/I- L.

MANY .,,BODY EFFECTS METALS

by

Graham Simpson

A thesis submitted for the degree of

Doctor of Philosophy

August 1972 Department of Physics, Imperial College, University of London.

ABSTRACT

The effects of electron scattering on the

amplitude of the de Haas van Alphen oscillations in metals

are considered. Following Luttinger's treatment of the

interacting electron gas, the oscillatory magnetisation is

derived from a general expression for the thermodynamic

potential for electron-phonon systems. Within certain well

defined approximations, valid under most experimental

conditions, the expression for the oscillatory magnetisation

is found to resemble that for free electrons, except that

within an integral the free electron energies are modified

by the full self energy.

This result is applied to the dHvA effect in

mercury. The experimental observation that the cyclotron

mass has no temperature dependence despite a low energy

phonon mode and strong coupling is shown to be consistent

with the standard theory of electron-phonon interactions.

The mathematical analogy between phonons and spin

fluctuations is exploited to predict the form of the dHvA

amplitude in nearly ferromagnetic metals, and the case of

localised spin fluctuations is considered in detail. The

results of this analysis are compared with experiments on

AlMn. Finally, the formulation is shown to apply to dilute

magnetic alloys and expressions are found for the amplitude

in these alloys and are shown to account for anomalous

experimental results in ZnMn and CuCr.

CONTENTS Page No.

CHAPTER 1 INTRODUCTION

1. The de Haas van Alphen Effect. '5

2. The Effects of Scattering. 7

3. Interacting Electron Theory. 9

CHAPTER 2 ELECTRON-PHONON INTERACTIONS AND THE DE HAAS VAN ALPHEN EFFECT

1. Introduction. 12

2. The Thermodynamic Potential. 14

3. The Electron-Phonon Interaction in Metals. 23

4. The dHvA Amplitude_ in Mercury. 29

5. Nearly Ferromagnetic Systems. 35 Appendix. 39

CHAPTER 3 THE DHVA EFFECT IN NEARLY. MAGNETIC DILUTE ALLOYS

1. IntrOduction. 43

2. Localised Spin Fluctuations at Finite 50 Temperatures and Fields.

3. Comparison of Theory and Experiment in AlMn. 62

Appendix A. 69

Appendix B. 71

CHAPTER 4 THE DHVA EFFECT IN DILUTE MAGNETIC ALLOYS

1. Introduction. 75

2. Perturbational Calculation of Conduction 81 Electron Self Energy.

3. dHvA Experiments in Dilute Magnetic Alloys. 90

4. Application of the Theoretical Results. 95

5. Recent. Developments. 103

CONCLUSION 106

REFERENCES 109

ACKNOWLEDGEMENTS

The author would like to acknowledge the

supervision of Stanley Engelsberg during the first part

of this work, and of the help and encouragement of

Martin Zuckermann throughout this work.

An S.R.C. Studentship was held during 1967-1969,

and a teaching assistantship at the University of

Massachusetts in 1969-1970; both of which are gratefully

acknowledged.

The author wishes to thank Mrs. Beryl Roberts

for taking on the typing of this thesis so competently

,-at short notice, and his wife, Val for her unfailing

support.

CHAPTER 1

INTRODUCTION

1. The de Haas van Alphen Effect

Oscillations in the diamagnetic susceptibility

with field strength at low temperatures were first

observed in bismuth by de Haas and van Alphen in 1930 (1),

the effect being named after these co-discoverers.

Peierls (2) and later Landau (3) gave the now generally

accepted explanation of the effect and good agreement

between theory and Shoenberg's more detailed experiments

on bismuth (4) was obtained.

Landau considered the diamagnetic behaviour of

a free electron gas due to the interaction of the field with

the orbital angular momentum of the electrons. He found

that the electrons perform orbital motions about the

magnetic field direction, the period of the orbit being

2rt/t0c, where We isthe cyclotron frequency. Further the

electronic energies are quantised perpendicular to the

field, and as the field is increased these quantised levels

pass through the Fermi energy and depopulate. This process

causes the free energy of the system, and consequently the

susceptibility, to. be periodic in (1/H), where H is the

magnetic field. The Landau energy levels are given by

(n + 1/2).N.0c. where n is an integer and tt.0c. is the cyclotron

energy. In order that the oscillations should be seen, the

thermal broadening of the Fermi level must be small in

comparison with the quantised energy splitting and hence the

de Haas van Alphen (dHvA) effect generally occurs at low

temperatures and in high fields. Agreement with the bismuth

experiments was obtained by assuming that the electrons have

an effective mass of approximately 1/10 and that the Fermi

e orgy is ery cm.11, r.nrrc.cn nnAing to about in-5 p1pctrons

per atom contributing to the effect. Such effects are, of

course, easy to obtain when the Fermi surface spills over

a Brillouin zone boundary, producing small pockets of electrons

or holes. Consequently by 1952 the dHvA effect had been

observed in some 11 polyvalent metals (5) whilst attempts

to see the effect in monovalent metals were unsuccessful.

With the advent of higher field magnets and apparatus to

obtain very low temperatures, together with more

sophisticated experimental techniques, the dHvA effect has

been observed in the monovalent metals, including the noble

metals. Indeed, the dHvA effect has now been observed in

practically all the commonly studied metals.

The reason why the interest in the dHvA effect has

been so intense since about 1950 is that the effect contains

a great deal of useful information about the electron states

in the material under study. In the original theoretical

explanations of Peierls and Landau, the conduction electrons

were taken to have ellipsoidal energy surfaces in momentum

space and the period of oscillation was found to be

proportional to the Fermi energy. Onsager in 1952 (6)

considered the case when the energy surfaces have arbitrary

shape. By rather general arguments, he was able to show that

the period of oscillation is inversely proportional to the

extremal area of cross section of the Fermi surface

perpendicular to the field. Lifschitz and Kosevich (7) then

made use of Onsager's result to give an explicit formula for

the dHvA magnetisation, in which the effective masses and

the Fermi energy are replaced by various geometrical

properties of the Fermi surface. The first application of

the dHvA effect as a tool for the systematic study of a Fermi

surface is due to Gunnerson (8) who has worked out the shape

and size of small pockets of the Fermi surface of aluminium.

It is not our intention to dwell on this, the major use of

the dHvA effect, except to note that the effect has been

remarkably successful in measuring the geometrical properties

of Fermi surfaces to great accuracy. There are extensive

reviews on these results, for instance, Shoenberg (9) and (10)

and Craeknell (11). The theoietieal -ex.1v(101i of Lhe dHvA

susceptibility will be described in full later.

2. The Effects of Scattering

It is clear that if electrons are unable to make

complete orbits about the magnetic field due to scattering

processes, then the dHvA effect will be diminished. This

idea was made quantitative by Dingle in an important paper

in 1952 (12). Dingle considered each Landau level to be

broadened to a Lorentzian shape of half-widthIlrt where

is the conduction electron relaxation time for the particular

orbit in question. (In fact, as Dingle recognised,'t is

twice the relaxation time; the error occurring because

classical mechanics was used). The consequence of

introducing the line width is to produce a term which reduces

the dHvA amplitude in such a way as to be equivalent to a

rise in temperature. The effective change in temperature x

is given by 08X = t/^c and x is universally known as the

Dingle temperature. This result was of immediate use in

interpreting the series of measurements made by Shoenberg (10)

and, as Shoenberg pointed out, enabled a discrepancy between

the field and temperature variations of the amplitude in

bismuth (4) to be cleared up.

The great importance of Dingle's work, however, is

that it showed that the dHvA effect could provide more

information than just cross-sectional areas of the Fermi

surface - it could provide data on conduction electron

lifetimes at the Fermi surface. Unfortunately, of course,

there are drawbacks; the amplitude is not so easy to measure

accurately as the period of oscillation, nor is it so

reproducible. Further, the scattering effects are averaged

around an orbit and it is only very recently that people

have attempted to invert Dingle temperature data to obtain

the conduction electron lifetime as a function of position

on the Fermi surface (13).

After oingle, a series of papers (14, 15, 16, 17)

followed, which have considered the problem in more detail.

The first, Williamson, Foner and Smith (14) extended Dingle's

treatment to arbitrary Fermi surface shapes utilising the

method of Lifshitz and Kosevich (7). Bychkov (15) attempted

a more rigorous treatment using the methods of quantum

field theory. However, his treatment was restricted to

free electronsj-function interactions and no quasi-bound

states. As it now seems that there were mistakes in the

treatment and Green's function were used, the work has

largely been ignored by Western scientists. In fact, the

approach is valid and has recently been used by Mann (17)

in considering the effects of point defects and is related

to the approach presented here, except that it attempts to

account for the effect of significant concentrations of

impurities. By contrast, Brailsford's work (16) is more

transparent to the experimentalists and avoids the explicit

use of quantum field theory. Brailsford supposes that, as

a result of scattering from impurities, the conduction

electron energy levels are shifted by an amount4(e), a

function of energy. This shifted energy, which we recognise

as the real part of the conduction electron self energy

evaluated in perturbation theory, is inserted into the Landau

'formula at an appropriate point. Brailsford then analytically

continues the energy shift into the complex plane and makes

use of the Kramers-Kronig relation to obtain the corresponding

energy width'''. He then shows that, for& and[ being slowly

varying functions of energy, the shift LS contributes a change

in the phase of the oscillation and the width 17 corresponds to Dingle's parameter x. Brailsford also contributes a

useful discussion on the relation between resistivity

relaxation time V and the relaxation time inferred from the

Dingle temperature -co. Apart from the point already mentioned

that To is an average over an orbit which is often only a small part of the Fermi surface, "rz. and Te differ in the relative importance of small and large angle scattering. For

resistivity, small angle scattering is relatively unimportant

due to the weighting factor (1 - cose), (18), but every

scattering event contributes equally to "'C D. The relation

between XI) and 17e depends then on the distribution of

scattering through angles e, and in general we should expect TD 4 re .

ct

3. Interacting Electron Theory

The work that has been described so far has all

dealt with the case of impurity scattering where the energy

shifts and line widths are small and smoothly varying

functions of energy. We will now discuss the effects of

interactions between the conduction electrons on the dHvA

effect.

Independent electron theory leads to a sharp cut-

off in momentum space of the ground state distribution

function - the locus of the points of discontinuity being

called the Fermi surface. In the case of free electrons the

Fermi surface is spherical and when the potential due to the

crystal lattice is added, it becomes distorted in a way which

is well understood in principle, if difficult to calculate in

practice. In this picture, the excited states of the electrons

are produced by removing them from the occupied states

immediately below the Fermi surface and placing them in states

immediately above. These excited electrons and their

corresponding holes account remarkably well for the properties

of metals in a great variety of experiments, including the

dHvA effect.

This agreement is, at first sight, very surprising;

the Coulomb interactions between electrons have been totally

ignored yet we know that these interactions are of the same

magnitude as the kinetic energies of the electrons.

Consequently one might suppose that the interacting electrons

would have a modified distribution function and, in particular,

there is no longer a discontinuity, that is that the Fermi

surface is no longer well defined. In a series of papers,

Luttinger and Ward (19) and Luttinger (20, 21) have treated

the electron-electron interaction to all orders of

perturbation theory. They have shown (20) that the Fermi

surface does exist and that many properties, the dHvA effect

amongst them, can be seen to be described by the same

expressions as in the independent particle picture, except

that the original single particle energies are replaced by

renormalised quasi-particle energies.

10

We will describe Luttinger's discussion of the dHvA effect in more detail since the ideas in 'it lead to the

formulation of the dHvA susceptibility which is central to

this thesis. Luttinger and Ward (19) produced a diagrammatic

perturbation theory for the thermodynamic potential for a

system of electrons interacting via the Coulomb potential.

They were able to obtain, by somewhat complex arguments, an

expression which relates the thermodynamic potential, U).) to

the full one electron Green's function. Luttinger (21) then

considered the electron self energy m in a magnetic field and demonstrated that the oscillatory part of the self energy

( tos4 ) - arising from the discreet set of Landau levels - is

small under the condition that iStoc<<JA. (Au is the Fermi energy).

He further neglected the temperature dependence of the self

energy - this will not be allowable for the systems we are to

consider. Luttinger then treatedAas a functional of the self

energy and expanded the expression as a Taylor series about the

non-oscillatory part of the self energy ( ). Making use of

the property thata. is stationary with respect to variations liar., it is clear that the first correction to the expression

using r. is of order ( Eric,) 2, which is negligible. Then, by analysing the oscillatory contributions to the individual terms

in the expression fora, Luttinger was able to show that the — t-i

expression reduces to flo‘c..= -1161- 4%. (- (IOW ) 1 hi oft whereelk(W) is th.e full Green's function evaluated usn7;h:

non-oscillatory part of the self energy. It is then

' straightforward to show that the expression for J10„ reduces to the conventional one for non-interacting particles i.e.

= up( (p.—E14)/keT)) , where, however, theEK are the renormalised quasi-particle energies. The

susceptibility is given by the second derivative of-a. with

respect to field and hence will have the same form as the free

particle dHvA susceptibility. We note that this result is only

valid for the oscillatory susceptibility; the constant Landau

diamagnetic susceptibility would be much more affected by the

interactions.

11

In this thesis we will follow Luttinger's lead

and consider the dHvA effect in some interacting systems.

Chapter 2 will present arguments similar to Luttinger's

for the electron-phonon system which lead to a general

expression for the dHvA susceptibility. This formulation

will be compared with previous studies of similar problems,

and will be applied to mercury which is a strong-coupling

electron-phonon system. Comparisons will be made with

experimental results on this metal. Finally we will exploit

the mathematical anology between phonons and spin

fluctuations in strongly paramagnetic metals to predict the

form of the dHvA amplitude in these systems.

In Chapter 3 we will discuss the problem of the

dHvA effect in dilute alloys, where the impurity is nearly

magnetic. The Anderson model is used and recent treatments

of localised spin fluctuations at zero temperature are

extended to finite temperatures and magnetic fields. The

results are compared with recent experiments on AlMn, an

alloy which has been shown to be adequately treated by the

localised spin fluctuation concept. Finally in Chapter 4

we tackle the similar problem when the impurity has a well

defined magnetic moment. These alloys exhibit the well known

resistance minimum known as the Kondo effect, and Kondo's

s-d Hamiltonian is• used in our calculations. The full

electron self energy is calculated in an external magnetic

field to third order in perturbation theory, and the resulting

dHvA amplitude is applied to the two examples of Kondo alloys

for which there are experimental results, that is CuCr and

ZnMn.

12

CHAPTER 2

ELECTRON-PHONON INTERACTIONS AND THE DE HAAS VAN ALPHEN EFFECT

1. Introduction

The effects of the interaction of the conduction

electrons with the lattice vibrations on the dHvA amplitude

are studied in this chapter. We start by reviewing the ways

in which the electron-phonon interaction can be observed and

then, in section 2, show in detail what information is

contained in the dHvA effect. In the following section we

develop the standard theory of the electron-phonon interaction

in metals by calculating the conduction electron self energy

at a series of discrete points on the imaginary frequency

axis. We then apply the theory of sections 2 and 3 to

calculate the temperature and magnetic field dependence of the

amplitude in mercury and compare the results with experiment.

Finally, in section 5, we use some general arguments to

predict the qualitative behaviour to be expected in nearly

ferromagnetic systems.

