i v e r s i t y of Minnesota, 1970 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November, 1973 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver 8, Canada Date i i ABSTRACT The temperature dependence of the exchange s p l i t t i n g of the energy bands i n ferromagnetic i r o n has been studied at low temperatures by c a r e f u l l y examining the de Haas-van Alphen frequency associated with the minority-spin electron 'lens' sheets of the Fermi surface as a function of temperature between 1° and 4°K. A high r e s o l u t i o n phase measurement technique revealed that the v a r i a t i o n of the lens frequency over t h i s temperature range was l e s s than one part i n 10^ and was v i r -t u a l l y i d e n t i c a l to that measured for the corresponding e l e c t r o n lens 4 i n molybdenum. By contrast, a v a r i a t i o n of one part i n 10 would be expected f o r the i r o n lens on the basis of a l i t e r a l i n t e r p r e t a t i o n of the Stoner model, i n which the exchange s p l i t t i n g of the energy bands i s proportional to the magnetization at a l l temperatures. The absence of any s i g n i f i c a n t change of the frequency with temperature gives strong evidence that the magnetization i n i r o n decreases almost e n t i r e l y by spin-wave e x c i t a t i o n s and that spin waves have n e g l i g i b l e e f f e c t on the exchange s p l i t t i n g . The l a t t e r conclusion i s consistent with a recent theory by Edwards for the electron-magnon i n t e r a c t i o n i n i t i n e r a n t - e l e c t r o n ferromagnets. On the basis of Edwards' theory the experimental technique described i n t h i s d i s s e r t a t i o n can be used to systematically study the s i n g l e -p a r t i c l e magnetization i n m e t a l l i c ferromagnets without any interference from spin-wave e x c i t a t i o n s . In p a r t i c u l a r our experimental r e s u l t s y i e l d an upper bound for the s i n g l e - p a r t i c l e magnetization i n i r o n at low temperatures. A new set of low-frequency de Haas-van Alphen o s c i l l a t i o n s has been studied i n i r o n and the experimental r e s u l t s obtained so f a r suggest the o s c i l l a t i o n s o r i g i n a t e from very small e l l i p s o i d a l surfaces centered on the points N of the B r i l l o u i n zone. This r e s u l t points to a p a r t i c u l a r ordering of the energy bands at N, a feature of the band structure that cannot be predicted r e l i a b l y from f i r s t p r i n c i p l e s . TABLE OF CONTENTS i v Page ABSTRACT i i TABLE OF CONTENTS i v LIST OF FIGURES AND TABLES v i ACKNOWLEDGMENTS v i i i Chapter I. INTRODUCTION . 1 §1.1 The Nature of the Ferromagnetism i n 3 d - T r a n s i t i o n Metals . 1 §1.2 An O u t l i n e of the Present I n v e s t i g a t i o n 11 I I . THEORETICAL BACKGROUND 13 §2.1 The de Haas-van Alphen E f f e c t as a Tool to Study the Temperature Dependence of the Exchange S p l i t t i n g 13 §2.2 The Temperature Dependence of the Exchange S p l i t -t i n g and of the de Haas-van Alphen Frequency . . . 23 I I I . EXPERIMENTAL PROCEDURE 30 §3.1 I n t r o d u c t i o n . 30 §3.2 Sample P r e p a r a t i o n 32 §3.2a The Growth of Iron Whiskers 32 §3.2b The T e s t i n g of Iron Whiskers 34 §3.2c E l e c t r o p o l i s h i n g Procedure 38 §3.3 Magnetic F i e l d . . . . . . . . 39 §3.4 Cr y o s t a t and Sample Rotator 41 §3.5 Sample O r i e n t a t i o n 46 V Chapter Page §3.6 Modulation F i e l d Technique and Electronics . . . . 48 §3.7 Numerical Analysis 52 IV. RESULTS AND THEIR INTERPRETATION 56 §4.1 Experimental Results 56 §4.2 Theoretical Interpretation 62 §4.2a Ferromagnetism i n a Narrow Band 65 §4.2b General Electron-Electron and Electron-Magnon Interaction Functions 78 §4.2c Physical Interpretation . . . 84 §4.3 Discussion of Other Effects 87 V. ADDITIONAL INVESTIGATIONS. . 92 §5.1 Observation of a New Low-Frequency de Haas-van Alphen Term i n Iron: the v - o s c i l l a t i o n s . . . 92 §5.2 Some Cyclotron E f f e c t i v e Masses. 105 §5.3 Quantum O s c i l l a t i o n s i n the Spin Magnetization . . 108 VI. CONCLUSIONS 113 §6.1 The Main Results of the Present Investigation. . . 113 §6.2 Suggestions for Further Study 115 APPENDIX I: The Dependence of the Extremal Cross-Sectional Areas of the Fermi Surface on the F i e l d B_ . . . . . 118 APPENDIX I I : Effects that Give Rise to Small S h i f t s i n the 1 de Haas-van Alphen Phase with Temperature . . . . . 123 REFERENCES 134 LIST OF FIGURES AND TABLES v i Figure Page 1.1 The energy bands for iron based on the APW calculation of Wood (1962). 4 1.2 The majority (t) spin and minority (4-) spin Fermi sur-faces for iron [A.V. Gold, J. Low Temp. Phys. (in press)] 7 2.1 Orbit quantization in a magnetic f i e l d for a free-electron gas 15 2.2 Schematic sketch of the energy bands in iron along the TH line (based on the analysis of the Fermi surface and the band structure given by Gold e_t a l . (1971) in which e - e = 2.0 eV) 19 T r 2.3 The temperature dependence of the magnetization for iron at low temperature 27 2.4 The energy bands for iron along the TH line in the neighborhood of the minority electron lens. . . . . . . . 29 3.1 Sketch of portions of the dHvA magnetization showing possible phase shift with temperature. 31 3.2 Figures (a) and (b) show the Landau domain structure in a [100] iron whisker in the absence, and in the presence of an external f i e l d 35 3.3 Diagram showing sample holder and the central part of the cryostat t a i l between the pole faces of the electro-magnet . 42 3.4 Diagram of the sample holder and t i l t mechanism used to make accurate in situ alignment of the sample i n the magnetic f i e l d 45 3.5 Block diagram of the basic electronics used to study the dHvA effect by means of the field-modulation tech-nique 50 V l l Figure Page 4.1 Temperature dependence of the [111] dHvA frequency f o r the electron lens . 57 4.2 F i g . (a) shows schematically a s p i n - f l i p or Stoner e x c i t a t i o n ; an e l e c t r o n i n state (k,-t-) i s t r a n s f e r r e d to a state (k+£,4-) 70 4.3 Magnetic e x c i t a t i o n spectrum for the Hubbard Hamiltonian i n the random phase approximation. . . . . . 72 5.1 Recorder tracings of the low-frequency o s c i l l a t i o n s i n i r o n whisker samples f o r the following o r i e n t a t i o n s : (a) 12° from [111] normal to the (110) plane, (b) [100] and (c) [111]. . . . . . . . . . . . . . . . . 93-95 5.2 The magnetic induction __ i n an e l l i p s o i d a l sample. . . . 98 5.3 Figure (a) shows pos s i b l e dHvA frequency branches i n the (110) plane expected f o r nearly e l l i p s o i d a l pockets along the TH l i n e of the B r i l l o u i n zone. . . . . 100 5.4 This f i g u r e gives a family of p l o t s of the r e l a t i v e amplitude of the v - o s c i l l a t i o n s f o r f i e l d d i r e c t i o n s i n the neighborhood of the [111] axis and i n the (110) plane. . 102 5.5 Cyclotron mass p l o t s f o r £ and v - o s c i l l a t i o n s at [111]. . 106 5.6 The density of states of a free e l e c t r o n gas i n a magnetic f i e l d corresponding to the Landau quantum tubes shown i n F i g . 2.1a 109 A l . l Figure (a) shows a r b i t r a r y + and 4- sheets of the Fermi surface before ( s o l i d l i n e ) and a f t e r (dashed l i n e ) the a p p l i c a t i o n of the f i e l d B_. . . . . . . . . . . 119 Table 5.1 Summary of the observed de Haas-van Alphen frequencies i n i r o n „ . 97 ACKNOWLEDGMENTS It i s a sincere pleasure to thank Professor A.V. Gold, under whose supervision t h i s work was c a r r i e d out, for h i s close and personal i n t e r e s t at a l l times and for much h e l p f u l advice and encouragement. I also wish to express my gratitude to Dr. P. Schmor f or assistance with the experimental apparatus and d a t a - c o l l e c t i o n system during the i n i t i a l stages of t h i s work, and to S. Feser, Dr. J . Carolan and Dr. P. Holtham for many stimulating discussions. A [111] molybdenum sample and a [110] i r o n whisker used i n a part of t h i s study were provided, r e s p e c t i v e l y , by J. Hoekstra & J.L. Stanford (Iowa State U n i v e r s i t y ) and R.V. Coleman (University of V i r g i n i a ) . Their contributions are s i n c e r e l y appreciated. I am e s p e c i a l l y indebted to Professor D.M. Edwards (Imperial College, London) f o r taking an i n t e r e s t i n t h i s problem and for very informative communications regarding the t h e o r e t i c a l i n t e r p r e t a t i o n of the experimental r e s u l t s . The research described i n t h i s d i s s e r t a t i o n was generously supported by a grant from the National Research Council and I am g r a t e f u l to the Uni v e r s i t y of B r i t i s h Columbia f o r f i n a n c i a l assistance i n the form of a Ki l l a m Graduate Fellowship. F i n a l l y i t i s a sincere pleasure to thank my wife Gerie f o r her help during various stages of t h i s study, for her s k i l l f u l preparation of the fi g u r e s and the typing of t h i s t h e s i s . Chapter I. INTRODUCTION la §1.1 The Nature of the Ferromagnetism i n 3d-Transition Metals In t h i s thesis we describe an experimental study i n i r o n which has been c a r r i e d out i n an attempt to shed more l i g h t on the nature of the ferromagnetism i n the 3 d - t r a n s i t i o n metals. We s h a l l begin with a b r i e f i n t r o d u c t i o n to the theory of ferromagnetism i n s o l i d s and then discuss the d i f f i c u l t i e s with the current theories which motivated the present i n v e s t i g a t i o n . It was recognized soon a f t e r the discovery of quantum mechanics h a l f a century ago that the ferromagnetic alignment of the spins of the va-lence electrons i n a s o l i d i s a consequence of the combined e f f e c t of the P a u l i exclusion p r i n c i p l e and the Coulomb i n t e r a c t i o n . On account of the exclusion p r i n c i p l e , electrons with p a r a l l e l spins avoid each other i n t h e i r motions more e f f e c t i v e l y than would p a i r s of electrons having a n t i -p a r a l l e l spins; consequently the t o t a l e l e c t r o n i c i n t e r a c t i o n energy of the s o l i d would be expected to be smallest when a l l the spins of the valence electrons are aligned. However t h i s ferromagnetic state i s only r a r e l y the preferred one because the remaining e l e c t r o n i c energy not counted above (eg. k i n e t i c energy) i s i n fa c t lowest when the spins are paired, as i n the paramagnetic s t a t e . Whether or not the ground state l i s ferromagnetic therefore depends on the r e l a t i v e importance of the Coulomb and k i n e t i c energies mentioned above. The search for a general q u a n t i t a t i v e theory of ferromagnetism has been hindered p r i m a r i l y by the w e l l known d i f f i c u l t y of t r e a t i n g Coulomb 'c o r r e l a t i o n s ' , a r i s i n g from the tendency of electrons (of both spins) to avoid each other i n t h e i r motion. Progress has been made by consid-ering separately the two l i m i t i n g cases i n which c o r r e l a t i o n e f f e c t s are 2 t r a c t a b l e . In the Heisenberg theory ( c f . M a t t i s 1965) the magnetic e l e c -trons are assumed to be l o c a l i z e d around the i n d i v i d u a l l a t t i c e s i t e s i n the c r y s t a l and at l e a s t some c o r r e l a t i o n e f f e c t s are contained essen-t i a l l y i n the assumption of l o c a l i z a t i o n . On the other hand i n the i t i n e r a n t - e l e c t r o n theory ( c f . Stoner 1938, Wohlfarth 1953, Edwards and Wohlfarth 1968) the magnetic e l e c t r o n s are f u l l y i t i n e r a n t and c o r r e -l a t i o n s are expected to p l a y a secondary r o l e owing to the h i g h k i n e t i c energy of the e l e c t r o n s . The d i f f i c u l t i n t ermediate case where the magnetic e l e c t r o n s are i t i n e r a n t and c o r r e l a t i o n s are important i s s t i l l an unsolved problem but a great d e a l of progress i n t h i s area has been made i n recent years ( c f . Hubbard 1963, Kanamori 1963, H e r r i n g 1966, Wang, Evenson and S c h r i e f f e r 1969, Murata and Doniach 1972, Edwards 1973, 1973a). The Heisenberg model has been used w i t h great success to account fo r the ferromagnetism i n r a r e e a r t h metals and i n s u l a t o r s where the magnetic e l e c t r o n s are known to be w e l l l o c a l i z e d . On the other hand i n the 3d-ferromagnets Fe, Co and N i as w e l l as i n weakly ferromagnetic compounds such as N i - A l , ZrZn- and Sn.In the s i t u a t i o n i s s t i l l uncer-t a i n . I t i s now w e l l known that at l e a s t some of the magnetic e l e c t r o n s i n these metals are i t i n e r a n t ( c f . H e r r i n g 1966) and the i t i n e r a n t -e l e c t r o n theory mentioned e a r l i e r has i n f a c t been able to account f o r a number of p r o p e r t i e s observed thus f a r i n these ferromagnets; f o r example, the non-integer number of e l e c t r o n s per atom r e q u i r e d to g i v e the observed magnetic moment and the d e t a i l e d f e a t u r e s of the Fermi su r f a c e . Most of the m e t a l l i c ferromagnets however a l s o e x h i b i t proper-t i e s such as s p i n waves and l o c a l i z e d moments (formed by i t i n e r a n t e l e c -trons) which a r i s e from c o r r e l a t i o n e f f e c t s ignored i n the i t i n e r a n t -3 e l e c t r o n theo r y . These and other p r o p e r t i e s t h e r e f o r e r e q u i r e a s t rong c o r r e l a t i o n theory f o r t h e i r i n t e r p r e t a t i o n . A l though the i t i n e r a n t - e l e c t r o n theory i s not expected to g i ve a complete d e s c r i p t i o n of the 3d-ferromagnets i t does seem to serve as a reasonable s t a r t i n g p o i n t i n the d i s c u s s i o n of these m e t a l s . In t h i s t h e s i s the i t i n e r a n t - e l e c t r o n or Stoner theory w i l l be de f i ned as e s s e n -t i a l l y a Hart ree-Fock theory i n which the r e l e v a n t mo lecu l a r f i e l d p a r a -meter con ta ins c o r r e l a t i o n e f f e c t s as g i ven f o r example by Kanamori (1963). The t h e o r e t i c a l foundat ions of t h i s theory w i l l be rev iewed i n §4.2 and here we w i l l summarize on l y the main p o i n t s . I f we ignore the Coulomb c o r r e l a t i o n s between the conduc t ion e l e c t r o n s i n a meta l then the t o t a l Hami l ton i an reduces to a sum of one-e l ec t ron Hami l ton ians and the Schroedinger wave equa t ion f o r a s i n g l e e l e c t r o n can be w r i t t e n as t , 2 9 *nk " e n k ^nk > 1 ' 1 where V ( r ) i s some app rop r i a t e p o t e n t i a l which has the p e r i o d i c i t y of the l a t t i c e , are the one-e l ec t ron energy l e v e l s l a b e l l e d by the band index n and wavevector k which i s r e s t r i c t e d to the p r i m i t i v e B r i l l o u i n zone. Given V ( r ) i t i s p o s s i b l e to compute the energy l e v e l s e -_ by some app rop r i a t e app rox ima t i on . F i g . 1.1 shows the r e s u l t of such a c a l c u l a t i o n f o r bcc i r o n c a r r i e d out by Wood (1962) u s i ng the augmented-plane-wave (APW) method. A l though the band s t r u c t u r e i s q u i t e complex the main f ea tu re i s the set of narrow bands i n the cente r of the f i g u r e which o r i g i n a t e s p r i m a r i l y from the f i v e atomic 3d s t a t e s . In a d d i t i o n there i s a broad q u a s i - p a r a b o l i c band ( h y b r i d i z i n g w i t h the d-bands) o ro o l-h •TJ PJ 0. 0> Figure 1.1 The energy bands f o r i r o n based on the AFW c a l c u l a t i o n of Wood (1962). 5 which o r i g i n a t e s p r i m a r i l y from the 4s-atomic o r b i t a l s . For the noninteracting system of electrons at T = 0°K every energy l e v e l l y i n g below some energy e_ w i l l be occupied by two electrons of t . opposite spin and a l l l e v e l s above e w i l l be empty. The surface i n k r space with e . = __ i s the Fermi surface for the nth band and __ i s the r nk F — F Fermi energy. Iron has 8 electrons/atom outside the closed argon s h e l l and at T = 0°K these electrons would f i l l the one-electron bands up to the energy marked e_ i n F i g . 1.1. Iron i n t h i s non-interacting e l e c t r o n r l i m i t would of course be paramagnetic (or diamagnetic). When the e l e c t r o n - e l e c t r o n i n t e r a c t i o n i s properly taken i n t o account the ferromagnetic state may become e n e r g e t i c a l l y favorable, as discussed e a r l i e r . According to the Stoner theory each e l e c t r o n experiences an e f f e c t i v e 'magnetic' f i e l d (or molecular f i e l d ) a r i s i n g from i t s mean i n t e r a c t i o n with a l l other e l e c t r o n s . This f i e l d , which i s believed to be proportional to the magnetization, can remove the spin degeneracy of the bands and l i f t the 4- spin (minority) bands above the + spin (majority) bands i n the ground s t a t e . The energy separation between the bands i s c a l l e d the exchange s p l i t t i n g A, and within the framework of t h i s type of theory A i s proportional to the magnetization. In d e s c r i b i n g the f e r r o -magnetic state f o r t h i s model with r i g i d l y - s p l i t bands, i t i s convenient to use a s i n g l e set of bands f o r both the + and 4- spins and to draw i n separate Fermi l e v e l s e. and e. (measured r e l a t i v e to the bottom of the T 4 conduction bands) f o r the + and 4- e l e c t r o n s , r e s p e c t i v e l y . * The ferromag-n e t i c state f o r i r o n may then be described by the band structure i n F i g . 1.1 with the two Fermi l e v e l s e. and e, drawn i n s o l i d h o r i z o n t a l l i n e s . The T 4 p o s i t i o n i n g of the two l e v e l s i s consistent with the t o t a l number of 8 valence electrons per atom and the *In t h i s t hesis the + and 4- Fermi l e v e l s w i l l always r e f e r to the energies e + and measured r e l a t i v e to the bottom of the conduction bands, as defined i n the text. It should be emphasized, however, that ( i n equilibrium) both the + and 4 spin electrons on the Fermi surface have the same energy or chemical  p o t e n t i a l which could be defined as n = e_ - A/2 = e, + A/2 at T=0°K. 6 observed magneton number of 2.2 obtained from measurements near T = 0°K. In more sophisticated Hartree-Fock c a l c u l a t i o n s ( c f . Wakoh and Yamashita 1966, Duff and Das 1971 for i r o n ; Connolly 1967 f o r n i c k e l ) the exchange s p l i t t i n g A i s not s t r i c t l y constant as assumed above but v a r i e s somewhat with band index and wave vector k_. However the main features of the ferromagnetic band structure agree with those of the simple r i g i d - s p l i t t i n g model as far as the bands near the Fermi surface are concerned. In the Stoner model outlined above, the magnetic electrons are f u l l y i t i n e r a n t and give r i s e to a well-defined Fermi surface. Further-more since the exchange s p l i t t i n g i s large (A = 2eV i n Fe) the sheets of the Fermi surface for the + and 4- spin electrons are expected to be quite d i f f e r e n t . These p r e d i c t i o n s of the Stoner p i c t u r e have received rather s t r i k i n g support from recent Fermi surface studies i n both Fe and Ni (Fawcett and Reed 1963; Hodges et a l . 1967; T s u i 1967; Gold 1968, 1968a; Gold et a l . 1971; Baraff 1973). In p a r t i c u l a r the experimentally deter-mined features of the Fermi surface of iron (Gold et_ a l . 1971) agree rather c l o s e l y with the Fermi surface obtained by r i g i d l y s p l i t t i n g Wood's bands i n the manner described e a r l i e r (see F i g . 1.1). In F i g . 1.2 we show the various sheets of the Fermi surface for the t and 4- e l e c -trons i n i r o n which are consistent with the de Haas-van Alphen data of Gold et a l . (1971). The Fermi surface d i f f e r s only i n small d e t a i l s from that predicted by the i and 4- Fermi l e v e l s i n F i g . 1.1, Although the Fermi surface data obtained so far point to the correctness of the split-band model as a reasonable s t a r t i n g d e s c r i p t i o n for the ground state of the 3d-ferromagnets, the nature of the band structure as the temperature i s raised from the neighborhood of absolute o ro o l-h <-6 V> 09 0> Figure 1.2 The majority (+) spin and minority (4-) spin Fermi surfaces f o r i r o n [A.V. Gold, J. Low Temp. Phys. ( i n p r e s s ) ] . The surfaces centered at T or along the TH l i n e s enclose occupied e l e c t r o n i c states and those centered at the points H enclose empty states (at T = 0°K). The Fermi surfaces predicted f o r example by Wood's APW c a l -c u l a t i o n can be determined by drawing the calculated band structure along a large set of r a d i a l l i n e s beginning at V and terminating on the boundary of the B r i l l o u i n zone ( i n F i g . 1.1 only two s p e c i a l r a d i a l l i n e s are shown, TH and TN). The f (or 4) Fermi surface i s defined by those points i n k space where bands in t e r s e c t the + (or 4-) Fermi l e v e l . Fe: majority spin (t) Fe: minority spin ( . ) 8 zero to the Cu r i e p o i n t T_ and beyond i s not yet f u l l y understood. According to the Stoner theory, the energy s p l i t t i n g A i s p r o p o r t i o n a l to the magnetization and should t h e r e f o r e decrease to zero a t T . Th i s means that as the temperature i s r a i s e d above T = 0°K the p o s i t i o n s of the + and 4- Fermi energies i n F i g 1.1 would s h i f t down and up, r e s p e c t i v e l y , r e s u l t i n g i n changes i n the d e n s i t y of s t a t e s at the Fermi l e v e l s as w e l l as changes i n the dimensions of the Fermi s u r f a c e . However t h i s p i c t u r e might not be a r e a l i s t i c one because, as was pointed out e a r l i e r , the Stoner theory does not f u l l y take i n t o account the Coulomb c o r r e l a -t i o n s which are b e l i e v e d to be strong f o r the d - e l e c t r o n s i n the i r o n s e r i e s . In f a c t there i s some evidence ( c f . F r i e d e l et a l . 