Louisiana State UniversityLSU Digital Commons

LSU Historical Dissertations and Theses Graduate School

1975

Spin-Orbit Coupling Effects in Band Structure ofFerromagnetic Transition Metals.Manjit SinghLouisiana State University and Agricultural & Mechanical College

Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses

This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion inLSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contactgradetd@lsu.edu.

Recommended CitationSingh, Manjit, "Spin-Orbit Coupling Effects in Band Structure of Ferromagnetic Transition Metals." (1975). LSU HistoricalDissertations and Theses. 2896.https://digitalcommons.lsu.edu/gradschool_disstheses/2896

INFORMATION TO USERS

This material was producad from a microfilm copy of tha original document. While the most advanced technological meant to photograph and reproduce this document have bean used, the quality it heavily dependant upon tha quality of tha original tubmitted.

Tha following explanation of technique! it provided to help you understand markingt or patterns which may appear on this reproduction.

1. Tha sign or "target" for pagas apparently lacking from tha document photographed it "Misting Pageisl". If it was possible to obtain the misting page(s) or taction, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity.

2. When an image on tha film is obliterated with a large round black mark, it it an indication that the photographer suspected that tha copy may have moved during exposure and thus causa a blurred image. You will find a good image of tha page in the adjacent frame.

3. When a map, drawing or chart, etc., was part of tha material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand comer of a large sheet and to continue photoing from left to right in equal tactions with a small overlap. If necessary, sectioning is continued again - beginning below tha tin t row and continuing on until complete.

4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could bo made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced.

6. PLEASE NOTE: Some pages may have indistinct print. Filmed as received.

Xerox University Microfilms300 North ZoobltooOAnn Arbor, Michigan 40100

76-12,941SINGH, Manjlt, 1945-

SPIN-ORBIT COUPLING EFFECTS IN BAND STRUCTURE OF FERROMAGNETIC TRANSITION METALS. \The Louisiana State University and jAgricultural and Mechanical College Ph.D., 1975 Physics, solid state

i

Xerox University Microfilms, Ann Arbor. Michigan woe

THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED.

SPIN-ORBIT COUPLING EFFECTS IN BAND STRUCTURE OF FERROMAGNETIC TRANSITION METALS

A DISSERTATION

Submitted to the Graduate Faculty of the Louisiana State University and

Agricultural and Mechanical College in partial fulfillment of the

requirements for the degree of Doctor of Philosophy

inThe Department of Physics and Astronomy

byManjit Singh

M.S., Louisiana State University, 1971 December, 1975

To my parents and my wife

ACKNOWLEDGEMENTS

The author is deeply indebted to Professor Joseph Callaway for suggesting the problem and for his guidance during the course of this investigation. He wishes to express his gratitude to Dr. J. C. Kimball for numerous advice and help. He is very grateful to his colleague,Dr. R. A. Tawil, for giving permission to use the results of energy bands of iron on which this work is based. It is his pleasure to thank his colleagues Drs. J. Rath, W. Y. Ching, R. A. Tawil, and in particular, Dr. C. S. Wang, for their help at various stages of this work. He would like to thank Mrs. Martha Prather for the efficient typing of the manuscript. He is grateful to the Computer Research Center for the computational facilities provided by them.

A financial assistance pertinent to the publication of this dissertation from Dr. Charles E. Coates Memorial Fund of the LSU Foundation donated by George H. Coates is gratefully acknowledged.

ii

TABLE OF CONTENTSPage

ACKNOWLEDGEMENTS iiLIST OF TABLES ivLIST OF FIGURES V

ABSTRACT viCHAPTER Is INTRODUCTION 1CHAPTER II*. SP IN-ORB IT COUPLING AND THE TIGHTBINDING METHOD 7

A. Spin-Orbit Coupling and the BandStructure of Iron 7

B. The Fermi Surface 15C. Optical Conductivity 19

CHAPTER III: g SHIFT OF CONDUCTION ELECTRONS INFERROMAGNETIC METALS 26

A. Perturbation Term in the Hamiltonian inthe Presence of Constant Magnetic Field 27

B. Matrix Elements of Perturbation in theTight Binding Method 29

C. Calculation of the g Shift in Ironand Nickel 42

CHAPTER IV: Conclusions 51REFERENCES 55APPENDIX: TRANSFORMATION PROPERTIES OF WAVE FUNCTIONEXPANSION COEFFICIENTS UNDER THE ACTION OF CUBIC MAGNETIC POINT GROUP OPERATIONS 60TABLES 65FIGURES 76VITA 89

iii

LIST OF TABLESTables

I. Energy Levels Including Spin-Orbit Coupling at Symmetric Points

II. Predicted de Haas-van Alphen Frequencies in mG Compared with Calculations Without Spin Orbit Coupling (Ref. 1) and with Experiments (Refs. 2, 3)

III. Interband Optical Conductivity Tensor and the Joint Density of States

IV. Calculated Values of Various Terms in the g Shift for the Case of Iron and Nickel

iv

Page

65

66

67

75

78

79

8081

82

83

84

85

86

87

88

LIST OF FIGURES

Self-Consistent Energy Band for Iron Including Spin Orbit Coupling Projected Density of States for Majority SpinProjected Density of States forMinority SpinTotal Density of StatesFermi Surface Cross Section in the(100) PlaneFermi Surface Cross Section in the (110) Plane Centered About r Fermi Surface Cross Section in the(110) Plane Centered About H Fermi Surface Cross Section in the(111) Plane Through VFermi Surface Cross Sections in the (111) Plane Through H The Real Part of a From 0.2 ev to 7.0 evThe Imaginary Part of uu from 0 toxy7 ev

ABSTRACT

A previous self-consistent linear-combination-of- atomic-orbitals calculation of energy bands in ferromagnetic iron has been extended through the inclusion of spin-orbit coupling. The exchange interaction is incorporated ac­cording to the Xa method. The Fermi surface is described in detail and compared with the results of measurements of the de Haas-van Alphen effect, and of magnetoresistance anisotropy. The interband contribution to the optical- conductivity tensor was computed using matrix elements determined from wave functions including spin-orbit coupling. Both diagonal and off-diagonal elements of this tensor have been obtained. Expressions are derived in the tight binding method, for the shift in the g-factor of conduction electrons, produced due to spin-orbit interaction. Values for the g shift are calculated for iron and nickel using the results of band structure calculations that include spin-orbit coupling effects.

vi

CHAPTER I INTRODUCTION

Spin-orbit coupling is of major significance in thedetermination of the Fermi surface and optical propertiesof ferromagnetic transition metals. It leads to the

37-43existence of magnetic anisotropy, anomalous Hall36effect and shift in the g (spectroscopic splitting)

21 22factor in ferromagnetic spin resonance experiments. 'Band of up and down spins are hybridized, and most of the accidental degeneracies which are present when spin-orbit coupling is neglected are removed. As a result there can be substantial changes in the connectivity of the Fermi sur­face. The symmetry group of the crystal in the presence of ferromagnetic exchange and spin-orbit coupling is reduced in comparison with a state in which the possibility of spin polarization is neglected.6 '^ The conductivity tensor is not diagonal,31 and magneto-optical effects are produced.

Attempts have been made to study spin-orbit inter­action effects in the band structure of ferromagnetic

37-43transition metals for more than thirty years. Much ofthis work, however, has been based on oversimplified tightbinding model of the d band structure. Other investigationshave employed interpolation schemes designed to fit empiricalinformation concerning the band structure, magnetic

45 46properties and Fermi surfaces. ' We are not aware of a1

2

previous first-principle band calculation for iron which includes both exchange splitting and spin-orbit coupling effects. A model band structure has, however, been

gdeveloped by Maglic and Mueller in which spin-orbitcoupling is added to an interpolation scheme based on theKorringa-Kohn-Rostoker (KKR) band calculation of Wakoh and

gYamashita. In our work, the spin-orbit interaction isdetermined from the self-consistent potential used in theband calculation of Tawil and Callaway.1 Our results showsome differences in the effects of spin-orbit coupling onthe Fermi surface of iron in comparison with the results ofMaglic and Mueller.

It has been observed that plane-polarized lightreflected from a ferromagnetic metal becomes ellipticallypolarized with its major axis rotated from the original

29 30direction of polarization. ' This phenomenon is known as the magneto-optic Kerr effect. Macroscopically, the rotation angle is related to the absorptive part of the off-diagonal element of the conductivity tensor through Maxwell's field equations22' a s shown in Section C of Chapter II. The spin-orbit interaction in the solid provides a means of coupling the magnetic dipole associated with electron spin to the electric dipole transitions produced by incident light. This is the accepted physical basis of magneto-optic effects in ferromagnets. Micro­scopic theories of magneto-optic Kerr effects have been

3

32-34given by a number of authors. The expression for theabsorptive part of the off diagonal element of conductivity tensor are derived and the calculated results for it are compared with the experimental ones in Section C of Chapter II.

The shift of the g factor from the free-electron value observed in spin-resonance experiments arises from spin- orbit interaction with the periodic potential of thecrystal. The theory of this effect was initiated by

56 59Yafet, reformulated and generalized by Roth and byBlount®^ and recently simplified and corrected by de Graaf

51 55and Overhauser and by Moore. According to de Graaf and51Overhauser, the expression for the g shift in metals with

inversion symmetry can be divided into three parts. Thefirst part represents the contribution of the spin-dependentpart of the velocity operator, namely, — ^xVv , to the

4m cexpectation value of the z-component of the orbital angular momentum and is given by

(1.1)

where v(r) is the crystal potential and ^ is the wave function of a conduction electron. The second part is the relativistic contribution

where p is the momentum operator. Finally, the last part is given by

where S stands for terms containing integrals over thesurface of the cellular polyhedron. These expressions are

52-55calculated by a number of authors for alkali metals where the conduction electrons are s electrons. All these three terms are of the same order of magnitude for these metals. In ferromagnetic metals, the presence of d electrons in the conduction band makes the contribution of the last term, namely, the expectation value of the z-component of angular momentum very large compared to other terms. The work on the calculation of the g shift in ferromagnetic metals has been confined to calculating the expectation value of z-component of orbital angular momentum, using the wave functions which are eigenfunctions of the one-electronHamiltonian containing spin-orbit interaction effects.