The electron-phonon interaction in metals plays an

important role in determining many of the low temperature

electronic properties, the most dramatic of these being the

phenomenon of super-conductivity. Besides this, the effects

in normal metals are significant and can be roughly

characterised by an increase in the effective mass of the

conduction electrons which can be as much as a factor of 2.5

in lead and mercury. Under the usual experimental conditions

(frequency much less than the Debye frequency and temperature

much lower than the Debye temperature) those electrons

interacting with phonons have energies close to the Fermi

energy, and a quasiparticle picture applies. It is then

possible to show that the enhanced effective mass would be

observed in measurements of the electron specific host,

cyclotron resonance frequency and the temperature dependence

of the dHvA amplitude (22). On the other hand the

enhancement would not be seen in the static conductivity,

anomolous skin effect, spin susceptibility and the period of

the dHvA effect.

1.3

If the system is probed with a frequency comparable

to the phonon frequencies or the temperature is raised,

higher energy electrons are involved and the quasiparticle

picture breaks down, causing the effects observed in the

various experiments to differ. Under these circumstances,

theoretical treatments must consider the full energy and

temperature dependence of the electron self energy. The

problem of cyclotron resonance in large magnetic fields has

been considered by Scher and Holstein (23) and the experiment

has been performed on lead and mercury by Goy and Weisbuch (24).

Not a great deal of information can be gleaned from this

particular experiment, however, since the imaginary part of

the self energy and hence the damping of the signal is very

large at high frequencies.

Grimvall (25) has worked out in detail the temperature dependence that would be observed in specific heat

and cyclotron resonance experiments on lead and mercury. He

made the important step of using the phonon density of states

weighted by the scattering matrix element eW F(w) inferred

from superconducting tunneling experiments (26). Consequently

his results have direct relevance to experiments on these two

materials. Experimentally, it is almost impossible to

separate the deviation from the linear temperature dependence

into the electron-phonon and pure phonon contributions.

Grimvall has proposed that the effects could be observed by

making careful measurements of the difference between the

specific heat in the superconducting and normal phases. The

temperature dependence in cyclotron resonance is more easily

observed and Grimvall's effects have been seen in zinc, lead,

mercury and cadmium (27, 28, 29, 30).

In this chapter we will discuss the temperature and

magnetic field (i.e. frequency) properties observable in the

dHvA effect. The reason for tackling this problem is twofold;

firstly the system has been studied by two different authors

(31, 32) with conflicting results and secondly, apparently

anomolous results for the dHvA amplitude in mercury have been

reported (33). We will discuss the earlier work in the body

of the chapter.

14

2. The Thermodynamic Potential

To describe a system of electrons interacting with

phonons in a magnetic field we will use Luttinger and Ward's

approach (19) to thermodynamic perturbation theory, which

has been reviewed in chapter 1. For electrons interacting

through the Coulomb potential, Luttinger and Ward gave the

following expression for the thermodynamic potential

11.= - keT G.,;` (011)1 ,c(wri) G-,,(011) (2.1) k, nor

where G k(W!,) is the full retarded Green's function evaluated

at the points itOn = (2n + 1)1tikai on the imaginary frequency

axis. (For a discussion of quantum field theoretical methods

in solid state physics, reference should be made to the book

by Abrikosov,- Gor'kov and Dzyaloshinskii (34)). Dyson's

equation (34) relates the proper self energy 1: (0) to the

Green's function :-

- €4- IKttoor (2.2)

where k 2- /4 , and/, is the Fermi. energy. It is not

A4 possible to write-a! in explicit form, but Luttinger and Ward

show that it has the property that

kaT lit(LOtt) VIA)4) (2.3)

o- If we define the corresponding unrenormalised Green's function

Gi t1(14r,) by G°6011) = Cion —cr then we see that

61/. =~ " kaT -.[Giojotiv Ejon)1-11 6 T (kin)Ame x

that is, thataISE=I0when Dyson's equation holds. We could regard this stationary property of Si_ as a variational principle for deteriuininy the coLrect self ene.cyyl:. We will not pursue this but the stationary property will be used lat.er.

Abrikosov et. al. (34) state that equation (2.1) has a general

applicability and is not restricted to the two-particle

15

interaction of Luttinger and Ward's treatment. It is

sufficient to understand thatili has the functional property

in equation (2.3).

It is possible to write& explicitly for the

electron-phonon system and Abrikosov et. al. and Eliashberg

(35) have given different but equivalent expressions for.n...

These can be written

.11.= —115"T -ell[—Gi;( (011)] ;flan) k (tan) if

Ical f D4.-1 (43,4) Tr4(0111) ,(WPn)1 q,.

k ckie' Cxt (Lon) -Diji4)01) Gc,..,t,(Loh—Lor0 (2.4)

The first term in the expression is of the same form as for

electron-electron interactions and the second is a similar

term for the phonon contribution, whilst the last is a cross

term. WW6) is the full phonon Green's function evaluated at the points Wm = 2.mkra1 T and TTNis the proper phonon self energy. The detailed form of the electron-phonon

interaction will be discussed later when the electron self

energy is derived. For the present discussion it is enough

to use the definitions

T1.1,644,) %tt, k5-r to, Gic (Loil) Git_ ii# (4)/1-14,)

Eg(t4a) = kaT 5:c Dc (0m) (2.5)

where q is the electron-phonon coupling constant. This

result is due to a theorem of Migdal (36) that the vertex

corrections are small, being of the order of the square rooi.

of the ratio of the electron mass to the ionic mass. Further,

there is a Dyson equation for the phonons

(2.6)

16

We write the expression foril as

— kaT E. GCS 6,411 4.. Yijon) Gbc (01%) AI (2.7)

where'll! contains the second and third term in equation (2.4).

If we use the result for the phonon self energy,X reduces to

Then, considering variations,

8-11! = '507 11. tri,(14,„) Ytift,othistr'& -MI, (444) iPik)hi,

0)1E (Wrst)R6)13 Gic--$6)11"14) -1-G1446144GLA it,,ibit) ki

By re-arranging and changing variables we find

= —(kaT) E %;,- Nun%) civt..„(wriorK) i§G,(Loro

Ek (tort) Gk (1.4)11) ) lc, n

which is the property required by Abrikosov et. al. Moreover,

it is straightforward to show that the thermodynamic potential

also has the property that b.11/8Z=0.

We will now consider the diamagnetic properties of

the electron-phonon system in the presence of a large magnetic

field. Within a Hartree-Fock effective mass approximation,

the non-interacting electron energies are found to be (37)

Ek 17 — --9-7/ty. 4. (1.4.1/2.M6k#WS/ 141114—'/4 • (2.8)

The magnetic field H is chosen to define the z-axis, ar=till

indicates the electron spin, 1411 is the Bohr magnetonettZgte , where Mo is the bare electron mass and g is the conduction

electron g-factor. In the present analysis we will assume

that the part of the Fermi surface considered can be described •

by a single effective_mass m. The extension to a general

Fermi surface shape has been done fOr non-interacting electrons

by Lifshitz and Kosevich (7) and a similar course can be

adopted for electrons interacting with phonons; however, this

would only be useful if we have a model for the wave vector

dependence of the

level in equation

Lx Ly is the area field.

electron-phonon interaction. Each quantised

(2.8) has a degeneracy d = (11)1-1 L L . where airk

of the system perpendicular to the magnetic

Consequently, the sum over the momentum k entering

the argument of the functions in equation (2.7) for-41., must

be regarded as the sum over k and the integers l in a magnetic

field. Now, the quantisation of the energy levels is the

direct source of the dHvA oscillations and any function

depending on the energy levels will have an oscillatory part.

However, we are able to show that the oscillatory part of the

electron self energy is small compared with the non-oscillatory a. part by a factor of (tWiLitt , which is small under

experimentally realisable conditions in metals. We will show

this explicitly in a second order perturbation theory

calculation in an appendix to this chapter. In his analysis of the

electron-electron interaction, Luttinger further neglects the

temperature dependence of the electron self energy, however

this is not possible in our case, since the characteristic

phonon energy is much smaller than the Fermi energy. Indeed,

we are particularly looking for the temperature dependent effects.

We write 2:kW . 1 (ci(LO) + I: k ( lose where

2:(6) is the non-oscillatory part of the self energy and

rklm contains the oscillatory terms. Using the smallness of ros, , we expandilas a functional Taylor series about Awe.) and make use of the stationary property of4 Then,

kb-r d.Z • ti E G`0} (I)41'(t)

coo +s G,(0

cLQ.1 ie ) 1(7..(0) +0(T- \I-

k 4-osci kvt

V' where VT is the Green's function obtained from the

kt, tole

9.8

non-oscillatory part of the self energy /166'),' We now follow

Luttinger and consider the oscillatory contribution to Ai .

He shows that, since contributions to the oscillations can

only come from the poles of the Green's function ca - that

is that G'* still has an oscillatory part due to its dependence on ekvtie - the leading oscillatory part of -12.1 is

obtained by taking this contribution just once from each

diagram. The result of this procedure is that

t T Ct. E

o) (WO GE (IQ 0sc. t kt

cancelling against a similar term in the expression for .... We can see that this process gives the correct result for

electron-phonon interactions by reference to the explicit

'expression for in equation (2.4). The phonons do not

couple to the magnetic field and hence the first pair of terms

have no oscillatory part. The oscillations from the final

term arise from taking a contribution from each electron

Green's function in turn, and, making use of the equation (2.5)

for the electron self energy, we just obtain Luttinger's

result. Hence, in summary

.11°sc. kEtTd. 211, G (Lon) 1/4,,n, cr e •osc. park' • ( 2 . 9 )

(From now on we shall drop the superscripts and subscripts

referring to oscillatory parts).

At this stage, it is useful to make a comparison with

the work of Fowler and Prange (31). Their aim was to test the

notion that, at sufficiently high energies, the renormalisation

effects on the electrons will become diminished and eventually

go to zero, leaving "bare" electrons. The interactions with

the lowest characteristic energy and largest effects would

appear to be that between electrons and phonons and accordingly,

they have investigated the effects of large magnetic fields on the

electron-phonon interaction as seen in the dHvA amplitude.

They start their analysis from an equation for the number of

electrons in terms of the electron Green's function; in order to

obtain the thermodynamic potential they then integrate with

11 311.

respect to the chemical potential ( N = at, ). In consequence their work is equivalent toil being equal to Jim in equation

(2.9). Hence the method is only correct for the oscillatory

part, and then only under the condition that Lu_<icro .

Wilkins and Woo (32) in a treatment restricted to quasi-

particles actually used T-4 as a starting point and integrated

twice with respect to/4.th obtain the thermodynamic potential.

We now utilise equation (2.9) to formulate an

expression for the dHvA magnetisation in an interacting system

that satisfies these restrictions that lead to equation (2.9).

Firstly, we note that Luttinger made use of the particular

properties of the imaginary part of the electron self energy

to show that a quasi-particle picture applied in the dilvA

effect. We will keep the treatment general. We make use of

the theorem, E i>cts(e40-1 F(1) (It

and the contour of integration isshown in figure (2.1) giving

.SL = 4 E ct 0-I An [1. ek (1)1, .111.

kvgicr t»

We distort the contour C' , enclosing the real z-axis and use

the analytic property 1: (X *AS) = 1:(X) 1160 to give

eD

cob( (ex+ IT - tick) i.ri,,(30

4,t,c E ty)

-cto itlitio

{).- I arC (

1 )k (1. X- 6k (x) cx)

„ rm, (2.10)

We now follow the standard process (37) to eliminate the discrete

Landau levels; firstly we use the Poisson summation formula (38) et) - oh

. Fa) r(,)[i + Z z D.' Cos ( arc nr) co 0 Ent

2.0

.11■11, NIMee. ••••11. ■•■■■• ■••■■• ■■■•11 WOO. •••■■•••• •■•■••■•■

.1=1■11•11 1■111111. 0.11=1111W - arm. •■••

Re 7_

.1111.0.10

Figure 2.1

The Contour of Integration Enclosing the Poles of the Fermi Function

dcf 6-2111404 -rnt Afqx))

No (2.15)

21

which becomes, if F(J) is real, ob

FN:: Re 101 F()L.t + 2. E RAI( 4r14'rS)1 eco a (2.11)

Then, defining the variable ka. by 1%1-kill = a/110+4)cl , we obtain

rciCtk d.l<2 Of (e, trarctlui v(x) victo V (2101*

E tr _ evc e- (2.12)

it a"

where V is the volume of the system. The integrals over

momenta can now be converted to energy integrals by the

substitutions

24A 2;(1i, and COS2-9

k:

1C:4 kJ!: •

Then,

-1054

V

eb t cb

(2...g1itar.Trirdekt(cose) (64 P. d_1( cu4. amtim IA tt kx-e-trifro -Too)

-op tot

("I)T RIAT (45-iv (41A)(1— cost e)) -kwc, (2.13)

The integral over cosG is done by the method of stationary phase, since the argument in the exponential is always large :

S <Cos 0 i/dXF [- tia: (6-irt) COStel = ., St

ed. -- 4- 0 (s,--9,-) lir (ii-4),.:1 ix . ,

Substitution of this result into equation (2.13) yields

JIDSC=2[411\3411.(60.11*Reotf E41)T411(4211117A- k Ur- irs 72- itu3'..

ob 4stirc to A)e.

X cbt I ke,

( t.• -

IV

-4 ..iv

art+an r (-1)

e.-1.4 al-1 -V4 (2.14)

We now integrate by parts on and neglect the end-poi4t

contribution of the integrated term, giving

-Roc (42Vc. )312. E It( (e_ox..0-' V 2tv't t r r1.1 -kw, 4/ 1 tt

-ab

TT ti 4-/

2.2.

The integral one can now be performed, resulting in

‘la

*P24 '21 (421..0c..); Ref_ ' .€14p (21V t otA r4 V ae t T T61 1AM tO4 * TN)

x 2!2.< (ex4 0-1 124(1,1zi h)ce (x- %64) 11(41

*V, -00 (2.16)

We are now able to see that this result is of the same form as

that for non-interacting electrons except that the single

particle energies are modified to include the full self energy

effects in the oscillatory exponential. We note however that,

unlike the non-interacting case, the integral is not readily

transformed to contain the derivative of the Fermi function.

For the discussion of the electron-phonon system, we may now

sum over the-two values of electron spin since the interaction

is not spin dependent. For spin-dependent interactions, as we

shall see later, the sum over spins leads to an observable

change of phase. Performing the sum we obtain „f

419, /c fr (41"4.1311- Re. t ( :c.") cos rc r ) V Tt21 f=1 -4 .4c. 4 t 2.00,

x cox e,P44+ ft? [41:;',.(x- + r tio nt - A 0b (2.17)

We see that the "spin-splitting" factor cos (TTS/N40) does not

contain the enhanced electron mass, nor indeed does the

argument of the oscillatory exponential. The "spin-splitting"

factor is a well known observable feature - if the band

effective mass is such that the argument of the cosing is equal

to an odd multiple of 1X/2, the signals from two spin states

interfere to give zero amplitude. The measurement of this

factor can give values for the electronic g-factor (39) and

changes in the g-factor upon alloying have been observed

(40, 41).

This result for the oscillatory part of the

thermodynamic potential can be used to obtain the oscillatory

magnetisation by differentiating with respect to field. In

taking the derivative, the dominant contribution comes from the

z3

afar term e

(144.014. he: Re

tgo (Qsart, \ cos/run m el) UI niklk ,rth. tw, — 130 .

x ctiv. xcr {till (4- Zix) 4.1. tot.