1961; Herr i n g 1966; Wang, Evenson and S c h r i e f f e r 1969) that these c o r r e l a t i o n s b r i n g about the formation of l o c a l i z e d moments which are ferromagneti-c a l l y ordered f a r below T_. The r e d u c t i o n i n magnetization i s then a consequence of the i n c r e a s i n g d i s o r d e r of the l o c a l i z e d moments (eg. by s p i n wave e x c i t a t i o n s a t low temperature), and i s i n c o n t r a s t to the simple Stoner p i c t u r e i n which A i s p r o p o r t i o n a l to the mag n e t i z a t i o n (Mott 1964, S l a t e r 1968, H a r r i s o n 1970). Presumably the r e a l s i t u a t i o n l i e s i n t e r m e d i a t e between these two extreme p o i n t s of view. A d e t a i l e d t h e o r e t i c a l d i s c u s s i o n (appropriate to low temperatures) of the Stoner model as w e l l as a model i n c l u d i n g more c o r r e l a t i o n e f f e c t s w i l l be given i n §4.2 i n the i n t e r p r e t a t i o n of the experimental r e s u l t s . The experimental evidence regarding p o s s i b l e energy s h i f t s i n the d e n s i t y of s t a t e s upon passing through T_ has been summarized and c a t e -g o r i z e d i n a recent review by Fadley and Wohlfarth (1972). Data from the MSssbauer e f f e c t , o p t i c a l a b s o r p t i o n , and the c h a r a c t e r i s t i c energy-l o s s spectrum, f o r example, have been i n t e r p r e t e d i n terms of the band 9 pi c t u r e i n which energy s h i f t s occur as i n the Stoner theory. On the other hand, data from u l t r a v i o l e t and x-ray photoelectron spectroscopy i n N i and Fe, r e s p e c t i v e l y , suggest energy s h i f t s which are considerably smaller than predicted by the Stoner model. In two recent papers (not included i n the Fadley-Wohlfarth review), Rowe and Tracy (1971) have argued that the absence of s h i f t s i n the photoemission spectra f o r Ni may be due to a f o r t u i t o u s c a n c e l l a t i o n of two competitive e f f e c t s , while McAlister et a l . (1972) have reported the observation of s t r u c -tures i n the soft x-ray emission spectra f o r Fe which do appear to s h i f t by about 0.5 eV on passing through T . More r e c e n t l y Mook, Lynn and Nicklow (1973) have reported that the sharp drop i n the spin-wave i n t e n s i t y i n n i c k e l at about 80 meV appears to be independent of the temperature over a wide range below and above T c. The sudden drop i n i n t e n s i t y i s believed to be the r e s u l t of the spin-wave e x c i t a t i o n crossing a continuum band of Stoner s i n g l e - p a r t i c l e e x c i t a t i o n s (defined i n §4.2a), which would be expected to s h i f t with temperature i f the band s p l i t t i n g were to vary i n a manner pr o p o r t i o n a l to the magnetization. Unfortunately i t would take us too f a r a f i e l d to give a more de t a i l e d discussion of the physics underlying each of the e f f e c t s mentioned above. However our b r i e f summary of the r e s u l t s makes i t cl e a r that the t h e o r e t i c a l and experimental s i t u a t i o n regarding the temperature dependence of the exchange s p l i t t i n g i s s t i l l very much uncertain. In order to shed more l i g h t on t h i s problem we have looked f o r e f f e c t s of p o s s i b l e changes i n the exchange s p l i t t i n g on the dimensions of the Fermi surface of i r o n by c a r e f u l l y examining the de Haas-van Alphen (dHvA) frequency as a function of temperature. F u l l d e t a i l s w i l l be given i n the following chapter but we emphasize here 10 that i n contrast to the photoemission studies mentioned above, i n which the e l e c t r o n system i s strongly perturbed i n the measurement process and i n which the state of the surface of the sample plays a major r o l e , the dHvA e f f e c t has the advantage of being an equilibrium phenomenon which i s i n s e n s i t i v e to surface d e t a i l s and which permits the study of h i g h l y -selected groups of e l e c t r o n i c states rather than the e n t i r e Fermi sea. On the other hand, dHvA o s c i l l a t i o n s can be detected only at tempera-tures which are low (T ^  4°K) when compared to the Curie point (T = 1043°K), so that the changes i n the ferromagnetic magnetization are extremely small. Nevertheless we w i l l see that the s e n s i t i v i t y of the dHvA e f f e c t can be made s u f f i c i e n t l y high to y i e l d s i g n i f i c a n t informa-t i o n about the behavior of the exchange s p l i t t i n g even i n the r e s t r i c t e d range of l i q u i d - h e l i u m temperatures. 11 §1.2 An O u t l i n e of the Present I n v e s t i g a t i o n In Chapter I I we b r i e f l y review the de Haas-van Alphen e f f e c t and d e s c r i b e i n d e t a i l how i t can be used to study the temperature dependence of the exchange s p l i t t i n g A. In p a r t i c u l a r we w i l l r e l a t e the change i n the de Haas-van Alphen frequency to the change i n A as a f u n c t i o n of temperature. Chapter I I I w i l l be concerned w i t h the d e t a i l e d experimental proce- . dure and the numerical a n a l y s i s which was used to measure s h i f t s i n the dHvA phase w i t h temperature to b e t t e r than one thousandth of a c y c l e , i n the presence of i n t e r f e r i n g s i g n a l s and n o i s e . In Chapter IV the f i n a l r e s u l t s of the experiment w i l l be presented and compared w i t h the p r e d i c t i o n s of the simple Stoner model. Then the experimental r e s u l t s w i l l be i n t e r p r e t e d on the b a s i s of the Fermi l i q u i d theory proposed by Izuyama and Kubo (1964) and extended by Edwards (1973). The i n t e r a c t i o n f u n c t i o n s appearing i n the phenomeno-l o g i c a l Fermi l i q u i d theory w i l l be evaluated f i r s t f o r a simple micro-scopic model based on the Hubbard Hamiltonian. Then the i n t e r a c t i o n f u n c t i o n s w i l l be evaluated f o r a more general case by the phenomenolo-g i c a l approach due to Edwards (1973), extended s l i g h t l y to take i n t o account the e f f e c t of magnetic a n i s o t r o p y . The p h y s i c a l i n t e r p r e t a t i o n and important consequences of the f i n a l r e s u l t s w i l l be d i s c u s s e d . This chapter w i l l f i n a l l y end w i t h a d e t a i l e d d i s c u s s i o n of other e f f e c t s which i n p r i n c i p l e could g ive r i s e to small apparent s h i f t s i n the de Haas-van Alphen frequency w i t h temperature. In Chapter V the r e s u l t s of some other i n v e s t i g a t i o n s w i l l be summarized. These i n v e s t i g a t i o n s i n c l u d e the study of a set of new low frequency de Haas-van Alphen o s c i l l a t i o n s i n i r o n , the measurements of some cyclotron e f f e c t i v e masses, and the study of quantum o s c i l l a t i o n s i n the spin magnetization which, as we w i l l show, should e x i s t i n an i t i n e r a n t ferromagnet. F i n a l l y i n Chapter VI we summarize the main conclusions of t h i s i n v e s t i g a t i o n and o f f e r suggestions for further study. Chapter II THEORETICAL BACKGROUND 13i §2.1 The de Haas-van Alphen Effect as a Tool to Study the Temperature  Dependence of the Exchange Splitting We begin this section with a brief description of the de Haas-van Alphen (dHvA) effect which consists of an oscillatory variation (with magnetic field) of the diamagnetism of the conduction electrons in a metal. The dHvA effect results from the quantization of the helical orbits of the conduction electrons in a magnetic f i e l d and can only be observed at high magnetic fields and low temperatures when the electrons are able to complete many orbits before being scattered, and when the thermal broadening of the Fermi surface i s not large compared to the energy difference between successive quantized orbits. Since the theory for the dHvA effect has been described in many text books and review articles (cf. Gold 1968) we shall only give a brief outline of the main results and in a form which i s appropriate either for an ordinary metal or for a ferromagnet in which the spin degeneracy has been removed. The motion of a Bloch electron in a f i e l d I_ can be described by the Lorentz force equation* CWc) - " f (V^ ) x B , where k and e._ are the wave vector and associated one-electron energy, and i s the group velocity of the electron. It follows from this *Electromagnetic units w i l l be used throughout this thesis. 14 equation that an el e c t r o n w i l l move around an o r b i t i n k space which i s formed by the i n t e r s e c t i o n of the constant energy surface with a plane normal to the f i e l d B.. The area enclosed by the o r b i t must be quantized so that the allowed o r b i t s l i e on so - c a l l e d Landau tubes i n k space which are shown schematically i n F i g . 2.1 (for free e l e c t r o n s ) ; the outer s p h e r i c a l surface corresponds to the Fermi surface enclosing f i l l e d (at T=0°K) e l e c t r o n i c s t a t e s . The energy spacing between Landau tubes increases with B so that by varying B i t i s p o s s i b l e to change the • density of states and therefore the free energy of the ele c t r o n gas. The dHvA magnetization i s then obtained from the negative gradient of the o s c i l l a t o r y part of the free energy with respect to the f i e l d . The r e s u l t s of the d e t a i l e d c a l c u l a t i o n can be summarized as follows. Each extremal c r o s s - s e c t i o n a l area A(B) of the Fermi surface l y i n g i n a plane perpendicular to the magnetic induction 15 gives r i s e to an o s c i l l a t o r y magnetization M(B) having a frequency i n B ^ given by the Onsager r e l a t i o n F(B) = j f i "  A& • 2- 1 The o s c i l l a t o r y magnetization associated with the extremal area > of the t or 4- spin electrons i s given by the harmonic s e r i e s M(B) = | [B - i 8 F 3F F 30 — F s i n 6 3<{> 2.2a I K x I ~\TL s l n t 2 7 T r ( f + ? " Y) T f-1 • n o i-h T. C-OQ 0) Figure 2.1 Orbit quantization i n a magnetic f i e l d f o r a f r e e - e l e c t r o n gas. Allowed states l i e on c y l i n d e r s i n k space and are occupied for those portions of the cylinders within the Fermi surface. The spacing between allowed o r b i t s i n k. space increases with increasing f i e l d strength. m A CQ .CM 16 /\ _\ -S Here (B, 9_, _>_) i s a set of orthogonal u n i t vectors when i s expressed i n s p h e r i c a l polar coordinates. D i s defined by -1/2 D - - _ f _ ( _ _ £ _ , 1 / 2 | _ _ L | 4TT m* h 3k,, h 2 where k., i s the component of k p a r a l l e l to B, and m* = -— The — _.TT ae q u a n t i t i e s I and K r give the reduction of the r t h harmonic due to the temperature broadening of the Fermi surface and due to the impurity broadening of the quantized Landau l e v e l s r e s p e c t i v e l y , and are defined by the following equations * r sinh rX(T) [rX(T)] 2.2c and K_ = exp [- rX(T_)] , 2.2d 2 m * k B T where X(T) = 2TT • •, T i s the temperature and T_ i s the e f f e c t i v e impurity or Dingle 'temperature' which i s r e l a t e d to the broadening of the Landau l e v e l s . F i n a l l y y i n Eq. 2.2a i s a constant, the minus or plus I T / 4 i s used according to whether the extremal area ^ i s a maximum or minimum and ? i s a phase quantity which depends on the o r i e n t a t i o n of the spin of the relevant electrons with respect to the induction B^ . When a magnetic f i e l d i s turned on, electrons are transferred from the + spin to the + spin Fermi surface so that we would expect the extremal area for an + spin surface to increase and that f o r a 4- spin surface to decrease with f i e l d . To f i r s t order i n B the r e s u l t i n g change i n the phase i n the argument of the sine function i n Eq. 2.2a i s indepen-dent of f i e l d and i s given by for + spin electrons 2.3 fo r -1- spin electrons where X. and X. are the o v e r a l l density of states of the t and 4- spin T 4-electrons at the Fermi surface and g * i s the appropriate g-factor which here i s assumed to be i s o t r o p i c (the same for a l l conduction e l e c t r o n s ) . Eq. 2.3 i s obtained from an a n a l y s i s given i n Appendix I where the depen-dence of the extremal area on 1$ i s evaluated for the general case i n which the g-factor i s a n i s o t r o p i c and renormalized by many-body i n t e r -actions. When X, i s equal to X, the q u a n t i t i e s i n the square brackets i n Eq. 2.3 are equal to one and Eq. 2.3 then reduces to the standard expression f o r t, i n normal (nonferromagnetic)metals ( c f . Gold 1968). We also wish to emphasize that on account of Eq, 2.3 the dHvA phase for an + (or +) spin extremal area of the Fermi surface i n a ferromagnet contains information about the r e l a t i v e d e n s i t i e s of states of the + (or 4-) spin electrons at the Fermi l e v e l . The r a t i o X./X. i s a T T quantity of considerable i n t e r e s t which has not yet been measured i n any r e l i a b l e way i n i r o n ( c f . Walmsley 1962, Belson 1966). We now wish to write a s i m p l i f i e d form of Eq. 2.2a which w i l l be 4 m o V v -g*m* 2 / j 4m  lX. +X* o f 4-18 use f u l f o r l a t e r reference. Since under the conditions of our ex p e r i -ment the quantity X(T) i s always greater then 2 and because only the component of M p a r a l l e l to J3 has been measured we then f i n d that to a good approximation the fundamental term (r = 1) i n the o s c i l l a t o r y magnetization associated with the extremal area A i s given by the simple formula M = A(T,B) s i n + <f>) , 2.4a where A(T,B) = D X(T) exp [-X(T + T_)] 2.4b and <fr = 2 (c - y) T TT/4 . 2.4c Having presented the main equations d e s c r i b i n g the dHvA e f f e c t we now consider i n d e t a i l how i t can be used to study the temperature dependence of the exchange s p l i t t i n g A. In F i g . 2.2 we show schemat-i c a l l y the energy bands f o r i r o n along the TH l i n e of the B r i l l o u i n zone. The drawing i s based on Woods bands (shown i n F i g . 1.1) together with the Fermi surface information (Gold et_ a l . 1971). At T=0°K the -r and -I- Fermi l e v e l s e. and e, are indicated by the s o l i d h o r i z o n t a l l i n e s . As the temperature i s increased any change i n the band s p l i t t i n g A can be described i n terms of a change i n the r e l a t i v e p o s i t i o n s of the two Fermi l e v e l s (as suggested by the dashed l i n e s i n F i g . 2.2) which Face of Page 19 Figure 2.2 Schematic sketch of the energy bands i n i r o n along the TH l i n e (based on the a n a l y s i s of the Fermi surface and the band structure given by Gold et_ al_. (1971) i n which E - E. = 2.0 eV). The electron lens i s a small pocket of the Fermi surface defined by those points i n Ic space where two p a r t i c u l a r i n t e r s e c t i n g d bands cross the 4- Fermi l e v e l . This f i g u r e gives only two Ic points (along the TH l i n e ) on the lens surface. In order to construct the o v e r a l l shape of the lens i t i s necessary to draw the band structure along many r a d i a l l i n e s i n k space t i l t e d up to 10 or 20 degrees from the TH l i n e . 19 20 would then r e s u l t i n changes i n the dimensions of the various sheets of the Fermi surface. Then any change i n an extremal area j r would be r e f l e c t e d i n a change i n the phase F(B)/B of the dHvA o s c i l l a t i o n s given by Eq. 2.4a. Of the various a v a i l a b l e sheets of the Fermi surface i n i r o n (cf. F i g . 1.2) the most s u i t a b l e one for the present i n v e s t i g a t i o n i s the 4- el e c t r o n 'lens' whose center l i e s on the TH-line i n F i g . 2.2. This lens and i t s f i v e symmetry-related counterparts (see F i g . 1.2) are believed to give r i s e to the three £-frequency branches i n the dHvA e f f e c t (Gold et a l . 1971). The _ branches coalesce to give j u s t a si n g l e frequency when the induction 13 i s p a r a l l e l to the [111] d i r e c t i o n , where the o s c i l l a t i o n s can be detected up to about 5°K, and without major complications a r i s i n g from the presence of any neighboring frequen-c i e s . By comparison a l l other dHvA frequencies i n i r o n which can be observed up to 4 or 5°K turn out to be quite d i f f i c u l t to separate r e l i -ably from each other. An a d d i t i o n a l advantage i n studying the e l e c t r o n lenses i s that s i m i l a r lenses appear i n the Fermi surface of molybdenum [which happens to resemble the + Fermi surface of i r o n rather c l o s e l y (Gold 1971)] so that i t i s pos s i b l e to make d i r e c t comparison of any temperature dependence i n the s i z e of the i r o n lenses with that for a metal f or which A = 0. The p o s s i b i l i t y of making t h i s comparison i s invaluable because small changes i n the s i z e of the Fermi surface as a function of temperature can i n p r i n c i p l e a r i s e from various other e f f e c t s [eg. thermal expansion, ordinary s h i f t s i n the chemical p o t e n t i a l (the 2nd term i n Eqs. 2.10a,b), e t c . ] . Although our estimates of these e f f e c t s sug-gest that they should be small under the conditions of our experiment (see §4.3) they should i n any event be roughly s i m i l a r f o r i r o n and molybdenum. 21 The temperature dependence of the [111] lens frequency has been studied i n both Fe and Mo between 1 and 4°K by c a r e f u l l y looking for s h i f t s i n the phase F/B over t h i s temperature range and at a f i x e d value of the f i e l d B. The high s e n s i t i v i t y of t h i s dHvA method i s l a r g e l y due to the fac t that F/B i s a large number. For the i r o n --frequency at [111] we have F = 4.13 MG so that for B = 40 kG the phase F/B i s about 100 c y c l e s . Since i n the experiment i t has been p o s s i b l e to measure phase changes somewhat smaller than 1/1000 of a c y c l e (up to 4 or 5°K), t h i s means the dHvA e f f e c t i s s e n s i t i v e enough to reveal f r a c t i o n a l changes i n the c r o s s - s e c t i o n a l area of the lens which are as small as 6A/A = 10 On the other hand i n §2.2 we w i l l show that i f the temper-ature i s changed from 1° to 4°K the simple Stoner theory p r e d i c t s f r a c -t i o n a l changes of the order 6<*M- = 1.6 x 10 \ which i s at l e a s t an order of magnitude greater than the r e s o l u t i o n of the experiment. We thus have at our di s p o s a l a technique with which the p r e d i c t i o n of the Stoner model can be subjected to quite a stringent t e s t , despite the fa c t that the experiments must be performed at l i q u i d helium tempera-tures . We end t h i s s e c t i o n with some preliminary comments on the crux of the experimental problem, which i s to measure phase s h i f t s to better than one thousandth of a cyc l e over a temperature range i n which the si g n a l changes i n amplitude by three orders of magnitude i n the presence of noise, background si g n a l s and other frequencies. In order to achieve t h i s kind of r e s o l u t i o n i t was f i r s t necessary to optimize the s i g n a l -to-noise r a t i o and t h i s was done by gi v i n g considerable a t t e n t i o n to the preparation of high-purity i r o n 'whisker' c r y s t a l s and by c a r e f u l l y reducing various sources of noise and background s i g n a l s i n the ex p e r i -22 merit. In a d d i t i o n a phase measurement technique was developed which suppressed numerous sources of systematic errors such as small t i l t s of the sample with temperature, magnetic f i e l d d r i f t s , e l e c t r o n i c phase s h i f t s , interference from background s i g n a l s and neighboring frequencies and so on. F i n a l l y i t was necessary to consider i n d e t a i l a l l e f f e c t s which could p o s s i b l y give r i s e to small s h i f t s i n the dHvA frequency i t s e l f as a function of temperature. In p a r t i c u l a r we w i l l f i n d that small apparent s h i f t s i n the frequency with temperature (and f i e l d ) are inherent i n the dHvA e f f e c t i t s e l f and t h i s w i l l be shown e x p l i c i t l y by r e d e r i v i n g the conventional expression for 3M/3t (the measured quantity) i n a s l i g h t l y more rigorous fashion. In the end we w i l l f i n d that a l l of the various extraneous e f f e c t s can be removed or taken into account n a t u r a l l y and that the r e s u l t s of the experiment can be explained i n an unambiguous fashion. 23 §2.2 The Temperature Dependence of the Exchange S p l i t t i n g and  de Haas-van Alphen Frequency In the simple Stoner theory the magnetization i s presumed to decrease with temperature by means of s i n g l e - p a r t i c l e (sp) e x c i t a t i o n s ; that i s by tra n s f e r of electrons from the top of the t spin bands to the 4- spin bands. On the other hand i t i s well known that the reduction i n the magnetization i n an i t i n e r a n t ferromagnet can also a r i s e from c o l l e c -t i v e or spin-wave (sw) e x c i t a t i o n s , which e x i s t on account of c o r r e l a -t i o n e f f e c t s ignored i n the Stoner theory. Therefore the t o t a l change i n magnetization i s given by an expression of the form 6M = 6M + 6M . 2.5 sp sw If 6M_w i s small compared to SM__, then the Stoner theory i s expected to give a good d e s c r i p t i o n of the properties of the ferromagnet at f i n i t e temperatures and i n p a r t i c u l a r we should expect the exchange s p l i t t i n g A to be proportional to the magnetization at a l l temperatures, that i s 6A 6M - , A ~ = M - ' 2 - 6 where the subscript o r e f e r s to T=0°K. Thompson, Wohlfarth and Bryan (1964) have generalized the Stoner p i c t u r e and have argued that even i f the spin wave magnetization <5Mgw i s not small the exchange s p l i t t i n g A should nevertheless follow the 24 t o t a l magnetization so that Eq. 2.6 would be v a l i d i n general. On the other hand Edwards (1973), applying the generalized Fermi l i q u i d theory of Izuyama and Kubo (1964), has pointed out that <$M_W may contribute only s l i g h t l y to the change i n the exchange s p l i t t i n g . We w i l l discuss the v a l i d i t y of these opposing points of view i n d e t a i l l a t e r i n the l i g h t of the experimental r e s u l t of Chapter IV. However f o r the purpose of comparison with experiment we w i l l i n t h i s s e c t i o n assume that the change i n the exchange s p l i t t i n g i s given by the Stoner equation (2.6) where 6M i s the t o t a l change i n magnetization. We now r e l a t e the change 6A(T) i n the c r o s s - s e c t i o n a l area of the lens to a change 6A(T) of the exchange s p l i t t i n g as a function of temper-ature. Within the framework of the split-band model for i r o n ( c f . F i g . 2.2) the number of -f- and + spin electrons (per atom) at temperature T should be given by the usual Fermi-Dirac i n t e g r a l s [e - e (T)]/k T n. (T) = f deJf(e) [e f + 1] T O -1 2.7a [e - E + ( T ) ] / k l n,(T) = 7 deX(e) [e + 1] T 0 -1 2.7b where /(e) i s the density of states (per atom) and e x(T) = e.. + 6e.(T), T T T e x(T) = e + + 6e +(T) are the + and 4- spin Fermi l e v e l s at temperature T (cf. the dashed l i n e s i n F i g . 2.2). At low temperatures we can expand the r i g h t hand sides of Eqs. 2.7a,b to obtain the expressions 25 2 -n +(T) = n +(0) + 6e +(T) + |- (k_T)Z/Y.» + . . . 2.8a 2 9 n +(T) = n +(0) + /f + 6e +(T) + | - ( k _ T ) Z X ' + . . . , 2.8b where X\ , X". , X. ', X*. 1 are the o v e r a l l d e n s i t i e s of states and t h e i r T T T + d e r i v a t i v e s at the . and 4- Fermi l e v e l s r e s p e c t i v e l y . Then from the constancy of the number of p a r t i c l e s n f(T) + n.