37 38Brooks and Fletcher considered the wave function foronly three states of the d band to calculate the g shift

39— 43 48 49for nickel and iron. Other authors ' ' took intoaccount all of the 5 states in the d band, and hybridiza­tion of s-p states with d states in order to calculate

5

the g shift. In all of these calculations, spin-orbit interaction is introduced in an approximate manner and some approximations of uncertain validity are made in the calculation of the matrix elements of orbital magnetic moment.

The accurate calculation of the g shift in ferro­magnetic metals requires the energy bands and eigenfunc­tions for these materials with exchange and spin-orbit interaction effects adequately included. These calcula­tions exist for nickel^* and they are attempted here for iron and are described in Ref. 61. Unlike all previous calculations, these calculations include the spin-orbit interaction in the first-principle calculation of energy bands in the tight binding method. The effect of the spin-orbit interaction was considered for both the d states and p states. The hybridization of s, p, d states has been included. In Chapter III, we have derived the expression for the g shift which is suitable for calculations in the tight binding or linear combination of atomic orbitals (LCAO) method. We have included the matrix elements of the magnetic moment when states are centered on different sites, unlike most of the other authors.

The emphasis of this work will be on the detailed study of spin-orbit interaction effects on the Fermi sur­face, the optical conductivity of ferromagnetic iron and the g shift in ferromagnetic iron and nickel. The plan of

6

this dissertation is as follows: In Chapter II Section A,the inclusion of spin-orbit coupling in the band structure of ferromagnetic iron is outlined/ and its effect on the Fermi surface and optical conductivity is included in the subsequent sections of the same chapter. Chapter III contains the derivation and the calculation of the g shift for iron and nickel. The work in Section B, concerning the derivation in the tight binding method of the matrix elements of perturbation term in the Hamiltonian due to uniform magnetic field, is mainly our own. Section C deals with the calculation of the g shift and the calcula­tion of each of the terms in 6g is discussed in detail.Our conclusions are stated in Chapter IV.

CHAPTER IISPIN-ORBIT COUPLING AND THE TIGHT BINDING METHOD

This chapter is divided into three sections. In Section A, we describe the introduction of spin-orbit coupling into the band structure of ferromagnetic transi­tion metals in the modified form of the tight binding

13method, which has successfully been applied to calculate the band structure of nickel,** iron,* chromium,*4 alumi­num, *^ and alkali metals*^'*® by Professor Callaway's group. The resulting band structure and density of states for iron are discussed in this section. Section B contains the description of the Fermi surface of iron. In Section C, optical conductivity of iron is discussed.

A. Spin-Orbit Coupling and Band Structure of iron

Spin-orbit coupling was introduced into the band structure calculation of ferromagnetic iron. The calcula­tion is based on a modified form of the tight binding method due to Lafon and Lin*^ in which the wave function for a state of wave vector k in band n, ^n (k,r) is expanded in a Bet of basis functions, 4>^(k,r). The $^(k,r) are in turn constructed from a set of localized functions

8

1* (k,r> ■ ±_ E e 14 V, (r-Rj (2.2)1 »TJ |i

where N refers to number of sites in the lattice and R^ isthe lattice vector. The functions (V^'str-R^) are situatedon the site R^ in the lattice. They consist of atomic wavefunctions for all states except 3d(Is,2s,3s,4s,2p,3p,4p)expressed as linear combination of Gaussian-type orbitals

18(GTO) determined by Watchers from a self-consistent field calculations for free atoms. The localized orbitals are given by

V ? > - <2 -3>

where K„ _(6,$) are the Kubic harmonics and the radial wave r ,mfunction can be represented by GTO

•+• Jt —1Rit<r) * Nii r e <2-4>

with the normalization constant given as

n - r2(2aii1'u ’5 i>fN»i L ra+m— J (2-5>In the cases where atomic wave functions were required,

the were linearly expanded in GTO

r 2

Rn l <r> “ \ anli Nti rl_1 • 1 <2-«>

9

where n is the principle quantum number. The values of18anii and aJli bave been tabulated by Hatchers. Five in­

dependent GTO * s were introduced for each of the five angular functions. The orbital exponents used in defining d functions were the same as given by Hatchers. The basis set thus consists of 38 functions for each spin: 4 for s-symmetry, 9 for p-symmetry and 25 for d-symmetry. The results of the band calculations for B.C.C. ferromagnetic iron with a lattice constant of 2.854 A°, using the spin- polarized potential was taken from the work of Tawil and Callaway.1 In these calculations the spin-orbit inter­action was neglected and the exchange potential was intro-

19duced according to the Xa approximation. The coefficientof the Slater exchange potential was taken as a * 0.64,as this choice gives the hole pockets around the H pointsin the Brillouin zone which are necessary to explain the

2 3results of de Haas van Alphen measurements in iron. 'These calculations were carried to self-consistency. De­tails of these calculations are given in the thesis of R. A. Tawil. These calculations were extended here through the inclusion of the spin-orbit coupling given by

Hs-°- " <3xVv)-5 (2.7)4m c

10

where V(r) Is the self-consistent crystal potential, the o fs are Pauli spin matrices and p is the momentum operator. Other relativistic effects were neglected as they do not affect the band structure in any significant way.

The introduction of spin-orbit coupling into the band calculation for a ferromagnet causes substantial compli­cations. First, it couples states of up (♦) and down (4-) spins, and leads to a doubling of the size of the Hamilton­ian matrix. The size of the Hamiltonian matrix in our case becomes 76x76. Moreover, the matrix elements become complex. Also, the symmetry group is reduced. The appro­priate group theory has been presented by Falicov and Ruvalds^ and Cracknell.7 The band structure depends on the direction of spin alignment. Separate band structure must be computed for each direction of spin alignment to be investigated. However, because of limitation of computer time, we have made extensive computations of energy levels for an assumed [001] direction of spin alignment and con­sidered a smaller number of k points for spins aligned along the [110], [111] and [211] directions.

The space group of a ferromagnet includes the following symmetry elements as pointed out by Falicov and Ruvalds as well as Cracknel1.

(i) lattice translations(ii) rotations about the direction of spin alignment n(iii) the product of these rotations with the inversion

11

(iv) combinations of these rotation and rotation- inversion with translation

(v) the product of time reversal which, by itself, is not a symmetry operation, and either a two-fold rota­tion about an axis perpendicular to the direction of spin

A> A

alignment n or a reflection in a plane containing the n axis. These operations are antiunitary.

The potential V(r) appearing in the expression for Hg0 is expressed in a Fourier series

where Kfl is reciprocal lattice vector.The matrix elements of Hg0 were calculated using the

basis set described before. It was found that the onlynon-negligible matrix elements of Hgo are those in the p-p and d-d blocks, with orbitals centered on the same site (central cell). The central cell matrix elements of Hgo have the form

The forms of the submatrices M and N for p-p and d-d blocks are given in detail in the thesis of C. S. Wang.

A

V(r) - L V(Kfl)e (2.8)

(2.9)

12

If the actual spin-orbit interaction given by Eq.(2.7) is approximated in the usual way

Hgo - £ L*S (2.10)

with an "atomic” spin-orbit parameter £ given by

r , _ 1 f 1 dVl5(r> 5? ?

then the strength of the spin-orbit coupling can be com­pared with that existing in the free atom. Using our potential (2.8) and atomic wave functions of Watchers toevaluate £, a value of 0.0043 Ryd was obtained which is

44somewhat larger than atomic value of 0.0035 Ryd.The Hamiltonian including exchange and spin-orbit

coupling was diagonalized at 729 points in 1/16'th part of the B.z. for a tOOlJ direction of spin alignment. The calculated band structure is shown along certain symmetry lines in Fig. 1. Some calculated energy levels at symmetry points are listed in Table I. Since the actual symmetry group for this problem does not permit a particularly informative classification of states, the states at symmetry points have been labelled in terms of the pre­dominant component, that is, neglecting the mixing of components of majority and minority spin. This labelling is meaningful since spin-orbit coupling is about 2-3

13

orders of magnitude smaller as compared to exchange splitting.

The band structure including spin-orbit coupling effects shown in Fig. 1 is, to a first approximation, the superposition of the majority and minority spin band structure computed without the spin-orbit interaction.The Fermi energies are almost identical. However, spin- orbit coupling removes most of the degeneracies present in such a picture and hybridizes states of opposite spin. In addition, the reduction of the symmetry group reduces the number of equivalent wave vectors in a star, so that energies at points such as N(*s,)j,0) and N(%,0,%) are no longer equal.

Our calculated energy bands are consistent with the group theoretical analysis of Falicov and Ruvalds® as

7extended by Cracknell. The following points deserve comment:

(i) Accidental degeneracies, which are present in a band structure not from symmetry requirements, may be permitted in symmetry planes perpendicular to the direction of spin alignment (10013 in our case), but are not required.

(ii) All degeneracies required by symmetry in the absence of spin-orbit coupling are removed (for example

A5'ri2)*The density of states was calculated from the energy

bands computed at 729 points in l/16fth part of the

14

20Brillouin Zone (BZ) using the Gilat-Raubenheimer method.The Fermi energy was determined by occupying states in order of increasing energy until 8 electrons per atom are accommodated. The spin projected and total densities of states are shown in Figs. 2-4. The total density of states shows a pronounced two peak structure as was the case without spin-orbit coupling. The two peak structure of total density of states is primarily a consequence of the exchange splitting. The Fermi energy falls in the region of the minimum between the peaks. The density of states at the Fermi energy is, for majority spin, 7.87/atom-Ryd and for minority spin, 5.65/atom-Ryd. This yields an electron specific heat coefficient

Nte^)

* 3 o 2of 2.34x10* joules/mole- K . A recent experimental resultfor this quantity is 4.74 x 10*3 joules/mol-°K^.3^ Aportion of the discrepancy can probably be attributed tothe effects of the electron-phonon interaction. Themagneton number was found to be 2.29 Mg which is to becompared with an experimental value of 2.12 m b as deduced

25 22 23from the measurements of magnetization and g ' or g,21,24 factors.

15

B. The Fermi Surface3Recent experiments on de Haas-van Alphen effect and

5magnetoresistance anisotropy have produced a fairly complete description of the Fermi surface of iron. In this section, we will compare our calculated Fermi surface with the experimental results. Most of our energy level calculations were performed with the [001] axis as the direction of spin alignment, which implies that we should obtain a Fermi surface cross section relevant to the experiment only in the (001) plane. However, the Fermi energy should depend only very weakly on the direction of spin orientation. He repeated the calculation for a sample of points lying in the (110) and (111) planes with the spin alignment direction perpendicular to these planes.The energies so obtained did not produce Fermi surface cross sections in those planes which differed significantly from those obtained in the same planes but with the field along [001]. Our calculated Fermi surface cross sections are shown in Figs. 5, 6, and 7. Numerical values for calculated and experimental cross sections are given in Table II.