-4 • (2.18)

Finally we transform the integral by closing in the upper

half-plane and by defining the analytic continuation of the

full electron self energy on to the imaginary axis

(2.19)

we obtain ebb

111.1°,2c - 2- (-,g' )112- 1411P-A- k T (-A Cos (ttrj) ) rt. kc ktaci a v.11

a. 40

eQ

K sift (kw -71) exf No. Viatnj „wt, 4 0 (2.20)

This solution for the dHvA magnetisation is consistent with

Fowler and Prange's solution (31),and hence we cannot agree

with Wilkins and Woo's (32) criticisms of the former's result.

We would emphasise that this result, apart from the final sum

over spins, is considered to have general applicability to any

system where perturbation theory (infinite order if necessary)

is valid and for fields such that the oscillatory part of the

electron self energy can be regarded as small.

3. The Electron-phonon Interaction in Metals

In order to calculate the electron self energy due to

interactions with phonons we will use the Bardeen and Pines

model (42) for the interaction. The result given in the appendix

to this chapter for 'Lhe self energy in a field evaluated in

second order perturbation theory, demonstrates that the non-

oscillatory self energy is not field dependent. The same

conclusion was made by Fowler and Prange. The Hamiltonian we

use is thus

H I-41-1 4- 14 • + too_ 1'6 ht- •

2.4

trict.,= 61,1 cteCtyr describes electrons in an effective

mass Hartree-Fock approximation.

Hey = x 44A describes the kinetic energy of the

phonons, 4. j10 is the bare phonon energy, and 7k, is the

polarisation of the phonons.

Hint: 51tikix C li d* Ck ( 4.°,410,6 kite, c

H 4. 117" c+ 14 C0 141Q, c

44fr ) a* 01,1cr• kcr ic)1/41,100,a'

KA 1* rti, is the square of the matrix element for the electron-electron

interaction and(ivis the matrix element for electron-phonon

interactions.

In considering the electron self energy we are

interested in excitations of energy . t03)4/,4.. . The Coulomb

interaction can lead to important screening and renormalisation

effects, but it does not lead to any interesting variations of

in the energy region of interest, which can be seen on

simple dimensional grounds. Quinn (43) has shown that, by

summing both interactions to all order in the RPA, one can

obtain an effective interaction between electrons. This divides

into two parts, the former being the usual screened Coulomb

interaction and the latter is an effective electron-phonon

interaction. The result is that Iv is modified to ill 6(1.1(4) where e61,114) is the dielectric function, is is modified in the

same way and the phonon energies are renormalised to

143(t- ie(i7w) .

The electron self energy due to electron-phonon

interactions is represented by the diagram :

_.4. 1)c1CLam) .•

E (to 1 •

1

G 1. 0.0 - 14 n

Figure 2.2

The Electron Self- Energy

•

2.5

where we have incorporated the coupling constant (lat„ in the

definition of the full phonon Green's function Dek(tor,) . We note that in general, the coupling constant is frequency

dependent through the dielectric function, but again dimensional

arguments enable us to use the static limit of the dielectric

function. Phonon corrections to the electron-phonon vertex

have been neglected following the work of Migdal (36). His

argument is roughly that since D04) falls off rapidly for

114),%1,1.0.D , only vertices in which the energy transfer 14.)n-141.01

is less than 44 can give appreciable contributions. It then

follows that the energies of the intermediate electronic states

must also be of order WD for the electron Green's functions to

have significant magnitude. This restriction leads to severe

limitations on the amount of phase space available for virtual

transitions and the vertex corrections are of order 19> =CATL which is small. Consequently the electron self energy can be

written

kcool:4(9T E, G (0 i) to kin',X it014x ki %-t i),(to -1) 4 4 * (2.21)

This is an integral equation for the self energy; however as a

consequence of Migdal's theorem, rt16.00 is essentially

independent of k in the region where it affects the integral,

implying that we can neglect it in the integral. Hence

equation (2.21) reduces to a quadrature, as was demonstrated

by Engelsberg and Schrieffer (44).

We will now perform these integrals in such a way as

to retain the detailed form.of the electron-phonon interaction'

and the phonon density of states. In this way we will be able

to use the experimental data for these quantities and avoid use

of simplifying models such as that of Frohlich. This method

was first adopted by Scalapino, Schreiffer and Wilkins (45) in

their treatment of strong-coupling superconductivity. We write

the phonon Green's function in a spectral representation ef#

th (1,,x 1.1.14 -v tiort, 461) (2.22)

b

For unrenormalised phonons the spectral density is a delta-

function SOL-V) , whilst for the renormalised case N (V) broadens and shifts. We make use of the relation

ticr E FRoo.= —..1— r(1.) 4 2rti. J c e -v%

where the contour of integration C was shown in figure' (2.1)

and we deform the contour Cr, taking account of the poles of

pciA6A,A). We obtain

ipo= E. 15 rtg of .1- vEici)-vt.i(v) icco+,(,) 1 A keA Aon-e v - 46)4 - Fie 4V

(2.23)

where is is the Fermi-Dirac function ( and TO) is

the Bose-Einstein function (e v.1)-) . The sum over k is transformed to an integral

E S cat' 1 cik i dg klu si4,0 ge (4101 (2e

A 1_1 where is the angle between k and IC

We can transform to energy variables and the phonon

momentum q, and taking a spherical average we find that

r two 4 t0) IditE olce 1 he IC) Sie x(v) Ittk;- j

where N(0) is the density of electron states at the Fermi energy

and kr is the Fermi momentum. Now, the phonon density of states is easily given in terms of the spectral density,

0 0

r i_ Its) 4,11,6)) §(e) (1) X 1 46 ion - e-v 40h...e4v ) (2.24)

(2.25)

and we follow Scalapino et. al. (45) by defining an effective

electron-phonon coupling constant c41(11) by

uF c42(.0) f:(v) 11.1.0 E di, 9: 4ju (11) (2.26) Srt 1,1* A

4340,, e v

Hence

(04,v) = (2.1160+i) + ct‘ ] ob

2.7

The quantity Ot1r(19 has been extracted from tunnelling data in

strong-coupling superconductors (46). Hence,

Eit(*n) z 164 Cell)) F(V) ilet [1. Ve) n(v) (e) 4 n(V) VOn-C-V itart -6 4 V 61

(2.27)

For the dHvA effect we require the analytically

continued function defined in equation (2.19). In the

expression for the dHvA magnetisation %;640 appears in an

exponential and the sum over frequencies ban is normally

truncated rapidly except at the lowest temperatures and highest

magnetic fields. Indeed, most experimental data is usually

analysed with only the first term of the sum; that is, the

temperature dependence is taken to be the exponential

exp (-2n9callkw,) . There is a useful and illuminating

transformation for (torl), when n is not too large.

Consider • oto

K(144 11) f de { I (6) ."(v) 'i(E) # "9 .40 C— V 1,0 - E 4.v

(2.28)

The constant terms in the numerator can be integrated directly,

since

•

• -4.4 ( IN 4- — 2-v r de --7----1(° • (to,- s,sixitzt, ,..€4,0

We close the integral in the upper half-plane and evaluate the .

residues at the poles of the Fermi function and of the two parts

of the denominator, ob

ni=o [4.20%+k)it

qq (2.29) kito„ 4,r)

e eNiu34-v)

Z8

....r.. - tR 1 1.07 f --I--- — 2. in (v) — 1 1 1, 0:29 145 6..%)114--P2' J . (2.30) P

We shift the variable in the sum,

(only) r: 14-v I ra= - Jev e rfv: ( 2")1.441

•

Now, the infinite sum is standard and we find that oo

It.V - 2.4.1(-0) -t = 2

Consequently the kernel K 631.1).0 simplifies to

KN,o) ~ illae + 2. it I 411:1 (a/i2Likeall1+

V

(2.31)

and the self energy becomes

44

5(4 11)r rCkg-T cti,, 2 41(17)1:(.9) + 2 Y.. /1. „,T ,Li /At *1 0 v fari I • (2.32)

A particularly important result obtains for the n = 0 term ob

(tt k8T .c civ 2 tt(v) F (v) kir (2.33)

that is, (1.047) is a linear function of temperature. The constantX can be shown to be given by

— .C) Art* I to 0=0

T=0 40, 9

as can be verified by differentiating equation (2.27). The

result (2.33) was first written down by Fowler and Prange (31),

hnt the full, result (2.32) is new and has proved to be most

useful in deriving the dHvA amplitude as a function of

temperature and magnetic field.

(2.34)

2.9

4. The dHvA Amplitude In Mercury

We will now consider the application'of this analysis

to a particular system. At the beginning of this chapter we

mentioned that on the baSis of quasi-particle arguments, the

temperature dependence of the dHvA magnetisation should be

enhanced by the electron-phonon interaction. Further, we

discussed the behaviour expected in certain other experiments

when the temperature or frequency is raised to be comparable

with phonon energies. Non quasi-particle effects have been

observed in cyclotron resonance experiments in several metals

(27, 28, 29, 30). There would seem to be no obvious reason why

such effects could not be seen in the dHvA effect.

The electrons and phonons are strongly coupled in

mercury which has a particularly low energy phonon mode

( ^-21°K) (26). From the experience of the effects in specific

heat and cyclotron resonance one might expect rather large

deviations from quasi-particle behaviour for temperatures

greater than about 4°K. However Palin (33) has performed

extensive dHvA experiments on mercury over considerable

temperature and magnetic field ranges and has observed no

variations from quasi-particle behaviour. Moreover, the Dingle

temperature, which is usually interpreted as a measure of the

scattering rate at the Fermi surface was found to be temperature

independent. These surprising results are shown to be in full

agreement with the standard theory of the electron-phonon

interaction discussed earlier.

It is instructive to see how the temperature effects

could arise. Equation (2.18) for the oscillatory magnetisation

contains the integral

06( e.M 0-1 Ur krLir ( X EIX) 4- ) . L

Viewed individually, the real and imaginary parts of the self

energy have large temperature dependencies when evaluated close

to the Fermi surface. Grimvall (25) has shown that the initial

slope of the real part .(T) changes by over 10% between T = 0°K

and 7°K and the imaginary part changes its temperature

• (2.35)

30

dependence completely in going from low to high temperatures

(4'1'3 for ka7c5t4W$ ^."1" for k‘T ):).01)). If we' were to

evaluate the integral by parts to give the derivative of the

Fermi function times some exponential function of the self

energy and treat this in the conventional way, we might expect

to see these temperature. variations. However, such an analysis

is not permissible since the exponential has a complicated

energy dependence. A quasi-particle treatment would have

permitted such a step and would have given erroneous results.

We also note that, if the wrong starting equation is used, one

can end up with the derivative of the Fermi function in

equation (2.18) as in Wilkins and Woo's treatment (32).

We now use the results (2.32) and (2.33) to analyse

the behaviour in mercury. Only the first harmonic r = 1 is

considered since the higher harmonics are exponentially smaller

and can be experimentally separated anyway. An amplitude, A,

which contains all the interaction effects is defined as

A E ft y, Loot, (tA)11))] n=o

For temperatures high compared to the cyclotron frequency, T te-k i.e.. x );> 1 where x = .2 only the first term in the

t.t4)t. summation will contribute significantly. Using the result

(2.33),

(2.36)

Consequently, at high temperatures, the amplitude is entirely

quasi-particle like. If we define am amplitude for free

particles of mass 111

A° = 2. [ skk. 232...S1') tk) (2.37)

th-- 4. LJ -1, — IA C .R A .Pn • n VV LAIC' Lempera,are iimiL oi ti is iuenLioai

to A°. For lower temperatures such that x. 1, higher order

terms will enter the summation (2.35) and will cause A to differ

from A°. We have not found it possible to obtain an analytic

form for the amplitude at all temperatures (even within an

Einstein model for the phonon spectrum), but can make some

qualitative remarks. From equation (2.32) we see that

—! x A Tz.. tZttke.-r ukg,-0) e rn

31

Von) ::(244.0 AniceT , and consequently the amplitude

op .11t (1.011 + X Van)

r:o i•e• A° at all temperatures. We would like to

consider the conditions under which the difference (A - A°)

could be maximised. The cyclotron frequency essentially fixes

the number of terms in the summation (2.35) that will contribute

significantly for any given temperature. We would like the

temperature to be a significant fraction of the phonon energy

so that the terms in the denominator of (2.32) are large. This

can be achieved by a low phonon energy (hence mercury is the

best candidate), large magnetic fields and a low cyclotron mass

orbit.

The analytically continued self energy has been

calculated numerically using the experimentally determined

phonon density of states for mercury (26), shown in figure (2.3).

The self energy, evaluated at several values of tOrt , is plotted

as a function of temperature in figure (2.4). The value of A obtained from the density of states data is 1.55, hence the

enhancement in the electron mass e is 2.55. The results for

the self energy were then used to calculate the amplitude

factor as a function of temperature and magnetic field using an

enhanced cyclotron mass m* of 0.183 which roughly corresponds

to the investigated by Palin (33). Two sets of results

are shown in figure (2.5) as functions of temperature. The

logarithm of the amplitude is multiplied by the "sinh

correction" factor (1 - JIN , which differs from 1 only

when x is small, and serves to keep in A° a linear function of

temperature over the whole range. The first case, for a field

of 40 kG, shows a maximum deviation of only 5% from a straight

line, and this occurs only at very low temperatures. This case

corresponds fairly closely with some of the experiments of

Pali who could detect no deviations from linearity over the

temperature range 1-10°K. The second case, also plotted in

figure (2.5), for a field of 100 kG shows larger deviations.

However, these deviations take the approximate form of a

straight line with only a slightly different slope (about 5%

0.4

02-

50 100 150 Ens (°K)

Figure 2.3

The product of the electron-phonon coupling constant and

the phonon density of states o(LF as a function of energy

for mercury, taken from the work of McMillan and Rowell (26).

O

111•••••

O

0

O

0

(14 O

33

Figure 2.4

The analytically continued electron self energy in mercury

evaluated at 04 = (2t.+ 1)ttkBT as a function of temperature

for n = 0, 1, 5, 10.

1190

In(Ac)

Figure 2.5

The temperature dependence of the amplitude of the dHvA oscillations in mercury : —D A, —

... 1rnx,,.. A ..... e. T. "JR

where X = U1%1- A tot . A cyclotron mass m4 = 0.183 is used. The dashed lines correspond to free-particle behaviour.

35

change). Consequently, in order to see the deviations

unambiguously, one would have to perform an exiperiment over a

broad temperature range up to and including the region where

the deviations disappear, which is just the region where the

signal is becoming very small. Palin has done such an

experiment (H = 82kG and T = 4-17°K), but still saw no

deviations from quasi-particle behaviour.

In summary, the dHvA effect in an electronphonon

system has its temperature dependence modified by the mass

enhancement. However, the enhancement does not change

appreciably with temperature, except in extremely high magnetic

fields. This is a consequence of the property that the

analytically continued full self energy, evaluated at the first

pole of the Fermi function, has only a linear temperature

dependence. Physically, the drop in the effective mass as the

temperature is raised is almost exactly cancelled by the

increase in the scattering by phonons. We note also that the

Dingle temperature is completely unaffected by the electron-

phonon interactions, and as a result is not a true measure

of the full scattering rate of the electrons at the Fermi

surface. Finally, we reiterate the well known result (21, 22,

31) that the oscillation frequency is not affected since the

electron self energy at the Fermi surface is negligible.