(T) = n +(0) + n +(0) , 2.9a and the i d e n t i t y 6e +(T) - 6e +(T) = _A (T ) , 2.9b i t follows from Eqs. 2.8a,b that f o r low temperatures „ 2 9 V + - V T Y + T ^ J 9 V + V 6 e . ( T ) = " ir^JT 6 A ( T ) " t < k B T ) X T 7 7 T • 2 - 1 0 b T T 4- T Using Eq. 2.10b f o r Se. we therefore f i n d that f o r a 4- surface i l _ M. = I _ _ * = . _ _ _ _ _ _ _ <SA(T) F A- A Se £4- ehF ^ + X / A t 4 - o m* r r 2 2 V + V TT" (kp T) ~7r __~_= » 2.11 eTiF 6 ^ B * ' /T, + X. ' 4- T 26 "h2 3A-where m* = - r — — i s the appropriate c y c l o t r o n e f f e c t i v e mass. 2lT d £ Eq. 2.11 gives the change i n the frequency a r i s i n g from any 5 A (T } change — with temperature. If t h i s change i s that given by the o Stoner theory (Eq. 2.6) then by i n s e r t i n g into Eq. 2.11 the numerical values m* = 0.71 m_ (§5.2), F = 4.13 MG, A_ = 2.0 eV (Gold et a l . 1971), Xj(X. - 0.7 (cf. Wood 1962, Duff and Das 1971), we obtain for T T T the + el e c t r o n lens i n i r o n _ _ _ _ 2 1 _ _ 2 1 2 F M ' o The v a r i a t i o n 6M/M_ (see F i g . 2.3) has been obtained from the data f o r 9M/9T measured by Legkostupov (1971) between 0.5° and 4.2°K and i n t e r -polated to an applied f i e l d H = 17 kG (the lower f i e l d value used i n the experiment; see §4.1). For the temperature i n t e r v a l of 1° - 4°K we then f i n d from Eq. 2.12 and F i g . 2.3 that the straightforward a p p l i c a t i o n of the Stoner model gives 6F/F = 1.6 x 10 4 ; as we have already pointed out, t h i s change i s more than an order of magnitude greater than the r e s o l u -t i o n of the experiment. For reasonable values for X. ' and X,1 the second term i n Eq. 2.11 i s estimated to be several orders of magnitude smaller than the f i r s t term at l i q u i d helium temperatures and has therefore been ignored. Eqs. :2.10a,b suggest that at l e a s t i n p r i n c i p l e the second term i n Eq. 2.11 could be eliminated e n t i r e l y by measuring changes i n the frequencies for two sheets of the Fermi surface, one for t electrons and the other f o r 4- electrons. In i r o n , unfortunately, there are no small T sheets which are s i m i l a r , i n terms of the strength of the dHvA s i g n a l , to the 4- lenses; the dHvA e f f e c t from the la r g e r t sheets can be Face of Page 27 Figure 2.3 The temperature dependence of the magnetization f o r i r o n at low temperature. The curve was obtained from the data f o r 3M/3T measured by Legkostupov (1971) between 0.5° and 4.2°K and in t e r p o l a t e d to an applied f i e l d of H = 17 kG. The magnitude of 3M/3T was determined from the emf induced i n a c o i l containing the sample as the temperature was vari e d p e r i o d i c a l l y at a frequency of about 10 Hz. 27 28 observed only at the lowest temperatures, so that a study over the range 1° to 4°K i s not p o s s i b l e . F i n a l l y we remark that Eq. 2.12 was obtained by using the measured value of m* which i s enhanced by electron-phonon and electron-magnon i n t e r a c t i o n s , whereas the quantity i n Eq. 2.11 i s presumably the bare band structure mass. In order to see t h i s , we r e f e r to F i g . 2.4 which shows expanded views of the band structure for i r o n along the TH l i n e and i n the neighborhood of the minority e l e c t r o n l e n s . The e l e c t r o n -phonon i n t e r a c t i o n renormalizes the band structure only i n the immediate v i c i n i t y of the Fermi l e v e l and gives r i s e to the jog shown i n an exaggerated way i n F i g . 2.4a f o r each band. Because t h i s jog i s t i e d to the Fermi l e v e l we see with the aid of F i g . 2.4b that the change &A-i n the c r o s s - s e c t i o n a l area of the lens due to a change 6e. i n the r Fermi l e v e l should not be a f f e c t e d by the electron-phonon i n t e r a c t i o n , 2 h 6 i . e . , the quantity m* = —— - — appearing i n Eq. 2.11 i s not renormalized T (cf. Prange and Sachs 1967). On the other hand the c y c l o t r o n mass measured f o r example from the amplitude A(T,B) i n the dHvA e f f e c t or i n cyclotron resonance experiments i s r e l a t e d to the a c t u a l slope of the bands j u s t at the Fermi l e v e l and i s therefore renormalized ( c f . Engels-berg and Simpson 1970). If the foregoing arguments also apply for the electron-magnon interaction,' we then conclude that the right-hand side of Eq. 2.12 should be divided by the electron-phonon-magnon renormaliza-t i o n f a c t o r . The information a v a i l a b l e so f a r (Gold et_ al_. 1971 and Chapter V) i n d i c a t e s that t h i s f a c t o r i s probably smaller than two for the various sheets of the Fermi surface i n i r o n . In n i c k e l the enhance-ment f a c t o r has been found to vary from about 1.4 to 1.8 (Goy and Grimes 1973). pa o CD o Hi ft) OP ro VO Figure 2.4 The energy bands for i r o n along the TH l i n e i n the neighborhood of the minority e l e c t r o n l e n s . The jog at the Fermi l e v e l , which arises from the electron-phonon i n t e r a c t i o n , i s t i e d to the Fermi l e v e l and therefore does not e f f e c t the change i n the area of the lens r e s u l t i n g from a change 6e i n the + Fermi energy. 29 Chapter I I I EXPERIMENTAL PROCEDURE 30a §3.1 Introduction The p r i n c i p l e of the experiment can be b r i e f l y described with reference to F i g . 3.1 which shows representative portions of the funda-mental waveform of the o s c i l l a t o r y magnetization at two d i f f e r e n t temper-atures T and T + 6T and d i f f e r i n g i n phase by the f r a c t i o n 6F/B of one c y c l e . Using Eq. 2.4b f o r A(T,B) we f i n d that the amplitude of the 3 [111] .-frequency decreases by a f a c t o r of about 10 when the sample i s warmed from 1° to 4°K i n a f i e l d B = 40 kG. The o s c i l l a t i o n s were studied over a l i m i t e d f i e l d region (at two d i f f e r e n t c e n t r a l f i e l d s ) and the phase information was derived from computer a n a l y s i s of the data. As mentioned e a r l i e r the phase s h i f t s were measured to an accuracy of better than one thousandth of a c y c l e between 1° and 4°K and i n the following sections we describe i n d e t a i l how t h i s accuracy was achieved and how various systematic errors were avoided. In §4.3 we s h a l l give a f u l l d i s c u s s i o n of other e f f e c t s which are inherent i n the measuring process but which could not be completely removed experimentally. It w i l l be shown that these e f f e c t s are of secondary importance. Face of Page 31 Figure 3.1 Sketch of portions of the dHvA magnetization showing pos s i b l e phase s h i f t with temperature. For T = 1°K and T + ST = 4°K the upper trace i s reduced by a 3 f a c t o r of 10 f o r the - - o s c i l l a t i o n s at [111] and f o r B = 40 kG. 32 §3.2 Sample P r e p a r a t i o n In order to optimize the s i g n a l - t o - n o i s e r a t i o and to reduce p o s s i b l e systematic e r r o r s i n the dHvA phase measurements th a t could a r i s e from any inhomogeneity i n the dHvA phase F(B)/B over the volume of the sample, we have devoted c o n s i d e r a b l e a t t e n t i o n to the prepara-t i o n , s e l e c t i o n and e l e c t r o p o l i s h i n g of hig h q u a l i t y i r o n c r y s t a l s . §3.2a The Growth of Ir o n Whiskers The i r o n samples used i n t h i s experiment were 'whisker' c r y s t a l s grown by a method f i r s t reported by Brenner (1956) i n which molten f e r r o u s c h l o r i d e i s reduced by f l o w i n g hydrogen gas at approximately 700°C. The f r e e i r o n atoms produced i n the r e a c t i o n 'condense' i n the bcc phase to form long and narrow 'whisker' c r y s t a l s which, when grown under f a v o r a b l e circumstances, are very pure and n e a r l y p e r f e c t i n s t r u c t u r e . The q u a l i t y of these c r y s t a l s was found to vary from run to run and the best r e s u l t s were obtained by a procedure that can be summarized b r i e f l y as f o l l o w s (see a l s o Dheer 1967). During a t y p i c a l run a pure i r o n boat* about 8" long and 1 1/2" wide i s f i l l e d w i t h approximately 200 gm of reagent grade FeCl-^H-O** and then placed i n s i d e a sealed 64 mm diameter vycor combustion tube *Made from 99.99% pure i r o n sheets purchased from A.D. MacKay & Co., 198 Broadway, N.Y. 38, N.Y. **Best r e s u l t s were obtained w i t h reagent grade FeCl-^H-O manufactured by J.T. Baker Chemical Co., P h i l l i p s b u r g , N.J. 33 f i t t e d i n a high temperature furnace. The temperature i s then ra i s e d slowly while any r e s u l t i n g water vapor i s c a r r i e d o f f by a flow of helium gas. At a temperature not exceeding about 480°C the helium flow i s stopped and high p u r i t y hydrogen gas i s introduced at a rate of 0.4 l i t e r s per minute. The temperature i s then r a i s e d to 710°C and the system i s allowed to reach equilibrium. The c r y s t a l s reach t h e i r maxi-mum s i z e ( t y p i c a l l y 1 cm long and 1/2 mm wide) i n approximately 15 hours time and are found to grow along the [100], and l e s s frequently along the [111] and [110], d i r e c t i o n s . The samples used i n the present i n v e s t i g a t i o n were [111] whiskers which were about 7 mm long and 0.5 mm t h i c k . Also [100] whiskers were used i n preliminary experiments and i n the experiments described i n Chapter V. The [111] molybdenum sample was a spark-cut c y l i n d e r s i m i l a r i n dimensions to the i r o n whiskers and was one of the c r y s t a l s used by Hoekstra and Stanford (1973) i n t h e i r recent study of the dHvA e f f e c t i n Mo. 34 §3.2b The Testing of Iron Whiskers The amplitude of the dHvA o s c i l l a t i o n s i s known to be reduced by i ) impurities and i i ) c r y s t a l l i n e inhomogeneities ( d i s l o c a t i o n s , mosaic struc t u r e , voids and so on). The r e s i d u a l r e s i s t i v i t y r a t i o , * which i s often used to gauge the q u a l i t y of a sample, gives information p r i m a r i l y about i ) and not i i ) . Since i i ) can give r i s e not only to a serious reduction i n the dHvA amplitude but also (as we w i l l show i n §4.3) to systematic e r r o r s i n the phase s h i f t measurements, i t was of i n t e r e s t to search for a general t e s t i n g procedure which could give information about both types of imperfections i ) and i i ) . Such a procedure was suggested to us by recent studies of the Landau domain structure i n i r o n whiskers reported by A r r o t t , H e i n r i c h and Bloomberg (1971 and p r i v a t e communications). In F i g . 3.2a we show the magnetic domain structure i n a [100] i r o n whisker i n zero external f i e l d . If we apply a small l o n g i t u d i n a l f i e l d H, the domain i n which the magnetization i s p a r a l l e l to H w i l l grow so that the Bloch w a l l w i l l be s h i f t e d to a new e q u i l i b r i u m p o s i t i o n as suggested i n F i g . 3.2b. By slowly varying H the domain wall can be moved continuously across the sample and i t i s p o s s i b l e to 'use' the domain w a l l to probe the i n t e r n a l s tructure of the c r y s t a l . Experimentally the domain w a l l i s v i b r a t e d by a small a.c. f i e l d h cos tot which i s superimposed on H and the o s c i l l a -tions of the wall give r i s e to an induced voltage that can be detected *Defined as the r a t i o of the r e s i s t i v i t i e s of the sample at room temper-ature (300°K) and at l i q u i d helium temperatures (4.2°K). In a f e r r o -magnet the measurement i s made i n a weak l o n g i t u d i n a l f i e l d ( p a r a l l e l to the current) i n order to magnetize the sample into a s i n g l e domain. Face of Page 35 Figure 3.2 Figures (a) and (b) show the Landau domain st r u c t u r e i n a [100] i r o n whisker i n the absence, and i n the presence of an external f i e l d . Figure (c) shows the out-of-phase component X Q u t of the amplitude of the wa l l v i b r a t i o n s i n a 1 kHz a.c. f i e l d f o r a poor ( l e f t ) and a good (right) sample. The r e s u l t f o r the good sample shows n e g l i g i b l e h y s t e r e s i s as H i s v a r i e d back and f o r t h from -20G to +20G. The general feature of the X _ versus H p l o t i n the good sample i s s i m i -out r ° r l a r to that reported by A r r o t t et. a l . (1971) . H = 0 (a) H > 0 (b) 36 by a small c o i l surrounding the sample. In a perfect c r y s t a l the amp-l i t u d e of the wall o s c i l l a t i o n s should vary smoothly (and reproduceably) as the wall i s moved from one side of the sample to the other ( i n a constant modulation f i e l d ) . On the other hand any c r y s t a l l i n e imper-f e c t i o n s might be expected to give r i s e to e r r a t i c and unreproduceable v a r i a t i o n s of the amplitude of the w a l l v i b r a t i o n s as a f u n c t i o n of H. Analogous conclusions may be expected to apply for [111] or [110] c r y s t a l s i n which the domain pattern i s not the simple Landau structure shown i n F i g . 3.2. A v a r i e t y of samples have been tested by t h i s a.c. technique which was found to be a simple and d i s c r i m i n a t i n g method of s e l e c t i n g the best i r o n c r y s t a l s f o r dHvA measurements. In F i g . 3.2c we show f o r comparison a sketch of r e s u l t s of measurements made on a poor and on a good [100] sample. Standard d.c. measurements have also been made of the r e s i d u a l r e s i s t i v i t y r a t i o s of a few representative samples (grown side by side with the best ones selected f o r the dHvA experiment) and the r e s i d u a l r e s i s t i v i t y r a t i o s were found to range from 1100 to 2000 when measured i n a weak l o n g i t u d i n a l f i e l d j u s t s u f f i c i e n t to saturate the whiskers. We end t h i s s e c t i o n with a d i s c u s s i o n on the p o s s i b i l i t y of using the a.c. technique described above to obtain some information about the impurity concentration i n [100] whiskers which have the simple Landau domain st r u c t u r e . A r r o t t ______ have shown that the o s c i l l a t i o n s of the domain wall (near the center of the c r y s t a l ) i n a modulation f i e l d h cos _ t are approximately harmonic and are characterized by an e f f e c -t i v e s t i f f n e s s k, an e f f e c t i v e damping b, and an e f f e c t i v e mass m which i s vanishingly small. The in-phase (X^_) and out-of-phase (X fc) 37 components of the amplitude of the o s c i l l a t i o n s of the domain w a l l should t h e r e f o r e be given by the expressions X. = F — 5 — - -l n 0 k 2 + ( _ _ ) 2 3.1a cob X ° U t = F ° "k2 + (cob) 2 3.1b where F q i s the amplitude of the d r i v i n g f o r c e . W i l l i a m s , Shockley and K i t t e l (1950) have shown that b i s i n v e r s e l y p r o p o r t i o n a l to the b4 2° r e s i s t i v i t y p so that i f we d e f i n e a r e s i s t i v i t y r a t i o R = — - = P 3 0 0 o b300° then by Eq. 3.1b we f i n d that p4.2° _ [(° Xout ]300° k . _ R = 7 — 1 i f co >> r- . 3.2 [ U Xout ]4.2° b Then by measuring the high frequency value of %. Q U t at 4.2°K and at 300°K the q u a n t i t y R can be r e a d i l y determined. The d e f i n i t i o n f o r R above looks very much l i k e the corresponding d e f i n i t i o n f o r the r e s i d u a l r e s i s t i v i t y r a t i o (Pg. 34). However i n the l a t t e r the r e s i s t i v i t y i s measured f o r c u r r e n t s t r a v e l l i n g p a r a l l e l to the i n t e r n a l f i e l d (which i s approximately 22 kG i n i r o n ) whereas i n the former case i t i s not d i f f i c u l t to see that the r e l e v a n t r e s i s t i v i t y i s that f o r eddy c u r r e n t s (induced by the domain w a l l v i b r a t i o n s ) which are i n f a c t t r a v e l l i n g t r a n s v e r s e to the 22 kG i n t e r n a l f i e l d and which can t h e r e f o r e experience a s i g n i f i c a n t magnetoresistance at low temper-38 a t u r e s . The q u a n t i t y R i s t h e r e f o r e always smaller than the t r u e r e s i -dual r e s i s t i v i t y r a t i o . In very pure samples R i s i n f a c t completely dominated by the magnetoresistance so that i n t h i s case R g i v e s essen-t i a l l y no i n f o r m a t i o n about the i m p u r i t i e s i n the sample. §3.2c E l e c t r o p o l i s h i n g Procedure A p r e r e q u i s i t e f o r the macroscopic i n d u c t i o n _B to be uniform throughout a homogeneous single-domain sample i s that the sample must be bounded by a surface of the second degree. Because a f i e l d inhomo-gene i t y can g i v e r i s e to a spurious temperature-dependent phase s h i f t as w e l l as a damping of the dHvA s i g n a l , i t was necessary to c a r e f u l l y e l e c t r o p o l i s h the i r o n whiskers i n t o a shape c l o s e l y resembling a long e l l i p s o i d of r e v o l u t i o n . This was accomplished by a technique described by Gold e_t al_. (1971) i n which the i r o n whisker i s held w i t h i t s a x i s v e r t i c a l by a s m a l l r o t a t i n g magnet and dipped i n e l e c t r o p o l i s h s o l u t i o n made up of 80 grams of chromium t r i o x i d e , 420 ml a c e t i c a c i d and 22 ml water. A current of about 20 to 100 mA i s used w h i l e the r o t a t i n g c r y s t a l i s s l o w l y withdrawn from the l i q u i d . The procedure i s repeated ( a l t e r n a t i n g the end of the c r y s t a l immersed i n the l i q u i d ) u n t i l the sample resembles c l o s e l y an e l l i p s o i d of r e v o l u t i o n . 39 §3.3 Magnetic Field When the applied f i e l d i s along the major axis of the el l i p s o i d a l sample the magnetic induction i s given by B = H + 4TT(1 - D)M , 3.3 where H is the applied f i e l d , D is the demagnetizing factor which is about 0.02 for our polished whiskers and M is essentially the saturation magnetization at the sample temperature. As shown in Fig. 2.3 the fractional change 6M/Mq in the temperature range between 1° and 4°K i s only about -7 x 10 ^ and the corresponding fractional shift in B i s typically half this amount. If we ignore this small temperature depen-dence of B we introduce an error in the measured phase F/B which is smaller than the resolution of the experiment. In any case the f i e l d B can be easily corrected for the small change in M with temperature. The applied f i e l d H was provided by a 15" Varian iron-core elec-tromagnet which could be rotated about the vertical axis and which produced fields up to 25 kG over the 2" pole gap used. The applied f i e l d was measured to an absolute accuracy of about 0.5% (which is roughly consistent with the uncertainty in the absolute value of the second term in Eq. 3.3) but to a relative accuracy of better than one part in 10^ by means of a Hall probe whose temperature was very closely regulated.* The temperature regulation provided by the Varian Mark II *This method was chosen instead of the more conventional nuclear magne-t i c resonance (NMR) technique because the high modulation f i e l d (approx-imately 600G peak-to-peak) required for the detection of the dHvA o s c i l -lations (§3.6) made reliable NMR measurements impractical. 40 f i e l d c o n t r o l system was found to be i n s u f f i c i e n t f o r our purposes and the H a l l probe was therefore enclosed by a small temperature-regulated jacket, the magnet was shielded from a i r currents by a p l a s t i c cover, the room temperature i t s e l f was regulated to ±1°C, and at l e a s t 24 hours were allowed for the magnet system to reach eq u i l i b r i u m . With these simple refinements the d r i f t i n the magnetic f i e l d could be kept below about 0.05 G at 20 kG f o r several hours at a time. A thermistor placed near the H a l l probe was used to monitor small d r i f t s i n the temperature of the probe; any change i n the thermistor.signal i n d i c a t e d that the f i e l d was d r i f t i n g s l i g h t l y . During an experimental run the magnetic f i e l d was swept slowly through several cycles of the dHvA o s c i l l a t i o n s , during which time the H a l l voltages (along with the dHvA signal) were d i g i t i z e d and c o l l e c t e d on magnetic tape. The phase of the dHvA o s c i l l a t i o n s , as computed at a given reference f i e l d , was always found to be the same at the beginning and end of a run to an accuracy of better than one-thousandth of a c y c l e . 41 §3.4 Cryostat and Sample Rotator In p r e l i m i n a r y experiments the sample was i n contact w i t h the l i q u i d helium and the temperature was v a r i e d by pumping. U n f o r t u n a t e l y the v a r i a t i o n s i n the helium vapor pressure were found to g i v e r i s e to a small (0.01°) temperature-dependent t i l t of the c r y o s t a t which i n t r o -duced e r r o r s i n the phase measurements. In order to avoid these e r r o r s as w e l l as any p o s s i b l e phase s h i f t s due to the pressure dependence of the €-frequency, a l l measurements reported here were c a r r i e d out at e s s e n t i a l l y constant pressure i n the c r y o s t a t shown s c h e m a t i c a l l y i n F i g . 3.3. The c e n t r a l part of the c r y o s t a t c o n s i s t e d of a s t a i n l e s s s t e e l dewar which contained the sample holder and helium exchange gas at a pressure of about 0.5 t o r r . This inner can was submerged i n a 1°K l i q u i d helium bath and the temperature i n s i d e the can could be conven-i e n t l y r a i s e d above 1°K (up to 10°K or higher) by means of an e l e c t r i c a l heater R. . The temperature was s t a b i l i z e d by a c o n v e n t i o n a l feedback c o n t r o l l e r and measurements of the sample temperature were made to an accuracy of ±0.01°K by a carbon r e s i s t o r R^ and independently by means of the known temperature dependence of the amplitude of the e - o s c i l l a -t i o n s . A check on the temperature homogeneity over the sample volume was made by v e r i f y i n g that the same f i e l d dependence f o r the amplitude was obtained when the sample was i n d i r e c t contact w i t h l i q u i d helium at the same temperature. In a d d i t i o n we emphasize that no evidence could be found f o r temperature-dependent t i l t s of the sample or of the c r y o s t a t as a whole. In p a r t i c u l a r the phase measurement r e s u l t s showed no h y s t e r e s i s upon repeated warming and c o o l i n g of the sample; a l s o , Face of Page 42 Figure 3.3 Diagram showing sample holder and the c e n t r a l part of the cryostat t a i l between the pole faces of the e l e c -tromagnet. The outer dewar and r a d i a t i o n s h i e l d as w e l l as the modulation c o i l s attached to the pole t i p s are not shown. and R- are 1/8 watt, 33f_ (at room temperature) Ohmite r e s i s t o r s . 