The Fermi surface calculated here is quite similar to the case where spin-orbit coupling was neglected, except for some alterations caused by the removal of accidental degeneracies and some hybridization of states of different spins. There are eight pieces of Fermi surface to be

16

discussed.I. Large electron surface centered about V. This

involves states of predominantly majority spin, except near N where there is some hybridization with minority spins.The calculated areas of the cross sections of this surface in the (100) and (110) planes are in good agreement with experiment. The calculated (111) cross section is slightly small. The area in this plane is reduced from that pre­dicted in the absence of spin orbit coupling as a result of hybridization with a large, predominantly minority spin hole pocket (VIII) near N.

II. Major hole surface around K. These arms extendin the H-N directions. In the (100) plane, the armshybridize with the hole ellipsoids around N, however,in the (110) and (111) planes there is an additional mixingwith the large hole surface (V) around H. These lattersurfaces do not mix in the (100) plane. This mixing leadsto the possibility of open orbits in the [110] direction,which are required by the interpretation of magneto-

4 5resistance experiments. 'III., IV. Intermediate and small hole pockets about H.

These involve states of majority spin. Our calculated values for the cross sections of these pockets are uniformly much smaller than the values attributed to them in the interpretation of the de Haas-van Alphen measurements.This indicates that our calculated position of the level

17

j ' is too close to the Fermi energy.V. Large hole surface around H. This is composed

mainly of minority spin states. The cross sections of this surface are closed in the (100) and (111) planes; how* ever, we find that hybridization with the hole arms in the (110) plane leads to open curves. Since closed orbits around this surface are actually observed in this plane,we conclude that magnetic breakdown must occur across a small gap produced by spin orbit coupling. A reasonable agreement between theoretical and experimental cross sectional areas is found. The cross section in the (111) plane calculated with spin-orbit coupling is somewhat too small, whereas the calculation without spin orbit coupling gives a result in substantially closer agreement with experiment. It is possible that magnetic breakdown restores an orbit similar to that calculated in the absence of spin orbit coupling. There is actually no contact between surface (V) and the electron ball (VII), but the separation is too small to show in Fig. 5.

VI. Central (minority spin) electron surface about T. Our results for the cross sectional areas of this surface are consistently larger than the experimental values.

VII. Electron ball (minority spin) along A. Our calculations predict a small ball which almost touches the hole octahedron (V) whereas the de Haas-van Alphen measure­ments indicate that the ball is larger and intersects the

18

central minority spin surface. This is one of the most serious disagreements between our calculation and experi­ment. If the spin splitting were reduced by approximately 0.05 Ry which is approximately 33% of exchange splitting, this surface would have approximately the correct shape and size. This would permit magnetic breakdown between surfaces III and V when the field is along the [112]

3direction, as is observed. The dimensions of the hole pockets (III, IV) near H would also be improved. In addi­tion, the somewhat too large value we obtain for the mag­neton number would be reduced, although the amount of this reduction (to 1.98) is too large. We do not find any hybridization in the (100) plane between the electron ball and surface I, although mixing is present in other planes.

VIII. Hole pockets around N. These surfaces arepredicted by our calculation and some other first principles

9 12studies. ' They are composed of both majority and minority spin states. Spin orbit coupling causes a marked reduction in the size of these surfaces through hybridiza­tion with the hole arms (II) . The existence and properties of these pockets depends on the relative position of the p and d like levels near N. However, our basis set is probably more adequate for d states than for those of p symmetry and it is possible that a calculation in which separated orbitals rather than atomic functions were em­ployed for states of p symmetry might lead to some

19

reordering of levels.The magneto-resistance measurements indicate clearly

the presence of open orbits in the [100] and [110]4 5directions. ' It was pointed out above that our model

of the Fermi surface provides for such orbits in the [110]direction. To obtain open orbits in the [100] directionit is necessary for us to suppose that magnetic breakdownestablishes paths close to the (100) plane involvingsurfaces II and V (or V, VII, and I). Additional openorbits in the [110] direction can be established similarly.Since we do not obtain overlap of the electron ball (VII)and the central hole surface (VI), we do not have as obviousa mechanism for the [100] orbits as is present in the model

2of Gold et al. Further, we have no evident explanation for the apparent pressure dependence of the [100] open orbits.*

C . Optical Conductivity

We have calculated the diagonal and off-diagonal inter­band conductivity of iron. The results will be presented in two cases:

1) Including a phenomenological constant relaxation time T.

2) In the limit T**-* so that the band states are sharp.

The general expression for the conductivity tensor in

20

50case (1) is obtained from the Kubo formula

„ _ i Ne2 ,1 v 2ie2 ^0ae<“ > - 1OTI7TT ‘r ’-* ' 5k tk

U 0

* - Z -----------1------------ 7 * * ("* ”5 , ) +| w2 (k) - (u+i/T) % t *n nl

(2.12)

The sum over Ik includes occupied states only and that over nk includes unoccupied states only.

The quantities ir®n etc. are the Cartesian components of the matrix element vector defined by

"nj “ T T - { (P + ^ 7 5*W(E)) l y i ^ l d 3;

(2.13)

in which & is the volume of the unit cell and U is the cell periodic part of the Bloch function. The contribution of the spin orbit contribution to matrix elements was negli­gible; hence in practice Trnj was replaced by pn in our calculations. The other quantities in the expression for aag(u>) are: N is the electron density, T is the relaxationtime, here assumed to be a single constant, wn is the

21

energy difference between bands n and &,

- K"1 [ E ^ M - E ^ O O J (2.14)

*and m is the optical effective masses which is defined by

where

-<777 >0,6 * sae + fa £, Re ■ (2-16>m^(k) ' nrt ni

The ordinary optical properties of ferromagnetic ironare determined by the diagonal components of o (we havea ** a ft a , z being the direction of spin alignment) . xx yy zz

The first term in o _ is the usual Drude formula withaBrelaxation time T. In the limit in which the band states are sharp (T-*-«) , we obtain the familiar expression for the real part of the conductivity for positive frequencies

2 2 Rela <u>)] - 1t«e— « (a,) + -52- E I <"?„)

2 (m ) ik nk *n nlaa 0 u(2.17)

The off diagonal components of conductivity vanishexcept for a m —o * In this case, we have xy yx

22

(w+i/T)(2.18)

The sharp limit of this formula is, for positive frequen cies

If the matrix elements in these expressions are treated as constants, the conductivity is proportional to the joint density of states.

The optical conductivity of iron was calculated in­cluding the k dependence of all matrix elements both in the sharp limit and with the inclusion of a relaxation time. The numerical integration was based on 729 points in 1/16'th of the Brillouin Zone. Since the contribution of the spin orbit coupling term to the matrix elements rt was negli­gible, we replaced by pnJ throughout the calculation. This implies that the off diagonal conductivity should be regarded as being produced by the modification of the band wave functions produced by the spin-orbit interaction.

Our results for the real part of (*xx from 0.2 to 7 ev are shown in Fig. 8 . The solid curve is the interband contribution in the sharp limit (T*«). The dashed curve includes a constant relaxation time Jrf/T «* 0.3 ev and an empirical Drude term.

2

0 U

in which the constant have been taken to be a * 6.4 xo1 0 ^ sec-1 and T'« 9.12 x 1 0 * ^ sec., as determined by Lenham and Treherne.2** We have also computed Re(<*zz), which is not the same as Re(o ) in the present case. However, the differences are quite small and are notsignificant on the scale of this graph. Experimental

27 28observations of Henze1 and Gebhardt, Yolken and Kruger,29and Johnson and Christy are shown for comparison in Fig.

8 . There is reasonably good agreement between the curve containing the relaxation time and the experimental results in regard to general shape and the position of maximum, which is the only significant structure observed. We are not aware of observations in the low-energy region (Hu < 0.5 ev) which could show whether sharp structure is present.

The maximum in the computed conductivity near 2.5 ev results from transitions between the nearly parallel pairs of exchange split bands in the vicinity of the Zone face (see particularly the directions H-P-N in Fig. 1). The transition would be forbidden in the absence of spin-orbit coupling. Even though the matrix element is relatively small, the joint density of states is quite large, and a large spike in the conductivity is produced. However, at an excitation energy of 2.5 ev, lifetime broadening effects

24

are apparently quite appreciable, and the Bharp structure is broadened into a smooth peak. The relaxation time mentioned above, 0.3 ev # seems to reproduce the width of the experimentally observed peak.

If our interpretation of the observed peak in the optical conductivity is correct, it indicates that the band calculation has given approximately the correct spin splitting between bands of predominately majority and minority spins.

The absorptive part of the off-diagonal elements of the conductivity tensor can be determined from measurements of the ferromagnetic Kerr effect. A plane-polarized light reflected from a ferromagnetic metal becomes elliptically polarized with its major axis rotated from the original direction of polarization. The rotation angle can be related to the absorptive part of the off diagonal elements of the conductivity tensor through Maxwell*s field equa­tions by'**'3*

Afl " i4TT Gvv£! - Re I ----------- n-- 1 (2.21)2 a) (n+iK) ((n+iK) -1)

where (n+iK) is the complex index of refraction.This effect involves spin-orbit coupling in an

essential way. We have calculated the off diagonalelement c of the conductivity tensor. Our results for xy[u Im(o )] are shown in Fig. 9 both in the sharp limit xy

25

(T*®), and including a constant relaxation time H/T ■0.3 ev. He have not included any interband contribution,which would simply shift the calculated curves by a

32constant. There is a moderate degree of agreement between theory and observation in regard to the magnitude and general trend of the data. He note that the most prominent feature of the experimental results, the peak around 1.7 ev, is also found in our calculations. The transitions which contribute to this peak are not well localized, but appear to come from a fairly large region in the vicinity of N. The minimum present in the calcu­lated conductivity around 5.7 ev seems to result from transi­tions from the minority-spin lower s-p band to the minority - spin d bands above the Fermi surface.

The calculated values for the real parts of °x x (u)and o (u)> in the energy range from 0.01 ev to 8.0 ev are z zshown in Table III. Also shown in the table are calculatedvalues for the interband part of imaginary component of[hi a (w) ] and the values for joint density of states for xythe same energy range.