5. Nearly Ferromagnetic Systems

It is now well known that there are electron systems

which have very large paramagnetic susceptibilities without

becoming ferromagnetic; these systems ( He3 , Pd for instance )

are known as nearly ferromagnetic. The tendency to

ferromagnetism is described by an exchange interaction which

favours parallel alignment of electron spins. This is opposed

by the Pauli principle which raises the kinetic energy of the

system when electron spins are aligned. If the lowering of the

potential energy due to exchange is greater than the rise in

the kinetic energy, then the system will become ferromagnetic.

In Pd, however, where the large susceptibility indicates a

strong exchange interaction, the time averaged lowering in the

36

potential energy is not quite sufficient to overcome the rise

in the kinetic energy. However, it was recognised by Doniach

and .Engelsberg (47) and Berk and Schrieffer (48) that, for short

periods of time, the system will have small ferromagnetic

regions, causing fluctuations in the net spin about zero. As

the system approaches a hypothetical ferromagnetic instability,

the spin fluctuations would persist for longer times and extend

their range until the system developed a macroscopic time

averaged net moment.

The effects of the spin fluctuations on the electronic

properties have been considered by several authors (47, 48, 49)

and the results, in the first approximation, bear strong

resemblance to the effects of phonons. The analogy is due to

the fact that both spin fluctuations and phonons are Bose-like

excitations and in both cases the electron self energy has a

weak momentum dependence. One consequence is that the spin

fluctuations (like phonons) do not affect the static

susceptibility and hence do not change the ferromagnetic

instability condition

I E 1 Ar (0) = (2.38)

where I is the semi-phenomenonological exchange energy constant

and - N(0) is the density of states at the Fermi surface for a

single spin. Another result is that the electron mass can be

regarded as being enhanced (47), affecting the electronic

specific heat for instance.

Luttinger's (21) arguments for neglecting the

temperature dependence of the electron self energy in the dHvA

effect are again invalid, since the energy scale for spin

fluctuations (ii/T), where is the lifetime of the

fluctuations, is small compared to the Fermi energy. In fact

(47) rr

+1/ and tle. 0. 9 2 Pd . et - I As in the electron phonon system, we do not include the magnetic

field dependence of the electron self energy. A complete

treatment would have to avoid this simplification - Oder (50)

31

has studied the field dependence of the specific heat of the

alloy NiRh and has found that the spin fluctuation behaviour

is suppressed in a field of 94kG. In chapter 4 we will treat

more fully the magnetic field behaviour of spin fluctuations

localised to a transition metal impurity in a simple metal host.

Within a Hartree-Fock approximation, the Hubbard

model (49) for the magnetic behaviour of transistion metals

gives rise to a contribution to the single particle energies in

a magnetic field representing the effects of a molecular field,

which can be written I/2 (Ar • ) Here Mr is the number of electrons with a given spin orientation o'. We may obtain

( Alicr-Nr) from the RPA susceptibility (47)

° = (

spar - 2 Ago 1. ...1* I —

(2.39)

(2.40)

(2.41)

Now the magnetisation M =.1r.H is also given by

so that we can identify

(Afer cr 9/3. it4. 14 2 Arto)

We can now combine.the exchange term with the single particle

paramagnetic term g.1.4 to give an enhanced term

I' al pa

The sum over spin in the dHvA derivation may now be performed to

give the enhanced spin-splitting factor

cos (rt ,s, lit 1' I

(2.42)

Hence, we see that the spin-splitting factor is enhanced by

the same amount as the susceptibility (2.39).

313

As mentioned earlier, in the present treatment, we

will ignore the magnetic field and hence spin dependence of the

electron self energy. Then, the analysis follows the same

course as the electron-phonon case with the spin fluctuation

contribution to the self energy replacing that of the electron

phonon interaction. Within the quasi-particle approximation

analogous to the electron-phonon case the temperature

dependence of the dHvA amplitude will be enhanced by the same

effective mass as that which appears in the specific heat.

where 0( 41.2/2,4m4) nz o x 2- 4 TI P associated Laguerre 'function.

and

3q

Appendix

In this appendix we will show that our results are

compatible with a second order perturbation theory calculation

that includes the full field dependent self energy. In doing

so we will be able to demonstrate that, within an Einstein

model for the phonon spectrum, the non-oscillatory part of the

electron self energy is not field dependent.

We use the Einstein model for simplicity; within this

model the integrals can be done analytically. Using the Landau A A

gauge Agt-04,0)0) where A is the vector potential, the

interaction Hamiltonian is written

Wiwi E ,4) (g. + a+ kz)V3- At ifIkelz ($.

The notation for the electron and phonon operators is that used

in the main part of the chapter. Mcilk) is the matrix

element of a plane wave between the magnetic field oscillator

eigenfunctions ON), where

_ ir--T--- ;ix tat

u 1

12 er He, 0-72*) (A. 2)

Here, a.2"=-*cie" 1 nx and H t. (x) is a Hermite

polynomial. We are then able to evaluate the matrix element,

and excluding the momentum conserving S.-functions the modulus

of M is

(A.1)

(e:)2 Cq4 M.e %I/ 2:4 I:)( e ot (tO

,,e( N

e1-2) (Co I

(A. 3)

is an

The shift in the thermodynamic potential to second

order in perturbation theory is described by the diagram :-

Figure 2.6

The Second Order Shift in the Thermodynamic Potential

where the full lines represent the unrenormalised electron

propagators in the magnetic field and the dashed line

represents the phonon propagator which conserves energy and

momentum. Algebraically,

A ji.(1). L Ly (42-wc. E E I flu, 4.01 1. 2.

411.k 141111 kv

?ON G 1:1441' 04n1- Lon 1)1 — 002' (A.4)

A simplification occurs in the case of our simple model; since

the coupling constant and phonon frequency are independent of

momentum, the only place where the variable qj occurs is in the matrix element. The sum over ciy is readily done when we

recognise that it is just the orthogonality integral for the

associated Laguerre polynomial (38)

Int 1 )1 2- = L., x 1- .2tr* • (A.5)

The expression for the shift in the thermodynamic potential is

now in a similar form to that of equation (2.9), except that we

have double summations to deal with. Nevertheless, we are able

41

to perform a very similar analysis, using the Poisson summation

rule twice and transforming the momentum variables as before.

We obtain co

jp) (10-)2. E E I I + kE (-0 suc-4 cos( a ) rz

oo ,„4„./ + 1) Cos (artrel) cos ( as eel) I

111.14=1 tw‘ two, ( t4hi- Atoft -e to) -to 1. (41 (A . 6 ) -

It is clear that the first term in the brackets is

the contribution to the thermodynamic potential in zero magnetic

field and does not give rise to dHvA oscillations. The second

term is analysed by performing the sums over n1 and n2. by

converting to contour integrals in the customary way, giving

6.-11 a = E. alf z f 1)T' cos( 2nie.i) 4r ttaL

dl I ( 1- 4- %IQ 4k') + 4% (0o)) , (A. 7 ) el+ el-140 et.t.t4).

Now, we refer back to equation (2.9), which was the result of

the analysis on the contributions from the various terms in-n..

We can expand the term inside the logarithm to second order and

we find

to (0 ko. (4212:4) Lx L3 rk (4:141%) %tit, it.On 61/41).t.q. • (A.8)

•

If we now convert the sum over n to an integral and treat the .

summation over 1 by the Poisson formula, we are able to see that

the two expressions are equivalent since the electron self energy

is given by

E 634

E " • 1 - +9t(4o) (ei) en WO .c IC . (A.9) iLon - 61- wo ktan -e+ Lot,

Hence, we have identified the second term in equation (A.6) as

the contribution arising from the non-oscillatory part of the

self energy.

. ".

42-

Finally, we can see that the third term must

correspond to the oscillatory part of the self"energy. It's

magnitude may be estimated by doing the integrals over cos

and cost)/ by the method of stationary phase (see equation (2.13)) which gives rise to a result (t114:9 112- smaller than

the single integral occurring in the second term. The neglect

of the oscillatory part of the self energy is thus seen to be

justified, at least within this model.

43

CHAPTER 3

THE DHVA EFFECT IN NEARLY MAGNETIC DILUTE ALLOYS

1. Introduction

In this chapter we will describe the localised spin

fluctuation (LSF) theory, based on the Anderson model, of the

formation of a magnetic moment in a dilute alloy and its

application to the dHvA effect. Particular reference will be

made to A1Mn which has been shown to be well described by

present treatments of LSF and on which dHvA experiments have

been performed.

This introduction contains a description of the

Anderson model of transition metal impurities in a simple metal

host and a discussion of theoretical attempts to describe the

approach to magnetism within this model. Also experimental

results for AlMn are described and interpreted in terms of the

LSF theory. In section 2 we extend the LSF theory, evaluated

in the Random Phase Approximation (RPA), to finite temperatures

and magnetic fields, in order to treat the dHvA effect in

nearly magnetic dilute alloys. Section 3 outlines dilvA

experimental results and compares these with the present

theoretical results. The chapter concludes with appendices to

sections 2 and 3.

Historically there have been two quite distinct

approaches to the problem of localised magnetic moments in

dilute alloys. One assumes the existence of a well defined

spin on the impurity site and then considers the interaction of

the spin and the host conduction electrons. The interaction is

characterised by an exchange energy J and gives rise to the

phenomenon known as the Rondo effect (74). This effect is the

anomolous rise in the resistivity as the temperature is lowered,

seen in certain alloys. Along with the resistivity behaviour,

Rondo's theory and its.extensions predict- a Curie-law

susceptibility and a "giant" thermopower. Experimental results

from several alloys have been fitted to the theoretical

expressions successfully and the above behaviour is generally

44

taken as characterising a magnetic alloy. Kondo showed that

the large effects seen at low temperatures are due to an

instability against the formation of a quasi-bound state, in

which the spins of the impurity and the conduction electrons

become correlated and anti-parallel. The Kondo effect will be

described more fully in the following chapter.

The second approach, which strictly encompasses the

first, does not assume the existence of the local moment, but

attempts to describe its formation. The first person to tackle

the problem in this way was Friedel (51). He observed that,

since transition metal impurity d-state energies often lie

within the conduction band, the impurity state would not be

truly localised. He introduced the concept of the virtual bound

state which contains a strong admixture of conduction electron

states, giving it a finite width Li. Anderson (52) recognised

that the basic interaction responsible for the formation of a

local moment is the Coulomb interaction U between electrons of

opposite spin on the impurity. If the impurity site is

occupied by an electron of a given spin, then an electron of

opposite spin is repelled, causing the impurity to be magnetic.

This tendency towards magnetism is opposed by the admixture

effect which broadens the impurity state. The role of the two

parameters U and A can be seen clearly in figure 3.1.

Anderson's model (52) formalises the above statements.

He introduces the impurity as a localised extra orbital,

representing the d-state of the transition metal ion, in the

conduction electron gas. He includes a mixing term which gives

rise to Friedel's virtual bound state and a Coulomb repulsion

term U Itttn4 , where liAr are the number operators for the

d-electrons of spin cr . The model does not consider any direct

electron-electron interactions in the host and thus is

inappropriate for alloys based on transition metals. Also there

is no'explicit mention of an impurity electron-conduction

electron exchange interation; however it has been shown that the

Anderson model does give rise to an antiferromagnetic exchange

which appears to dominate the ferromagnetic contribution for

transition metal impurities (60).

45

Figure 3.1

Density of states distributions illustrating the role of

the Coulomb repulsion U and the d-state width A . The numbers of electrons Octet) and <11,0 occupying the

states are denoted by the unshaded nortions below the Fermi

energy (The figure is taken from Anderson (52)).

Anderson himself (52) gave a solution to the

Hamiltonian in the Hartree-Fock approximation x:/hich reduces the

problem to a one electron problem in which the number averages

<new.) have to be determined self-consistently. He found that

the magnetic behaviour is determined by the ratio (U/4), and

for a given d-electron energy there is a critical value of (U/4)

at which the system becomes magnetic. The most favourable

condition for magnetism is when the virtual level lies self-

consistently at the Fermi energy and is Urn& = 1. For the

self-consistent field approximation to have any validity the

relaxation time of the virtual level into the continuum (kit%)

should be short compared with the interaction time (ik./U), i.e.

U/6. <C 1. Consequently, Anderson's solution is inaccurate near

the predicted position, and in fact overestimates the tendency

to magnetism. In the Renormalised RPA, to be described later,

this condition become modified.

If the impurity does acquire a magnetic moment then

the alloy exhibits a Kondo-like resistivity and a Curie-law

susceptibility whilst if the impurity remains non-maglietic the

alloy does not show the characteristic Kondo minimum in the

resistivity, and the susceptibility is Pauli-like. A smooth

change between these properties has been observed in 6 - CuZn

with iron impurities (53) indicating that the sharp transition

of the H.F. approximation is unphysical. The limitation of the H.F. treatment is that it omits the dynamics of the situation -

there should be a continuous change from the non-magnetic case

where the moment has an infinitesimally short lifetime to the

fully magnetic case where the lifetime is infinitely long. The-

time dependent magnetic moments were called spin fluctuations by

workers investigating magnetism in strongly paramagnetic

metals (47, 48).

Rivier et. al. (54) have calculated the dynamic

susceptibility contributed by the impurity in the Anderson model

by methods closely analogous to the band spin fluctuation

treatments mentioned in the previous chapter (47,48). They view

the Coulomb repulsion as an attraction between a localised

electron and a localised hole of opposite spin and consider the

41

effect of multiple scattering of this localised electron-hole

pair to form a localised spin fluctuation. The lifetime ( Tsi )

of the spin fluctuation increases with the magnitude of u/rul and diverges to infinity at the H.F. magnetic limit U/tti& = 1.

A renormalised theory would suppress this divergence, but we

might expect the unrenormalised treatment to have some validity

in the non-magnetic region. Some remarkable results appear when

the transport properties of the dilute alloy are calculated (55).

For instance the residual resistivity at low temperatures

( kisT< Vcss.) is schematically e = ( — A Tsl-f where

(6 and A are constants, which is very similar to the resistivity of a dilute magnetic alloy due to Nagaoka (79) in

the"spin-compensated state" at temperatures below the Kondo

temperature. Also, at high temperatures the resistivity has

close similarity to Abrikosov's result (56) and has the

characteristic logarithmic dependence of a magnetic alloy above

the Kondo temperature. In summary, for low temperatures

( kii1"<tirsf) the alloy exhibits non-magnetic properties; as

the temperature rises the alloy appears magnetic in the usual

sense. These results led Rivier and Zuckermann (55) to suggest

that the spin fluctuation temperature ( Ts.; iN/k01:ss ) and the

Kondo temperature ( Tis ) are equivalent and that the non-

magnetic state of the Anderson model is equivalent to the spin

compensated state of the Kondo model. Indeed, Schrieffer (57)

had earlier conjectured whether some alloys normally considered

to be non-magnetic were in fact magnetic alloys with high Kondo

temperatures.

Experiments on AlMn (58), which has the typical non7

magnetic temperature independent electronic contribution to the

susceptibility (63), revealed that the impurity resistance does

indeed have a T7- dependence at low temperatures. The question

which was then asked about AlMn was : is AlMn in a non-magnetic

state modified by spin fluctuations or is it in the spin- cnmpansa4-nel lnw 4-cm1pcirAi-rtr ct4-a-Fem. of = mngnafin pllny nr

indeed are these states equivalent? The last choice was ruled

out by analysis by Hamann (59) who showed that the Kondo effect

is not included in the RRPA treatment of the Anderson model. As

the Anderson and Kondo Hamiltonians have been shown to be

48

equivalent in the large U limit (60) there has been considerable

effort to formulate a treatment of LSF which does include the

Kondo effect. The first two choices can be resolved if the

ration (U/4 ) is determined. The Coulomb splitting of the

d-electron state of Mn in Ag has been determined optically (61)

to be about 5eV; this energy should not change greatly for

different host metals.