43 int e r c h a n g i n g the r o l e of r e s i s t o r s R.. and R_ by making R_ the heater and R. the sensor d i d not s i g n i f i c a n t l y a l t e r the measured phase; and f i n a l l y , as we w i l l show i n §4.1, the change 6F/F f o r the e l e c t r o n l e n s i n Mo was found to be smaller than one part i n 10"* (the r e s o l u t i o n of the experiment) between 1° and 4°K, i n agreement w i t h the t h e o r e t i c a l e x p e c t a t i o n f o r an o r d i n a r y metal where A = 0. The l a t t e r r e s u l t v e r y s t r o n g l y suggests that there was no s i g n i f i c a n t change i n the l e n s frequency i n Mo due to any p o s s i b l e t i l t of the sample w i t h temperature. The same c o n c l u s i o n should apply even more r i g o r o u s l y f o r i r o n which was st u d i e d i n the same apparatus and f o r which any change i n the l e n s frequency a r i s i n g from a given t i l t i s expected to be about four (or more) times s m a l l e r than the corresponding change i n Mo. Although a l l measurements reported here were made usin g our low f i e l d and high homogeneity pole f a c e s , the exchange gas c r y o s t a t sketched i n F i g . 3.3 was designed so t h a t i t could a l s o be used i n c o n j u n c t i o n w i t h our h i g h - f i e l d pole faces where there i s i n s u f f i c i e n t space between the pole t i p s to accomodate both the helium baths and the exchange gas c r y o s t a t . In that case the helium can shown i n F i g . 3.3 i s removed, the c y l i n d r i c a l outer dewar and r a d i a t i o n s h i e l d are reduced i n diameter, and the exchange gas c r y o s t a t i s cooled to low temperatures by means of a thermal l i n k to a 1°K l i q u i d helium r e s e r v o i r l y i n g above the pole faces (not shown i n F i g . 3.3). I n t h i s c o n f i g u r a t i o n the sample could be cooled down to temperatures as low as 1.5°K ( i n weak modula-t i o n f i e l d s ) and then r a i s e d above t h i s v a l u e as before by means of an e l e c t r i c a l heater. F i n a l l y we wish to d e s c r i b e a sample holder and r o t a t o r , shown s c h e m a t i c a l l y i n F i g . 3.4, which was used to make accurate i n s i t u 44 alignment of the samples i n the magnetic f i e l d f o r the experiments reported i n Chapter V. The sample holder and r o t a t i o n mechanism was designed to f i t i n s i d e a l i q u i d helium can having a 7/16" inner diameter, and to t i l t the sample and pick-up c o i l by a few degrees i n a plane normal to the plane of r o t a t i o n of the magnetic f i e l d . R e f e r r i n g to F i g . 3.4 the r o t a t i o n of the c o n t r o l s h a f t i s converted by means of the o f f s e t p i n i n t o a t i l t i n g motion of the sample holder about the a x i s of the p i v o t p i n which i s secured to the outer housing. The r o t a t i o n b acklash was found to be q u i t e s m a l l (< 0.1°) s i n c e the b a l l of the o f f s e t p i n was machined f o r a very c l o s e f i t i n the groove of the sample h o l d e r . Furthermore, as i t moves i n i t s groove, the b a l l of the o f f s e t p i n tends to sweep any accumulated s o l i d a i r out of i t s path so th a t the mechanism does not bind at low temperatures. Face of Page 45 Figure 3.4 Diagram of the sample holder and t i l t mechanism used to make accurate i n s i t u alignment of the sample i n the magnetic f i e l d . Front Right Side 46 §3.5 Sample O r i e n t a t i o n When the magnetic f i e l d i s t i l t e d away from the [111] d i r e c t i o n the t o t a l o s c i l l a t o r y magnetization a s s o c i a t e d w i t h the e l e c t r o n l e n s e s may be represented by the equation 3 2TTF. M = _ A.(T,B) s i n (—rr^ + <.) . 3.4 J - l J where the s u b s c r i p t s 1, 2 and 3 r e f e r to the three branches of the . - o s c i l l a t i o n s . I t i s only when B_ i s p a r a l l e l to the [111] d i r e c t i o n that we have A.. = A_ = A_ and F^ = F_ = F_. Since ( f o r the n e a r l y e l l i p s o i d a l e l e c t r o n lenses) the c y c l o t r o n e f f e c t i v e mass m*^  a s s o c i a t e d w i t h the j t h branch should be roughly p r o p o r t i o n a l to the corresponding frequency , i t f o l l o w s that i f 15 i s not o r i e n t e d p r e c i s e l y along the [111] a x i s the three amplitudes i n Eq. 3.4 w i l l i n general have s l i g h t l y d i f f e r e n t temperature dependences (see Eq. 2.4b f o r A(T,B)). T h i s means that f o r a mis o r i e n t e d c r y s t a l the t o t a l o s c i l l a t o r y magnetization M, as measured over a l i m i t e d f i e l d range, would appear to be a s i n g l e p e r i o -d i c term but w i t h a net temperature-dependent phase (or frequency). From the known frequency v a r i a t i o n of the three l e n s branches i t i s estimated i n §4.3 that an apparent phase s h i f t of about one part i n 10^ would r e s u l t , upon warming the sample from 1° to 4°K, i f the m i s a l i g n -ment of Ji from the [111] d i r e c t i o n were approximately ±1.0° f o r molyb-denum (and approximately ±4.0° f o r i r o n ) . In order to avoid these p o s s i b l e systematic e r r o r s , the magnetic f i e l d was a l i g n e d to l i e p a r a l l e l to the [111] a x i s to an accuracy of 47 about ±1/4° i n Mo and ±1/2° i n Fe. F i r s t the sample was mounted i n a compensated pick-up c o i l f i x e d to the cryostat, and then back r e f l e c t i o n x-ray photography was used to ensure that the [111] axis was i n the plane of r o t a t i o n of the electromagnet to approximately ±1/4°. The electromagnet was then rotated to the f i n a l p o s i t i o n using the dHvA e f f e c t i t s e l f to adjust the alignment. The [111] ax i s was f i r s t located approximately by making use of the f a c t that the beat pattern produced by the three e - o s c i l l a t i o n s must disappear when i s along the [111] d i r e c t i o n . The f i n a l adjustment was made to an o v e r a l l accuracy of ±1/4° f o r Mo and ±1/2° f o r Fe from considerations of the peak to peak amplitude, which must be a maximum when ]_ i s p r e c i s e l y p a r a l l e l to the [111] d i r e c t i o n where a l l three £-oscillations add coherently. 48 §3.6 Modulation F i e l d Technique and Electronics The G.-oscillations were studied by the conventional f i e l d -modulation technique i n which a weak low-frequency magnetic f i e l d h s i n cot i s superimposed on the d.c. induction B. The dHvA magnetiza-ti o n then varies i n time and the induced emf i n a pick-up c o i l surround-ing the sample i s proportional to -ft ^ = dt" [ A ( T * B + h s i n ^ s i n <B 4- h l i n cot + * ) ] 3 " 5 a = ? 2ncoJ (p) [A(T,B) s i n + cb + SI) ] s i n (ncot + ^ ), n=l B 2 2 3 > 5 b 2 where J i s the nth Bessel function of the f i r s t kind and u = 2irFh/B . n Eq. 3.5b i s v a l i d i f h i s small compared to B (cf. Gold 1968) and gives the Fourier components of dM/dt which are i n each case proportional to the dHvA magnetization M(T,B) (except for a constant phase s h i f t nTr/2). Phase sensitive detection was used to extract the amplitude of the 2nd or 4th time-harmonic of the induced emf, using a modulation frequency between 40 and 50 Hz (which was low enough to ensure complete penetra-tion of the modulation f i e l d into the sample). The modulation f i e l d was provided by Helmholtz c o i l s mounted on the pole faces of the elec-tromagnet and the amplitude h could vary up to about .360G and was chosen to maximize (as much as possible) the appropriate Bessel function i n Eq. 3.5b. The pick-up c o i l was a compensated design consisting of 3200 turns of 46 AWG copper wire wound i n one d i r e c t i o n plus an additional 1500 turns 49 wound i n the reverse d i r e c t i o n . The exact number of reverse windings was chosen (to the nearest turn) so as to minimize the d i r e c t pick-up from the uniform modulation f i e l d . The output of the pick-up c o i l was fed v i a a low noise matching transformer i n t o a low n o i s e d i f f e r e n t i a l a m p l i f i e r and then to a phase s e n s i t i v e d e t e c t o r f o r demodulation. The detected v o l t a g e was c o l l e c t e d i n analog and d i g i t a l form on a s t r i p chart recorder and magnetic tape, r e s p e c t i v e l y . The b a s i c e l e c t r o n i c s was the same as that reported by P. Schmor (1973) and i s shown i n b l o c k form i n F i g . 3.5. The d e t e c t i o n of s i g n a l s somewhat s m a l l e r than 10 v o l t s was r e a d i l y achieved by m i n i m i z i n g v i b r a t i o n s of the sample and pick-up c o i l i n the magnetic f i e l d , by c a r e f u l l y compensating the p i c k -up c o i l to r e j e c t emf's a r i s i n g from sources other than the sample, and by a p p r o p r i a t e l y s h i e l d i n g the l o w - s i g n a l c i r c u i t r y (the experiment was c a r r i e d out i n a m e t a l - s h i e l d e d room). Prec a u t i o n s were taken to avoid i n t r o d u c i n g temperature-dependent s h i f t s i n the phase of the measured G - o s c i l l a t i o n s which could a r i s e e i t h e r from the e l e c t r o n i c system or from the presence of background s i g n a l s i n the data. In p a r t i c u l a r i t was necessary to avoid any e l e c -t r o n i c f i l t e r i n g which could i n v o l v e d i f f e r e n t i a t i o n or i n t e g r a t i o n of the demodulated (dHvA) v o l t a g e f o r the purpose of removing i n t e r f e r i n g s i g n a l s . Because the amplitude A(T,B) of the dHvA magnet i z a t i o n i s f i e l d and temperature dependent, i t f o l l o w s that any d i f f e r e n t i a t i o n or i n t e g r a t i o n of M w i t h respect to B would r e s u l t i n a s i g n a l having a net temperature dependent phase ( s i m i l a r e f f e c t s not a r i s i n g d i r e c t l y from the e l e c t r o n i c s are discussed i n §4.3). Background s i g n a l s were essen-t i a l l y e l i m i n a t e d by paying p a r t i c u l a r a t t e n t i o n to t h e i r source and any necessary f i l t e r i n g of the data was c a r r i e d out by the numerical a n a l y s i s Face of Page 50 Figure 3.5 Block diagram of the basic e l e c t r o n i c s used to study the dHvA e f f e c t by means of the field-modulation tech-nique. The low noise transformer, d i f f e r e n t i a l ampli-f i e r and phase s e n s i t i v e detector were Princeton Applied Research models AMI. 113 and 124, r e s p e c t i v e l y . Modulation Coils Sample and Pick-up Coll UA 2 UA Low Noise Transformer Low Noise Amplifier High Power Amplifier Precision Low Frequency Oscillator UA Band Pass Filter dHvA signal Phase Sensit ive Detector Frequency Multiplier x n n u A Reference Signal Amplitude of the nth harmonic of the dHvA signal Strip Chart Recorder Interface and Magnetic Tape Recorder Computer Analysis 51 described i n the following s e c t i o n . Field-dependent background si g n a l s could o r i g i n a t e from the ferromagnetic sample i t s e l f and from the modulation f i e l d which became very s l i g h t l y d i s t o r t e d (0.1% harmonic d i s t o r t i o n ) due to magnetic i n t e r a c t i o n s with the pole faces of the electromagnet. The background s i g n a l s , appearing whenever the saturation magnetization i n the sample was not homogeneous and oriented p a r a l l e l to the applied f i e l d H, were reduced by means of the sample preparation technique described i n §4.2. The other field-dependent background s i g n a l , a r i s i n g from the d i s t o r t i o n of the modulation f i e l d by the i r o n pole faces of the electromagnet, was most conveniently eliminated by improving the e f f e c t i v e s i g n a l compen-sation of our pick-up c o i l . A second small c o i l was wound outside the compensated c o i l described e a r l i e r and the induced emf was passed v i a a low noise transformer to a phase s h i f t e r and attenuator and then subtracted from the voltage of the f i r s t c o i l at the d i f f e r e n t i a l ampli-f i e r . By c a r e f u l l y adjusting the amplitude and phase of the voltage from the second c o i l i t was p o s s i b l e to reduce the unwanted background voltage by a f a c t o r of 20 or more over the required f i e l d range of about 3 kG. 52 §3.7 Numerical Analysis The phases of the recorded . - o s c i l l a t i o n s were determined from a computer a n a l y s i s s p e c i f i c a l l y designed to help remove noise, r e s i d u a l background si g n a l s and other dHvA components. The data f i r s t underwent a preprocessing i n which the record length was standardized to an i n t e g r a l number of dHvA cycles and the d.c. l e v e l as well as the f i e l d dependence of the amplitude of the o s c i l l a t i o n s were removed. . Both Fourier a n a l y s i s and a least-squares f i t t i n g procedure, described below, were then used to compute the phase of the o s c i l l a t i o n s r e l a t i v e to a f i x e d f i e l d . A f t e r preprocessing, the data from a s i n g l e f i e l d sweep may be represented by an equation of the form y(x) = a^ cos 2TTFX + a- s i n 2TTFX + d(x). 3.6 The parameter x measures 1/B r e l a t i v e to a reference 1/B_ and v a r i e s from -Ax/2 to +Ax/2, where Ax i s the record length i n the r e c i p r o c a l f i e l d . In Eq. 3.6 the f i r s t two terms combined are proportional to the fundamental term of the dHvA magnetization with frequency F and phase 0 = ( l / 2 i . ) t a n ~ 1 ( a 1 / a - ) cycles at x = 0. The l a s t term d(x) i n Eq. 3.6 represents a small d i s t o r t i o n a r i s i n g from noise, other dHvA components i and backgrounds. From Eq. 3.6 the cosine (1) and sine (2) transforms of y(x) have the general form a. Ax Y l ( f ) = ~1— [ d i f f ( f ~ F ) + d i f f ( f + F ) J + ° l ( f ) 3 , 7 a 53 a „ A x Y 2 ( f ) = - | — [ d i f f (f - F) - d i f f (f + F)] + D-(f) , 3.7b where d i f f (f) i s the usual d i f f r a c t i o n function (1/TrfAx) s i n (TrfAx) and D . ( f ) , D_(f) are the respective transforms of d ( x ) . In the Fourier a n a l y s i s technique and Y^ were c a l c u l a t e d numer-i c a l l y from the data f o r f = F, and the measured phase (at each temper-ature T) was determined from the formula 1 -1 Y 1 ( F ) 9 M(T) = ^ tan . It can be shown from Eqs. 3.7a,b that the d i f f e r e n c e [0^.(T) - 0(T)] changes by l e s s than 0.001 c y c l e between 1° and 4°K provided that the f r a c t i o n ( F ) I + |D_(F)| 2 2 1/9" [Y. Z(F) + Y 2 Z ( F ) ] 1 / Z changes by l e s s than about 0.01 over t h i s temperature range, assuming that Ax i s kept e s s e n t i a l l y constant. The v a r i a t i o n s i n D^(F) and D 2(F) due to e l e c t r o n i c noise were simply averaged out at each temper-ature and the v a r i a t i o n s due to the presence of any other dHvA component were made small by choosing a s u f f i c i e n t l y long record length (about 5 cycles i n Fe), by using i n some cases a Gaussian 'window' which reduced the side bands or the d i f f r a c t i o n function of the unwanted frequency, 54 and by taking FAx equal to an integer so that (F) and D_(F) are iden-t i c a l l y zero for a l l the harmonics r = 2, 3, . . . o f the £-oscillations. In the l e a s t square f i t t i n g technique the data y(x) was f i t t e d to a function of the form y f i t ( x ) = a 1 cos 2TTFX + a. s i n 2TTFX + a- cos 2T T F ' X + ct^ s i n 2T T F ' X + a_x 3.8 and the measured phase was determined by the formula 1 -1 " l 8 M(T) = .=- tan •"•(—) . M 2TT va_ The l a s t three terms i n Eq. 3.8 correspond to the dominant part of the d i s t o r t i o n d(x) (other than e l e c t r o n i c n o i s e ) , and the values of a-, ct^ and ct_ could also be used to determine roughly the accuracy i n the phase s h i f t measurements by Fourier a n a l y s i s (and i f necessary the estimated 3rd and 4th terms i n Eq. 3.8 could be subtracted from the data before Fourier a n a l y s i s ) . In i r o n , F' was set equal to 1.42 MG, the value of a weak neighboring frequency 'rediscovered' during the course of the measurements (see §5.1), whereas i n molybdenum the 3rd and 4th terms i n Eq. 3.8 were not required. As i n the Fourier a n a l y s i s approach, the interference from noise was reduced by averaging up to 15 determinations of the phase at each temperature and 9 as a function of T was determined for several close values of F and F' to ensure that any uncertainty i n F and F' did not e f f e c t the phase-shift r e s u l t s . Furthermore, as an 55 a d d i t i o n a l check that i n t e r f e r i n g s i g n a l s did not play a s i g n i f i c a n t r o l e i n the f i n a l r e s u l t s both the F o u r i e r and c u r v e - f i t t i n g analyses were repeated f o r various f i e l d i n t e r v a l s ( i . e . , for d i f f e r e n t center f i e l d s B q and record lengths Ax). The e n t i r e phase an a l y s i s program was thoroughly pretested with synthetic data, and the two methods of a n a l y s i s , Fourier and l e a s t squares, were always found to give the same phase r e s u l t s f or the measurements reported here i n both i r o n and molybdenum, to an accuracy of about 0.0003 c y c l e between 1° and 2.5°K and. 0.001 c y c l e up to 4°K (at which temperature the s i g n a l was only a few times stronger than the background n o i s e ) . Chapter IV RESULTS AND THEIR INTERPRETATION 56_7. §4.1 Experimental Results Measurements of the temperature dependence of the phase of the £-oscillations i n Fe and Mo were made at two d i f f e r e n t applied f i e l d s and the corresponding r e s u l t s for the frequency s h i f t s are given i n F i g . 4.1. Each point i s obtained from an average of 5 to 15 measure-ments and the error bars give the standard d e v i a t i o n . The data i n F i g . 4.1 show that the lens frequencies for Fe and Mo are both essen- t i a l l y independent of the temperature to an accuracy of about one part i n 10 5 between 1° and 4.2°K. By way of comparison we also show i n F i g . 4.1 the curve expected f o r the Fe lens frequency on the basis of a l i t e r a l i n t e r p r e t a t i o n of the Stoner model (Eq. 2.12) i f M i s regarded as the t o t a l change i n the magnetization, i . e . , a r i s i n g from both spin waves and s i n g l e - p a r t i c l e e x c i t a t i o n s . In accordance with the discussion at the end of §2.2 t h i s curve should be scaled i n proportion to the r e c i p r o c a l of the electron-phonon-magnon renormalization f a c t o r which i s expected to have a value between one and two i n i r o n . The observed frequency s h i f t s are quite i n s i g n i f i c a n t (and of opposite sign) when compared to the s h i f t s predicted by the Stoner analysis given i n §2.2, and we conclude that the exchange s p l i t t i n g does not vary with temperature i n proportion to  the t o t a l magnetization. We defer u n t i l §4.3 a d i s c u s s i o n of p o s s i b l e o r i g i n s of the small negative frequency s h i f t s observed and we s h a l l be concerned here and i n the following section with the i n t e r p r e t a t i o n of our fundamental r e s u l t s . If we write the change i n the t o t a l magnetization as the sum of contributions from Stoner s i n g l e - p a r t i c l e e x c i t a t i o n s (sp) and from Face of Page 57 F igu re 4.1 Temperature dependence of the [111] dHvA f requency f o r the e l e c t r o n l e n s e s : 0 Fe H = 17 .0 kG ; A Fe H = 21.0 kG ; 0 Mo H = 19.5 kG; A Mo H = 23.0 kG. The s o l i d curve g i v e s the f requency s h i f t expected f o r the Fe l enses i f the exchange s p l i t t i n g i s p r o p o r t i o n a l to the t o t a l magne t i za t i on at a l l temperatures and t a k i n g m* i n Eq. 2.11 to be the observed c y c l o t r o n mass. 57 T(°K) 58 spin-waves (sw), then our r e s u l t s show that i t i s i n c o r r e c t to w r i t e — = (6M • + 6M )/M , 4.1 A sp sw 7 o ' o i . e . , the s i n g l e - p a r t i c l e and spin-wave c o n t r i b u t i o n s cannot be simply added together as f a r as the change i n the band s p l i t t i n g i s concerned, which i s c o n t r a r y to the c o n c l u s i o n a r r i v e d at by Thompson, Wohlfarth and Bryan (1964). On the other hand very r e c e n t l y Edwards (1973) has shown from an a n a l y s i s of the coupled system of q u a s i p a r t i c l e s and spin-waves, that the c o r r e c t r e l a t i o n f o r SA/A should be of the form o k_,T | A = ( a 6 M + B p _ 5 M s w ) / M o t 4 > 2 0 o where the dimensionless q u a n t i t i e s a and 8 are of order u n i t y . R e c a l l i n g that the exchange s p l i t t i n g A q i s approximately 2.0 eV, then at l i q u i d helium temperatures the f a c t o r k_-T/AQ i s of the order of 10 ^  so that the s p i n wave c o n t r i b u t i o n to <$A/AO i n Eq. 4.2 i s n e g l i g i b l y s m a l l . Furthermore, evidence from the low temperature magnetization data (Argyle et a l . 1973, Legkostupov 1971, Aldred and Froehle 1971) i n d i c a t e that f o r i r o n ( a l s o N i and Co) the spin-wave e x c i t a t i o n s are dominant so that <5Mg_ << <5M_w (except at very low temperatures, T < 1°K, and high f i e l d s , H , 30 kG, when the s p i n wave energy gap discussed i n §4.2 becomes imp o r t a n t ) . Consequently, f o r i r o n , Edwards' Eq. 4.2 p r e d i c t s a n e g l i g i b l y small change 6A/A_ i n the band s p l i t t i n g at low  temperatures, which i s c o n s i s t e n t w i t h the dHvA r e s u l t . 59 Because at low temperatures the spin-wave c o n t r i b u t i o n to <5A/AQ i n Edwards' r e s u l t (Eq. 4.2) can be ignored, we may then write A M o o The change i n the s i n g l e p a r t i c l e magnetization i s defined by the expression 6M <$n_(T) - 6n, (T) 4.4 Sp _ +_ +• M. n (0) - n,(0) ' 0 T + where 6n_(T) = n_(T) - n A ( 0 ) , 6n,(T) = n,(T) - n,(0) and n. and n, are T T T T T T T + the t o t a l number of T and + spin conduction electrons (per atom). Then from Eqs. 2.8a,b f o r n.