CHAPTER IIIg SHIFT OF CONDUCTION ELECTRONS IN FERROMAGNETIC METALS

For a conduction electron the shift in the value of MgH, the spectroscopic splitting factor in the ferro­magnetic resonance exponents from the free electron value(2.0023), arises from spin-orbit interaction with the periodic potential of the crystal. The g-factor should be distinguished from the magnetocnechanical ratio g*, measured in Einstein-de Haas or Barnett experiments. The g shift depends on the orbital magnetic moment of the metal in a magnetic field. Some recent experimental work has demonstrated the presence of a small but non-negligibleorbital contribution to the magnetic moment of iron,

21,23.24cobalt, nickel and their binary alloys. This small orbital electronic moment makes g slightly more than 2 and hence accounts for the shift in g.

In this chapter we first derive the expression for the g shift in the tight binding method and then use it to calculate the value for iron and nickel and finally compare it with experimental results. In Section A we derive the expression for the perturbation term in the Hamiltonian giving rise to the g shift. Section B contains the derivation of the expression for the matrix elements of the perturbation term using the tight binding wave functions. Section C deals with the calculation of the terms for the g

26

27

shift given in Section B r for the case of iron and nickel and then comparison is made with experimental values.

Section APerturbation Term in the Hamiltonian

in the Presence of Constant Magnetic Field

The Hamiltonian Hq of an electron in a periodic potential is

H <= P2/2m+V(r) + — 0 . (VVx$) (3.1)° 4m c

where the last term is the spin-orbit interaction term.In the presence of a steady magnetic field 5, the

Hamiltonian becomes

2 2H - H + £- w-X + — - n - e*(r) (3.2)° 2mc

where A(r) and $(*) are vector and scaler potentials due to magnetic field and tt is given by

£ = $ + Ji-y (SxVV) (3.3)4mc

The form of the Hamiltonian in the presence of mag­netic field is obtained by substituting

t + - £<r) for ? in H . c o

28

We choose the gauge

■+ i -► -*•A - ^ B x r

and

<Mr) = 0 (3.4)

Neglecting the third term in H as no effect proportional 2to B are observed in normal fields, the Hamiltonian H in

this gauge becomes

H - H0 + g UB (3.5)

where y0 = " Bohr magneton.D 4RICFor a z-directed magnetic field, the perturbation term

in H is given by

i uB B{(rxf)2 + [rx(cxVV)J} (3.6)4mc

The shift 5g in g is given by the average over the occupied portion of the Brillouin Zone of the expectation value of the perturbation term divided by u b b where ST is total spin of the system.

The expectation value of each of the two terms in(3.6) are derived in the next section using tight binding

29

wave functions

Section BMatrix Elements of Perturbation in Tight Binding Method

The calculation of g shift requires the calculation of the matrix elements of each of the two operators in Eq. (6), the first one being the (rxP) operator. This operator is unbounded and non-periodic. Matrix elements of this operator using Bloch functions contain derivatives of delta functions. To get around this difficulty for the calculation of the matrix elements of operator it is convenient to use the wave packet representation in which as will be seen later, the derivatives of delta functions will be adequately taken care of.

For a wave packet centered about a point kQ , we have

where iMk,r) is the Bloch function for state k, given in the tight binding method as

(3.7)

and

V r_V ■ I ci<k> V * - V (3.8)

30

where C^(k) are the expansion coefficients from band cal­culation described in Section A Chapter II and V^'s are the GTO*s . The function f(k-k ) is the localized function and

*w Ocan be represented by a Gaussian

f (k-k ) * (f^)3/4 e ° (' (3.9)% “v Q 7T

where a should be large to make f a localized function and the factor (2at/Tr) ^ 4 normalizes i|<W (ko ,r) .

The matrix elements of I*2 operator between the functions defined by (3.7) is given by

-ik *R ik'*Ru ve M ev

<*W (k ,r)|(rx?) |*W (k ,r)>

I 7 7 7 T fd3kfd3k' f*(k-k >f(k'-k > I(2 IT) * * U,

(<Uk [r-(RM-Rv)][t(r+Rv)x?)z |Uk ,(r)> +

<Uk (r)|[(r+Ru)x?]z |Uk ,[r-(R^-R^)]>} (3.10)

In the first term within {} in (3.10), the origin is displaced such that the wave function on the right side of the operator is centered at the origin and in the second term within {} it is the wave function on the left which is centered at the origin. This method of splitting the terms into two halves results in the simplification of

the final expression as will be clear later. The point worth noticing here is that L2 operator does not have the translational symmetry of the lattice.

Consider the first term in (10) and concentrate on the part

ik-R ik* *RK ‘ - < v v >

[(r+Rv)x?lzUk ,(r)d3r

which can be rewritten as

i(k'-k)-Rv -ik-R. t *,1 juk (r-R£)(rx?)2

i(k*-k)*R -ik- R„V „ I

e

Uk ,(r)d3r +

(3.11)

Using the relations

32

Equation (11) simplifies to

-ik*R“XJC * R ffi(k*-k) E e 1 JuJtr-R^) (rxP)2 Uk , (r)d3r

+ i^^lr t«(k'-k)] Px (k,k*> - -jIt [6 (k'-k) 1 py <k,k')>y

(3.13)

where

Px (k,k') - E e i' "* fuk (r-R^) px Uk , (r)d3r (3.14)

Substituting (13) into the first term of (10), we get

| |d3k|d3k ' f*(k-kQ )f (k'-ko ) 6(k'-k)Ee ^ ^ ^ ‘ (r-R^

LzUk ,(r)d3r + | |d3k d3k* f*(k-kQ )f(k'-kQ)

{a£r 15 (k'-k)]px (kfk*) - -£r t« (k'-k)]py (k,k*)} (3.15)

After integrating with respect to k', the first term in (15) becomes

k p k | f (k-ke)|2 £ e “ lZ V * > d3r

where

L_(k) * E e Z I

1 |u*(r-R|,)Lz Uk (r)d3r

expanding L_ (k) in power series about k and substitutinga O

The odd terms in the expansion of l>z (k) about the point kQ make the integrand an odd function of k and hence the integral for those terms vanishes. Only the first three terms in the expansion are important for localized wave packets. Keeping only these terms we get for (3.17)

2for |f(k-ko)| the expression defined by (9) we get

j lim

(3.17)

34

because in the limit of large a, the second term vanishes.L-(k ) is given by £ o

w - j e R* K 0 <r‘R*)Lz uko (r)a3r

Using (18) Equation (3.15) becomes

i f * ^ 4, 3Juko (r- V (rxP»z °k„(r)

| Jd3kjd3k ’ f*(k-kQ )f (k*-ko) i-^r [6 (k'-k))px (k,k')

■£r [fi(k'-k)]P <k,k’)> (3.19)

integrating second term in (19) w.r.t k' gives

5 |<J3k | f (Jc-k0) |2{^|r [py Oc.k*)]k ._k - (Px tk,k')]k..k )sc y

+ 1 f*<k- V {9 % If(k'*ko )]k ,-k py (k)

* lf<k'_ko>]k'-k p* (k) (3.20)y

where

db- IPy(k'k,)Ik'-k - E « 4 f°k(r‘R1,Py * k lUfc'(r> W * rX *

35

and

***£ f * 3py (k) - I e * jUk <r‘V py Uk (r)dJr

Defining E*x (Jc) « ■jjp- lpy ) 1 k i-Jc and similarly Fy (k) . Using these definitions, the first term in Eq. (3.20) becomes

* |d3k[f(k-ko)|2 [Fx (k)-Fy (k)] (3.21)

The integration in (21) is carried out after substitutingthe first three terms in the expansion of F (k) and F (k)x yabout k . The result iso

7 I V V W 1

± " i k o * R i r • a u ^ ( r )

2 E 6 Z ft/ * vU- \i / 1< H o (r-Ri>Py TTT— !k-ko d £

f * 3U. (r) 3- K j r- V px "3TC“ lk“ko d ?} <3’22)

After using (6)( the first term of (3.20) becomes

. . 9C (k) 3C. (k)7 ^ W ‘"a^r'k-k,, piJ(ko> - T V - ' k - k , , plj'V>

(3.23)

36

Pij(ko> ■ I e-iko-R* jv* (ir-Rt)Py (r)Vj (r)

To evaluate the second term In (20), use is made of Eqs. (8) and (9). The integration over k is accomplished by expanding the wave function expansion coefficients C^(k) and momentum matrix element components p^(k) in power series about kQ , and keeping only the first three terms in the expansion. The contribution of the second term in (20) is

+ 4 I C*(k ) C.(k )i,j j °

(3.24)

Using the results of Eqs. (19), (20), (23), and (24), the first term in (10) becomes

7 A V W V I ° ik°'H | v * ( r - R |l) ( ? * P ) I V j ( r > d 3ri # j Xi

4 * 3C.<k) 3C.(k)+ * °i(ko> { 5<x 'k-ko PiJ <ko> Sfiy lk-kQ plj(ko)>

, . <*J 3 p J . (k)+ * E Ci(k0 )C j (k0) [ ------- I t — Ifc-k.

if j y x

3C*(k) „ 3C*(k)■ I * cj (ko> {-5E— lk-k P i j < V * -5E— I k-k P?j‘V >

x | j x o y o

(3.25)

The second term in (10) is treated in a similar manner and yields the following expression

1 * f * 4, + 1£ £ Ci<ko ,Cj (ko } Z e JVi (r)(rxP)2 V j (r-R^)dJr

i t 3 &

38

In Eq. (3.25) ^(kQ ) where O is some operator isgiven by

**i Jc * R f°ij (kQ ) - I e ° ~l O Vj(r)d3r

while in (26) it is given by

iko'R* f * 3°ij(ko> - ? * JVi (r) ° Vjtr-R^d3;X

Equations (25) and (26) together give the expectation value of Lg for the wave packet defined by (7), (8) and (9).