However, Schrieffer and Mattis (62) have considered

the effect of Coulomb correlations on the impurity site and

found, when the d-state lies at the Fermi level, that the

Coulomb repulsion U is reduced to U ,0 UV/044/ ) The

calculation is only valid in the low density limit, but can be

considered to indicate that in general U < U. One might expect

that the optical experiment, being almost instantaneous, would

measure the full repulsion U, whilst macroscopic experiments,

which are generally time averages, would sense the reduced

repulsion U. The dHvA effect would fall into the latter

category so long as the time of a cyclotron orbit is much greater

than the spin fluctuation time i.e. iitOqii41. From this point

we will refer only to a Coulomb repulsion U which we will assume

has been corrected for intra-atomic correlations.

The level width in AlMn was found to be O.2eV in the

specific heat measurement of Aoki and Ohtsuka (63). These

values appear to place AlMn firmly in the magnetic regime. This

situation was not to last long however, as Hargatai and Corradi

(64) have shown that the measured impurity contribution to the

electronic specific heat is characterised by an effective level

width, calculated in a partially renormalised RPA, which is

larger by an order of magnitude than that assigned by Aoki and

Ohtsuka.

Hargatai and Corradi's theory (and its modification by

Paton and Zuckermann (65)) follows the earlier work .on LSF but

attempts to *e.cnr,=14.qo the theory by assttmiassuming a simple

dependence for the d-electron self-energy, namely 7;(X) = (1-zI )X,

zI is then determined self-consistently. The details will appear

as the zero temperature and field limit of the work presented

later in this chapter. The renormalisation parameter z4 increases

44

as the ratio U/tt& increases and suppresses the divergence of

Tss at VILA = 1. Hargatai and Corradi showed that, in the

specific heat experiment is replaced by 201 where z, is

about 10. In fact, Paton and Zuckermann showed this to be an

overestimate since Hargatai and Corradi did not renormalise"tsi

and, using their approximation, have obtained an internally

self-consistent set of values for the parameters of the RRPA

from a variety of experiments. They have found that In A1Mn

U/tt& = 0.93, Z4 = 1.9 and Ts = 1857°K, showing that AlMn is in

the Hartree-Fock non-magnetic regime with relatively long lived

spin fluctuations.

A1Mn remains the most closely studied nearly magnetic

alloy, and the only one where a complete set of parameters for

the impurity state has been determined. As we have argued, these

parameters have to be known before it can be decided whether the

RRPA can be applied. A partial list of these alloys is :-

A1Cr (63), CuCo (64), AuV (67), ZnFe (68), and ThU (69). The

spin fluctuation temperatures of CuCo and ThU have been estimated

as 530°K and 1000°K respectively.

We will now briefly discuss the possibilities of

observing LSF effects in the dHvA effect and the experiments on

A1Mn by Paton (70). In the previous chapter we made some quite

general predictions about the effects of band spin fluctuations

based on the mathethatical analogy between phonons and spin

fluctuations. It would seem that the analogy would still have

some validity for a system with localised spin fluctuations.

However, one would not expect the analogy to be as close, since

the conduction electrons interact with the localised spin

fluctuations only indirectly via the hopping interaction with

the d-state electrons.

In analysing the dHvA effect results in A1Mn, Paton

has taken the standard formula for the oscillatory magnetisation

without interaction effects. The Dingle temperature, being

related to the scattering rate at the Fermi surface, is assumed

to be proportional to the impurity contribution to the resistivity.

Also the cyclotron mass which appears in the temperature

dependence of the magnetisation is equated to the enhanced

50

specific heat mass according to the ideas presented in the

previous chapter. It has been shown in chapter 2 that such a

procedure is only valid in very limited circumstances, namely

the energy dependence of the conduction electron self-energy is

simple, the magnetic field is not large with respect to the

characteristic energies of the scattering, and the temperature

and field are such that only a single term in the frequency sum

is required. In fact, we will show that the dHvA experiments in

A1Mn do satisfy these conditions to a good approximation.

2. Localised Spin Fluctuations at Finite Temperatures and Fields

We now present the theory of localised spin

fluctuations at low temperatures ( Ttrs ,0 ) and for magnetic

fields such that ( ittaH<<kil b, ) following the partially renorm-

alised Random Phase Approximation treatment at T = H = 0 due to

Hargatai and Corradi (64). We will use the Anderson model for

transition metal impurities in a simple metal host and, for

simplicity, will consider only a non-degenerate impurity d-state.

The extension to the degenerate state was indicated by Anderson

(52) and analysed by Klein and Heeger (71).

The Anderson model Hamiltonian can be written in

second quantised form as

H „ if C+10. C Ethr + V itd.1:11.04 k o cr

L Vic ( ct, cat, + c00. Cicr) ka cr (3.1)

where %-", Ckix are the creation and annihilation operators for

the state of momentum k and spin o" ecta,Cctcr is the

number operator for the state (d,cr ) and the electron energies

in a magnetic field H are

€1;cr Et, 8/0404 E e tc — Cr ) (3.2)

e tr 6d, 9/2. /An H sp- cot — , (3.3)

51

In the following analysis we will take the conduction

electron and impurity d-electron g-factors to Le equal to 2. U is the effective intra-atomic Coulomb repulsion energy and

Vkit is the matrix element describing the "hopping" of a

conduction electron into the impurity state and which causes

the d-state to have a finite energy width. Using the equations

of motion technique (72) Anderson has shown in the Hartree-Fock

approximation that the d-electrons have modified energies

Edo. Sticr -VV<'Wet.' with a width given by 6 rc<1.Vxd11) .(G) where <IV keit V.), is the average of the square of the matrix t element over all states k and ?(el is the density of

conduction band states. Along with Anderson, we will take c(e)

to be a constant in this calculation.

Hence,the d-state Green's function can be written

G:rt (t"i6) V°2 L (113k + (3.4)

The d-electron density of states td. 04.(e) is given by

c'd (6) Tr- 144 Ge7e4ii)] a (e Ecttiy- )

and consequently

A'

NO= el(e) Ct coV i -116" ) (3 . 5 )

Recalling that E(Lt. = Edo, U , we have a pair of self-.

consistent eqqations for <11/4) and 0110 which are treated

extensively by Anderson. Above a critical value of the ration

U/11,4 the solution is magnetic, that is OLIO 0 010

whilst below this value there exists only a single solution

<il.d.1)=(i1.d.4.,)= 1/2. The most favourable case for magnetism occurs when the virtual lr:vp1 fallsself-cnnsi nt-ly at +414% Fermi

energy, that is, eci,. -U/2 and then for U/tLktb < 1 there is the single non-magnetic solution <;'&6,/1.) <11a,k)t and

= cr (3 . 6 )

52.

Under these conditions, the criterion for the existence of a

magnetic moment is u/ttt, ), 1. Measurements indicate that E01, is close to 0 in Al based alloys (G3).

The transverse dynamic susceptibility in the absence

of the Coulomb repulsion is taken to be the Hartree-Pock term

shown in figure 3.2 and is given by (59)

/ • Z 41.1;LA k "T G Wm) w ot, "4 et, r%

where ton = 2. kaT w h, (2.11.4- rt. kaT

(3.7)

Then we follow the work on nearly ferromagnetic metals (47, 48)

and sum the set of diagrams shown in figure 3.3 corresponding to

RPA or dynamic Hartree-Fock approximation to include the effects

of the intra-atomic Coulomb repulsion. The diagrams sum to give

X-4"(w) O c(43)

I - X;#(14)

At zero frequency and with no, applied magnetic field,

r;+(o)=1/Ith. and we see that Anderson's magnetic instability

criterion UM& = 1 corresponds to a pole in the susceptibility.

At non-zero frequency we will approximate the behaviour of the

dynamic susceptibility by a single pole on the imaginary

frequency axis (59, 73).

The Hartree-Fock dynamic susceptibility (3.7) is

evaluated at T = 0 in Appendix A with the results,

{ 12.4 lz.0:41( k+ 4)-11, _14` ‘) 1 2.4 )

(3.9)

fx-+ (0) = arctaftpt_, ) 0 it h. (3.10)

Equation (3.8) shows that can be a rapidly varying

function of 'W whilst V46.0 may have only a slow

variation and it is sufficient to expand Xc";+(tO about LA = 0;

• (3.81

53

Figure 3.2

The transverse dynamic susceptibility in the absence of

the Coulomb repulsion in terms of the d-electron Green's

functions.

• • 1. • •

xo

_ (IX°

Figure 3.3

The dynamic susceptibility in the RPA in terms of 1(0 and

the Coulomb repulsion U which is represented by the dotted

lines.

54

further, to retain the single pole approximation we take only

the imaginary part of the ti.) dependence of Ir(4 .

Schematically if 1V(W) = Xc. (1 +i•tttt) ), where 0( is an expansion co-efficient to be determined later, then

r ito) (1÷4.,41,0)

1— WX0(1 4 (It La)

40 +4, ( I U "Co) sot

W.+ i:Ts

60) now has the same form as that in zero field (59), where

d = VaL and the magnetic field dependent spin fluctuation

temperature Ts will be calculated self-consistently later.

Simple analysis shows that reversal of the sign of h has no

effect on r+60 ; hence r+(4.)) = 10-(40 and whilst d and Ts a

will be functions of magnetic field, they will not depend on the

sign. Also we should note that by using a single pole

approximation with the pole on the imaginary axis, we are

precluding any formation of a permanent magnetic moment by the

field. (The field required would be such that h = b , whilst we

are interested only in fields such that h6.4tii ).

We will now calculate the d-electron self energy and will consider the contribution of the "ladder" diagrams shown

in figure 3.4. again in analogy with the band spin fluctuation

theories (47, 48). The self energy is given by

(3.11)

•

14. Gck i3 +4.10h 4, 4, Oh) , "trr • X-4 ( (3.12)

This expression is then transformed in the usual way to give a

contour integral

.n --t d 2. q(z) G-ce (z. 4 43m) r 4 ) (3.13)

where g (z) is the Bose-Einstein function. The contour of

integration is shown in figure 3.5. Now 00= lot.

has a cut along the real axis and Gcla(Z-tioDin)

14- Lon, - h &

SS

...■•■••1110

Figure 3.4

The d-electron self energy due to interaction with localised

spin fluctuations which are taken to he the sum over the

ladder diagrams shown in the first part of the figure.

••■••■•■■■• ••••••■• 1•■■■ ONIONS. ■••••■•

Jr" La

Figure 3.5

The contour of integration in equation (3.13) . The Green's function GrGI.' z m ) has a cut at Im z = —.tom and the susceptibility 'k (z) has a cut along the real axis.

S7

has a cut at Im z = -0O3„ ; deforming the contour gives

Etrato)=.91. fax irpowa(x4114 on .0 -x46c-i6)3 d et. x I d ant -t, -}‘ 3(X-4tOn.3 X."-+6(-itbra)[Gici-17(X4i6) (3.14)

But g(x-lons ) = -f(x), where f(x) is the Fermi-Dirac function.

We will calculate the temperature dependence of the self energy

by using the Sommerfeld expansions, valid for low temperature :-

a) 0

dbX I (10 + (k 8'

-a 6 TX x40 (3.15)l 06 0 1 CtX ()0 (x) d X 1(x) 13.4:1 (k9-1 §

xr_o • At T = 0 then

0 41.6itbilk) "")_;--)1. d.x G ..t2(x-Cual )[`)(:4"(x4.;b) `Xet-41X-i6)) 2rci.

-I- Vix-Com) (:(31;7(x411) - Cqr(X .)1 )

leading to

(3.16)

(3.17)

rtrntailt) -4.. -a. ,Pvti (-411 +I (tom 41 A, a lek-(wkit,--tsfi (-114.1-Ts )÷ criviI,(4144L-vrs) - is

1 tn(40m*TO) I ,Ai ti(tor44.-rs) ) (3.18)

Vh-L(Wriftt.4-r) -1.11 ai.-4.04%-A+Ts) 4.11+` with

00)=-021-tf Th. z (A—is) arctall(cr..6)1 4 41 w ill 4- (6,-T01- Ts2- II 4 itt--re .3 (3.19)

We see that there is the expected equal and opposite shift in

the two spin states which disappears at zero field.

SS

The equation for the self energy (3.14) is evaluated

for low temperatures by using the Sommerfeld expansions (3.15)

and (3.16) about the zero temperature value (3.18). The

resultant expression is then expanded in a Taylor series in the

magnetic field to order h2. Finally, following Paton and

Zuckermann (65), we adopt the linear approximation for the

energy dependence

Ed (W)ZG: (0) 4- Re. (LI x to (A) Lor,o

We obtain for the d-electron self energy :

(3.20)

E;(7.)) Urtnct {- jet (Tirsr + 6( y2-t- ( '1)1 to rs I )

(3.21)

with the factors given by

2 4 FITS 11'74 7. 4 (" TS (6--rs)

t_i__ f„,61, 8:= (4410CDTsj k s si /It;

+ t 1 4- ( 41/Ts + &-Ts) (Q-TSB 3- 1- (4-1.0

(3.22)

Forcing the self energy to be of this simple form and using

Dyson's equation we are able to achieve self consistency fairly

readily. We recall that the d-state Green's function can be

written in terms of the self energy via Dyson's equation

Galit434-i0 = (14 4 ti* + -`E:(w+ii6)) -1 (3.23)

and we can write the frequency dependence of the self energy as

(3.24)

Sq

then

Gr(w) 2, (4)4 (rki (3.25)

where

and A'= -1•2_ • (3.26)

Using equation (3.12) we can then obtain the renormalised

d-electron self energy by the substitution :-

r: (0) td ( h--) ki I k *Nij (3.27)

where the parameters 71 , T and zi are determined self

consistently by the following equations

I

2rV7.1 A

tjj-i ci'( 2-ct 'Ts' (3.29)

I ' 21 =1 Y '_'.4 ( /

2 rt z, i where 0( , (Si , 4)1 , 6 , r are of the same form as q , p , cp I

with / ,,e1-+ Ai, 4/ respectively.

It now remains to determine ci and TS . We recall that in the RPA

— U V (W) (441's

In order to retain the dominant pole approximation, we expand

VW about to = 0 and take only the imaginary term in the expansion .-

/ txrci-441( II) qt. I A

rr t tr 01'4 41P- (3.31)

rr (I 4-1.0“47) , (3.32)

(3.28)

61(3-* Y. 4- it4-1 k'r 1 (3.30)

li 4, 40) V64 (w)

60

Then (4.1

L.A. 4-46.0) U2X061(141-4-(1-kilAor) wt.+ Ts'

' WXott hence

1 (3.33)

It ( 414.1t9

eN7'1,7 1* expanding in powers of k.. we have

= t_At I + ( Uu

from equation (3.31) and

(3.34)

We generalise Suhl's expression (65) for Ts in terms of the

Coulomb interaction U to allow for the effects of a magnetic

field, obtaining

1124211+ 11(1 4 kB1- 01 GIP(siOrt) Grd!(1.14) (3.35) U 62- 1%. ct

e. ( I 4- 2111. I U %o) (3.36)

where X0 is the static susceptibility calculated using the

renormalised Green's functions (equation (3.25)). Again,

expanding in powers of h. and T we find

3 6! 3 le Tr el* e (3.37)

We have now obtained a set of eight self consistent

expressions for the d-electron self energies in the so called

Renormalised RPA, valid for magnetic fields and temperatures -

small compared with the energies A and kT respectively. '1e

move on to consider the conduction electrons which experience

the spin fluctuations on the impurity site when they hop in and

out of the d states. This process is depicted in figure 3.6.