(T) and n,(T), Eqs. 2.10a,b for Se_(T) and 6e,(T) T T T T and Eq. 4.3 we obtain the standard r e s u l t (for a weak ferromagnet l i k e iron) that f o r low temperatures <5M -§_ = _ __ T 2 M M o o 4.5a where f o r the model described here (see also Edwards 1973) M o _-(n+(0> - n ^ O ) ) ^ " 1 * V 1 ) " ^ ' 4.5b 60 From the experimental r e s u l t s f or 6F/F (corrected for the small e f f e c t to be discussed i n §4.3) together with Eq. 2.11 f o r 6F/F and Eqs. 4.3 and 4.5a we determine an upper l i m i t where p i s the electron-phonon-magnon enhancement f a c t o r f or the e l e c t r o n lens which was discussed at the end of §2.2. The parameter p i s a number between one and two and a i s i d e n t i c a l l y equal to one f o r the model f o r i t i n e r a n t ferromagnetism discussed i n the following s e c t i o n ; therefore the f a c t o r p/ct i s expected to be of order u n i t y , The estimate for |A/M q| from the dHvA data, as interpreted using Edwards' formula Eq. 4.2 or 4.3, i s of the same order of magnitude as that obtained from the curve f i t t i n g analyses of the magnetization data of Argyle et_ a l . A/M < ^ x 10 o' a 8 ( °K) -2 (1963) above 4°K, g i v i n g A/M = (1 ± 3) x 10"° (°K) -2 and of Legkostupov (1971) below 4°K, g i v i n g A/M q = (11.5 ± 3.5) x 10 (°K) -2 In a r r i v i n g at our upper l i m i t we have ignored the small c o n t r i b u t i o n to Eq. 2.11 a r i s i n g from the s h i f t i n the chemical p o t e n t i a l with temp-erature, which i s estimated to be n e g l i g i b l e i n the temperature range of our experiment (unless a very large gradient i n the e l e c t r o n i c density of states happens to e x i s t j u s t at the t or I Fermi l e v e l ) , 62 §4.2 T h e o r e t i c a l I n t e r p r e t a t i o n As pointed out by Edwards (1973) the temperature dependence of' the exchange s p l i t t i n g can be conveniently studied ( for low temperatures) by means of Landau's theory for Fermi l i q u i d s , as generalized by Izuyama and Kubo (1964) to include spin-wave (magnon) e x c i t a t i o n s . The basic assumption i n t h i s phenomenological theory i s that the e l e c t r o n i c energy E of the c r y s t a l i s a f u n c t i o n a l of the e l e c t r o n - q u a s i p a r t i c l e occupation numbers n(ka) and the magnon-quasiparticle occupation numbers u(q), where k and a give the wavevector and spin d i r e c t i o n of the e l e c t r o n -states and q_ l a b e l s the magnon s t a t e s . Then at low temperatures E can be expanded i n t o the form E({n(ka)},{u(q)}) = E + Z e (ka) 6n(ka) + I tiu (q) u(q) — o . o — — 0 +1/2 1 f (ka.k'a') 6n(ko) 6n(k*a') ka e e k'a* + E f (ka,q) <5n(ka) u(q) . em — — ka _ + 1/2 1 f (__,_.') v(s) u(q') + . . . , 4.6 where E q i s the ground state energy and 6n(ka) and u(q_) are the number of thermally excited electrons (or holes) and magnons, r e s p e c t i v e l y , The other parameters i n Eq. 4.6 are the usual Taylor expansion c o e f f i -c i e n t s defined by the formulae 63 8E e (ka) = -—.? . o — ' 3n(ka) 4.7a 9E two (q) = , ° o ^ 9u(__) 4.7b 3 2E 3e (ka) f .lea lf'aM = - 0 ~ ee^- ; 8n(ka) 3n(k'a') 9n(k'a*) 4.7c 8 2E 9e (ka) 3two (q) o — o -1-fem (- a ,^ ) 3n(ka) 3u(__) 9u(__) 9n(ka) 4.7d 9 2E 3tia) (q) f fa a'"i = o _ o mm^'1 ; 9u(__) 3u(__') 9u(__*) * 4.7e The q u a s i p a r t i c l e energies e(k_,a) and two(q_) at f i n i t e temperatures are equal to the change i n the t o t a l energy E i f v i r t u a l q u a s i p a r t i c l e s are introduced i n statesQc, a) and <_, r e s p e c t i v e l y . Therefore from Eq. 4.6 we f i n d e(ka) = e (ka) + Z f (ka,k*a') 6n(k'a') 0 k'a' e e + E fem(h?>2) U(SL> 4.8a hco(£) = hcoo(c_) + E (__,£') u(__') + E f (ka,__) u ^ . 4.8b 3.' ka In these equations E q (ka) and two (__) are i d e n t i f i e d as the q u a s i p a r t i c l e energies at T=0°Kand f , f and f are the e l e c t r o n - e l e c t r o n , ee em mm ' electron-magnon and magnon-magnon i n t e r a c t i o n functions. Given f and ee 64 f , then from the known Fermi d i s t r i b u t i o n function f o r 6n(ka) and the em — Bose d i s t r i b u t i o n function f o r u(__) i t i s pos s i b l e to determine (from Eq. 4.8a) _(k f) and e(k + ) and therefore the exchange s p l i t t i n g A as a function of temperature. The e n t i r e problem i s thus centered around the determination of f and f . The Fermi l i q u i d theory i t s e l f does ee em J not provide these parameters and i t i s necessary to resort, to a micro-scopic theory or experiment f o r t h e i r evaluation. In the following s e c t i o n f and f are determined f o r a simple microscopic theory. ee em r v . J Then i n §4.2b these w i l l be obtained by the more general (though pheno-menological) approach due to Edwards (1973). 65 §4.2a Ferromagnetism In a Narrow Band In t h i s s e c t i o n we consider the ferromagnetism i n a p a r t i a l l y f i l l e d and narrow energy band characterized by the non-interacting e l e c t r o n energies {EJ-}> which are assumed known.* Although t h i s s i n g l e band model i s not d i r e c t l y a p p l i c a b l e to i r o n (which has several p a r t i a l l y f i l l e d d bands), i t does nevertheless provide considerable p h y s i c a l i n s i g h t on the general properties of the e l e c t r o n - e l e c t r o n and e l e c t r o n -magnon i n t e r a c t i o n . * * If we define (j>(r) as the nondegenerate atomic state associated with our narrow band then the Bloch state i j . ^ can be wr i t t e n approximately i n the t i g h t - b i n d i n g form _ 1 / 2 i - ' ^ i * k __ = N a *j e • _ " _ i ) » 4 -where the sum runs over a l l atomic s i t e s i . Roughly speaking, an elec t r o n i n t h i s Bloch state moves through the c r y s t a l by hopping from one atomic o r b i t a l to the next. If the atomic o r b i t a l s are f a i r l y w e l l l o c a l i z e d around each atom the Coulomb i n t e r a c t i o n between any two electrons i n the band w i l l be r e l a t i v e l y small except when they happen *Following usual convention we w i l l denote the one-electron band energii i n the absence of i n t e r a c t i o n s by a n < * the q u a s i p a r t i c l e energies i n the presence of i n t e r a c t i o n s by e(k_,a) (as defined by Eq. 4.7a, 4.8a). **We also note that the model has been used with some success to study the ferromagnetism i n n i c k e l (whose d bands are nearly f u l l ) . 66 to be on the same atom. In that case the Hamiltonian f o r the e l e c t r o n s i n the band should be given approximately by the expression ft = E E, n, + I E n. _ n. , , 4.10 ka — — where n. = a . a, and n. = a . a. are the u s u a l occupation number ka ka ka xa l a l a operators f o r the e l e c t r o n s i n Bloch s t a t e (k,a) and i n the atomic o r b i t a l s t a t e on the i t h atom, r e s p e c t i v e l y , and I i s the Coulomb i n t e r -a c t i o n energy f o r two e l e c t r o n s on the same o r b i t a l . The f i r s t term i n Eq. 4.10 i s the Hamiltonian f o r n o n - i n t e r a c t i n g e l e c t r o n s and the second term represents the t o t a l Coulomb i n t e r a c t i o n s f o r p a i r s of e l e c t r o n s on the same atom; on account of the P a u l i e x c l u s i o n p r i n c i p l e these e l e c t r o n p a i r s must have a n t i p a r a l l e l s p i n s . The Hamiltonian (Eq. 4.10) was obtained from more r i g o r o u s arguments by Hubbard (1963) and has been used e x t e n s i v e l y to study Coulomb c o r r e l a t i o n s among i t i n e r a n t e l e c t r o n s . Here we s h a l l o n l y d i s c u s s the Hamiltonian w i t h i n the g e n e r a l i z e d Hartree-Fock and random phase approximation (HFA and RPA). In the s t r a i g h t f o r w a r d Hartree-Fock approximation we ignore Coulomb c o r r e l a t i o n s and assume that two a n t i p a r a l l e l e l e c t r o n s have the same p r o b a b i l i t y of c o e x i s t i n g on the same atomic o r b i t a l as they would i f they were n o n - i n t e r a c t i n g . Then from Eq. 4.10 the energy of the e l e c t r o n gas i n the HF approximation i s found t o be EHF = E e k n ( - 0 ) + N a 1 n-r n+ ' 4 - 1 1 ka — where n(ka) i s the occupation number f o r the Bloch s t a t e ( k , a ) , n and 67 n, are the number of + and + spins per atom, and N i s the number of 4- a atoms i n the c r y s t a l . From a perturbation expansion of the energy (for the Hubbard Hamiltonian Eq. 4.10) Kanamori (1963) has shown that, at l e a s t for low ele c t r o n d e n s i t i e s , some e f f e c t s of Coulomb c o r r e l a t i o n s can be accounted f o r by simply replacing I i n Eq. 4.11 by an e f f e c t i v e i n t e r a c t i o n I which i s always smaller than I and which i s given approximately by I e f f W » 4 , 1 2 2 e E F where e i s the Fermi l e v e l (measured from the bottom of the band) f o r the noninteracting electrons and W i s the width of the band. The phy s i c a l i n t e r p r e t a t i o n of t h i s r e s u l t can be described as follo w s . When I i s lar g e , electrons w i l l avoid being on the same atom so that the t o t a l Coulomb energy w i l l be reduced from that computed i n the Hartree-Fock approximation. At the same time however the k i n e t i c energy of the electrons must increase since the volume of the c r y s t a l a v a i l a b l e to each el e c t r o n i s smaller than that i n the noninteracting case. When !-*•<*> t h i s increase i n k i n e t i c energy then corresponds to the e f f e c t i v e magnitude of the i n t e r a c t i o n and I ej£ then depends only on the property of the band and not on I. Subst i t u t i n g the appropriate I ££ i n Eq. 4.11 and r e w r i t i n g 1 t v 2 1 . .2 68 we then obtain n, N a T e f f , ,2 h - 1. e n(k,a) ^ (n. - n,) + constant ka - 4 + r 4.13a / ey(e)de + /* ex(E)de - (n - n , ) 2 U O £f T T N + constant. 4.13b This r e s u l t , which i s not simply the Hartree-Fock approximation, provides the basis f o r the Stoner theory which we described i n the i n t r o d u c t i o n to t h i s t h e s i s . * The c r i t e r i o n f o r ferromagnetism i n the ground state can 2 2 be determined by r e q u i r i n g that [d E/d(n. - n.) ] < 0 for (n. - n,) -> 0; T Y T Y from Eq. 4.13b t h i s leads to the w e l l known Stoner cond i t i o n JfUJ I e f f > 1 , where e p i s the Fermi energy of the noninteracting system. From our d e f i n i t i o n of the q u a s i p a r t i c l e energies and i n t e r a c t i o n f u nction f _ g i n the Fermi l i q u i d theory, i t follows from Eq. 4.11 (with I replaced by I e f f ) that i n the Stoner theory 3E £ ( k- o ) = anlkaT " ek + n-a ^ f f ' 4.14a so that f (kn k ' d M = 9 £ ( k q ) _ T e f f . 4.14b *Stoner was the f i r s t to propose, on phenomenological grounds, an expression s i m i l a r to Eq. 4.13 for a free e l e c t r o n gas. 69 where -a r e f e r s to the spin o r i e n t a t i o n opposite to a and 6 , i s r a ,-a zero unless a' = -a. The exchange s p l i t t i n g between the t and 4- quasi-p a r t i c l e bands i s then given by A = _(k+) - e(k-r) = (n - n,) I , 4.15 — — T + err i . e . , A i s proportional to the s i n g l e - p a r t i c l e magnetization i n t h i s approximation. The excited states i n the Stoner theory are p a r t i c l e - h o l e e x c i t a t i o n s defined by V = a +. . , a. V , 4.16a _L k+q_,a ka o ' where . i s the Stoner ground st a t e , a +, , . a, are p a r t i c l e - h o l e o k+q_,a ka r c r e a t i o n operators; we w i l l be i n t e r e s t e d e s p e c i a l l y i n s p i n - f l i p e x c i t a t i o n s so that a = + and a 1 = +. A s p i n - f l i p or s o - c a l l e d Stoner e x c i t a t i o n corresponds to a t r a n s f e r of an e l e c t r o n from an occupied state (k+) to a previously empty state (k+q_,4), i n the same band (see F i g . 4.2a). Such an e x c i t a t i o n reduces the net magnetic moment by an amount 2u_ and increases the t o t a l e l e c t r o n i c energy by fi_>(q_) = e(k+£,4-) - _ 0 _ T) = - _ k + A . 4.16b The l a s t r e l a t i o n follows from Eqs. 4.12a, 4.15. For a parabolic band, t i 2 2 with e, = * k » the e x c i t a t i o n energy h_(_i) reduces to Face of Page 70 Figure 4.2 F i g . (a) shows schematically a s p i n - f l i p or Stoner e x c i t a t i o n ; an el e c t r o n i n state (k_t) i s tr a n s f e r r e d to a state (k+g.,4-)- In F i g . (b) the + and 4- s p i n Fermi spheres are superimposed and each dashed l i n e represents a zero energy Stoner e x c i t a t i o n , which corresponds to the t r a n s f e r of an e l e c t r o n from the + to the 4- spin Fermi surface. f Spin 70 i Spin ( b ) 71 -h2 ? fiw(__) = - ^ j - - [2k-£ + q ] + A , 4,16c and because the i n i t i a l state (k+) must be occupied and the f i n a l state (k+q_, 4-) must be empty the wavevector 1c must s a t i s f y the conditions |k| < k + and |k+qj > k^, where k^ and k + are the r a d i i of the t and + Fermi spheres as shown i n F i g . 4.2a. For each q there e x i s t s , i n general, a large set of solutions tiw(c_) which give r i s e to the shaded continuum on the hto(<_) versus q graph i n F i g . 4.3. The upper boundary l i n e on t h i s graph i s obtained by choosing k«_c_ to be the maximum allowed value +k.q so that T [ ^ ( i ) ] m a x = l m * ( 2 V + £12) + A Also f o r q < ( k + - k + ) the lower boundary l i n e i s found by choosing to be the minimum allowed value -k.q so that T For the values of q greater than k. - k, the l a s t equation above does not hold since the s u b s t i t u t i o n of k_'<_ by -k^q v i o l a t e s the requirement |k + __| > k^. The lower boundary l i n e f o r k^ - k^ < q < k^ + k^ a c t u a l l y f a l l s on the q axis and each e x c i t a t i o n l y i n g on t h i s l i n e corresponds to a tr a n s f e r of an electron from the + spin Fermi surface to the 4 spin Fermi surface, with zero e x c i t a t i o n energy (see F i g , 4,2b), When q Face of Page 72 Figure 4.3 Magnetic e x c i t a t i o n spectrum for the Hubbard Hamil-tonian i n the random phase approximation. Both the t and 4- spin bands are assumed to be- p a r t l y f i l l e d (weak ferromagnetism) so that Stoner e x c i t a t i o n s of zero energy, but large momentum q, are allowed. hu A 73 becomes greater than k + k the lower boundary f o r Tiu.(q) increases i n T T energy since t r a n s f e r s from the i spin to the 4- spin Fermi surfaces are no longer p o s s i b l e . In order to obtain c o l l e c t i v e or spin-wave e x c i t a t i o n s , which are believed to e x i s t i n an i t i n e r a n t ferromagnet, i t i s necessary to take into account Coulomb c o r r e l a t i o n s i n a better approximation than i n the Stoner theory, i n which each e l e c t r o n experiences e s s e n t i a l l y a mean f i e l d due to a l l other e l e c t r o n s . The approach which i s most frequently used i s analogous to that applied to the study of plasmons i n normal metals and has been often c a l l e d the random phase approxima-t i o n . In t h i s approximation we assume that the wavefunction f o r a c o l l e c t i v e e x c i t a t i o n can be w r i t t e n as a l i n e a r combination of Stoner ( s p i n - f l i p ) e x c i t a t i o n s of the general form • V = E c. a +, . , a. . . , 4.1 _ k — —^—' — 0 where k i s assumed to l i e i n s i d e the t Fermi surface and k+c_ i s outside the 4- Fermi surface. The c o e f f i c i e n t s c^ and the energy !._>(__) of the e x c i t a t i o n s can now be determined ei t h e r by a v a r i a t i o n a l method, or by the l i n e a r i z a t i o n of the equations of motion, i n which we use the Hubbard Hamiltonian (Eq. 4.10). The r e s u l t of such c a l c u l a t i o n s (cf. Herring 1966) i s that the e x c i t a t i o n energies fiu.(__) of the f e r r o -magnet must s a t i s f y the equation N = E n ( k t ) ( l - n(k+q,4)) L e f f k "k+£ E k + A - h_(g) 4.18 74 where N i s the number of atoms i n the c r y s t a l and I r c has replaced a J e f f r the parameter 1 i n the Hubbard Hamiltonian i n order to account for higher order c o r r e l a t i o n e f f e c t s . It i s w e l l known that t h i s equation has two types of solutions which we discuss with reference to F i g . 4.3. F i r s t there i s a large set of solutions tia)(__) given approximately by Hio(c_) = e f e - e k + A , 4.19 which are the energies of the s p i n - f l i p or Stoner e x c i t a t i o n s discussed e a r l i e r , and which give r i s e to the shaded continuum i n the tico(_j) versus q graph i n F i g . 4.3 (assuming a parabolic band). For s u f f i c i e n t l y l a r ge I g££ there w i l l a l s o . e x i s t , f or each c^ , a s o l u t i o n that corresponds to a c o l l e c t i v e (magnon) mode whose wavefunction c o n s i s t s of a s p e c i a l l i n e a r combination of many s p i n - f l i p e x c i t a t i o n s . For small q the spi n wave energy i s small and the r i g h t hand side of Eq. 4.18 can be expanded i n powers of 1/A. The r e s u l t f or the magnon energy i s ( c f , Herring 1966) hto(£) = Dq 2 , 4.20a where n 1 v .n(kt) + n(kr) ~ _ .2 n(kt) - n(kr) * _ .2, = (n - n )N I 2 ( ^ ' V £k A ^ " ^ ' V ^ K - h + a l c — — 4.20b It i s now p o s s i b l e to compute, within the framework of the RPA, the electron-magnon i n t e r a c t i o n . The quantity of i n t e r e s t f e f f l(ka,__) i s equal to the change i n the e l e c t r o n q u a s i p a r t i c l e energies e(ko) (Eq. 4.14) due to the i n t r o d u c t i o n of a magnon i n state c^ . On account 75 of equation 4.7d, f (ka,cj) i s also equal to the change i n the magnon energy "ho)(q_) a r i s i n g from a change i n the q u a s i p a r t i c l e d i s t r i b u t i o n n(ka). Therefore f can be computed d i r e c t l y from Eqs. 4.20a,b which give the e x p l i c i t dependence of hu)(c_), or more p r e c i s e l y D, on the q u a s i p a r t i c l e d i s t r i b u t i o n n(ka). By d i f f e r e n t i a t i n g Eqs. 4.20a,b we f i n d • IMS 9n(ka) N (n. - n.) — a T T v 2 ek - f £k ) 2 rn(k't) + n(k'Q » 2 N a ( n t " V k' _ 2(n(k» - n (k'Q) - ,2 4.21a where £ = +1 or -1 f o r a = + or 4-. A s i m i l a r i n t e r a c t i o n f u n c t i o n a (for a = +) was discussed by Herring (1966) and also by Izuyama and Kubo (1964) (who obtained the i n t e r a c t i o n function from a Green's function method) i n t h e i r c a l c u l a t i o n f o r the temperature dependence of tico(c_) using Eq. 4.8b. I f we now assume a parabolic band and n^ >> n^ then Eq. 4.21a reduces to the simple form f e m C£°>4> V a 1 1 _ 5 C t / M L (1 - {„> 4.21b where D i s the value of D at T = 0°K and e. i s the t spin Fermi l e v e l O T measured from the bottom of the band. The important feature of the 2 r e s u l t (Eqs. 4.21a,b) i s the q dependence of f which holds also f o r em more general models, provided there i s no magnetic anisotropy. The 76 effect of the latter w i l l be discussed in §4.2b. From Eq. 4.14b for fg.Oca.k'o') , Eq. 4.21b for f (ka,c_) and Eq. 4.8a for _(k,a) we can now compute the change in the exchange spli t t i n g A with temperature from SA = 6e (k+) - 6e ( k r ) 4.22 where 6e(ka) = e(ka) - £_(__). The calculation i s straightforward and the result i s - A SM k_T SM 6A _ sp , - B sw A M A M o o o o 4.23 where for our model a = 1 and 3 = - £ 3 £(5/2) 2 .(3/2) V 3 A 5 A ;  K 5 A ; The quantities .(3/2) and £(5/2) are Riemann's zeta functions defined by coo = 1 ^ T 7 y - - 3 - ^ - , r(z) o u . » e - 1 and the spin-wave magnetization i s given by 6 Msw • f} = U(-> = -2p_.(3/2) 4TTD 4.24 where u(cj) = (e - 1) i s the Bose d i s t r i b u t i o n f u n c t i o n f o r magnons, u i s the Bohr magneton and £2 i s the volume of the c r y s t a l . Our f i n a l r e s u l t (Eq. 4.23) i s of the same form as that g i v e n by Edwards (Eq. 4.2) whose parameters a and $ are a l s o of order u n i t y . We a l s o note that B depends on k_ so th a t s p i n waves can i n p r i n c i p l e d i s t o r t the + and 4- q u a s i p a r t i c l e bands as the temperature i s v a r i e d . 78 §4.2b General E l e c t r o n - E l e c t r o n and Electron-Magnon In t e r a c t i o n Functions In t h i s s e ction we discuss the general properties of the i n t e r a c t i o n functions f o r i r o n i n the manner given by Edwards (1973), extended s l i g h t l y to take into account the e f f e c t of magnetic anisotropy. The l a t t e r introduces a term i n the i n t e r a c t i o n function f which i s em independent of q, leading to a term i n 6A/A o which has the same temper-ature dependence as 6M sw From a general treatment of long wavelength spin-waves i n a c o n t i -nuous medium, Herring and K i t t e l (1951) have shown that the magnon energies are given approximately by a d i s p e r s i o n law of the form fico(q) = e + Dq 2 , 4.25a where e i s an energy gap for the e x c i t a t i o n of magnons. For the simple model, without anisotropy, discussed i n the l a s t s e c t i o n E was i d e n t i c a l l y z ero and D was given by Eq Q 4•20b» The energy gap e a r i s e s from the f a c t that i n order to create a magnon, which reduces the net magnetic moment of the c r y s t a l by an amount gu , a minimum energy B £ g = g y B H e f f 4 ' 2 5 b i s required, where H i s an e f f e c t i v e magnetic f i e l d given by an expression of the form (Argyle et a l . 1963) H = H + H, + H ef f d a 4.25c 79 The f i r s t term i s the applied f i e l d and i s a demagnetizing f i e l d (proportional to the magnetization) which r e s u l t s from surface magnetic poles as w e l l as the volume magnetic poles a r i s i n g from the nonuniform magnetization created by the spin wave i t s e l f . The l a s t term i n Eq. 4.25c i s an e f f e c t i v e anisotropy f i e l d which i s r e l a t e d to the f i r s t cubic anisotropy constant by K H a = - 4 / 3 * r > 4 - 2 5 d assuming the magnetization l i e s i n the [111] d i r e c t i o n ( cf. Chikazumi 1964); H a r i s e s from the f a c t that, on account of the s p i n - o r b i t in t e r -a c t i o n , the magnetization has a preferred d i r e c t i o n i n the c r y s t a l even i n the absence of an external f i e l d . To determine the approximate form of the electron-magnon i n t e r -a c t i o n f e i Q j we now consider the general dependence of the parameters M, and D i n Eqs. 4.25 on the s i n g l e p a r t i c l e magnetization SM^. The magnetization M i s defined by M = M + 6M + 6M , 4.26 o sp sw and the magnetization dependence of the anisotropy constant i n the Zener theory i s given by v m h_ _ M A K ° ~ CM~] » 4 ' 2 7 1 o 80 w i t h m^ = 10 f o r cubic c r y s t a l s ( c f . Chikazumi 1964), For i r o n K l e i n and K n e l l e r (1966) have found that below room temperature Eq. 4.27 gives f a i r agreement w i t h the experimental r e s u l t s f o r the temperature depen-dence of K^, but w i t h 3 < m_ < 6, For the purpose of making an order of magnitude estimate we s h a l l c o nsider m as a parameter having a v a l u e between one and ten, and we assume M to be the t o t a l magnetization as i n Eq. 4.26. The spin-wave c o e f f i c i e n t D can be r e l a t e d to the magnetization from c o n s i d e r a t i o n s of the neutron s c a t t e r i n g data of S p r i n g f e l l o w (1968) which i n d i c a t e that at low temperatures D i s given approximately by an equation of the form D = D - D,T2 . 