For Hermitian as well as translationally invariantoperator O such as p , p , etc.x y

E Ci ( V Cj (V 1 e Jvi (r-»i)0 V j(r)dJrif j

* ik * R. r * -- I C, (k )C,(k ) E e ° R V. (r)0 V. (r-R, )dJr i(j i o j o t I i J I ~

(3.27)

Using (27), some of the terms cancel out when (25) and(26) are added together to yield the valuet

<i(/W (ko,r)f (rx?) z\ i(»W (ko,r) >

39

i it “ik * Rj r ^± Z C< (k jc., (k„) Z e ° * ]Vi (r-RJl) (rxP)2 i,j 1 ° A

ik • R. o IV. (r)d3r + i Z C*(k )C, (k ) £ e j ~ 2 i,j 1 ° 3 ° t

jv*(r> (rxf)z V^r-R^d3; + | C*(*o)

3C.(k) v 3C.(k)1 lk-kQ pij(ko) Sky lk-kc pij(ko)]

. 3C*(k) 3C*(k)" 2 i^j Cj (ko ) 1 3kx Ik-ko Pij<ko> " 3ky lk«k0

p^j tJcQ ) ] (3.28)

Using the relationship

* 3C.(k) *[ * W i t — W P i j ' V >J- 9 J X O

3C,(k)- ,E . I F — I k-k cj«o> P i j ' V (3‘29)1,3 x o J

Cq. (28) simplifies to

<*w (ko ,r) |Lz K W (kD ,r)>

* f * 3- Real { Z ci (k0)Cj (ko) 1 e JVi(r~Rl}V j ( *if j A

40

3c. (JOImag. { E C 1 (k<>) [ lk Plj <ke>

X r J X O

3C,(k)- s F - i k - k (3-30>y o

The second term in (6) is

^ 7 “B b '<* 3; + y 77>°z - ' * V » V W 1 <3-31)

Only the first term in (31) will contribute. The expectation value of (x + y using wave functions given by (7) is

. 3C . (It) -ik -R,* Im»9- * * Ci (ko) 1 -Jfc— lk-k„ f *1 f j x o t

★ , 3C.(k) -ik -R-V ^ V 7y °z V j (r)d 5 + - 5 ^ — 'k-kQ \ e

^(r-R^) f j ° z vj(r)d3^ (3.32)

combining (30) and (32), the various terms in the g shift are

41

^ n 1' “ R fReall1 j Cn i(V Cn j< V * e’ ik° "* j v ^ l 1

(rxp)z (r)d r] (3.33a)

3C (k)«9n2’ * * R Ima9t ^ Cn i(V ( - ) k-k0 P i j 'V ^ n i ^ o 1

5C j (k)(— 21-— ) .‘ 3ky Jk=ko Pi j (ko>] (3.33b)

5g-ik.

n 4mc'Z

i f jZI R‘ H <r-Ri

(x iz 3Vj3x y 3y z (r)d3r] (3.33c)

tA\ u * 3 C n 4 t k ) “ i k o ‘ R l

{9n = ■ i ? Iraa,t C n i < v ( 8kx ,i‘-ko t e

Ivi (r-Ri> I? °z v r>a3;i (3.33d)

42

a V .(r)d3r z j (3.33e)

where subscript n refers to band index.In addition to above terms de Graaf and Overhauser

suggested that there is a relativistic contribution to 6g which is of considerable importance in alkali metals. This relativistic contribution to the g shift is given by

where p is momentum operator. This term arises because the spin precession frequency of a moving electron is smaller

Section CCalculation of g Shift in Iron and Nickel

The average value of the g shift over the Fermi distribution is calculated for each of the terms in Eq.

term gives the largest contribution. This term was cal­culated for all the occupied bands at each of the 729 k points in iron and 1357 points in nickel in 1/16'th part of

(3.34)

2 1/2than that of one at rest by the factor (1-S )

(33) for both ferromagnetic iron and nickel. The

the Brillouin Zone. The term containsn

V^'s are the GTO's which are the product of a radial part and an angular part, the latter being the linear combina­tion of suitable spherical harmonics. Since spherical harmonics Y^'s are eigenfunction of Ljj, hence we have

L2 (r) = A Vj , (r) (3.36)

where A is some constant and the angular part of Vj'(r) is a Kubic harmonic of some other symmetry. Using this result in (36)

-iko'Ri f * 3L e ° * jVi (r-RJl)Lz V j (r)dJr - A Sij((ko) (3.37)

where S ^ t(k0) are overlap integrals given by

-i k 0 *Ro f # 3Sijr(ko ) “ 1 e Ivi (r-RJl)Vjf (r)dJr (3.38)

2 2In Eq. (36) if the j orbital is of xy, yz, xz, x -y ,2 23z -r , x, y, z, s symmetry then the j* orbital is of

x2-y2 , xz, yz, xy, 3z2-r2, y, x, z and s type symmetry respectively and the constant A takes the values -2i, -i, +i, +2i, 0, +i, -i, 0 , 0 respectively.

In the overlap integrals (38), the summation is carried to convergence which occurs after 50 neighbors when

44

both orbitals i and j' are of s or p-type and after 5-6 neighbors when both are of d type symmetry. The matrix of Lg for d-d, p-p, s-d, s-p and p-d block is non zero while for s-s, d-s and p-s blocks it is zero. The matrix elements of Lz along with the wave function expansion co­efficients Cni*s are used to calculate the contribution to 69^ ^ at various k points in the zone. At each k point, the contribution from all the occupied bands is summed.The average contribution to 6g for this term is calculated for all the occupied k points and is given by

<«9U ) > * qjOTrr I «gi1) <k) W(k) (3.39)SUMWT „fk<1/16 nocc.

where W(k) is the weight factor of a k point in the 1/16* thirreducible part of the BZ and SUMWT * sum of weight factorsof all the equally spaced k points in the zone.

(2)The term for 6g ' contains k and k„ derivatives ofn « ythe expansion coefficients as well as momentum matrix elements. The expressions for the momentum matrix elements using GTO are given in the thesis of C. S. Wang. To estimate the contribution of this term a coarser mesh size, 2ir/4a, was used for the calculation of this term as compared to 2ir/16a for the first term. This mesh size gives 27 and 4 3 equally spaced points in the irreducible wedge of the BZ for the case of iron and nickel respectively. To calculate the kx-derivatives of the C ^ ’s at some k

45

point, the Cn^'a at the two neighboring points separated by 2n/16a from the given point, one on either side of the given point on the x-axis, were used. If any of the two neighboring k points does not lie within the first Brillouin Zone, it was brought back to first BZ by transla­tion of a reciprocal lattice vector. On the other hand, if any of the two neighboring k points does not lie in the 1/16'th irreducible wedge of the Brillouin Zone, the wave function expansion coefficients for those points were cal­culated from the C ^ ' s of the corresponding k point in the irreducible part of the zone, using the transformation properties of bloch function under unitary and antiunitary operations of the magnetic point group of the Cubic crystal, described in the Appendix. Suitable phase factors were supplied for wave function of each of the bands to make the variation of the C .'s smooth. This task wasvery complicated in that part of the zone where band crossings occur. Assuming that the variation of Cn^'s is linear for small interval Akx, the derivative is given by

(3.40)

Similarly k^ derivatives were found. These derivatives of Cn^'s along with the matrix elements of the components of

46

)) and expansion coefficients C ^ ' swere used to calculate the contribution $9^ ^ at each ofthe selected points in the zone for all the occupied bands.The derivatives associated with bands whose wave functionschange abruptly at some region in k space, are large in

(2 )that region and hence contribution to <5g in that case is large. At a given k point 6g^ for some of the occupiedbands has positive value and for others negative value.Adding the contribution of all the occupied bands at most of the points yields a small value. The average value over the occupied portion of the Brillouin Zone for this term yields a very small value for iron as shown in Table IV.Not too much quantitative significance should be attached to the value of this term because the number of sampled k points in the BZ used for the calculation of this term was fairly small and the approximation of linearity of wave function expansion coefficient may be inaccurate. Some error is also introduced by the ambiguity in making the behavior of C ^ ' s smooth in the region at which derivatives were calculated. Qualitatively our estimate has shown, how­ever , that the contribution of this term for iron compared to the first term is extremely small.

For the case of nickel we ran into problems in thebeginning as the value for this term was dominated by the contribution of a single band at the point 2?r/4a (003) .This particular band was very close to the Fermi energy.

momentum (p*j(h^,p^,(k

47

# 2 \The average value for (g using this result came out tobe about half the value for 6g ^ while that particularband was contributing about 70% to the total value of<5gv ' . To improve the value for the 2Tr/4a (003) point, wesubdivided the bigger cube with its center at that pointinto eight small cubes and calculated the derivatives atthe centers of those minicubes. He calculated the contri-

(2)bution to 6g^ for these minicubes using the derivativevalues at the center of these minicubes. The average ofthese values was used in place of the original value which

(21was too large. The average value of the 6g term, overthe occupied portion of the BZ was calculated for nickel.

(2)The average value for 6g term in nickel did not turn out to be extremely small compared with 6g ^ term as was the case for iron* Since the wave functions in some region change a lot as one goes from one point to another in the BZ, the number of sampled k points may not be large enough to get a very good estimate for this term. Compared to the difficulties involved in the computation of thisterm, we believe we made a good attempt in estimating it.

(31The third term in (33) giving 6g^ contribution is expected to be small because of the factor l/4mc . In the calculation of this term for iron, all other integrals in the matrix elements of (x + y except those whenthe two orbitals are centered on the origin, were considered

48

negligible. The contribution due to this term was so smallin iron that it was not calculated for nickel.

The contribution of the 5g and fig^ terms aren nzero as the largest contributing factor in the matrix elements of 3v/9x and 9v/9y when both oribitals are centered on the site at the origin, is zero in this case.

The relativistic contribution arises because the spin-precession frequency of an electron at relativistic velocity is different from the one at rest by the factor (l-82)*^2 , where 8 * V/C. This relativistic term is given by:

6g"(k) j-y L*<k,r)p2az *n (k,r)dT2m c o

This term is quite small as the following rough esti­mate shows. It will not be calculated precisely. Instead we approximate it by

6g£(k) = - <T>U*(k,r)oz ^n (k,r)di c '

A

where <T> stands for average kinetic energy for band n . The average value for (6g)H for a conduction electron is

2(fig)" » - —y <2SZ> x (average K*E for a conduction electron) c

Here <2SZ> stands for the expectation value of spin magnetic

49

moment. The value for the average kinetic energy (K*E) for a conduction electron was evaluated and ia given by

Fe 11.73 RydNi 15.09 Ryd

A value of 2.29 p^ and 0.62 were obtained for the spin magnetic moment pg *■ <2Sg> for iron and nickel respectively, from the result of band calculations. These values are somewhat larger than the values of 2.12 p_ and 0.56 p_ asO D

deduced from the measurement of magnetisation and g or g* factors. The value for (6g)" are:

-0.00072 for Feand

-0.00023 for Ni

The contribution of the various terms are shown in Table IV. The total shift 6g is obtained by summing the contributions of all the terms and dividing the sum by the total spin angular momentum. The calculated values are

6g = 0.058 for Fe- 0.129 for Ni

50

The experimental values for the g shift are 0.09 and 0.18 for iror?\nd nickeIrrespectively.