Dyson's equation gives us for the conduction electron Green's

function

G (Li) vr (IZ)

- C. I vi -G074)) Gov tit4)

caot.,,r(L)

rr(43) G (

(3.33)

hence

:60) = c IV ( + Qt IcT + L 4)-1 (3.39)

where c is the concentration of impurities. Now we have written

(equation (3.25)) Vr Ctti) (s- ri L r2.) + (1.

so that

Nom) b zi4tr(iorn) = (a -c+zt ton) er ri) (3.40)

(b4-;41.1 43m) (h_T,yu

As has been argued in the previous chapter this self energy can

be inserted into the expression for the dHvA amplitude in the

same way as was done for the electron-phonon self energy.

Vkd. Vdk

ki ct. ts- r lc/a'

Figure 3.6

A representation of the scattering of the conduction

electrons by the impurity potention Vkd into and out of

the d-state represented by the double line.

2..

3. Comparison of Theory and Experiment in A1Mn

As briefly mentioned earlier, Paton (70) has

performed dHvA experiments on AlMn, in an attempt to observe

LSF effects. The measurements were carried out at temperatures

between 1.1 and 4.2°K in fields up to 60kG on alloys with

concentrations as high as 445 p.p.m. In particular, the

orbit in the third Brillouin zone was studied.

The amplitudes were analysed using the standard

Lifschitz-Kosevich formula (7) and the increase in the period,

effective mass and collision parameter were plotted as

functions of impurity concentration. All three quantities were

found to vary linearly with the Mn concentration, indicating

that no impurity-impurity interaction effects were occurring.

The variation in the period of the oscillations fits the

predictions of the rigid band theory well. Paton estimates the

effect of the resonant d-states on the Dingle temperature by

substracting the known Dingle temperature of AlZn alloys (Zn

having the same relative valence) and finds that the effect is

1.38 x 10-2 °K/p.p.m. Further, he finds that the Dingle

temperature has no temperature dependence within experimental

accuracy.

Paton then makes use of theoretical expressions

for the resistivity and specific heat enhancement to analyse

his Dingle temperature and effective mass results. Brailsford

(16) has shown, under certain circumstances discussed in

Chapter 1, that the collision parameters inferred from

_ resistivity and the dHvA effect are related by a simple constant

of order 1. By comparison with Boato et. al's (58) resistivity

result, Paton determines this constant K to be 0.86. Then,

using the zero temperature residual resistance (i.e. with no

spin fluctuation effects) calculated by Klein and Heeger (17),

Paton obtains the expression for the Dingle temperature as

Tt, = c Kott(o)

(3.41)

zh where Zh is the number of conduction electrons per atom in

aluminium, e (0) and e(o) are the density of states at the

Fermi surface for the impurity and the host respectively.

In analysing the effective mass, Paton uses

the result indicated in Chapter 2 of this thesis, that the

cyclotron mass should be enhanced by the same factor as the

specific heat. Hargatai and Corradi (64) have calculated the

specific heat enhancement for LSF and hence Paton's equation

for the change in cyclotron mass, ignoring electron-phonon

effects which should remain constant, is

---. 1.1. G eLt2) Z (3.42) NA, ( o)

By making use of the self-consistent equation for zi at zero

temperature and field (65), he obtains values for zi , 4 and

Ts which are in good agreement with other experimentally

determined values.

We now shove, under the particular experimental

conditions, that Paton's extrapolation of theoretical results

from other experiments is valid. The conduction electron self

energy obtained by us is given in equation (3.40) where zt ,

Et and 'EL are functions of field and temperature. These parameters are plotted in figures 3.7, 3.8 and 3.9 as functions

of the reduced variable (h/A ) and (T/Ts). We see that zi

rises and Ts drops in an increasing magnetic field consistent

with an increasing tendency to magnetism. At the highest

fields and temperatures in Paton's experiments (h/A) = 3.6x10-3

and(T/Ts) = 2x10-3; consequently the changes from zero

-temperature and field are completely unobservable. This disappointing result, whilst not unexpected, is not obvious

since the temperature dependence is seen in the resistivity at

low temperatures. However, it should be noticed that 14

is of the same order of magnitude as h and so the effects of

the magnetic field are significantly renormalised.

In 11111n, then, the conduction electron self

energy can be approximated as

%Tho) =[( —12. tom) 4. 4. cr t)] (3.43) At

64

Figure 3.7

Renormalisation parameter zi as a function of the'reduced

variables h/4 and T/Ts using data appropriate to AlMn,

that is U = 0.91eV, = 0.31eV.

3T5 T.

I • 1 0.02, 0.04- 0.06 0.08 0.1

65

10 -ae,v

E2. InFIO UMW .111=MMI MON. lama& •••••• mom. 0.10111r •■■•■•

— t. 0

-.1.6

Figure 3.8

111.1■1■111 411011.••■111111. 111•••■■•■•••■• ••■••••■••••11 T=0

Tr

Lo41•1•11•1111 OW" •■•••••• M=11111 .11••■ WWII& 4111MINI/

The field and temperature dependence of the parameters El

and r2. under the same conditions as Figure 3.7.

0.03 0.06 0.04. 0.02,

0.170

Ts (eV)

T=0

0.166

0.162

0.160

(11513

0.156

Figure 3.9

Field and temperature dependence of the spin fluctuation

temperature Ts under the same conditions as Figure 3.7.

67

We now make use of results of appendix B where we have formally

analysed the effects of a spin-dependent self energy on the

dHvA amplitude. For most of the experimental range

so we may make use of the simplified results (B.19) to (B.22).

4vr.lkal 2.1% ci - 4. ff kir) toc. Eh. 0'

e e,

ti44. rt.%

(3.44)

0.11111 ••■••

0

(3.45)

(3.46)

and 4 . (3.47) nitri3 (1 — MI6

Since A. Tt c (o) and (4(0) , we may write tt

= eato)

, leading to a Dingle temperature 51 7 eco

kax, c u(0) (6-10 (3.48)

et0 which should be compared with Paton's expression (3.41). We

see that under the experimental conditions, apart from an

unimportant numerical factor, his expression is in agreement

with the dHvA expression (3.48). The mass enhancement is simply

= 1 c ti(2) z1 (3.49)

et° in agreement with the predictions of Chapter 2. Further there

is an effective g-shift hypothetically observable at a spin-

splitting zero given by equation (3.47). The treatment does

not produce a term which would cause a spin-splitting zero to be

smeared into a minimum.

It has been shown that the LSF effects in A1Mn

are essentially unaffected by temperatures and magnetic fields

achieved in the dHvA experiments. It is this fact which allows

62

one to extrapolate from the leading terms of theoretical

results describing other experiments. If the fields and

temperatures were higher, the conduction electron self energy

would change as predicted in this work and the dHvA effect

would sample these changes differently from other exneriments.

At present there are no other nearly magnetic alloys with

sufficient data on which to work. For simple metal host systems,

there seems little hope of finding an alloy where

simultaneously the energies A and Ts are considerably smaller

than in AlMn and the ration U/tuS. remains less than 1. Hence

it is not considered fruitful to predict quantitative effects

beyond those drawn in figures 3.7 - 3.9.

69

Appendix A

We give here a derivation of the results (3.9) and

(3.10) for the Hartree.-Pock transverse dynamic susceptibility

in the absence of the Coulomb repulsion, defined in equation

(3.7).

We recall equation (3.7)

eX;41401)= kBT E iton 14) net) 41.1. (A.1)

where the frequencies are given by 4.)4 = 24t.ctkal. and

Wal = (2nli. 1)(0(EtT . The d-electron Green's functions are defined

as

Ga 3-(0±16) = IA) 4- GA: (.6)-% •

We again make use of the relation

F(401,4).= icia ("0-1 F() , nn KIT t

(A.2)

(A.3)

where the contour C is shown in figure 3.5. We distort the

contour C to Cr avoiding the cuts in Green's functions (A.2).

Hence we obtain, writing 5(u) as the Fermi function, oe

rolkk40.= —1- 2tr 1 °kW L 4 -

f(to-lkoto Gt(w-ion)CGalt'(a4710 -G-C(0.4031 (A.4)

Now f(4)-1010 = (1.45) , and confining the treatment to zero

temperature we have

+atiOn) vt*f tAI G,(Lia4 iikh) rG -aA'(0-16 )1 °

4-G: (to- [ GO.,+( L4416) - (SAO) (A.5)

10

We split the integrand into partial fractions and perform the

_integrals to obtain

xtt-Tio ) = J (3() 40=1.1....L.-)11{L(L4)ri4&)-k)i 'tt't Lit% ii(Ott4211) L 4- k / 2.6 iA4- L

and hence, by continuation on to the real axis,

)(0+(to+101m 1.1 I 4( (4)- ‘416 It1lto-Zt+2Zh k 6-14

and at zero frequency,

)(-0 +((;) = arctan( ty6) 1%.

being the required results.

I tjw.k4i6,11. 1,4) 14- MN, it..tui tm

(A. 6)

71.

Appendix B

In this appendix we give a formal derivation of the

effects of a spin dependent self energy on the dHvA amplitude. used

The results of this analysis are both in the present chapter and

in the next on Kondo alloys.

The amplitude of the dHvA oscillations can be written

schematically using equation (2.20),

znzcitte" .111 (tOrt 4:5(r14) A(1-0) Re. ho't f e:Qc- 2_ e, c. n=o

(B.1)

(B.2)

It is useful to write the two spin contributions as the sum of

spin independent and spin dependent parts :-

7-7- +E) e'L (4'44)

and + (4h- e) Then by simple manipulation,

I 2 -"(2./M COS+ + 2.te si4‘.43)}1 A(1-1,1) Rei e

(B.3)

(B.4)

(B.5)

The oscillation frequency is thus shifted by 6 and the modulus

of the amplitude is given by

1A041-01 to 2. ( 4%7- cos"-+ + siAi-4)112- • (B.6)

If € is zero, we recover the conventional formula (111F1

to Dingle (12), where m represents the usual amplitude factor

and cos , the spin splitting factor. However, if 6 is

non-zero, the amplitude cannot fall completely to zero and the

spin splitting zero is smeared out.

72

We now relate the parameters itn1e(to and 6 which we

have introduced, to the self energy terms. Let us split the

self energy V.(I,t)h) into the spin dependent and independent parts

in the following way;

(t44) = 51T "fir

Cr+Kr R ) kilt) I 2 (B.7)

where we will drop the explicit frequency dependence for

simplicity and the superscripts R and I denote real and

imaginary parts respectively. This expression is inserted into

equation (B.1) with the following results :- et 2.tt 640 5 a)

Re (1 = cos(Ithlt= e R COS 511 mo n=0 titk

00 .„„ (Lo_, siqtr it)E e, ttat, " Sittl. ( 411 5f) , (B.8) nco

16 INN, (le) = Cr Si% (no) to ea- t(t44.5/° COS (t c.51"

41C 641Ti cr) COS(TEM )t. s"°‘ R 21-1 V%) (B.9)

mo nto 'Mk I

If we further define

(L T [ e (E A) 4(Tr4 (Ict)

and ,stre,t r In% ti r (.1.c)

(B. 10)

(B.11)

then the parameters of equations (B.3) and (B.4) are given by

a. 4 Ct. E. ate-a 4. 2.. Z.

and 4?.= = + 44. . (B.12) 2.

?3

Hence, we see that the amplitude (B.6) is a somewhat complex

function of the various parts of the self energy and little can

'be said about it without resource to a computer to do the

summation over frequenCies.

Fortunately, most dHvA experiments are performed under

the conditions that21-tik 1. is larger than unity. (

If this is

ttOc.

the case, the summation converges rapidly and it is a good

approximation to consider only the first term, n = 0. Then we

have

Re 2rt t (t r-* cos (*Oa - ZIT 541 rk,),,OCkel6 5;.)

'40 tact

and

(B.13)

Irri (icy) siAA,(11m %ssr m0 -

leading to ■ aekal. / n 5:

41t 1/1 e toAh.

2stvr ( 2 rt 6 Q. *6 tioc. e c

roc (tticaT 5Z) (B.14)

(B.15)

(B.16)

and

4> Tr= ( — 21AaW

, ( ,p +) t

t(,), T. II

If we utilise the decomposition of the self energy defined in

equation (B.7), these results can be simplified to

2(1%kiT - /rt 5in iht = E: Sikk, e. -NA. cos k (

, . 5...-* ) tit4c, (B.19)

melb. ONO

- al" 2ti Sig sitio,‘

it ttst

icta c. (B.20)

1q.

and

" 1?--"x ma /4814

an 5I'm twc,

We note that the spin independent term of the real

part of Vr(1411) appears as the conventional Dingle parameter,

whilst its spin dependent counterpart provides the difference

in amplitudes which keeps the total amplitude from falling to

zero. From the imaginary part of , the spin independent term

is absorbed into the oscillation frequency, whilst the spin

dependent term gives rise to an effective g-shift in the spin

splitting factor.

75

CHAPTER 4

THE DHVA EFFECT IN DILUTE MAGNETIC ALLOYS

1. Introduction

In this chapter we will consider dilute alloys from a

complementary viewpoint to that used in the previous chapter;

that is, the impurity is taken to have a well defined'magnetic

moment. These magnetic alloys exhibit the low temperature

resistance minimum known as the Kondo effect and we shall use

the model employed by Rondo (74) to attempt to describe the

dHvA effect in such alloys. The remainder of the introduction

is concerned with the theory of the Kondo effect and related

phenomena, firstly when there is no magnetic field and secondly

when a field is applied. The following section describes the

present calculation of the conduction electron self energy in

the presence of a magnetic field to third order in perturbation

theory. The third section contains a discussion of the dHvA

experiments in some dilute magnetic alloys together with previous

theoretical treatments. Using the formulation presented in

chapter 2 we apply the results of our perturbation calculation to

the dHvA effect in the fourth section. In the final section

recent developments and possible extensions of the theory are

discussed.

(a) The Kondo Effect.

The s-d model was first proposed by Zener (75) for

ferromagnetic transition metals, in which it was assumed that

the d-electrons are localised on the atomic sites and the

s-electrons are itinerant. Zener then considered an exchange

interaction between the s- and d-electrons to try to explain the

metallic magnetic properties. This model is appropriate to the

case of a transition metal magnetic impurity in a simple metal

host. The Hamiltonian for the model has been derived by

Kasuya (76). It contains an exchange term which can be written

as -JSt, where J is the exchange energy and S andtare impurity and conduction electron spins respectively. J is an

approximation to the wavevector dependent function J 111 ),

which in turn is the exchange part of the matrix element of the

76

Coulomb interaction between the conduction electron and impui-ity

states.

The exchange energy J is normally taken to be a

constant parameter, but it can in fact, be related to the more

physical parameters of Anderson's model. This was done by

Schrieffer and Wolff (60) for the case U/ b, 1 by effectively

making an expansion in ( 4/u ). The relations between the magnetic behaviour in the Anderson model and in the s-d model is

an unresolved subject; however, some progress has been made using

the localised spin fluctuation concept described in the last

chapter.