4.28 o 1 This temperature dependence i s c o n s i s t e n t w i t h the form of f e m ( k a , q ) given f o r the simple model i n the previous s e c t i o n (Eq. 4.21), which 2 suggests that D^T a r i s e s p a r t i a l l y from s i n g l e - p a r t i c l e e x c i t a t i o n s between bands of the same s p i n and p a r t i a l l y from s p i n - f l i p e x c i t a t i o n s , The l a t t e r part should be roughly p r o p o r t i o n a l to the change i n the s i n g l e - p a r t i c l e magnetization so that we may w r i t e 6M SD _ sp . O Q — = ™ '- t 4.29 where 6D = D - D q, and m^ i s an unknown parameter which Edwards has set equal t o u n i t y , f o r the purpose of making an order of magnitude estimate. The electron-magnon i n t e r a c t i o n f u n c t i o n can now be estimated 81 from Eqs. 4.26, 4.27 and 4.29 (for M, and D respectively) which allow us to relate hu)(__) to the single particle magnetization. From "hio(c[) defined by Eq. 4.25 we then find (approximately) fem<^*> = HlkoO " N (n°- n ) ^ V ^ a ^ a + V + V o ^ > 4» — a t + 30 where i s +1 or -1 for a = + or +, respectively. The electron-electron interaction function f (ko.k'a') w i l l be ee — — assumed to be independent of _k and k', so that the interaction between quasiparticles is characterized by three numbers f e e(++)» f _ e ( Y Y ) and f (++) = f ( r + ) . ee ee In the Stoner model discussed in §4.2a the change in the exchange spl i t t i n g 6A was defined as 6e(k-t-) - SeOcf). However in the general Fermi-liquid theory the energies e(k.,a) are defined only near the Fermi level so that i n a ferromagnet e(k+) and e(k+) for a given k_ cannot both be meaningful quantities. Following Edwards we therefore define a differential quantity 6A by <5A = SeQc^) - 6e(1^+) , 4.31 where k^ and k^ are wavevectors close to the + and + spin Fermi surfaces respectively. Then the temperature-dependent exchange sp l i t t i n g can be defined as A = A + 6A , o 82 where A q is the parameter determined operationally as the splitting of the noninteracting energy bands required to obtain the observed magne-tization and Fermi surfaces at T = 0°K. With these definitions in mind we may now substitute f and f ee em from Eq. 4.30 into Eq. 4.8a for e(kcx) and determine <5A/A from Eq. 4.31. — o The calculation is completely analogous to that given by Edwards (1973) and the final result is * A 6 M o - k_T <5M 6M oA _ sp / B sw _, sw A ~ a M M - +T-M— • 4.32a o o o o o where N o - f 9 - [fC++) - y (fC++) + f(++))] 4.32b eff 1 _ 3m_ C(5/2) Z(5/2,e /k_T) B ~ ~2~ C £ ( 3 / 2 ) Z(3/2,e /k_T) ] 4 ' 3 2 c g B gUB[(tna - 1) H a + Hd] Y A * 4.32d o The functions Z are those defined by Argyle et a l . (1963) and appear because we have assumed a gap e for the magnon energies. The first two terms in Eq. 4.32a were those obtained by Edwards and a and g are quan-tities of order unity (in the Stoner theory a is exactly equal to one). The new term appearing in Eq. 4.32a has the same temperature dependence as the spin-wave magnetization and arises from the presence of the aniso-tropy and demagnetizing fields H and H. which depend on the single a d p a r t i c l e magnetization. Since for i r o n the quantity I (m - 1) H + H,, I ' a a d 1 3 i s of the order 10 G i t follows from Eq. 4.32d that y i s of the same order of magnitude as k^T/A q and therefore both of the spin-wave terms i n 6A/A are n e g l i g i b l e at l i q u i d helium temperatures. 84 §4.2c P h y s i c a l I n t e r p r e t a t i o n The p h y s i c a l i n t e r p r e t a t i o n of the r e s u l t t h a t magnons have n e g l i g i b l e e f f e c t on the band s t r u c t u r e f o l l o w s from our p i c t u r e of a s m a l l q spin-wave as being a wave-like disturbance of the d i r e c t i o n of the l o c a l m a g n e t i z a t i o n , which propagates through the c r y s t a l , , For an i t i n e r a n t e l e c t r o n ferromagnet, t h i s e x c i t e d s t a t e can be viewed c l a s s i c a l l y by imagining t h a t each e l e c t r o n (or q u a s i p a r t i c l e ) , w h i l e t r a v e l l i n g through the c r y s t a l , c o n t i n u a l l y v a r i e s i t s s p i n o r i e n t a t i o n so that i t i s always approximately i n phase w i t h that of neighboring e l e c t r o n s . Although a spin-wave reduces the o v e r a l l magnetization of the c r y s t a l , w i t h i n a r e g i o n much s m a l l e r than the magnon wavelength (2ir/q) the magnitude of the l o c a l magnetization i s e s s e n t i a l l y the same as that at T=0°K. For low val u e s of to(<_) , the frequency a t which the l o c a l magnetization v e c t o r precesses about the e q u i l i b r i u m d i r e c t i o n , the constancy of the magnitude of the l o c a l m agnetization leads us to expect a n e g l i g i b l e e f f e c t on the band s t r u c t u r e as proved e x p l i c i t l y by the t h e o r e t i c a l c o n s i d e r a t i o n s of the previous s e c t i o n s . As was emphasized i n §4.1 the dHvA r e s u l t s p o i n t to the co r r e c t n e s s of the above model f o r spin-waves i n an i t i n e r a n t ferromagnet. A l t e r n a -t i v e l y , i f we assume the v a l i d i t y of Edwards' theory from the s t a r t then the dHvA data can be i n t e r p r e t e d as p r o v i d i n g d i r e c t evidence t h a t the dominant l o w - l y i n g e x c i t a t i o n s i n i r o n (probably a l s o N i and Co) are spin-waves r a t h e r than Stoner e x c i t a t i o n s . As we pointed out e a r l i e r , t h i s c o n c l u s i o n has a l s o been a r r i v e d a t from c o n s i d e r a t i o n s of the temperature dependence of the magnetization which i s expected to c o n t a i n 2 a term p r o p o r t i o n a l to T from Stoner e x c i t a t i o n s and terms p r o p o r t i o n a l 3/2 5/2 to T and T ( i g n o r i n g anisotropy) from spin-waves at 85 low temperatures. The curve f i t t i n g analyses of the magnetization 3/2 data give evidence for a dominant T term i n both i r o n and n i c k e l , which i s consistent with the spin-wave theory. However i n weak i t i n e -rant ferromagnets, such as N i ^ A l and Z r Z ^ , the separation of the 2 3/2 5/2 magnetization data i n t o Stoner T terms and spin-wave T and T terms has not been achieved i n any convincing way (cf. deChatel and deBoer 1970). As emphasized by Edwards (1973) the temperature depen-dence of the dHvA frequency (Eq. 2.11) gives d i r e c t information about the change i n the s i n g l e - p a r t i c l e magnetization without in t e r f e r e n c e from spin-waves (Eq. 4.2).* Therefore the dHvA method provides for the f i r s t time, at l e a s t i n p r i n c i p l e , an unambiguous way of separating the spin-wave and s i n g l e - p a r t i c l e c ontributions to the magnetization. Such a separation would be of s p e c i a l i n t e r e s t i n view of the present ambi-guity regarding the t h e o r e t i c a l d e s c r i p t i o n of weak i t i n e r a n t ferromag-nets. Edwards and Wohlfarth (1968) have proposed a theory i n which Stoner e x c i t a t i o n s play a dominant r o l e i n these metals, whereas i n the f l u c t u a t i o n theory of Murata and Doniach (1972) , the d i s o r d e r i n g process as T c i s approached i s due s o l e l y to magnon-like e x c i t a t i o n s . We end t h i s s e c t i o n with a few comments about the nature of the band structure at high temperatures approaching the Curie point, The foregoing t h e o r e t i c a l d i s c u s s i o n was based on the Landau theory of Fermi l i q u i d s and therefore the r e s u l t for the temperature dependence of the exchange s p l i t t i n g applies r i g o r o u s l y only at low temperatures where the magnon and Stoner elementary e x c i t a t i o n s are well defined. However i f *As noted i n §2.2 the second term i n the expression for 6F/F i n Eq. 2.11 can i n p r i n c i p l e be eliminated e n t i r e l y from separate measurements of 6F/F for + and I sheets of the Fermi surface. 86 the magnetization i s reduced at high temperatures by spin f l u c t u a t i o n s (which correspond to magnons at low temperatures) rather than by Stoner-l i k e e x c i t a t i o n s , then i t may be pos s i b l e to extend our low temperature reasoning i n the following way (as proposed f o r example by S i l v e r s t e i n 1970). At high temperatures electrons may be moving through a s e r i e s of f l u c t u a t i o n regions each characterized by a l o c a l magnetization which has a magnitude close to the T=0°K value; the average magnetization of the c r y s t a l however could be close to zero. Such f l u c t u a t i o n regions would be s t a b i l i z e d f o r short time i n t e r v a l s by c o r r e l a t i o n e f f e c t s . For example, on account of the P a u l i p r i n c i p l e a 4- spin e l e c t r o n would tend to scatte r away from an + spin f l u c t u a t i o n region more r e a d i l y than would a f spin e l e c t r o n ; consequently, such a region would tend to be self-perpetuating. The e l e c t r o n i c density of states f o r the c r y s t a l would then be an average over a l l f l u c t u a t i o n regions, and apart from a broadening, the t o t a l density of states c l o s e l y resembles that at T=0°K. This i s i n contrast to the simple Stoner model i n which the reduction of the exchange s p l i t t i n g with temperature would cause the up and down spin bands to merge at T_ r e s u l t i n g i n pronounced changes i n the t o t a l density of stat e s . As we pointed out i n Chapter I, however, the experimental evidence i s s t i l l too ambiguous to determine the v a l i d -i t y of t h i s f l u c t u a t i o n model f or high temperatures. 87 §4.3 D i s c u s s i o n of Other E f f e c t s We end t h i s chapter w i t h a d i s c u s s i o n of the p o s s i b l e o r i g i n s of the s m a l l frequency s h i f t s i n F i g . 4.1. These s h i f t s appear to be roughly independent of the s t r e n g t h of the a p p l i e d f i e l d f o r Mo, whereas f o r Fe s i g n i f i c a n t l y l a r g e r s h i f t s are observed f o r H = 17 kG than f o r H = 21 kG. We begin by c o n s i d e r i n g a number of p o s s i b l e mechanisms f o r the observed frequency s h i f t s which are a l l based on the f a c t t h a t an appar-ent s h i f t i n the frequency w i l l r e s u l t i f the dHvA magnetization c o n s i s t s of a sum of s e v e r a l components which are out of phase w i t h each other and whose amplitudes have s l i g h t l y d i f f e r e n t temperature dependences. I t t u r ns out t h a t a l l of these e f f e c t s g i v e frequency v a r i a t i o n s which depend approximately l i n e a r l y on the temperature [assuming t h a t the temperature dependence of the i n d i v i d u a l amplitudes i s as i n Eq. 2.4b f o r A(T,B)], and the t o t a l (apparent) frequency s h i f t i s given by the f o l l o w i n g e x p r e s s i o n : 6F F = [1 + u-J V y ) Jn(y) - (• 2TTF Am* . 2 s i n ty B m* J K 2 cos ty ± 1 2-rrFAB 2TTAF B 2ir k_,m*B tie(2iTF)2 6T . 4.33 The f i v e terms i n s i d e the bracket w i l l be l a b e l l e d i , i i , i i i , i v and v and t h e i r meaning can be summarized as f o l l o w s (the d e t a i l e d proofs are given i n Appendix I I ) : 88 i & i i ) In the conventional d e r i v a t i o n of the dHvA formalism leading up to the expression for dM/dt (Eq. 3.5b), the f i e l d dependence of the amplitude A(T,B) i s u s u a l l y ignored when compared with that of the r a p i d l y varying s i n u s o i d a l function sin(2-rrF/B + <fi). This approxi-mation i s made at the following two steps: ( i ) , i n the d i f f e r e n t i a t i o n of the o s c i l l a t o r y part of the f r e e energy of the conduction electrons with respect to f i e l d to obtain M, and ( i i ) , i n the Fourier decomposi-t i o n of dM/dt to give Eq. 3.5b. If these two steps i n the c a l c u l a t i o n s are performed r i g o r o u s l y we f i n d that the dHvA frequency has a small (apparent) temperature dependence given r e s p e c t i v e l y by the f i r s t two terms i n Eq. 4.33 ( J n ( u ) i s the same Bessel function appearing i n Eq. 3.5b). In p r i n c i p l e a s i m i l a r frequency s h i f t can also a r i s e at the stage i n the usual dHvA theory where the o s c i l l a t o r y c o n t r i b u t i o n s to the f r e e energy are summed f o r p a r a l l e l s l i c e s through the Fermi surface; however t h i s s h i f t i s zero f o r an e l l i p s o i d a l Fermi surface, and should be very small indeed for the e l e c t r o n lenses discussed here. i i i ) The t h i r d term i n Eq. 4.33 gives the frequency s h i f t expected i f the d i r e c t i o n of the magnetic induction i s misaligned s l i g h t l y from the [111] axis of the sample (the o r i g i n of t h i s s h i f t was discussed i n §3.5). On account of the alignment procedure o u t l i n e d i n §3.5 only the misalignment 9|_ i n the d i r e c t i o n perpendicular to the (110) plane turns out to be relevant. The q u a n t i t i e s appearing i n Eq. 4.33 are 3m*. D Am* = 8_ 39_ and [111] [111] 89 where m*^  and ¥^ r e f e r to the doublet frequency branch for the e l e c t r o n lenses.* The plus or minus sign i n Eq. 4.33 i s used according to whether cos ty i s p o s i t i v e or negative, r e s p e c t i v e l y . It turns out that each of the various q u a n t i t i e s l i s t e d above (8__ , [ 3m* /30__ | , & [ 3FQ/36__ |) can be estimated quite r e l i a b l y from the a v a i l a b l e data. i v & v) The l a s t two terms give the approximate frequency s h i f t r e s u l t i n g from an inhomogeneity i n the f i e l d B. As we pointed out e a r l i e r , s u c h inhomogeneity can be s i g n i f i c a n t i n a ferromagnetic sample i f the surface i s not exactly of the second degree. A Lorentzian d i s t r i b u t i o n (over the sample volume) has been assumed f o r both the magnitude B and f o r the i n c l i n a t i o n of 15 with respect to the [111] a x i s ; the l a t t e r gives r i s e to a d i s t r i b u t i o n i n the frequency over the sample volume. In Eq. 4.33 AB and AF give the c h a r a c t e r i s t i c h a l f broadening of the f i e l d strength and the frequency. Since the amplitude of M i s also effected by a f i e l d inhomogeneity (cf. Shoenberg 1969) i t was possible to estimate an upper l i m i t f o r the magnitude of the l a s t two terms i n Eq. 4.33 from measurements of the dHvA amplitude as a function of f i e l d strength. This was accomplished by d e r i v i n g an e x p l i c i t r e l a -tionship between the frequency s h i f t and the amplitude change due to a given f i e l d inhomogeneity. Upon i n s e r t i n g the various known or estimated values (Appendix II) of the parameters into Eq. 4.33 we f i n d that f o r Mo only term ( i i i ) i s *As we pointed out e a r l i e r the s i x symmetry r e l a t e d lenses give r i s e to three d i f f e r e n t frequencies for an a r b i t r a r y d i r e c t i o n f or _3. When Ii_ l i e s i n the (110) plane, two of the three frequencies are degenerate and give r i s e to the s o - c a l l e d doublet frequency branch (see Gold et_ al_. 1971). 90 important whereas for Fe terms ( i ) , ( i i ) , (iv) and (v) are a l l s i g n i -f i c a n t . For Mo Eq. 4.33 f i n a l l y gives the r e s u l t <5F -5 ^ = -0.20 x 10 I r which i s e s s e n t i a l l y independent of the f i e l d strength and i s of the correct sign and magnitude to explain the observed frequency s h i f t shown i n F i g . 4.1. For Fe, on the other hand, Eq. 4.33 gives the r e s u l t 6F F ( -0.45 x 10~ 5T f o r H = 17 kG -0.15 x 10~ 5T f o r H = 21 kG, which v a r i e s with f i e l d p r i m a r i l y because of term ( i i ) ; because of the l i m i t a t i o n on the amplitude of the modulation f i e l d , i t was not po s s i b l e to make u large enough to achieve J ' n ( u ) = 0, i . e . , corresponding to the f i r s t maximum of J ( u ) . The cal c u l a t e d v a r i a t i o n s of SF/F for Fe are n of the correct sign and order of magnitude to account f o r the Fe data i n F i g . 4.1, and i n p a r t i c u l a r f o r the f a c t that the 17 kG data points l i e below the 21 kG ones. We therefore conclude that the general features of the small frequency s h i f t s i n F i g . 4.1 for both Mo and Fe can be described rather well by the e f f e c t s which have been included i n Eq. 4.33. A number of other e f f e c t s have also been investigated i n d e t a i l and a l l were found to be i n s i g n i f i c a n t under our experimental conditions; these include the 2 higher order T term i n Eq. 2.11 (assuming that the t o t a l density of states does not vary s i g n i f i c a n t l y with energy within about k T of the 91 Fermi l e v e l s ) , thermal expansion, e f f e c t of electron-phonon i n t e r a c t i o n , Shoenberg magnetic i n t e r a c t i o n and e f f e c t s associated with the approach to the quantum l i m i t [eg. quantum o s c i l l a t i o n s i n the chemical p o t e n t i a l ; c f . Wampler and Springford (1972)]. Numerical estimates of these e f f e c t s are also given i n Appendix I I . Chapter V. ADDITIONAL INVESTIGATIONS §5.1 Observation of a New Low-Frequency de Haas-van Alphen Term i n  Iron: the v - O s c i l l a t i o n s In the e a r l y stages of t h i s research some new dHvA frequencies were observed i n i r o n and i n t h i s section we describe t h e i r properties and probable o r i g i n . Some evidence for the existence of these low frequencies had already been reported by Gold (1964); however the evidence was not conclusive since the impulsive f i e l d technique used to study the dHvA e f f e c t at that time was hot i d e a l l y s u i t e d to inves-t i g a t e o s c i l l a t i o n s of such a low frequency. The f i r s t r e a l l y convincing evidence for the existence of these o s c i l l a t i o n s was reported independently by Angadi and Fawcett (1972, 1973), by Lonzarich and Gold (see Gold 1972), and by Coleman, Morris and Sellmeyer (1973). The precise numerical values and o r i g i n of the v-frequencies have not yet been f u l l y determined and we s h a l l summarize here the information obtained so f a r i n our whisker studies. The dHvA measurements were c a r r i e d out i n i r o n whisker samples with the apparatus described i n Chapter I I I . The v - o s c i l l a t i o n s were found to c o n s i s t of e s s e n t i a l l y three close frequency branches, two of which are degenerate when 15 l i e s on the (110) plane, and a l l three appear to be degenerate when JB i s p a r a l l e l to the [111] d i r e c t i o n . This, conclusion was also a r r i v e d at by Angadi and Fawcett (1973) . In F i g . 5.1a, b and c we show examples of the v - o s c i l l a t i o n s for 15 i n an a r b i t r a r y d i r e c t i o n , f o r B_ i n the (110) plane ( i n p a r t i c u l a r , p a r a l l e l to the [100] d i r e c t i o n ) , and f o r IJ p a r a l l e l to the [111] a x i s . The numerical values of the frequencies were determined by Fourier a n a l y s i s (for increasing and decreasing f i e l d sweeps) and the r e s u l t s obtained Face of Page 93 Figure 5.1 Recorder tracings of the low-frequency o s c i l l a t i o n s i n i r o n whisker samples for the following o r i e n t a t i o n s : (a) 12° from [111] normal to the (110) plane, (b) [100] and (c) [111]. The Fourier a n a l y s i s of the data reveals three close v-frequencies f o r (a), two for (b) and only one f o r ( c ) . The f a s t e r o s c i l l a t i o n s i n these figures a r i s e from the _ and n. (at [100]) frequency branches. Values of the applied f i e l d H are quoted at the ends of the f i e l d sweeps ( i n kG). 96 i n [100], [111] and [110] whiskers for the respective symmetry d i r e c t i o n s are given i n Table 5.1. The errors quoted i n Table 5.1 correspond essen-t i a l l y to the half width of the d i f f r a c t i o n function (defined i n §3.7) i n the frequency spectrum obtained by Fourier a n a l y s i s . Experimental errors a r i s i n g from the uncertainty i n the value for B and from any e l e c t r o n i c phase s h i f t s due to f i l t e r i n g were estimated to be l e s s than 0.5%. The whisker samples (mounted on the adjustable sample holder shown i n F i g . 3.3) were oriented i n the magnetic f i e l d to better than 0.5 degrees. The amplitude of the v - o s c i l l a t i o n s was found to be stronger for the upper frequency branch at [100] (by a f a c t o r of two or three) and f o r the lower branch at [110] (by a f a c t o r of eight or ten). The r e l a t i v e l y weak amplitude of the upper branch at [110] accounts f o r the large uncertainty quoted i n Table 5.1 f o r t h i s f r e -quency term. The other low frequency o s c i l l a t i o n s i n i r o n , a r i s i n g from the el e c t r o n lenses and necks (cf. F i g . 1.2), were also studied and the r e s u l t s i n the symmetry d i r e c t i o n s , summarized i n Table 5.1, agree with those previously reported by Gold et a l . (1971). The dHvA data c o l l e c t e d f or a r b i t r a r y f i e l d d i r e c t i o n s (so that B^  was not p a r a l l e l to the whisker axes) could be used to r e v e a l the number of frequency components present but could not be used to compute accurate values for the frequencies themselves. The reason f o r t h i s has been discussed by Gold et^ al_. (1971) and can be understood with reference to F i g . 5.2 which shows schematically an e l l i p s o i d a l sample with the applied f i e l d H_ at an angle 6 with respect to the c r y s t a l a x i s . Due to the shape anisotropy the ferromagnetic magnetization M i s not par-a l l e l to H except f o r 6 = 0 or 0 = y. In a d d i t i o n , the angle x between M 97 Low Frequencies i n Iron Or i e n t a t i o n & Branch [100] ]J ( E l l i p s o i d s at N?) (Electron Lenses) F (MG) (Present  Investigation) 7] (Neck) F (MG) (Gold et a l . 1971) not reported not reported 3.87 ± 0.02 4.15 ± 0.02 5.05 ± 0.03 [111] \J ( E l l i p s o i d s at N?) (Electron Lenses) 1.42 ± 0.02 4.15 ± 0.02 not reported 4.13 ± 0.03 [110] y ( E l l i p s o i d s at N?) £ (Electron Lenses) not reported not reported 3.91 ± 0.03 4.12 ± 0.03 Table 5.1 Summary of the observed de Haas-van Alphen frequencies i n i r o n . The [110] frequency values were determined i n a [110] i r o n whisker provided by R.V. Coleman, U n i v e r s i t y of V i r g i n i a . Face of Page 98 Figure 5.2 The magnetic induction 15 i n an e l l i p s o i d a l sample. H i s the external f i e l d and M i s the magnetization vector. I t can be shown that 15 (which i s the vector sum of H, 4TTM and the demagnetizing f i e l d not shown i n the fi g u r e ) i s p a r a l l e l to M. XI 99 and the whisker axis increases with increasing f i e l d H, so that the t o t a l f i e l d 15 changes i n magnitude and o r i e n t a t i o n as the strength of the applied f i e l d i s v a r i e d . The d i r e c t i o n of B as a f u n c t i o n of H and 0 can be evaluated numerically by r e q u i r i n g that the magnetic energy of the c r y s t a l i s a minimum ( c f . Schau and Stone 1967). For the long t h i n whiskers used i n t h i s experiment a computer an a l y s i s revealed that for 9 = 20°, x i s expected to vary from 4° to 14° as H i s increased from 2 kG to 22 kG (a t y p i c a l f i e l d range required to separate the v-frequency components by Fourier a n a l y s i s ) . Because of t h i s v a r i a t i o n of x with H, d i f f e r e n t extremal c r o s s - s e c t i o n a l areas are selected during the f i e l d sweep, consequently the precise frequency values away from the symmetry d i r e c t i o n s were not determined i n t h i s i n v e s t i g a t i o n . We now consider, on the basis of the experimental r e s u l t s obtained so f a r , the p o s s i b l e o r i g i n s of the v-frequencies. In F i g . 5.3a,b we show schematically the frequency branches on the (110) plane expected fo r pockets centered on the TH l i n e s and f o r e l l i p s o i d s centered at the points N of the B r i l l o u i n zone. If each e l l i p s o i d i s roughly symme-t r i c a l about i t s major axis then the four branches i n F i g . 5.3b become e s s e n t i a l l y two branches, which are s i m i l a r to those f o r the TH-pockets shown i n F i g . 