The magneto mechanical ratio g' of a material which is measured in Einstein, de Haas or Barnett experiments is defined as the ratio of its total magnetic moment to its total angular momentum. Thus

a. _ " V ” !*9

In the presence of spin-orbit coupling, the expectation valueof L>z , the orbital angular momentum is non zero, leading tog* value less than 2. Our calculation yields for g* valuesof 1.945 and 1.885 for iron and nickel, respectively. Theexperimental values of g* for these metals are 1.93 and

241.84, respectively.

CHAPTER IV CONCLUSIONS

The effect of spin-orbit interaction in the band structure of ferromagnetic iron has been investigated using the tight binding method. It appears that a band calculation employing a local exchange potential in combination with spin-orbit coupling is capable of accounting for the general features of the iron Fermi surface and the optical-conductivity tensor. Although numerous discrepancies in detail exist, there is a large degree of general agreement between theory and experiment. In view of the success of these®1 and earlier calculations of our group,1'11,14 it seems that band theory is capable of providing a basically satisfactory account of the properties of transition metals.

The results of the calculation of real part of diagonal element of the conductivity tensor indicate that the band calculations given approximately the correct spin splitting between bands of predominately majority and minority spins for iron. This contrasts with the situa­tion in nickel,11 where a similar calculation yielded an exchange splitting perhaps 60% larger than the actual value (which is, however, not precisely known). In the discussion of the Fermi surface of iron it was pointed out that agree­ment between the calculated and experimental Fermi surfaces

51

52

would be improved in some respects by a reduction of ex­change splitting. This indicates a contradiction with the results of optical conductivity calculations.

The results for the calculation of the g shift due to conduction electrons in iron is approximately 2/3 of the experimental result. This indicates our calculated value of the orbital magnetic moment in this metal is small compared with the experimental value. In the case of nickel, the agreement with the experimental value of g shift is somewhat better.

The calculations can be improved in several ways. A better agreement with the experimental Fermi surface results is possible if the band calculation is repeated using that value for a f the exchange parameter, which gives the correct experimental size of the hole pocket (surface IV) around the H point. This will also result in the improve­ment of the somewhat large value for the spin magnetic moment we obtained and it could possibly result in the extension of the electron ball (surface VII) along the A direction towards the T point, resulting in the overlap with the central electron surface VI. This overlap is necessary to explain the experimental pressure dependence

4of the [100] open orbits as pointed out in Chapter II.An increased variational freedom in the trial wave

functions would lead to some improvement in the results. This could be achieved if all the basis functions were

53

chosen to be individual GTO. Recently, the band structure of Li, Na^® and A l ^ have been investigated using basis functions consisting of individual GTO. Successful results were obtained after repeating the whole calcula­tions several times with different sets of GTO. In principle, a basis set consisting of individual Gaussian orbitalB with large exponents that are capable of repro­ducing the atomic core states and one or two small exponents to allow sufficient variational freedom in the conduction s and p-like states is more likely to give a satisfactory result. The small exponents are subject to the computa­tional restriction that the eigenvalues of the corresponding overlap matrices cannot be negative or unreasonably small.

A further improvement would be to include more con­duction states, such as 4f for transition metals. The resulting eigenfunctions would be a better approximation to the exact solution of the Schrodinger equation. The hybridization between the 4f and s, p and d states will undoubtably improve the results of energy bands and con­ductivity tensor. In the conventional LCAO calculation, where the expressions for integrals between higher states are obtained by successive differentiations, the problemmay seem to be too complicated to be considered. The

62modification, made by Chaney and Norman, to separate the variables in cartesian coordinates and to take bionomial

54

expansions in the integrand, drastically reduces the complexity of the problem.

REFERENCES

1. R. A . Tawil, and J. Callaway, Phys. Rev. B 1_, 4242(1973).

2. A. V. Gold, L. Hodges, P. T. Panousis, and D. R. Stone, Int. J. Magn. _2, 357 (1971).

3. D. R. Baraff, Phys. Rev. B 8 , 3439 (1973).4. M. M. Angadi, E. Fawcett, and M. Rasolt, Phys. Rev.

Lett. 32, 613 (1974).5. R. V. Coleman, R. C. Morris, and D. J. Sellmyer, Phys.

Rev. B 8, 317 (1973).6. L. M. Falicov and J. Ruvalds, Phys. Rev. 172, 498

(1968).7. A. P. Cracknel1, Phys. Rev. B 1_, 1261 (1970).8. R. Maglic and F. M. Mueller, Int. J. Magn. 1, 289

(1971).9. S. Wakoh and J. Yamashita, J. Phys. Soc. Jap. £1, 1712

(1966) .10. E. Abate and M. Asdente, Phys. Rev. 140, A1303 (1965).11. C. S. Wang and J. Callaway, Phys. Rev. B 9^ 4897

(1974) .12. J. H. Wood, Phys. Rev. 126, 517 (1962).13. E. Lafon and C. C. Lin, Phys. Rev. 152, 579 (1966).14. J. Rath and J. Callaway, Phys. Rev. B £, 5398 (1973).15. Y. M. Ching and J. Callaway, Phys. Rev. B 9, 5115

(1974).55

56

16. Y. M. Ching and J. Callaway, Phys. Rev. B 11, 1324(1975).

17. R. A. Tawil and S. P. Singhal, Phys. Rev. B 11, 699(1975).

18. A. J. H. Watchers, J. Chem. Phys. 52, 1033 (1970).19. J. C. Slater, T. M. Wilson, and J. H. Wood, Phys. Rev.

179, 28 (1969).20. G. Gilat and C. J. Raubenheimer, Phys. Rev. 144, 390

(1966).21. A. J. P. Meyer and G. Asch, J. App. Phys. 32, 3305

(1961).22. S. M. Bhagat and M. S. Rothstein, Sol. St. Comm. 11,

1535 (1972).23. R. A. Reck and D. L. Fry, Phys. Rev. 184, 492 (1969).24. G. G. Scott and H. W. Sturner, Phys. Rev. 184, 490

(1969).25. H. Danan, A. Herr, and A. J. P. Meyer, J. Appl. Phys.

39, 669 (1968).26. A. P. Lenham and D. M. Treherne, in Optical Properties

and Electronic Structure of Metals and Alloys, edited by F. Abeles (North-Holland, Amsterdam, 1966), p. 196.

27. E. Mensel and J. Gebhardt, Z. Phys. 168, 392 (1962).28. H. T. Yolken and J. Kruger, J. Opt. Soc. Am. 5j>, 842

(1965).29. P. B. Johnson and R. w. Christy, Phys. Rev. B £, 5056

(1974).

57

30. G. S. Krinchik and V. S. Gushchin, Zh. EKSP. Teor. Fiz. 56, 1833 (1969) tSov. Phys - JETP 29, 984 (1969)].

31. P. N. Argyres, Phys. Rev. fT_, 334 (1955).32. J. L. Erskine and E. A. Stern, Phys. Rev. B £, 1239

(1973).33. J. L. Erskine and E. A. Stern, Phys. Rev. Lett. 30,

1329 (1973).34. Herbert S. Bennet and E. A. Stern, Phys. Rev. 137,

A448 (1965).35. M. Dixon, F. E. Hoare, T. M. Holden, and D. E. Moody,

Proc. Roy. Soc. Lond. A285, 561 (1965).36. Robert Karplus and J. M. Luttinger, Phys. Rev. 95,

1154 (1955).37. H. Brooks, Phys. Rev. 58, 909 (1940).38. G. C. Fletcher, Proc. Phys. Soc. A67, 505 (1954).39. Shoji Ishida, J. Phys. Soc. Jap. 33, 369 (1972).40. Nobuo Mori, J. Phys. Soc. Jap. 2J_, 307 (1969) .41. N. Mori, Y. Fukuda, and T. Ukai, J. Phys. Soc. Jap. 37,

1272 (1974).42. N. Mori, T. Ukai, and H. Yoshida, J. Phys. Soc. Jap. 37,

1272 (1974).43. M, Asdente and M. Delitala, Phys. Rev. 163, 497 (1967).44. E, U. Condon and G. H. Shortley, The Theory of Atomic

Spectra (Cambridge U.P., Cambridge, 1959), Chap. 7,p . 210 .

58

45. L. Hodges, H. Zhrenreich, and N. D. Lang, Phys. Rev. 152, 505 (1966).

46. E. I. Zornberg, Phys. Rev. B 1_, 244 (1970).47. J. C. Slonczewski, J. Phys. Soc. Jap. 17, S-34 (1962).48. J. Friedel, P. Lenglart, and G. Leman, J. Phys. Chem.

Solids 25, 781 (1964).49. G. Allan, G. Leman, and P. Lenglart, J. de Phys. 29,

885 (1968).50. R. Kubo, J. Phys. Soc. Jap. 12_f 570 (1957) .51. A. M. de Graaf and A. W. Overhauser, Phys. Rev. 180,

701 (1969).52. A. W. Overhauser and A. M. de Graaf, Phys. Rev. Lett.

22, 127 (1969).53. A. M. de Graaf and A. W. Overhauser, Phys. Rev. B 2_,

1437 (1970).54. A. Bienenstock and H. Brooks, Phys. Rev. 136, A784

(1964).55. R. A. Moore and C. F. Liu, Phys. Rev. B 8 , 599 (1973) .56. Y. Yafet, Phys. Rev. 106, 679 (1957).57. F. J. Arlinghaus and R. A. Reck (to appear in Phys.

Rev. B).58. R. A. Moore, J. Phys. F: Metal Physics 5, 459 (1975).59. L. M. Roth, J. Phys. Chem. Solids 23^ 433 (1962).60. E. I. Blount, Phys. Rev. 126, 1636 (1962).

59

61.

62.

M. Singh, C. S. Wang and J. Callaway, Phys. Rev. B 11, 287 (1975).R. C. Chaney and F. Norman, Int. J. Quantum Chem.(to be published).