The low temperature resistance phenomenon was initially

tackled by Kasuya (76) and Yosida (77), treating the s-d

Hamiltonian to second order in perturbation theory. Both

treatments give a temperature independent contribution to the

resistivity. Kondo -(74) however, was able to show that the third

order term in perturbation theory does give rise to rather a

large energy dependence in the conduction electron relaxation

time. This energy dependence is logarithmic and contributes a

term in the resistivity proportional to ln(T), which diverges as'

the temperature approaches Tx , the Kondo temperature. The

origin of this divergence lies in the non-cancellation of

intermediate state Fermi factors due to the non-cummutivity of

the spin operators. The problem is thus a true many body problem

in that an electron being scattered is sensitive to all the other

electron states through these Fermi factors. Kondo's expression

can be fitted to the resistivity of many dilute magnetic alloys

over large temperature ranges, provided that the exchange energy

J is taken to be negative; that is, for antiferromagnetic

coupling. A negative J arises naturally out of the treatment of

Anderson's model (60).

The Kondo temperature mentioned above is defined by

the equation

ka Tk = ]) suKfl :A_ ] , (4.1) "1 '

where D is a cut off energy introduced by assuming a square

density of states p and N is the number of atoms in the crystal.

77

From this equation we see that Tx can vary over a large range

of temperatures for relatively small changes in J. Further

there will always be a non-zero divergence temperature, no matter

how small J becomes. This situation is reminiscent of the BCS

treatment of superconductivity (78), in which perturbation theory.

breaks down at the transition temperature to the condensed state.

This analogy was first taken up by Nagaoka (79), who sought to

solve the problem at low temperatures by non-perturbative methods.

Nagaoka reasoned that in analogy to the superconducting condensed

state, a correlated state between the localised and conduction

electron spins might exist at low temperatures. The transition

to this state could not, however, be sharp as the system has a

limited number of degrees of freedom and consequently thermal

fluctuations would be significant. Nagaoka considered high and

low temperatures separately and solved the equations of motion of

the conduction electron Green's function. At high temperatures,

he used perturbation theory and showed that it would reproduce

Kondo's result for the relaxation time, whilst at low

temperatures he solved the equations self-consistently by making

a simple ansatz for an unknown correlation function. 'The results

for the Green's function were used to calculate the resistivity

and specific heat.

Following Nagaoka, there have been a. series of papers

(80, 81, 82, 83) which have fully exploited his methods, to give

results for the conduction electron self energy over a continuous

temperature range. The physical properties calculated in these

works are in rather good agreement with experiment except at the

lowest temperatures, (84). Other authors have used rather

different methods to remove the low temperature logarithmic

divergence. Abrikosov (56), by using Feynmann diagrams, was able

to take perturbation theory to infinite order. To overcome the

difficulty associated with the spin operators, he represented

them by quasi-fermion operators. By summing a certain class of

diagrams and taking care to avoid contributions from unphysical

states, Abrikosov obtained a result for the resistivity which was

well behaved as the temperature goes to zero. However, the

divergence at Tx was still present. Suhl (85) has pointed out

that Abrikosov's result violates the analyticity requirement on

18 •

the vertex function. Suhl himself has developed a theory (86)

based on the Chew-Low scattering theory. When the resultant

equations are solved by successive approximations, Suhl's theory

reproduces the results of second order perturbation theory,

Rondo's and Abrikosov's results in ascending order. Rondo (87)

has shown that the full result of Suhl'S theory is equivalent

to the Bloomfield-Hamann (81) solution of Nagaoka's equations.

Further, Silverstein and Duke (88) have demonstrated that all

these results are only correct to the leading logarithmic terms,

which turns out to be satisfactory at intermediate and high

temperatures. On the other hand, as the temperature goes to

zero, the neglected terms become more important and presumably

account for the discrepancies between these theories and

experiment at low temperatures. Treatments based on a

variational ground state energy - notably Appelbaum and Rondo's

(89) - appproach the problem from the other end, by attempting

to extrapolate from T = 0°K. A full discussion of these

theories and the comparison between them and the finite

temperature theories in terms of a variety of experiments is

given in Heeger's recent review article (84).

(b) The Rondo Effect in a Magnetic Field.

The first attempts to understand the behaviour of a

dilute magnetic alloy in a magnetic field were in fact concerned

with the internal field seen by one impurity due to another.

The magnitude of the resultant interaction had been observed to

be large even in relatively dilute alloys and it was concluded

that the host conduction electrons played an active role.

Consequently the response of the electron gas to an impurity via

the s-d interaction was investigated by Yosida (77) and Kasuya

(76) to first order in perturbation theory. They found that a

polarised impurity gives rise to a conduction electron spin density

cr 0-) oc, Ftlt, tr. (4.2)

where itA, and kp are the Fermi energy and momentum respectively,

<SO is the time averaged z-component of the impurity spin and

"79

the function F(x) is given by

F(x)

X cosx x (4.3) X4.

This leads to the long range interaction between impurity spins

known as the RKKY interaction after the above two authors and

Ruderman and Kittel. Behringer (90) worked out the expected line

shapes and shifts in NMR due to the RKKY oscillations and found

good agreement with experiment in CuMn.

These first order perturbation theory calculations do

not allow for spin flip scattering which arises in third order

and hence do not include the Kondo effect. Fullenbaum and Falk

(91) examined the way in which the Kondo mechanism manifests

itself in the RKKY spin polarisation. The impurity spins have

to be at least partially polarised for there to be a net

conduction electron polarisation. This can be achieved by an

internal crystalline field, by sample preparation or by the

application of an external field. The last choice is the most

controllable and provides a way of studying the effects of

freezing out the spin-flip scattering as the field is increased.

Fullenbaum and Falk extended Nagaoka's equations to include an

external magnetic field and then decoupled them to give an

expression for the conduction electron Green's function to

order J2. They then used standard formulae to compute the

susceptibility of the system and the conduction electron spin

polarisation. The modification to the RKKY polarisation can be

written

RKKY 0rz(y)2. ( [t -4- lair AiK(2.k,,r)) . (4.4)

N

As the relevant experiments measure Ti(v) over just a few lattice spacings, the correction is small and has not been

clearly observed.

The field dependence of the transport properties has

been studied extensively by Beal-Monod and Wiener (92, 93). They

have calculated the conduction electron relaxation time to third

order in perturbation theory, allowing for potential as well as

exchange scattering. The magnetoresistance, Hall coefficient •

and thermoelectric power can then be computed from the

expression for the relaxation time. Their resulting formulae

are extremely complex but can be viewed simply as the product of

80

an impurity spin polarisation function and a Kondo series of •

the scattering amplitude. The relative roles of these two

factors depend essentially on the value of the ratio,

. When G00( ko. the most rapidly varying contribution is caused by the progressive freezing out

of spin-flip scattering due to the polarisation of the impurity

spin, whilst the logarithmic series is a relatively slow

function of magnetic field. In the opposite limit, when

1410*V , the impurity spin will be saturated and the

logarithmic terms will predominate. Also at high fields the

temperature in the argument of the logarithm is replaced by the

magnetic field. 136al-Monod and Wiener's expressions have been

fitted to data on CuMn and CuFe alloys with remarkable success;

particularly since CuFe has a Kondo temperature above the

temperature at which the measurements were made. However, most

perturbation studies imply that there is a boundary in the field-

temperature plane defined by (see equation (4.1)),

ra"w kaTk (4.5) 1Z.1

consequently in large fields perturbation theory is valid even

at low temperatures. The magnetoresistance for arbitrary

temperatures and magnetic fields was calculated numerically by

More and Suhl (94) using Suhl's S-matrix theory, with

qualitative agreement with experiment. They also confirmed that,

the higher the field, the lower the temperature range where

perturbation theory remains valid. Their numerical work

represents practically the only work on the Kondo effect in a

magnetic field at low temperatures, which is an indication of

the considerable algebraic difficulties involved in the problem.

The destruction of the symmetry between up and down spins appears

to make it impossible to cast Nagaoka's equations into the

Hilbert form (95) required for a full analytic solution (81).

To describe the dHvA effect we will require the full

conduction eledtron self energy in the presence of a magnetic

field. We will not be able to consider only small fields and

low temperatures as was the case in the previous chapter, since

the characteristic energy involved ka; is of the same order

81

as kJ and /445{-1 . In order to obtain analytic results we will consider only the region where perturbation theory applies.

B4al-Monod and Wiener's work gives results only for relaxation

time, that is, the imaginary part of the self energy, and so is

not sufficiently complete for our purposes. On the other hand

Fullenbaum and Falk do give a result for the full self energy,

but only to order J2, so that it does not include the Kondo

term which appears in the next order of perturbation theory.

Accordingly we have used Nagaoka's equations in a

magnetic field to calculate the full conduction electron self

energy to third order in J. To do this we have had to make some

plausible approximations for the conduction electron spin

polarisation in some of the higher order terms. The resultant

self energy gives a relaxation time in agreement with Beal-Monod

and Wiener, so long as a slowly varying denominator in their

result is ignored. However, our result does not agree with

Fullenbaum and Falk's second order self energy; the discrepancy

being due to the way in which they decouple the equations of

motion.

2. Perturbational Calculation of the Conduction Self Energy

We will calculate the equations of motion of the

conduction electron Green's function, using the s-d Hamiltonian

with an apllied magnetic field. The chain of equations is

broken by generalising Nagaoka's truncation procedure to the

case where the spin is not restricted to be 1/2 and where the

up-down symmetry is broken by the magnetic field. The results

that are valid in the perturbation region are then obtained by

treating the averages which correlate the spin and the conduction

electron density to first order in J. The self energy is then

extracted by straightforward algebra.

The Hamiltonian of the system under consideration is

14 =1: e c4 -4) s g". ( sr CI" S 4. c c S_ to' tor KT 0 / 2N for 7. ♦ b(cr te4. 7).6) ;a-

where c to- , c are the creation and annihilation operators foi.

tr,leicr

a conduction electron with wave vector k and spin W , ("O: is

shorthand for -ar). Sz , S4 and s. are the components of the

spin operator associated with the impurity. The single particle

energies are defined by

6 .kt 7,.7%. e**36./413 1-4 (4.7)

and t is the Zeeman energy gp4.43H. We take the conduction

electron and transition metal imnurity g-factors to be equal, as

this leads to a considerable simplification in the analysis. The

retarded one electron Green's function (34) is defined by

= < Cc,f,(0 C (0)] +)

(4.8)

where el(t) is the Heaviside unit function = 1 if t ›, 0

{ = 0 if t 4 0,

and IA,B)+ indicates the anti-commutator AB + BA. A(t) denotes

the Heisenberg representation of A and <H.') a statistical'

average. The Fourier transform, written as << MB >) , is defined by

ca tl* ‘Cw.le+la = 0 e G.** (0 Kie (4.9)

-0)

As shown by Zubarev (72), the equation of motion for <Z AIB

is

ti.)<< A113??. =-1 [A)141_ 1 eb1 + 1 [A/8]4- ›) . (4.10)

Hence the equation of motion for the conduction electron Green's.

function is

G critie(L0) = Stott + etca.)>

(4.11)

We form the commutators of c kfcr with the various terms in the

Hamiltonian and arrive at the result :

(1,0-e ) (10 6kle 21 rcr (LO) ttle 4.r+ (4.12)

11°- (t4) = Cle Sv tr Cso. S47. I C.+1„4

T'' (u) + a Cr (14 ki kk •

where

II 3

The equations of motion for the Green's functions Pa (w) and

Qcrkt (t...)) are evaluated in the same way to yield

t43 €KIe "t3o)Pa',643) E f‘c s S + ,, ist ihr vr - alco Sit C k6A Lot

E etr lette s Q T R

le eve. St4 e cta))

-- 2 <<C4;ZT Clitr Cki Szl c>)1

(4.15)

and

(tA)- etc/4.) `;CL, 4)) = <sz> Sic' /21N Rea sirs icte‘/}4. tr « 1 eicAl

43*/214 F li ‘r; VC C S 1C+

e/ kiC Ka")}

(4.16)

Now, as we assumed that the g-factors are equal, eit,e-crtao = 6k,0.

and so we can add equations (4.15) and (4.16) together directly

to give-the equation of motion for r"r,(4) : Kic

(w cif) rist;(14) = 0'<<z) 7/a t4= [s (S +t) Gkai.(14) - (16

—. {<<ctrce, coe ci+,7 cet. C lcie (qv)) XN ilk,

- tr Ct evir Citr ff ex,) Cietr ewe Sa.I C 1+04

—e"rcr eve elect se l c r( cr ) (4.17)

We follow Nagaoka in replacing the higher order Green's functions

by products of thermal averages and lower order Green's functions.

84

This procedure does not have any obvious physical basis, but

the results of Nagaoka's decoupling are in good agreement with

other treatments of the s-d model, as was discussed earlier. We

consider only those averages which conserve spin to be non-zero,

with the results ;

CkerCiscr Clef Sa' IC+ko)) ~ <C+C (0) - IT ea. kle ▪ c s (464) 4, IC tr".4 ICA., (4.18a)

C C S. 1 c# !L. Toe (L4)) -<c+- Pcr (43) iCie Ke a Ai* kW ku kle , (4.18b)

e C S 1e .< S t E

C irr Of 21 kg - Ctrr Clean' ZiG I- <Cut' Cki is%) ( tattr (0)

+ <c4"--c- 2/<s;) 63) Cr - , (4.18c)

C C S IC+ <C+.. C S )G1. <c S (4)) i& leo* lc me 41. K C itt! , (4.18d)

<<c:ccileciersitele*,}):■:.-<ciccict,yP e(0.1.<cl-tr ceesa)Giimico , (4,18e)

Inserting these results in equation (4.17) we obtain

(4-eteir)riL(0 = cr<S7.)610e -3.414 [S(S4-)G171.(14) r111:1(w)]

3. 14 I I f<c-t,ce,) < Cie Tktrki (43)

+1<C-tack/iv) < 4 cier)3 ?itct, (4) + 2 cr <Cte ex, Qtbrat(ta)

+[<Cle, Cticr$4.) - cve s1.)) G:„4„(w) ( ewe so

<c; 2 cr <c+ c ) (c4 Go' (to) 1 Li? K kti

(4.19)

\•

This compares with Nagaoka's equation (2.14), which mw formed a closed set of equations for Glr (1,0 and kV 1k K140

when combined with the equation of motion for GI' (0). This kk!

is not the case here, and in order to progress without undue

complexity we must approximate, Firstly we define the quantities

AA. r <c+ c kc ki

SdLa f(w) G:f 1,60 k(r / -41

to.-`E<CkiicrCiaSil = ALA dt.3 S(413:1.k (Q) ki„b.

a0 7

Cir cr Sz) = Trs.r c(L43 .(w) kix It be

where (b3) = (e 49 • Then equation (4.19) can be written more concisely as

(4.20)

(4.21)

(4.22)

(w-Gv0)11Z(4) = o<Sz) &kw -76.14 It(21mmar. -I) 1- :(0 k

+1s(s4t) 4- 2a-q,k,/-103.-Nic -2.1twz. r< .)J Gcria(0

(ter-1).t.a.-)Cr:k,(0) 4- (44,,,-1110C(w) + Nkic-'llorr)114:4.(Q)1 A (4.23)

We approximate by dropping the final three terms in

the above expression; each of the terms represents a

contribution of order J3 or greater to the self energy and is

small since the conduction electron polarisation is small

compared with that of the impurity spin. In his high temperature

solution Nagaoka set the correlation functions pier and qier

equal to zero, but in doing so did not regain Kondo's original

result. Consequently, apart from the neglected terms, we will

treat the correlation functions to first order in perturbation

theory, thus obtaining the self energy to third order.

and

(w) .= Cr 1St) &ice °12.1.4 {(L)- €kicr) (10-e k Cr) (113 " 6 ki0 - 2 %kio- 4S7.)