5.3a. The v-frequency values determined from the Fou r i e r analysis of the data can therefore be consistent with e i t h e r the e x i s -tence of nearly e l l i p s o i d a l pockets on the TH l i n e or approximate e l l i p -soids of r e v o l u t i o n at N. However, there i s some i n d i r e c t evidence to suggest that the l a t t e r choice i s probably c o r r e c t . F i r s t , at p r e c i s e l y the [111] d i r e c t i o n i t was found that the Dingle temperature T__ (determined from the f i e l d dependence of the dHvA amplitude) for the v - o s c i l l a t i o n s was about twice as large as that f o r Face of Page 100 Figure 5.3 Figure (a) shows pos s i b l e dHvA frequency branches i n the (110) plane expected f o r nearly e l l i p s o i d a l pockets along the TH l i n e of the B r i l l o u i n zone. Figure (b) shows the frequency branches expected f o r e l l i p s o i d s at the points N of the B r i l l o u i n zone. [100] [ m ] [110] 101 the G - o s c i l l a t i o n s , i n the same c r y s t a l . A pos s i b l e explanation f o r the unusually high value f o r i s that i n the [111] d i r e c t i o n there are two very close v-frequencies which give r i s e to a broad beat-pattern; over a l i m i t e d f i e l d range the dHvA magnetization would appear to be a s i n g l e p e r i o d i c term but with a d i f f e r e n t f i e l d dependence ( i . e . , Dingle temperature)from that expected f o r a s i n g l e frequency. I f t h i s i n t e r -p r e t a t i o n i s correct then the v - o s c i l l a t i o n s cannot o r i g i n a t e from TH-pockets which would give r i s e to only one frequency at the [111] d i r e c -t i o n . The second piece of i n d i r e c t evidence can be discussed with r e f e -rence to F i g . 5.4 where we have p l o t t e d the r e l a t i v e amplitude of the v - o s c i l l a t i o n s as a function of angle near the [111] d i r e c t i o n and at several d i f f e r e n t applied f i e l d s H. If the v-frequency branches coalesce to give exactly one frequency i n the [111] d i r e c t i o n we would expect the dHvA amplitude to be a maximum (as a function of angle) p r e c i s e l y i n the [111] d i r e c t i o n , as was found to be the case f o r the l e n s -o s c i l l a t i o n s i n i r o n ( i n the same c r y s t a l ) and also i n molybdenum. However from F i g . 5.4 we see that f o r the v - o s c i l l a t i o n s the r a t i o of the amplitude at [111] to the amplitude of nearby peaks v a r i e s with f i e l d and becomes l e s s than one at low f i e l d s . This r e s u l t suggests once again that at [111] there e x i s t two sets of o s c i l l a t i o n s of nearly the same frequency. The r e l a t i v e phase of the o s c i l l a t i o n s changes with magnetic f i e l d and the two o s c i l l a t i o n s add incoherently at the lower f i e l d s quoted i n F i g . 5.4; the l a t t e r i s consistent with the large observed Dingle temperature discussed e a r l i e r . F i n a l l y we point out that the unusually low amplitude of the upper v-frequency at [110] can also be understood i f the v - o s c i l l a t i o n s o r i g i -Face of Page 102 Figure 5.4 This f i g u r e gives a family of p l o t s of the r e l a t i v e amplitude of the v - o s c i l l a t i o n s f o r f i e l d d i r e c t i o n s i n the neighborhood of the [111] axis and i n the (110) plane. The r a t i o of the maximum to the minimum ampli-tude i s of the order of three f o r each of the p l o t s shown. H i s the applied f i e l d and the angles quoted are those between H and the [111] a x i s . 102 TJ Z3 CL < I TJ CD > [100] [111] 110] H I7.2 kG I4.7 kG 12J kG 9.5 kG 6.9 kG 2 0 c 10 c 0 10° 2 0 c 103 nate from e l l i p s o i d s at N, since we would then expect two close upper-frequency terms at [110] (see F i g . 5.3b) which could be nearly out of phase i n the f i e l d range of the experiment. The amplitude of the v - o s c i l l a t i o n s as a function of o r i e n t a t i o n near [110] has not yet been studied. The evidence given above, though not completely conclusive, does suggest that the v-frequencies most l i k e l y o r i g i n a t e from small e l l i p -soids at N and the data at the symmetry d i r e c t i o n s i s consistent with t h e i r major axes l y i n g p a r a l l e l to the PN l i n e s and being about 10% la r g e r than the minor axes. Hole pockets are expected to e x i s t at the points N i f the state N^' (which i s connected to r - ^ ' an& ^ 2 i n F i g . 1. l i e s above the 4- spin Fermi l e v e l . The band structure c a l c u l a t i o n s that have been c a r r i e d out so f a r i n d i c a t e that the ordering of the energy l e v e l s near N i s very s e n s i t i v e to the precise approximations used. Thus the band structure of Wood (1962) and Wakoh and Yamashita (1966), f o r example, suggests that N^' should l i e about 1 eV above the 4- spin Fermi l e v e l , whereas i n the c a l c u l a t i o n of Duff and Das (1971) the state N^' l i e s below the Fermi l e v e l by about 1 eV. The former band structures p r e d i c t e l l i p s o i d s at N which are about 10 or 20 times l a r g e r ( i n c r o s s - s e c t i o n a l area) than the ones we propose here, whereas i n the band c a l c u l a t i o n of Duff and Das the e l l i p s o i d s are absent. Our v-frequency r e s u l t s suggest that the N. ' state l i e s j u s t above the 4- spin Fermi l e v e l and at an energy which i s intermdiate between the two extreme values predicted by the band structure c a l c u l a t i o n s . F i n a l l y we wish to point out that the v - o s c i l l a t i o n s could i n p r i n c i p l e also o r i g i n a t e from small 'mixed-spin' pockets (near N) formed by the i n t e r s e c t i o n s of the t-spin hole arms (which extend along the 104 HN l i n e s i n F i g 1.2) and r e l a t i v e l y large 4--spin e l l i p s o i d s at N. This model however pr e d i c t s several extra dHvA frequency terms, a r i s i n g from the h o l e - e l l i p s o i d s and from the [110] necks of the mixed spin pockets, which have not been c l e a r l y observed.* Furthermore Mijnarends (1973a,b) has concluded from h i s p o s i t i o n - a n n i h i l a t i o n studies i n i r o n that any hole pockets at N must be quite small, at l e a s t four times smaller ( i n cr o s s - s e c t i o n a l area) than those predicted by the band c a l c u l a t i o n s of Wood or Wakoh and Yamashita. These considerations argue against the existence of mixed-spin pockets near the points N for i r o n ; however further studies are required before these pockets can be completely ruled out. *Baraff (1973) has t e n t a t i v e l y associated one of h i s experimental f r e -quency branches with large hole e l l i p s o i d s at N (having the same gene-r a l dimensions as those predicted by Wood's c a l c u l a t i o n ) . However more experimental evidence i s required to substantiate t h i s conjecture. 105 §5.2 Some Cyclotron E f f e c t i v e Masses In t h i s s e c t i o n we discuss the cyc l o t r o n e f f e c t i v e masses asso-ciated with the e. and v-frequencies at the [111] d i r e c t i o n . According to Eq. 2.4b f o r the dHvA amplitude A(T,B) we f i n d 2ir 2k . ln(A/T) = 7~ ^V 1 + constant , • 5.1 en B where the constant i s independent of temperature. It follows from t h i s equation that the cyc l o t r o n mass m* associated with each dHvA frequency can be determined from the slope of a p l o t of ln(A/T) versus T. The amplitudes f o r the £ and v-frequencies as a function of temperature were measured at the [111] d i r e c t i o n and the r e s u l t s at one f i e l d point are p l o t t e d i n F i g . 5.5. The e f f e c t i v e masses computed from the slope of these p l o t s are (0.51 ± 0.01)m f o r the v - o s c i l l a t i o n s and o (0.71 ± 0.01)mQ f o r the e-frequency. The l a t t e r r e s u l t i s about two times greater than that reported by Panousis (see Gold et a l . 1971) s who studied the temperature dependence of the amplitudes i n i r o n by means of the impulsive f i e l d technique.* In order to check the r e l i a b i l i t y of the e f f e c t i v e mass values reported here, we have repeated the measurements at a number of f i e l d points ranging from H = 10 kG to about 32 kG. In ad d i t i o n the measure-*In t h i s technique the sample i s f i r s t magnetized by a d.c. f i e l d of about 1 kG and then i s subjected to a time-varying f i e l d produced by discharging a bank of capacitors through a solenoid. The a.c. f i e l d t y p i c a l l y v a r i e s from zero to 150 kG i n about ten m i l l i s e c o n d s . Face of Page 106 Figure 5.5 Cyclotron mass p l o t s f o r £ and v - o s c i l l a t i o n s at [111]. The respective values of m* are (0.71 ± 0.01)_x and (0.51 ± 0.01)m . o 106 1 2 3 T ( ° K ) 107 merits were c a r r i e d out i n two separate whiskers and with various modu-l a t i o n f i e l d amplitudes. In a l l cases the e f f e c t i v e mass values were found to agree with each other. As an a d d i t i o n a l check on the apparatus, e f f e c t i v e mass measurements were also c a r r i e d out f o r the neck o s c i l l a -tions i n gold and f o r the lens o s c i l l a t i o n s i n molybdenum. The measured cyclotron masses were i n each case found to be i n good agreement with those previously reported. The discrepancy between our r e s u l t and that of Panousis has not yet been f u l l y resolved. One p o s s i b i l i t y i s that i n the impulsive f i e l d technique the i r o n sample was heated s i g n i f i c a n t l y during the f i e l d pulse;* i t can be shown that t h i s heating e f f e c t could indeed r e s u l t i n a low measured e f f e c t i v e mass, as observed by Panousis. In ordinary metals any eddy current heating during a t y p i c a l f i e l d pulse i s expected to be small (for long, t h i n samples). However i n a ferromagnet the heating could be s u b s t a n t i a l i f there e x i s t small domains of reversed magnetization which p e r s i s t even i n applied f i e l d s greater than those required to achieve t e c h n i c a l s a t u r a t i o n . As pointed out by Gold (1968) the presence of such microdomains could also explain the puzzling disappearance of the dHvA e f f e c t when modulation f i e l d s above about 25 kHz are used. In order to t e s t the above hypothesis i t would be necessary to repeat our m* measurements i n high frequency (several kHz) modulation f i e l d s . *The time rate of change of the a.c. f i e l d i n the i m p u l s i v e - f i e l d tech-nique i s about three orders of magnitude greater than that i n our low frequency field-modulation method. 108 §5.3 Quantum O s c i l l a t i o n s i n the Spin Magnetization In t h i s section we wish to show that i n an i t i n e r a n t ferromagnet the t o t a l spin magnetization may be expected to e x h i b i t small quantum o s c i l l a -tions which are p e r i o d i c i n 1/B. These o s c i l l a t i o n s can i n some cases be comparable i n magnitude to those i n the dHvA e f f e c t and could lead to an apparent deviation of the f i e l d dependence of the observed o s c i l l a t o r y mag-n e t i z a t i o n from that expected f o r the L i f s h i t z - K o s e v i c h theory ( i . e . , Eqs. 2.2). A q u a l i t a t i v e d e s c r i p t i o n of the e f f e c t w i l l be given before the formal a n a l y s i s . We begin the discussion by considering a small i s o l a t e d 4- spin pocket of the Fermi surface (for example one of the e l e c t r o n lenses i n i r o n ) . As we pointed out i n §2.1 the allowed e l e c t r o n i c states l i e on concentric Landau tubes (Fig. 2.1) and the density of states has an o s c i l l a t o r y s t r u c -ture as shown i n F i g . 5.6. As the f i e l d B i s v a r i e d the p o s i t i o n of the Landau tubes changes, and the r e s u l t i n g change i n the density of states to-gether with the assumed constancy of the number of p a r t i c l e s i n the i s o l a - ted pocket then lead to an o s c i l l a t o r y v a r i a t i o n i n the Fermi l e v e l e, i n Y F i g . 2.1 (Kaganov et a l . 1957), We now combine our small 4- spin Fermi surface with a much l a r g e r Fermi surface for + e l e c t r o n s , so that X. >> /f. . In e q u i l i b r i u m the chemical T Y p o t e n t i a l must be the same for both t and 4- electrons and the o s c i l l a t i o n s i n the 4- spin Fermi l e v e l e discussed i n the l a s t paragraph are 'damped' Y by the large r e s e r v o i r of majority c a r r i e r s (as i n the d i s c u s s i o n of Kaganov et a l . 1957 f o r ordinary metals). The reduction i n the v a r i a t i o n s of e. Y then implies that the number of c a r r i e r s i n the 4- spin pocket must undergo pe r i o d i c v a r i a t i o n s with f i e l d . This r e s u l t s i n an o s c i l l a t o r y behavior of the t o t a l spin magnetization. Face of Page 109 Figure 5.6 The density of states of a free e l e c t r o n gas i n a magnetic f i e l d corresponding to the Landau quantum tubes shown i n F i g . 2.1a. 109 110 In order to obtain the magnitude of this effect we proceed as follows. If e. and E , are the Fermi levels for the + and 4- electrons T 4' measured from the bottom of the respective bands then in equilibrium we must have A = ( e + - e + ) = I e f f ( n + - n +) , 5.2 where n and n are the number of + and 4 carriers per atom defined by n = 7 /(e,B) + o • ( £ - £ + ) / k _T e + 1 -1 de 5.3a and n. = n - n,, t 4- 5.3b Eq. 5.2 is the Stoner relation for the exchange splitting derived in §4.2a (Eq. 4.15) and (n, - n.) is proportional to the single-particle T 4-magnetization. Since the density of states >T(e,B) depends on B we see from Eqs. 5.3a and 5.2 that n and e vary with field; e, is essentially 4" 4- T constant since we assume X. » X. [and F,/B >> 1, where F, is the dHvA T \ T T frequency associated with the f Fermi surface (Kaganov et a l . 1957)]. If we cal l e and n the respective oscillatory components of e and 4 - 4 " v n then from Eqs. 5.2 and 5.3b i t follows that 4" e. = 2 n. I . 5.4 4- 4- ef f The oscillatory part of the number of particles can be determined directly from the Lifshitz-Kosevich (1955) theory; from their Eq, 3.4 for the number of particles as a function of field and Fermi level we 1 1 1 f i n d that for small £. + m*+B .» % = ^ i e + ~ ehF.N MdHvA,r ' 5 , 5 4- a ' where )f, i s the 4- spin density of states at the Fermi l e v e l and m* , F. Y Y T and ^ are the c y c l o t r o n mass, dHvA frequency and magnetization associated with the 4- Fermi surface, and N & i s the number of atoms per unit volume. The o s c i l l a t o r y part of the spin magnetization i s defined by M = -gn N u so that by e l i m i n a t i n g e from Eqs. 5.4 and 5.5 we s j i T a 15 T obtain the f i n a l r e s u l t ^ y ^ o MdHvA,+ + Ms,+ " F +/B ( 1 - 2 X + I e f f ) * The two o s c i l l a t i o n s +and M g + are i n phase and become comparable i n magnitude as the quantum l i m i t i s approached ( i . e . , as F^/B -*• 1 and gm*^=2mo) . The quantity ( 1 - Df^I^^) ^ i s the w e l l known exchange enhancement f a c t o r which i s always greater than u n i t y and can i n p r i n -c i p l e lead to a large M . P h y s i c a l l y t h i s f a c t o r a r i s e s from the f a c t j S , Y that when the magnetization i s v a r i e d ( i n some way), the exchange s p l i t -t i n g (Eq. 5.2) changes i n such a d i r e c t i o n as to enhance the o r i g i n a l change i n the magnetization. ( E s s e n t i a l l y the same enhancement f a c t o r also appeared i n Eq. 4.5 f o r SM^.) We note from Eq. 5.6 that i n p r i n -c i p l e i t should be p o s s i b l e to d i s t i n g u i s h between the contributions of MjHvA + and M g ^ because of the extra f a c t o r of B i n the f i e l d dependence 112 Eq. 5.6 can be generalized to describe a system with an a r b i t r a r y number of i and 4 sheets of the Fermi surface. If X. and X. are the over-T T a l l d e n s i t i e s of states at the Fermi l e v e l f o r the + and 4- electrons and i f the parameters m* . F and M,T, . are associated with a given a a dHvA,o sheet of the Fermi surface of spin a then the r e s u l t i n g o s c i l l a t o r y spin magnetization i s given by M s , a gm* /2m & a o F /B 0 X. - a M dHvA,a X + X r a - a 21 1 - e f f X " 1 + X'1 - a 5.8 where f;^ = +1,-1 f o r a = f I n obtaining t h i s r e s u l t the dHvA magne-t i z a t i o n a r i s i n g from a l l other sheets of the Fermi surface was assumed to be small. Eq. 5.8 i s s i m i l a r to Eq. 5.6 except that the exchange enhancement factor has been generalized to include cases where X. i s not T much la r g e r than X. . The f a c t o r X /(/ + X )arises e s s e n t i a l l y because + —o o —o both the and e Fermi l e v e l can vary with f i e l d i n general. *The t o t a l o s c i l l a t o r y magnetization M , + M.TT , , would appear to s, 4- dHvA, 4-have a net field-dependent Dingle temperature T__(B). Anderson, Hudak and Stone (1973) have i n f a c t observed a puzzling f i e l d dependent T_. fo r small pockets of the Fermi surface i n ferromagnetic cobalt. However i t i s too e a r l y to determine whether t h e i r f i n d i n g s can be explained by the theory proposed here. Chapter VI CONCLUSIONS 113$ §6.1 The Main R e s u l t s of the Present I n v e s t i g a t i o n By means of a high r e s o l u t i o n technique f o r measuring s m a l l s h i f t s i n the dHvA phase we have shown that the exchange s p l i t t i n g of the energy bands i n i r o n i s not p r o p o r t i o n a l to the t o t a l m agnetization at f i n i t e temperatures, c o n t r a r y t o the p r e d i c t i o n s of a l i t e r a l i n t e r p r e -t a t i o n of the Stoner theory (Thompson et a l . 1964). On the b a s i s of a Fermi l i q u i d theory, g e n e r a l i z e d by Izuyama and Kubo (1964) to i n c l u d e magnon e x c i t a t i o n s , Edwards (1973) has shown from general arguments that s p i n waves have n e g l i g i b l e e f f e c t on the exchange s p l i t t i n g A at low temperatures, so t h a t A i s e s s e n t i a l l y p r o p o r t i o n a l o n l y to the s i n g l e - p a r t i c l e magnetization. We have a l s o a r r i v e d at t h i s c o n c l u s i o n from a simple microscopic model based on the Hubbard Hamiltonian f o r a s i n g l e band. Furthermore Edwards' c o n c l u -s i o n was found to be e s s e n t i a l l y u n a f f e c t e d when magnetic a n i s o t r o p y was taken i n t o account. Our experimental r e s u l t s a r e c o n s i s t e n t w i t h Edwards' theory and w i t h the magnetization decreasing i n i r o n p r i m a r i l y by spin-wave e x c i t a t i o n s at low temperatures (the l a t t e r i s i n keeping w i t h the curve f i t t i n g analyses of the low temperature magnetization data as discussed i n §4.1). On the b a s i s of Edwards' theory the dHvA e f f e c t p r o v i d e s f o r the f i r s t time, at l e a s t i n p r i n c i p l e , an experimental way of unambiguously se p a r a t i n g the s i n g l e - p a r t i c l e from the spin-wave c o n t r i b u t i o n s to the magnetization at low temperatures. We have shown that such an e x p e r i -ment i s t e c h n i c a l l y f e a s i b l e and have o u t l i n e d the v a r i o u s precautions which are r e q u i r e d f o r i t s execution. In p a r t i c u l a r our dHvA data provide an upper bound f o r the (small) s i n g l e - p a r t i c l e magnetization 114 i n i r o n . As pointed out by Edwards (1973) the a p p l i c a t i o n of our technique to very weak ferromagnets (such as Ni_Al and Sc_In) should help to resolve the present ambiguities regarding the t h e o r e t i c a l d e s c r i p -t i o n of these materials (see §4.2c). Our study of a new set of low-frequency dHvA o s c i l l a t i o n s i n i r o n suggests that these o s c i l l a t i o n s o r i g i n a t e from small e l l i p s o i d a l sur-faces centered on the points N of the B r i l l o u i n zone. This conclusion points to a p a r t i c u l a r ordering of the energy l e v e l s near N and helps remove one of the remaining u n c e r t a i n t i e s concerning the band structure of i r o n . F i n a l l y we have shown that i n an i t i n e r a n t ferromagnet the t o t a l spin magnetization may be expected to ex h i b i t small quantum o s c i l l a t i o n s which are pe r i o d i c i n 1/B. I t was shown that these o s c i l l a t i o n s can i n some cases be comparable i n magnitude to those i n the dHvA e f f e c t and could lead to an apparent d e v i a t i o n of f i e l d dependence of the observed o s c i l l a t o r y magnetization from that expected f o r the L i f s h i t z - K o s e v i c h theory (Eqs. 2.2). 115 §6.2 Suggestions f o r Further Study As discussed e a r l i e r i n §4.2c i t would be of c o n s i d e r a b l e i n t e r e s t to apply the dHvA experiment described i n t h i s t h e s i s to the study of the s i n g l e p a r t i c l e magnetization i n very weak i t i n e r a n t ferromagnets such as N i ^ A l , Z r Z ^ , Sc^In and Pd(Co). S u f f i c i e n t l y pure samples of some of these m a t e r i a l s are becoming a v a i l a b l e so that the dHvA measure-ments may be p o s s i b l e i n the near f u t u r e . In our whisker s t u d i e s the values f o r the v-frequencies were d e t e r -mined a c c u r a t e l y only f o r the symmetry d i r e c t i o n s . In order to extend the measurements to a l l o r i e n t a t i o n s i t w i l l be necessary to use spher-i c a l or d i s k shaped samples (with 15 i n the plane of the d i s k ) so that the s u r f a c e a n i s o t r o p y i s not a c o m p l i c a t i o n (see §5.1). Furthermore i n order to e s t a b l i s h w i t h c e r t a i n t y the presence of four frequency branches i n the (110) plane i t would be u s e f u l t o i n c r e a s e the range of the magnetic f i e l d and t h e r e f o r e the number of o s c i l l a t i o n s that can be observed. U n f o r t u n a t e l y the i n d u c t i o n B i n the sample cannot be lowered below about 23 or 24 kG, at which p o i n t magnetic domains begin to form and g i v e r i s e to background s i g n a l s that are orders of magni-tude g r e a t e r than the dHvA s i g n a l . In the remaining f i e l d range 24 kG - °° there are only about 50 to 60 c y c l e s of the fundamental of the v - o s c i l l a t i o n s so that even i n p r i n c i p l e i t w i l l o n l y be p o s s i b l e to increase the frequency r e s o l u t i o n i n F o u r i e r a n a l y s i s by a f a c t o r of about two above that obtained i n t h i s i n v e s t i g a t i o n , unless higher harmonics of the v - o s c i l l a t i o n s can be observed. In the end the i d e n -t i f i c a t i o n of four frequency branches on the (110) plane may have to r e s t on i n d i r e c t evidence s i m i l a r to that given i n §5.1. 