APPENDIX

The band structure is obtained at those k points which lie in the 1/16'th irreducible part of the Brillouin Zone. However, to calculate the derivatives of expansion coeffi­cients, one needs the wave function at some k points not in this part of the zone. To obtain these wave functions we investigate the rotation properties of the wave function expansion coefficients under the operation of the ferro­magnetic cubic point group operations. For the spin align­ment direction of [001], this group contains 8 unitary operations which are ordinary rotations, inversion or the product of rotation inversion as discussed in Chapter I and 8 antiunitary operations which are the product of time reversal and some rotation or rotation inversion or reflections. Below are listed the 8 unitary operations:

1 . E XYZ2. C4Z YXZ3. C2Z XYZ4. C?Z YXZ5. I XYZ6 . S4Z YXZ7. az XYZ8. c+4Z YXZ

60

61

Th* transformation proportion of Bloch functions under the operation of these elements are

a 'f' (k,r) - (kfcT1*) (A.l>n n

and

a +n (k,r) ■ * ^n (aJt,r) (A.2)

where a is any of the above eight operations, and X is a constant of modulus unity. The Bloch function in the tight binding method is

* <k,r) - — S C .(k) £ • y V.(r-R) (A.3)n ^ i ni y i M

Making use of the properties (A.l) and (A.2), we get the rules for transformation of expansion coefficients Cni' s as

Cn;J(ak) - X Z Cnl (k)D1;J (a"1) (A.4)

where (a) is the rotation matrix defined by

o V± (r-Ry} - Vi (o_1r-Ry)

- Z Dij(a”1)Vj {r-oRw) (A.5)

#

62

The antiunitary operations of the ferromagnetic cubic point group are denoted by

6 - 3t (A. 6)

where t is the time reversal operation and 6 is any one ofthe following operations:

1. C2x XYZ

2. C2y XYZ

3. <Mu YXZ

4. C2b YXZ

5. aX XYZ

6. ay XYZ

7. ada YXZ

8. °db ?Xz

Under the action of B, the of Bloch functions are given by

0 ipn (k,r) * i|i* (K, B**1r)

transformation properties

(A. 7)

and

63

0 ¥n <k,r) - X (A.8)

where X is again a constant of modulus unity.For the Bloch function given by (A.3), the right hand

side of (A.7) becomes

* l ** f k , p r) ■ — Z C .<k)e w V .(6 r-R ) (A.9)n 4T i,w ni 1 “

If orbital i is of s or d symmetry, then

vi(r) - V i (r)

For orbital i of p symmetry

V* (r) - -V± (r)

as they are pure imaginary. Hence for orbitals of s or d-type we have

* 7 i * -ik*R ■,^ (k,p" r) * — Z C.(k)e w V. (B^r-R ) (A.10)n fit i,u ni 1 v

Also6 tk,r) - X i(in C— it)

- X — Z iZ C . t-6k)D..U ”1)> fit j.v i ni j±

”ik* Ry .1» V j O r-Rv ) (A. 11)

Using results (A.10) and (A. 11) we have the transformation properties for expansion coefficients for orbitals of s or d type symmetry

For the orbitals of p-type symmetry, we have the rela­tion

ENERGY LEVELS

Band r (ooo) MI1 1 N(j j 0)

12 -0.0795 <r!2*> 0.0404 (n 3+)11 -0.0796 -0.0023 tN1, + )10 -0.2184 <r25'+» -0.0616 (*V>9 -0.2207 (r2S, + ) -0.0700 (N1, + )8 -0.2217 <r2S, + > -0.0809 (NL+)7 -0.2513 (ri2+» -0.1403 (n 3+)6 -0.2521 <ri2f> -0.2363 (n 4+)5 -0.3827 (r25't) -0.2526 (N^)4 -0.3847 <r25.t) -0.3053 (n 2+)3 -0.3866 <r25,4) -0.3990 <NX+)2 -0.6592 (rx+) -0.4605 (n 2+)1 -0.7300 a y ) -0.5272 (N1+)

TABLE I AT SYMMETRY POINTS (Ry)

as roj i-* o 1 1 J 1 Pt2 I H (100)

0.0405 cn3+) 0.5789 (P4t> 0.5252 <H15+)0.0023 w lt+i 0.5768 (P4t) 0.5187 (H15+>0.0620 (n 4+) -0.0593 (P3+) 0.0215 <H25'*0.0700 m v f) -0.0593 (P3+) 0.0190 (h 25,+0.0806 (N^) -0.2341 (P3+) 0.0164 <h 25,+0.1401 (n 3+) -0.2342 (P3+> -0.1576 (H25'+0.2367 (N4t) -0.3070 (P4+) -0.1601 (H25,+-0.2525 (Nxt) -0.3072 (P4+) -0.1625 tH25,+0.3052 (n 2+) -0.3076 (P4+) -0.3481 (Hi2+)0.3991 (N^) -0.4458 (P4+) -0.3481 <■„♦)-0.4605 (n 2+) -0.4464 (P4+> -0.4896 tH^t)-0.5272 (N x+) -0.4470 (P4+) -0.4896 (H12+>

<T»Ul

66

TABLE II

Plane SurfacePresentResult Ref. 1

Exp. Ref. 2

Exp.Ref. 3 (+1%)

I 433 421 43611 321 267III 5.1 5.5 23.8 20.6

100 IV 3.5 4.6 21.0 15.0V 218 219 198VI 120 119 71VII 11.0 11.0 37VIII 43 72I 349 347 347 349TT (around N)

(around H)8.455

8.420

ill 8.4 8.2 33.4110 IV 2.8 3.7 12.0 12.3

v withbreakdown 167 172 145

VI 89 92 58VII 7.9 12.4VIII throu9h r through H

6323

7065

34

I 346 370 369+4 370__ through T

through H10.0207

9.6282

III 6.15 9.9 28.0 27.0111 IV 2.97 4.4 11.3 11.4

V 115 162 154 157VI 84 69 51.8 52.2v t t t through r through H

6525

7261

TABLE IIIInterband Optical Conductivity Tensor (T*») and the Joint Density of States

(ev)

Re(o ) xx(1015 sec"1)

Veia2x)

15 -1(10A3 sec A)

Im(uoxy)29 >2{10 sec )

J(u)(ElectronsV ato«-Fy I

1. 0.0136 5.9092 5.4857 0.0271 0.23372. 0.0680 3.3888 4.1042 0.1768 1.10313. 0.1224 2.1082 2.4513 0.5169 1.23674. 0.1769 1.3822 1.4860 0.3699 1.54805. 0.2313 1.4125 1.4431 0.1049 2.25106. 0.2857 1.4697 1.5802 0.4095 2.47507. 0.3401 1.3484 1.4398 0.3888 2.96178. 0.3945 1.4574 1.7215 0.9276 3.46969. 0.4490 1.2782 1.4304 0.8694 3.656110. 0.5034 1.1051 1.2117 0.6060 3.696711. 0.5578 1.1585 1.3359 0.4668 4.356512. 0.6122 1.3640 1.4554 0,7736 5.708213. 0.6666 1.5621 1.5407 0.9422 7.110614. 0.7211 1.7334 1.7176 1.2761 8.363215. 0.7755 1.7504 1.8039 0.6000 9.852716. 0.8299 2.0826 2.0681 0.6291 11.595517. 0.8843 2.7019 2.6704 0.4339 14.057818. 0.9387 2.8096 2.8487 1.9724 16.460019. 0.9932 2.9957 2.9905 1.2122 20.212520. 1.0476 3.5410 3.5886 2.1416 23.2144

TABLE III (cont'd)

* #u (ev) (1015 sec'

21. 1.1020 4.183322. 1.1564 4.685623. 1.2108 4.723724. 1.2653 4.266725 1.3197 4.054726. 1.3741 3.829727. 1.4285 4.088528. 1.4829 4.511529. 1.5374 4.829730. 1.5918 4.843831. 1.6462 4.849032. 1.7006 5.007333. 1.7550 5.078234. 1.8094 5.311335. 1.8639 5.431936. 1.9183 5.718037. 1.9727 5.681038. 2.0271 5.831039. 2.0815 5.616040. 2.1360 5.4317

(1015 sec"1)

4.35124.92874.56854.18704.07543.78074.10464.49414.81634.92864.99535.02085.01645.31045.37405.63255.72145.68955.56835.3997

Im(uo )' xy

(1029 sec"2)

1.47901.62303.35844.26734.67635.55684.83055.20486.92516.98005.22186.79215.16084.95024.99585.58434.07594.18392.75482.6470

j(u) /Electrons*\ atom-Ry /

27.440531.205731.959531.749832.883333.375635.627238.695341.464942.551242.716343.909846.497849.561952.221660.831865.877164.703862.759460.5377

TABLE III (cont'd)

* Hu(ev)

Re(0xx)

(1015 sec"1) 15(10i;> sec

41. 2.1904 6.0754 5.978142. 2.2448 7.3876 7.222843. 2.2992 9.1882 9.241144. 2.3536 13.4383 13.205045. 2.4081 6.4346 6.549046. 2.4625 6.9433 7.035347. 2.5169 6.7283 6.782048. 2.5713 6.2639 6.495649. 2.6257 6.1906 6.210350. 2.6802 5.9470 5.989251. 2.7346 5.2366 5.299652. 2.7890 5.2926 5.245953. 2.8434 5.1447 5.151554. 2.8978 4.9772 4.973355. 2.9523 4.8360 4.911256. 3.0067 4.7530 4.845957. 3.0611 4.3927 4.384058. 3.1155 4.5516 4.535259. 3.1699 4.3207 4.378060. 3.2244 4.4397 4.4094

Im(nKJxy)

29 -2(10" sec *)

2.8521 73.68931.9537 106.79685.7254 216.74809.1873 291.79323.0862 87.07521.3286 58.78702.5227 59.49722.7210 62.46902.4061 66.99043.1336 68.93333.4322 63.29613.3921 61.28433.1005 61.79335.0599 63.68833.8280 66.35821.6608 66.83943.8958 65.78614.5252 71.15722.5139 67.83102.8505 68.5806

TABLE III (cont'd)

I Ko>(ev) (1015 sec”1)

Re<aZ2)15(10A:> sec

61. 3.2788 4.0332 4.036962. 3.3332 3.9930 4.017663. 3.3876 3.9594 4.018564. 3.4420 3.7649 3.793365. 3.4965 3.7051 3.678266. 3.5509 3.7588 3.759467. 3.6053 4.0630 4.060168. 3.6597 3.9604 3.972069. 3.7141 3.7374 3.705170. 3.7686 3.4011 3.390171. 3.8230 3.1271 3.121172. 3.8774 2.9311 2.937473. 3.9318 2.7049 2.700174. 3.9862 2.6201 2.634975. 4.0406 2.6732 2.683676. 4.0951 2.6018 2.590077. 4.1495 2.4796 2.443778. 4.2039 2.6394 2.612679. 4.2583 2.5853 2.593580. 4.3128 2.5145 2.5393