1 1 6 The apparatus and technique used i n the present i n v e s t i g a t i o n can be applied d i r e c t l y to study of the pressure dependence of the Fermi surface i n i r o n . According to Eqs. 2.3, 2.4 the pressure d e r i v a t i v e of the t o t a l dHvA phase can be w r i t t e n as 9 / 2 i t F _ . A ^ 2 i t F r 9 F / 4 l T ^ 9 M i _. o _ 9 rR*m* 4x 3 F ( T + * ) = T [ F 3 ? - <—> M3? ] + 2 7 T ? a 3p"[-lm- X + X] ' O T + where a r e f e r s to the s p i n o r i e n t a t i o n of the electrons associated with the c r o s s - s e c t i o n a l area of i n t e r e s t and E = + 1 or -1 as a = + or 4-. a Since (3M/3p)/M can be obtained from other experiments,then the slope of the p l o t of 3(2TTF/B + <}>)/3p versus 1/B can provide the value of (3F/3p)/F. Furthermore, assuming the f a c t o r g*m*is only weakly depen-dent on pressure, the intercept of the p l o t gives i n p r i n c i p l e the pressure d e r i v a t i v e s of the r e l a t i v e density of states f o r the t or 4-s p i n c a r r i e r s at the Fermi surface. We a l s o point out that f o r a ferromagnet the quantity (3F/3p)/F w i l l have a c o n t r i b u t i o n a r i s i n g from the change i n the exchange s p l i t t i n g A with pressure ( c f . the f i r s t term on the r i g h t hand side of Eq. 2 . 1 1 ) . According to the Stoner theory t h i s c o n t r i b u t i o n should be proportional to the pressure d e r i v a t i v e of the magnetization. According to our d i s c u s s i o n i n §5.3, an i t i n e r a n t ferromagnet should e x h i b i t small o s c i l l a t i o n s (M_) i n the spin magnetization which could i n p r i n c i p l e be detected by a c a r e f u l study of the f i e l d dependence of the amplitude of the t o t a l observed o s c i l l a t o r y magnetization (M_ + M d H v A ) « As pointed out i n §5.3, M_ i s proportional to ( F / B ) " 1 and to the exchange enhancement f a c t o r . Consequently the possible e f f e c t s of M g should be investigated f o r small sheets of the Fermi surface ( i n high f i e l d s ) and preferably i n weak i t i n e r a n t ferromagnets where the exchange enhancement i s expected to be la r g e . 118 APPENDICES i 118a Appendix I: The Dependence of the Extremal Cross-Sectional Areas of  the Fermi Surface on the F i e l d B In t h i s section we consider a r b i t r a r y + and 4- sheets of the Fermi surface and determine the change i n a given c r o s s - s e c t i o n a l area r e s u l t i n g from the a p p l i c a t i o n of a f i e l d IS. The c a l c u l a t i o n w i l l be c a r r i e d out for general a n i s o t r o p i c g-factors and with many-body i n t e r a c t i o n ( i n the Fermi l i q u i d theory) included i n the i n i t i a l steps. Since we wish to consider systems with s p i n - o r b i t i n t e r a c t i o n we note that the appropriate one-electron states are not (generally) eigenfunctions of the spin oper-ator s/B_. In t h i s case i t i s customary to assign to each e l e c t r o n i c state a spin index a which i s + (majority) or 4- (minority) depending on whether the expectation value of (-s_'B) i s p o s i t i v e or negative, In the following a n a l y s i s the band indice s w i l l be suppressed and a l l summations over k w i l l be understood to be also over band number. Further-more a l l states i n a given (nondegenerate) band w i l l be assumed to have the same spin index (the g e n e r a l i z a t i o n to the case of spin-hybridized bands i s straightforward). As the f i e l d B_ i s turned on electrons are transferred from the 4-to the + spin Fermi surface, which r e s u l t s i n changes i n the dimensions of the Fermi surface as i l l u s t r a t e d i n F i g . A l . l . In general the chem-i c a l p o t e n t i a l w i l l change from ri to n_ and the new t and 4- spin Fermi surfaces w i l l be defined by the wave-vectors k s a t i s f y i n g the c o n d i t i o n e B(ka) = n B , A l . l where e (ka) are the q u a s i p a r t i c l e energies i n the presence of the a — f i e l d B_ which are given by e_(ka) = e(ka) - ^ g(ka) y_B + £ f(ka,k'a') 6n(k»a') . A1.2 — k'a Face of Page 119 Figure A l . l Figure (a) shows a r b i t r a r y + and 4- sheets of the Fermi surface before ( s o l i d l i n e ) and a f t e r (dashed l i n e ) the a p p l i c a t i o n of the f i e l d j i . In f i g u r e (b) we show a given c r o s s - s e c t i o n a l area & before and a f t e r the f i e l d i s applied. 119 120 In t h i s equation e(ka) are the zero f i e l d energies (defined by Eq. 4.14a i n the simple Stoner t h e o r y ) ; £ a i s +1 or -1 i f a i s + or +, r e s p e c t i v e l y ; g(ka) i s a k and a dependent g - f a c t o r which a l s o depends g e n e r a l l y on B_; and the l a s t term i n A1.2 i s the i n t e r a c t i o n term d e f i n e d i n §4.2. The l a s t term must be inc l u d e d s i n c e the q u a s i p a r t i c l e occupation numbers change when the f i e l d i s a p p l i e d . I f A k n i s the normal s e p a r a t i o n (which v a r i e s w i t h k) between the ol d and the new Fermi surface (see F i g . A l . l ) , then f o r sm a l l Ak n and fo r k on the new Fermi s u r f a c e we have e(ka) = n + |v, e ( k a ) | Ak , A1.3 so t h a t from Eqs. A l . l , A1.2 i t f o l l o w s t h a t n B - n = |vk e(ka)| Ak n - g ( k a ) y B B + E f(ka,k'a') 6n(k'a') . A1.4 — — k'a' The t h e o r e t i c a l a n a l y s i s of Prange and Sachs (1967) suggests t h a t when the s p i n - o r b i t i n t e r a c t i o n i s not l a r g e , (rig - n) should not be renormalized by the electron-phonon i n t e r a c t i o n (EPI) . In terms of Eq. A1.4 t h i s means the EPI a l t e r a t i o n of the v e l o c i t y V ke(k,a) p r e c i s e l y c a n c e l w i t h the EPI c o n t r i b u t i o n to f ( k a , k ' a ' ) . This p o i n t has a l s o been made by H e r r i n g (1966). The f i n a l r e s u l t s of t h i s c a l c u l a t i o n , which depend e s s e n t i a l l y on Eq. A1.4, w i l l then be independent of the EPI when the s p i n - o r b i t i n t e r -i a c t i o n i s s m a l l ( i n which case g(ka) w i l l have a value c l o s e to 2 when the o r b i t a l angular momentum i s quenched). Having made t h i s p o i n t we now assume f o r s i m p l i c i t y t h a t the i n t e r a c t i o n term can be combined w i t h g(ka) to g i v e a renormalized parameter g*(kcr). (In general the r e n o r m a l i z a t i o n f a c t o r i s a n i s o t r o p i c . ) We now compute n B - n from the requirement of constancy of the number of p a r t i c l e s . That i s , 121 f f dS y B B f S ng*0_a)dS AN « / Ak dS = (ru - n) / |„ ? , i + - ? - / -r^ , i = 0 , A1.5 where the i n t e g r a t i o n i s over both + and -I- spin Fermi surfaces. From Eq. A1.5 and the usual d e f i n i t i o n of the density of states we f i n d u BB <g*+>tff - <g%»£ n B - n = - - f ' f + ' ' . A1.6 + + where X and X are the o v e r a l l d e n s i t i e s of states at the t and + spin Fermi l e v e l s and < g * _ > i s a Fermi surface average / g*(ka)dS/|v e(ka)| a <8* > f j c/ln .,1^1 » A1.7 a / dS/ V, e(ka) a k — where the i n t e g r a t i o n i s over the a spin Fermi surface. Having obtained the changes i n the chemical p o t e n t i a l we may now use Eq. A l . l to compute the change 6^ of a c r o s s - s e c t i o n a l area of the a spin Fermi surface. This quantity i s defined by SA- =6 Ak, d l , A1.8 where the i n t e g r a l i s around the perimeter of the area & and Akj_ i s the normal distance between the old and new contours l y i n g i n a plane perpendicular to the induction B^  (see F i g . A l . l b ) . For small Ak, and for k on the new contour we must have e(ko) = r, + | v ke(ka)|^ Ak^  A1.9 122 where TV ke(ka)j i s the component of V^eCko) i n the plane of i n t e r e s t perpendicular to 15. The quantity Ak^, determined from Eqs. A l . l , A1.2, A1..6 and A1.9, can now be substituted i n t o Eq. A1.8 to give the f i n a l r e s u l t 6A = 9 * a V a 9e 2 £ g* ^ a B ca Al.lOa where /3e a r d i J |V, e(ka and g* C ( J i s an o r b i t a l average g*(ko)dl V ke(ka) ca ( d l " V, E (ka) I k — JL Al.lOb When <g*.>X, equals <g* >X. the second term i n Eq. A l . l O a vanishes and T T T T the expression f o r 6A- reduces to that for a normal metal given by Holtham (1972). Furthermore, i f g* i s i s o t r o p i c i t i s straightforward to show that 6M- gives r i s e to a s h i f t i n the dHvA phase given by Eq. 2.3, where 2TT 9e Eq. Al.lOa gives the change i n the c r o s s - s e c t i o n a l area i n a f i x e d plane perpendicular to the applied f i e l d . In general, however, the extremal c r o s s - s e c t i o n a l area, which i s the area of i n t e r e s t , need not remain i n the same plane as the f i e l d strength i s v a r i e d . I t can be shown that t h i s leads to an a d d i t i o n a l term for the change i n the extremal c r o s s -2 s e c t i o n a l area which depends on B and i s therefore ignorable at low f i e l d s . In accordance with the discussion given at the end of §2.2 t h i s e f f e c t i v e mass parameter should not be renormalized by the electron-phonon or (probably) the electron-magnon i n t e r a c t i o n s . 123 Appendix I I : E f f e c t s that Give Rise to Small S h i f t s i n the de Haas- van Alphen Phase with Temperature In t h i s appendix we give the derivations f or the various terms i n Eq. 4.33 which are l a b e l l e d from r i g h t to l e f t i , i i , i i i , i v and v. The fundamental component of the o s c i l l a t o r y part of the free energy of an el e c t r o n gas i n a f i e l d 15 i s given by an equation of the form (cf. Gold 1968) 2 fi = - ( 9 ? ? ) A(T,B) cos (^ F-+ <{.) , A2.1 where F i s the dHvA frequency r e l a t e d to the extremal c r o s s - s e c t i o n a l area of the Fermi surface by the Onsager r e l a t i o n (Eq. 2.1) and A(T,B) and <j> were defined i n Eqs. 2.4b,c. For convenience we rewrite A(T,B) into the form -1/2 -kir^+V A(T,B) = a TB L ' e * U A2.2 2 where b = 2TT k /eh and a i s a quantity independent of T and B. Eq. 2.4b a m* or Eq. A2.2 are v a l i d to a good approximation provided X(T) = b^— T D i s greater than about 2, which i s , i n f a c t , the case under our experi-mental conditions. (i ) More Exact D i f f e r e n t i a t i o n of the Free Energy The fundamental component of the dHvA magnetization i s given by (cf. H o l s t e i n et a l . 1973) 124 M = -V_fl . A2.3 — Ji 2 Normally the d i f f e r e n t i a t i o n of B A(T,B) i s ignored compared to that 2TTF of c o s ( — g — + <f>) i n Eq. 2.1. However i f we keep both terms then we f i n d that the component of M p a r a l l e l to 15 i s given to a good approximation by „ bm*(T + T ) + | B M = A ( T , B ) s i n <?f- + * + ^ £_) . A2.4 Eq. A2.4 d i f f e r s from the u s u a l expression f o r M (Eq. 2.4a) by an o r d i n a r i l y i g n o r a b l e f i e l d and temperature dependent phase term. This term gives r i s e to an apparent frequency s h i f t w i t h temperature as g i v e n by ( i ) i n Eq. 4.33 and which i s not completely i g n o r a b l e under our exper-imental c o n d i t i o n s . ( i i ) More Exact C a l c u l a t i o n of 4~ M. dt We now consider the harmonic expansion of Eq. 3.5a f o r the time r a t e of change of M i n the presence of a modulation f i e l d h s i n cot. The F o u r i e r expansion given i n Eq. 3.5b was obtained by i g n o r i n g the time dependence of the amplitude A(T,B + h s i n tot). However i f we expand the amplitude as w e l l as the argument of the s i n e f u n c t i o n i n Eq. 3.5a to f i r s t power i n h/B and c a r r y out the expansion of the s i n e f u n c t i o n by the a i d of the i d e n t i t y e x p ( - i u s i n y) = I J n ^ u ) exp(-iny) , A2.5 125 we f i n d that ^ i s given by = E 2ncoJ n(y) s i n (ntot + —-) n=l x A(T,B) s i n o y J ' (M) bm*(T + T- )' 2TTF , , . HIT , n K / V D B 2 J (u) 2TTF ' A2.6 where J ' (y) = J (y)• This r e s u l t d i f f e r s from Eq. 3.5b by the n dy n ^ J a d d i t i o n a l phase term which v a r i e s with temperature. The corresponding apparent change i n the frequency F with temperature i s then given by term ( i i ) i n Eq. 4.33. For small u the quantity y J ' n ( y ) / J n ( y ) approaches the value n so that the phase c o r r e c t i o n term can be considerable when dM detecting high harmonic components of In t h i s experiment y was chosen so that J ' n ( y ) , with n = 4, was as close as poss i b l e to zero. However on account of the high modulation f i e l d h required i n i r o n t h i s c ondition could not be achieved exactly. In f a c t i t was found that the d i f f e r e n c e i n the observed frequency s h i f t s f o r i r o n at 17 kG and 22 kG ( F i g . 4.1) could be accounted f o r quite w e l l from the known diff e r e n c e i n y j ' ( y ) / J (y) f o r the two measurements. For molybdenum n n J' n ( y ) could be made i d e n t i c a l l y zero i n the experiment since the modulation f i e l d h required to make J n ( y ) maximum was about one-fourth as large as that f o r i r o n . In order to check the v a l i d i t y of Eq. A2.6 f o r the modulation dM f i e l d s used i n this experiment, the Fouri e r expansion f o r (Eq. 3.5a) was c a r r i e d out exactly by computer. I t was found that the temperature dependent phase co r r e c t i o n given by Eq. A2.6 gave exce l l e n t agreement 126 with the computer r e s u l t s f o r a l l values of u and n tested. We should point out that Eq. A2.6 w i l l also contain an a d d i t i o n a l phase correc-t i o n which i s dependent on B but not on T. This term i s of no concern here and has not been included. ( i i i ) E f f e c t of Misalignment i n the Magnetic F i e l d As we pointed out i n §3.5 when the magnetic f i e l d i s t i l t e d away from the [111] d i r e c t i o n the t o t a l o s c i l l a t o r y magnetization associated with the el e c t r o n lenses may be represented by the equation 3 2TTF. M= Z A.(T,B) s i n (-r- 1 + <J>) , A2.7 3-1 J B where the subscripts 1, 2 and 3 r e f e r £-oscillations. It i s only when B i s that we have A^ = A^ = A^ and F^ = F 2 of B_ from the [111] axis we may write to the three branches of the p a r a l l e l to the [111] d i r e c t i o n = F-, For a given small t i l t F = F + AF. A2.8a 3 ° J m. = in + Am. , A2.8b j o J where F q r e f e r s to a reference frequency near the [111] d i r e c t i o n . I f we su b s t i t u t e these parameters into Eq. A2.7 we can rewrite the t o t a l dHvA magnetization into the form 2TTF M = A s i n (-r-2- + <j> + 0(T)) , A2.9a 127 where and A = A 2 + A 2 A2.9b c s -1 A *(T) = tan 1 ^ A2.9c c 2TTAF. A = E A.(T,B) cos —=-J- A2.9d c . , J B 3 2TTAF. A = E A.(T,B) s i n — L . A2.9e S j = l 2 B It follows f rom Eq. A2.9a that over a short f i e l d range M looks l i k e a s i n g l e p e r i o d i c term of frequency F q and with the net temperature depen-dent phase $(T). For a given ST we can determine 5$ by d i f f e r e n t i a t i n g Eq. A2.9c, assuming A_.(T,B) to be of the same form as A(T,B) given i n Eq. A2.2. The r e s u l t i s 5$ 6T b B 3 E Am. 3 A 2TTAF. A 2TTAF. C J S 1 o o ( A ) c A 2 + 1 \ A 1 s A 2~ _ \ °l \ °l _ A2.10 where A i s the value of A. corresponding to F and m . o J o o Under the conditions of our experiment Eq. A2.10 can be written i n rather simple form. According to the alignment procedure outlined i n §3.5 the sample was f i r s t oriented (by x-ray back r e f l e c t i o n photo-128 graphy) so that the (110) plane was c l o s e l y p a r a l l e l with the plane of r o t a t i o n of the magnetic f i e l d . The misalignment Q± of the [111] axis from the plane of r o t a t i o n was kept below about 1/2° for Fe and 1/4° for Mo. The electromagnet was then rotated c l o s e to the [111] axis by making use of the fac t that the beat pattern produced by the three e - o s c i l l a t i o n s must disappear when IJ i s along the [111] d i r e c t i o n . The f i n a l adjustment was made by l o c a t i n g the maximum of the amplitude (Eq. A2.9b) f o r the G - o s c i l l a t i o n s . In the end the misalignment of B. from the [111] d i r e c t i o n could be characterized by an angle 0± perpen-d i c u l a r to the (110) plane and an angle Q,, p a r a l l e l to the (110) plane. These two angles were not independent, however, because of the exp e r i -mental requirement that the amplitude A i n Eq. 4.29b must be a maximum. From t h i s consideration i t i s poss i b l e to eliminate 0n and then rewrite A2.10 in t o the simple form 6$ = -bAm* 6T B where (, 2 S l " t i> , A 2 . l l 2 cos tl» ± 1 ' * * - 4 | ^ | ana • . | i e, | ^ | The angle 0^  i s assumed to be p o s i t i v e , the subscript D r e f e r s to the doublet frequency and the d e r i v a t i v e s are evaluated at the [111] d i r e c -t i o n . The r e s u l t i n g apparent frequency s h i f t i s then given by term ( i i i ) i n Eq. 4.33. The change i n the frequency F^ with angle was estimated from the published data f or both Mo and Fe and the change i n m* was computed by 129 assuming that m*^  i s proportional to F^, as would be expected for an e l l i p s o i d a l Fermi surface. F i n a l l y the angle was taken to be 1/2° for Fe and 1/4° for Mo i n accordance with the alignment accuracy quoted e a r l i e r . The r e s u l t i n g apparent frequency s h i f t s between 1° and 4°K were found to be n e g l i g i b l e f o r i r o n (because SF^/DS^ i s small) but s u f f i c i e n t l y large i n molybdenum (for which aF^/dO^ i s estimated to be eight times l a r g e r than i n iron) to account f o r the observed sign and order of magnitude of the experimentally observed s h i f t s (see.§4.3). ( i v & v) E f f e c t of Inhomogeneities Any inhomogeneity i n the f i e l d B_ or the c r y s t a l s t r u c t u r e would r e s u l t i n a nonuniform d i s t r i b u t i o n of the dHvA phase F/B and the ampli-tude A(T,B) over the sample volume. If F and B are the average values o o of F and B, then f o r a given region of the c r y s t a l F = F + FT o 1 B = B + B, o 1 A2.12 where F^ and B^ are small qu a n t i t i e s which have some p r o b a b i l i t y d i s t r i -bution P(B 1,F^). For de f i n i t e n e s s we assume P ^ j F ^ ) i s the product of two Lorentzian functions P(B 1,F 1) TTAB 1 + AB 21 -1 TTAF 1 + _1 AF -1 A2.13 where AB and AF give the c h a r a c t e r i s t i c h a l f broadenings of F and B i n the c r y s t a l . The net dHvA magnetization of the sample i s then given by 130 M = l„ L„ A(T,B +13^ s i n — OO — OO 2TT(FO + ? X ) + B]_) + + P ( B 1 , F 1 ) d B 1 d F 1 A2.14 In the argument of the i n t e g r a l we have included the e f f e c t of the inhomogeneity not only on the dHvA phase (as i s u s u a l l y done) but also on the exponential part of the amplitude A(T,B) (Eq. A2.2). The c y c l o -tron mass entering the argument of t h i s exponential w i l l be assumed to be proportional to the frequency F as i n part ( i i i ) . It i s by i n c l u d i n g the e f f e c t of the inhomogeneity on the exponential f a c t o r i n A(T,B) that we w i l l obtain a net temperature dependent phase for M. Assuming AF and AB i n Eq. A2.13 are small, then M i n Eq. A2.14 can be r e a d i l y evaluated and the r e s u l t i s - > n 2 I T F O T, *T M = A(T,B q) e" W s i n [ — ° + $ - Q f^A] , A2.15a o o where 2TTF AB Q = 2 1 5 — • A2.15b B o o From t h i s r e s u l t we see that an inhomogeneity leads to a reduction i n the amplitude (see also Shoenberg 1969) and to a temperature dependent phase term i n M.. The r e s u l t i n g apparent frequency s h i f t i s then given by terms (iv) and (v) of Eq. 4.33. In order to estimate the parameter Q i n the temperature dependent phase term we exploited the fact that Q also appears i n the amplitude factor e ^. Eqs. A2.15 and A2.2 i n d i c a t e that 131 + constant , A2.16 where the constant i s independent of B q. The amplitude of the dHvA magnetization was measured as a function of B q and then the logarithm of the amplitude times J^B~ was p l o t t e d against B q With the aid of Eq. A2.16, the slope (over a l i m i t e d f i e l d range) then y i e l d e d an e s t i -mate of Q i n the experiment. The parameter T^ i n Eq. A2.16 was assumed to a r i s e from impurities i n the sample and was therefore ignored compared to the l a r g e r term due to inhomogeneities f o r i r o n . For molybdenum i t was found that Q was n e g l i g i b l e whereas for i r o n a value of about 1.7 was obtained by t h i s approach f o r the experimental value of B q i n the neighborhood of 40 kG. ln(Ae (W j) 2TTF AB bm*(T + T J + — — + 2TTAF U a B We end t h i s appendix with a b r i e f d i s c u s s i o n of other e f f e c t s which could i n p r i n c i p l e give r i s e to s h i f t s i n the dHvA phase but which were found to be i n s i g n i f i c a n t under our experimental condit i o n s . The temperature dependence of the 4- Fermi l e v e l e + given by the second term i n Eq.2.10b was discussed i n §2.2 and estimates show that the change i n frequency due to t h i s term i s of the order of one part i n 10 6 or 10 7 between 1° and 4°K. The thermal expansion of the c r y s t a l gives r i s e to changes i n the dimensions of the B r i l l o u i n zone as w e l l as d i s t o r t i o n s of the bands. The change i n the extremal area due to r i g i d s c a l i n g of the B r i l l o u i n zone i s given by 5A-/A- = -2 6L/L, where SL/L i s the l i n e a r expansion of 132 the c r y s t a l determined from the thermal expansion c o e f f i c i e n t . The change i n due to d i s t o r t i o n s of the bands can be estimated from the data of Svechkarev and Pluzhnikov (1973) f o r the pressure dependence of the Fermi surface of molybdenum. Their data show that f or Mo the change 6&/& i s f i v e times greater than that predicted by the simple s c a l i n g of the B r i l l o u i n zone. If we assume that t h i s f a c t o r of f i v e a lso applies to i r o n then the change i n the lens frequency between 1° and 4°K i s estimated to be 6F F 2.0 x 10 7 f o r Fe - 0.6 x 10 7 for Mo which i s n e g l i g i b l y small for both metals. We next consider the e f f e c t of the Shoenberg magnetic i n t e r a c t i o n s , As f i r s t pointed out by Shoenberg the dHvA magnetization M should be included i n the t o t a l f i e l d B entering the argument of the s i n u s o i d a l functions i n Eq. 2.2a f o r M(T,B). P h i l l i p s and Gold (1969) have shown that the s e l f - c o n s i s t e n t s o l u t i o n f or the fundamental component of the dHvA magnetization i s then given by the following approximate equation M = A s i n 2*F , -1 ^2 F A 2 ( T » ^ " ~B~ + * 1 t a n _ 2 A2.17 where B i s the n o n o s c i l l a t o r y part of the magnetic induction and A 2(T,B) i s the amplitude of the second harmonic of the t o t a l dHvA magne-t i z a t i o n . The values of A 2(T,B) estimated from the experimental data in d i c a t e that the apparent frequency s h i f t s a r i s i n g from the temperature dependent phase term i n Eq. A2.17 were l e s s than one part i n 10^ for both i r o n and molybdenum, between 1° and 4°K. 133 F i n a l l y we consider the e f f e c t of the o s c i l l a t o r y v a r i a t i o n i n the chemical p o t e n t i a l with f i e l d , which a r i s e s on account of the f i e l d dependence of the density of states of the conduction electrons i n the metal (Kaganov et a l . 1957). The o s c i l l a t i o n s i n the chemical p o t e n t i a l give r i s e to deviations i n the p e r i o d i c i t y of the dHvA magnetization (i n B ^) and since the amplitude of these o s c i l l a t i o n s varies with temper-ature they can lead to s h i f t s i n the dHvA phase between 1° and 4°K. 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