Im(<ja )^ T(m) fg^tronsV

,, .2 ' M atcn-Ry )(10 sec *)

1.6379 66.57263.8912 65.60583.3913 66.69750.8898 67.56371.5690 65.20461.7761 67.55653.1386 68.17613.1357 68.87364.4242 67.66531.2828 68.26511.0731 70.17221.8510 68.81363.0872 66.65283.2729 67.12833.5383 70.23644.2308 73.04863.9500 72.90404.1610 77.53824.0757 76.16024.0109 69.2663

TABLE III (cont'd)

# (ev)

Re (a )XX

<1015 sec-1)

R e ( a z z )(1015 sec

81. 4.3667 2.5719 2.567282. 4.4216 2.5321 2.525183. 4.4760 2.4967 2.501384. 4.5304 2.4967 2.488785. 4.5848 2.5076 2.488986. 4.6393 2.5806 2.574487. 4.6937 2.9245 2.945788. 4.7481 3.2512 3.277689. 4.8025 3.2368 3.213790. 4.8569 3.4877 3.512691. 4.9114 3.7974 3.824692. 4.9658 3.8903 3.897093. 5.0202 3.7237 3.718794. 5.0746 3.6039 3.609395. 5.1290 3.5344 3.541396. 5.1835 3.2958 3.302197. 5.2379 3.3489 3.327398. 5.2923 3.3607 3.351599. 5.3467 3.3991 3.3917100. 5.4011 3.4563 3.4564

Im(u)Ox y )29 -2

(10*y sec z), , /Electrons^(w)\ Atom-Ry /

3.9803 66.57333.9139 66.41893.8971 66.41183.7723 65.64742.8012 60.53522.9971 59.64373.1344 60.99782.7428 61.88803.5960 59.32303.7178 58.19601.7692 58.90571.2095 58.17042.3192 60.53651.8217 62.57170.5909 60.74450.3169 59.1815

-0.3502 59.4366-0.7314 61.2938-1.6169 59.5011-2.3004 59.4330

TABLE III (cont'd)

* Mu(ev)

Re(lJ*x>

(1015 sec"1}

Re(a22)

15 -1 <10A:> sec A)

lm(waxy)

29 -2 (10 sec )j(«)(Electrons

\ atom-Ry (

101. 5.4556 3.3140 3.3109 -2.3546 59.5541102. 5.5100 3.1050 3.1129 -2.0182 61.7527103. 5.5644 2.9967 3.0114 -1.1409 61.0821104. 5.6188 3.1884 3.1853 -2.8390 61.3995105. 5.6732 3.3003 3.2938 -4.0205 55.7577106. 5.7277 3.1911 3.1778 -3.4469 56.4813107. 5.7821 3.0258 3.0070 -2.8246 55.4004108. 5.8365 3.0859 3.0863 -3.6795 52.1811109. 5.8909 2.9980 2.9974 -2.7918 49.6489110. 5.9453 2.6994 2.6893 -2.1210 47.5803111. 5.9998 2.7944 2.7779 -0.9274 46.1011112. 6.0542 2.4964 2.4893 -1.4785 43.1742113. 6.1086 2.3120 2.3076 -0.9818 42.5749114. 6.1630 2.4807 2.4688 -1.2892 41.6462115. 6.2174 2.4033 2.4020 -1.0070 40.8434116. 6.2719 2.2797 2.2770 -0.6227 40.4793117. 6.3263 2.2064 2.2082 -0.4004 39.4453118. 6.3807 2.1954 2.1988 -0.2824 38.8735119. 6.4351 2.1399 2.1410 -0.2073 37.5916120. 6.4895 2.0730 2.0748 -0.0654 36.9071

TABLE III (cont'd)

* Jtofev)

Re {a )' XX

(1015 sec"1)

Refozz)

15(10i3 sec

121. 6.5440 2.0203 2.0192122. 6.5984 1.9750 1.9720123. 6.6528 1.9187 1.9174124. 6.7072 1.8624 1.8584125. 6.7616 1.8122 1.8089126. 6.8161 1.7702 1.7696127. 6.8705 1.7129 1.7131128. 6.9249 1.6583 1.6569129. 6.9793 1.6176 1.6172130. 7.0337 1.5975 1.5979131. 7.0882 1.5716 1.5735132. 7.1426 1.5392 1.5436133. 7.1970 1.4847 1.4853134. 7.2514 1.4434 1.4426135. 7.3058 1.4205 1.4190136. 7.3602 1.3848 1.3846137. 7.4147 1,3738 1.3730138. 7.4691 1.3723 1.3733139. 7.5235 1.3625 1.3665140. 7.5779 1.3575 1.3481

Im<wOxy)

(1029 sec-2)

-0.0147-0.05400.02940.15050.22390.27650.35810.40980.50310.54910.59250.64650.57190.76431.27511.14121.04561.52861.90711.2761

j (U) /E.lectrons\v \ atom-Ry j

35.054534.556834.446433.266431.718331.481631.042328.657327.642527.042527.362126.239725.604725.287324.921224.472024.218223.914323.798823.6590

TABLE III (cont'd)

t K<o (ev) (10^ sec

141. 7.6323 1.3245142. 7.6868 1.2905143. 7.7412 1.2846144. 7.7956 1.2466145. 7.8500 1.2217146. 7.9044 1.2178147. 7.9589 1.2041148. 8.0133 1.1820149. 8.0677 1.1459150. 8.1221 1.1337

Re(azz) Imtwa^)15 -1 29 -2

(lO10 sec A) (10 3 secJ(w) /Electrons;

\ atom-Ry j

1.3284 0.9353 23.38331.3063 1.7895 23.21141.2684 2.9345 22.80081.2458 1.8792 22.34261.2295 1.8933 21.81981.2297 2.0772 21.31031.2087 2.3193 21.08791.1822 2.4991 20.74481.1431 1.6762 20.47721.1294 1.2785 20.4052

75

TABLE IV

Term In Iron Nickelfig Results Results

fig(1) 0.0588 0.0448

fig(2) 0.0030 -0.0086

6g^3) 0.0003 not ocnc&ited

fig^ zero zero

6g*^ zero zero

(fig)" -0.00072 -0.00023

FIGURE CAPTIONS

Figure 1

Figure 2 Figure 3 Figure 4 Figure 5

Figure 6

Figure 7

Figure 8

Band structure of iron along some symmetry lines in the Brillouin zone. The spin alignment direction is [001]. States are labelled accord­ing to the symmetry of the larger spin component. The solid lines indicate states of predominately majority spin; the dashed lines, those of minority spin.Projected density of states for majority spin. Projected density of states for minority spin. Total density of states.Cross sections of the Fermi surface in the (100) plane. Portions of predominately majority spin are drawn with solid lines; predominately minority spin pieces are dashed. See text for labels.Cross section of the Fermi surface in the (110) plane (a) centered about r, (b) centered about H. Cross sections of the Fermi surface in (111) planes: (a) through T; (b), through H.The real part of the xx component of the con­ductivity tensor from 0.2 to 7 eV. The solid curve is the interband contribution in the sharp limit (t-*■<»). The dashed curve includes both a Drude term (see text) and a constant relaxation time of 0.3 eV. Experimental results are shown

76

77

as follows: --- , Rsf. 27; A, Ref. 28; O, Ref.29.

Figure 9 The Imaginary part of from 0 to 7 eV. Thexysolid curve is the interband conductivity in the "sharp" limit; the short dashed curve includes a phenomenological relaxation time H/T“0.30 eV. Experimental results are shown as follows: , Ref. 33; 0, Ref. 30.

ENERGY

(Ry)

0 . 2 0

0.00-

- 0.20

-0.40-

- 0.60 -

-0.80

¥

(0.0,0) (I flfl) (5.3.5) <5.5.0) (0.0,0)N

(OAD (±.0.±) (OAO) ( i i i ) ( i 0 i ) 2*2*2 '2* *2k IN UNITS OF Z w / q

vl00

DEN

SITY

OF

ST

ATES

(E

LEC

TRO

NS/

ATO

M

Ry) 64

MAJORITY SPIN56

48

40

32

24

10 -9 -8 -7 -6 -5 -4 -3 -2ENERGY ( I0-1 Ry)

Fig* 2

DENS

ITY

OF

STAT

ES

(ELE

CTR

ON

S/A

TOM

R

y)MINORITY SPIN

32h

I6r-

HO -9 -8 -7 -6 -5 -4 -3ENERGY (IO '1 Ry)

Fig. 3

BOTH SPINS

-9 -8 -7 -6 -5 ENERGY

82

Fig. 5

83

H

a t

H

Fig. 6a

84

Pig. 6b

85

t—iro i —iro c\J|rO<fcl°

*—i

z

Fig. 7a

86

Fig. 7b

01) [(»)

87

14.0

12.0

10.0

8.0

6.0MMbo o e o o

W“= 4.0© o o

2.0

0.2 2.01.0 3.0 4.0 5.0 6.0 7.0ENERGY (eV)

Pig. 8

68

E 0.0

3.0 4.0E N E R G Y (tV)

Fig. 9

VITA

Nanjit Singh was born on June 15, 1945 in Punjab, India. He graduated from Khalea High School, Gurdespur in 1960. He went to Government College Gurdaspur and got hiB B.Sc. degree from Punjab University, Chandigarh in 1964. In 1966 he received his M.Sc. degree in Physics from Panjabi University, Patiala. He worked as Instructor in the Physics Department of Panjabi University, Patiala for two years. He obtained his M.S. degree in Physics from Louisiana State University, Baton Rouge in August, 1971.He is now a candidate for the degree of Doctor of Philosophy in the Department of Physics and Astronomy.

89

EXAMINATION AND THESIS REPOET

Candidate:

Major Field:

T itle of Thesis:

M anJit Singh

Physics

SPIN-ORBIT COUPLING EFFECTS IN BAND STRUCTURE OF FERROMAGNETIC TRANSITION METALS

Approved:

ajor Professor and Chairman

Dean of the Graduate School

E X A M IN IN G C O M M ITTE E:

jd ZUf

Date of Examination:

December 2 , 1975