+ 3- 1..cr(1,4)

Defining

s - f s(s+0 4-2 ("kJ r "KN. 1)0 Ave 4'4S2' 3

(4.24)

and

Sicia — 41' I/

k'T (4.25)

the approximated expression for re (,3) is kW

(w-ek4)C(4) = l'''0810e+'72velkertGcr,dy.3) Mkt (w) (4.26)

Equations (4.12) and (4.26) now constitute a pair of closed

simultaneous equations for the two Green's functions G`r (to) kV and CM, which can be solved by standard, but rather tedious, algebraical techniques. The results are

r 14 S e-(4)) -37z cr4s1 F(sA)) cr<Sz.) Pite.64) t * S37(0) .+3> get (w) F (z)

(4.27)

GIc(164 = 45kki 7/M4 (w4 041) (1/4)-61")(0-eK01. 3- act4

[ d or - lor e(w) I 4- Tfr(63) - fi tr4Si.) RIZ) 1 1

i # 3-2.°14,4) t "V .1(r(L4) t 5u4 0(5-(4) F (14)

(4.28)

where tta"(w) = t elkc N k 14- ekr (4.29)

(4.30) k 14- ktr.

(rad) Ike.

and

N k to- C-140. N (4.31)

We must now evaluate e4(W) and % (0) in perturbation theory

so as to obtain a result for the conduction electron self energy

to third order in the coupling constant. It will be sufficient

to calculate 5 m in zero order, giving

fr(0 v.. -1,71 E Rev') "2. (4). aka.

(4.32)

We follow the approximation made by Hamann (80) in replacing the

Fermi function in the summation by its zero temperature value,

and broadening the pole by T. The quality of this approximation

was carefully considered by Hamann who concluded that it was

satisfactory. The exact integral is not difficult but has a

complicated result involving the digamma function; it agrees

with the approximation in the two limits 4».0 to- and ta<ckaT . Performing the integral we obtain

(1(0 ti(w+crta0 .tkir AZ.1) (4.33)

To obtain cer 104 to 0(J) we require the averages nkO. pkr

, and qkr

, and by the definitions (4.20) - (4.22) we must

evaluate the three Green's functions to the same accuracy. We

find,

G (0 = Ekki aka Oil" °leo.)

-i2r1 fzr -!Sx) (0- 6k (0- Kr) (4.34)

pcs- 6.4)) = —17Ltt scs+).-s..,:t) icr <si) (E„,0 -14)1, (4)- ei,o (4)-ek,0 (4.35)

and

4 z% 6ke (w_Gkia.)

T/IN (1'1 S7,.2) ( 4)- eke) (4)-6100

(4.36)

Then, by use of■■■• equations (4.20) - (4.02),

iCektr) 472. <sz> ( tr (4.37)

Z./2.{ (S(S4 -<S2' Ir(ekt-cf.430) 2(1. Sz>f(eica.)•i i**

(4.38)

and

az.) (€1(c) 4 Z/1 vr<S2.2.) (6kci -0.42o) (4.39)

Here, the function g0 is just the real part of gr defined in 1 equation (4.33).

Using these results, we have for e(14)

atr(w) = in ?4,1 s (s+ - aytm 0±,,0[1crsz (4(61/4 0 17(6,..)

-S (ewe) Vieki.))- [us+ -q11 (gle1/4.- (NO + (Elea qt0;)] (4.40)

Now equation (4.27) can be written formally as

(U) c 64-eKo

tcr44) (4.)— 6kaf ) . (4.41)

but since this is just Dyson's equation in perturbation theory,

we have that the conduction electron self energy is

C I 4. ze* (It-VC Celt0) J am t 4. T 5'11.441)

(4.42)

SPI

with car (k)) and (pb) defined by equations (4.32) and (4.40).

The momentum independence of this result is due to approximating

the interaction by a constant J.

. If the magnetic field is set to zero, (Si) = 0 and the result reduces to

"tr. Lit4) :: -tag 4t4+,

44-41 ÷ (Ls) (4.43)

which is Kondo's original result (74) for the relaxation time.

We note that Fullenbaum and Falk's solution (91) does not have

this property.

In summary, we have derived a result in third order perturbation theory for the conduction electron self energy

within the s-d exchange model. To be able. to achieve a simple

analytic solution some approximations have been made about the

magnetic behaviour of the conduction electrons in third order

terms. The result contains the averages <St) and <S7/: which

have not been calculated; however, B4al-Monod and Wiener (92)

have shown that it is a good approximation to consider the

impurity as a free spin when T ) Tx . We will use this result

for the self energy in considering the dHvA effect in dilute

magnetic alloys.

9 0

dHvA Experiments in Dilute Magnetic Alloys

We now discuss the experimental results reported over

the last eight years on two dilute magnetic alloys, ZiaMn and

CuCr. The interpretations put on these results will be

critically discussed in the light of the present work on the

dHvA effect and the theory of the Kondo effect in a magnetic

field.

The dHvA effect in ZnMn was first studied by Hedgcock

and Muir (96) in 1963 before Kondo's theory had been formulated.

However, it was realised by then that the properties such as

the resistance minimum and giant thermoelectric power were

intimately connected with the presence of a magnetic impurity.

Moreover, it had been demonstrated that a conduction electron

relaxation time which is sharply energy dependent near the Fermi

energy can explain such phenomena (97). In the light of Dingle's

work (12), Hedgcock and Muir reasoned that this interesting

behaviour should be reflected in the amplitude of the dHvA

effect. They chose to study ZnMn since it was known to exhibit

a resistance minimum and Zn a large amplitude dHvA effect.

The experiments were performed on a third zone needle

orbit of a very small effective mass in fields up to 5kG, and

since this orbit is ellipsoidal, the amplitude was analysed

using Dingle's free electron formula for the free energy. We

can characterise the amplitude of the oscillations by the

expression' ...001 AA

1A(1.4)111 B e (4.44)

where F(H,T) contains known field and temperature dependence

terms, B and O( are constants and x is the Dingle temperature.

Then, if the logarithm of the amplitude divided by F(H,T) is

plotted against the inverse of the field, the slope gives the

temn-raturc. Since the resistivity is a strong function

of temperature in dilute magnetic alloys, it would be expected

that the Dingle temperature should show some temperature

dependence. This is indeed the case, with the Dingle temperature

rising as the temperature drops. However, equation (4.44)

91

implies that even though the slope of the logarithmic plot is

temperature dependent, the intercept at 1/H = 0 should not be

so. When Hedgcock and Muir's results are extrapolated to

infinite field, the intercept is found to vary considerably with

temperature, in conflict with simple theory.

To overcome this discrepancy they inserted a

parameterised energy dependent relaxation time into Dingle's

formulation at a convenient stage. The relaxation time was

assumed to be constant in energy except for a small region

around the Fermi energy where it was set equal to zero. The

effect of this is to add a term to the free energy which is

temperature dependent. The corrected logarithmic plot then

gives both a slope and intercept which are independent of

temperature. Hence, a phenomenonolgical explanation of the

data was provided before knowledge of Kondo's theory.

In 1968 Paton and Muir (98) attempted to reinterpret

the earlier results on ZnMn in terms of Kondo's theory. Instead

of the parameterised relaxation time used earlier, they

inserted Kondo's result into the Dingle formulation. .'They used

Kondo's result directly, that is, the zero temperature limit

which diverges logarithmically at the Fermi energy. Despite

some evident mathematical errors in their treatment Paton and

Muir obtained an expression for the dHvA amplitude similar in

form to that of Hedgcock and Muir. The correction term has

sufficient temperature dependence to account for that of the

observed Dingle temperature, when the exchange energy J is set

equal to - 0.31eV. However, this value of J is not consistent

with that inferred from the temperature independent contribution

to the Dingle temperature which gives - 0.10eV. Paton and Muir

suggest that this smaller value of J might be due to the

magnetic field inhibiting the spin-flip scattering.

This approach was criticised by Nagasawa (99) who

stated that it was more appropriate to use the high temperature

limit of Kondo's result, that is in the limit WIAkkaT , since

the dHvA effect samples states only within an energy width kBT

around the Fermi surface. In this limit the relaxation time

becomes independent of energy and depends only on temperature.

q2

This gives rise to logarithmic plots of different slopes when

the temperature is varied, but the (1/H) = 0 intercept is not

temperature dependent, in contradiction with experiment. Holt

and Myers (100) have studied ZnMn more extensively and obtained

results for the amplitude which behave in a similar way to those

of Hedgcock and Muir. It is clear from the experimental results

that Nagasawa's relaxation time is inappropriate as it cannot

explain the crossing of the Dingle temperature curves. Also the

zero temperature limit used by Paton and Muir is certainly wrong

and a complete treatment should contain the full energy

dependence of the logarithm in the relaxation time.

More recently, Paton, Hedgcock and Muir (101) have

reported a weak magnetic field dependence in the Dingle

temperature on a needle orbit in ZnMn in fields up to 50kG.

The earlier work had been confined to fields less than 3kG and

no field dependence had been observed. The amplitude curves for

different temperatures were found to actually cross over, as

shown in figure 4.1, rather than the extrapolated curves from

low fields. Neither Paton and Muir's nor Nagasawa's treatments

can account for this phenomenon. Instead of using Kondo's

expression to derive a Dingle temperature the authors have

• utilised Brailsford's result (16) that the collision parameters

measured in the dHvA effect and in the resistivity are simply

related. The work on the electron-phonon interaction in

Chapter 2 shows that this result should be used with caution.

The authors have adapted Beal-Monod and Wiener's (92)

perturbational calculation of the relaxation time in a magnetic

field. The impurity spin was considered to be free and the

logarithmic magnetic field dependence neglected. The resultant

Dingle temperature can be written

x (tvr) = x o g ( t — <Sz)2.

(4.45)

where x0 and x (T) are the potential scattering and the Kondo

scattering terms respectively, and 0., is a constant. In the

absence of field term, the expression reduces to Nagasawa's

result; but the full expression provides the required crossing

of different temperature curves. However, after intersecting,

le=

lL

at. 0.3 1/14 (KG)

93

Figure 4.1

(a) ass ciq °IA

(b) ° K

The field dependence of the dHvA amplitude on a needle

orbit in a Zn 64 p.p.m. Mn alloy, using an equation similar

to equation (4.44). The figure is from Paton, Hedgcock and

Muir (101).

94

the curves become non-linear as (1/H) approaches 0 and converge

to the same intercept.

The other Kondo alloy to be investigated by the dHvA

effect is CuCr. Following Daybell and Steyert's (102) study of

the effects of temperature and magnetic field in degrading the

quasi-bound low temperature state in CuCr, Coleridge and

Templeton (103) undertook to study the system in the dHvA effect.

They investigated the amplitude of the dHvA signal on both neck

and belly orbits in fields up to 50kG. The neck orbit showed

normal behaviour, however the belly orbit showed a strong anomaly

as a function of field and temperature. Below 2°K, the Dingle

temperature showed a sharp maximum as the field was raised.

Since the Kondo temperature of CuCr is about 1°K, Coleridge and

Templeton were led to believe that they might be seeing the

break un of the quasi--bound state. One might expect an increase

in scattering as the correlated state is destroyed, followed by

a decrease as the magnetic field supresses the spin-flip

scattering, but this process would probably be rather more

gradual than the experimental result. Further, Coleridge and

Templeton were sceptical of this explanation as the effect was

seen clearly in a 30 p.p.m. alloy but was absent in a 15 p.p.m.

alloy.

Later, the same authors (40) realised that there was

an alternative and somewhat simpler explanation of their data.

As we have shown earlier, equation (2.13), the spin splitting of

the electronic Landau levels leads to a cosine factor which

modulates the amplitude. If the effective mass ratio is such

as to make the argument of the cosine an odd multiple oftt./2,

the amplitude will dip to zero. If the spin states are further

split by a constant energy,.then this phenomenon becomes a

function of magnetic field. The exchange splitting of the

conduction electron states had been seen in this way in PdCo

alloys by H8rnfeldt, Ketterson and Windmiller (41). They found

that small amounts of Co impurity suppressed the spin-splitting

zeros, which had moved to different directions where the

effective mass ratio. was different. This phenomenon can be

regarded as a beat between signals of slightly different

frequency or equivalently as a modified spin-splitting zero.

9 5

Coleridge and Templeton were able to fit their amplitude result

with such a modified spin-splitting factor. They pointed out

that the origin of the exchange energy was just the s-d

interaction used by Kondo and to a first order approximation is

cJS, where c is the impurity concentration. Using this exchange

energy, they calculated J to be 5.5eV, which is of the same

order as that inferred from the Kondo temperature of 19K (1.8eV).

It then becomes clear why the neck orbit shows no anomaly, since

the conduction electron states are p-like and J will be

considerably reduced. Finally, Coleridge and Templeton noticed

that at 1°K the modified spin-splitting zero is well defined but

as the temperature is raised the minimum becomes progressively

shallower.

4. Application of the Theoretical Results

We now use the results of the previous section to

present a description of the dHvA effect in Rondo alloys which

incorporates and extends the earlier treatments of the different

experimental aspects. The experimental facts to be co-ordinated

include the temperature and magnetic field dependence' of the

Dingle temperature as seen in ZnMn and the modified spin splitting

zero observed in CuCr.

The formulation for the dHvA effect for interacting

electrons presented in Chapter 2 is directly applicable to the

magnetic impurity problem when it is treated in finite order

perturbation theory. This result can be seen from the appendix

of Chapter 2, where it was shown that the effects of the

orbital quantisation on the electron propagators contained in

the self energy is to generate extra oscillatory terms which are .

of order (tiagt4)1/7. smaller than the leading term. The fact that

this result is independent of the particular form of the

interaction is somewhat disguised in the equations of motion used

here, but can be seen easily in the diagrammatic methods (e.g. -

Abrikosov (56)) which represent the self energy in terms of

electron and pseudo-fermion propagators. Consequently the dHvA

amplitude is given by equation (2.16) where the sum over spins

has been retained, since the self energy is clearly spin

dependent.

TC-3:4)) 1 + Cr (it.o (1) 2-1,4 2

51.z (44) and

ci

The effects of the conduction electron self energy on

the dHvA amplitude are characterised in the same way as in the

previous chapter. We split the self energy into four terms; the

spin dependent and independent components of the real and

imaginary parts respectively. We write

r (on) crikon)

(V+0'54 ) lrit t51- 4- r 'S m (4.46)

Then we find the dominat contributions from the self energy (4.42)

to be multiplying by the number of impurities cN

516,30 = T/2_ [ vt,1? s (s-v S<S1) 57.1 (41001 + SEr(itoo aN (4.47)

I + Tfri ( ion)

(4.48)

• (4.49)

The fourth term )1 tWo is of the order (J3) and has no

logarithmic term and hence is small. The logarithm has been

split in the following way

on) I (-it0 ) + Cr ct‘r n (40 ) a 2.

+kg-rr- +1 • 0 - L./yr 4144( 140 D' w 2(t)n+ kir) ) (4.50)

We recall the result derived in Appendix n i-o

Chapter 3 that the amplitude of the dHvA signal can be written as

(141-r) r o 1+. cost el- slut +] (4.51)

9 7

In the present analysis, we will restrict ourselves to

considering only the first term in the summation over

frequencies (equation (B.1)), as this is generally a good

approximation and is necessary to be able to obtain transparent

results. The computed results presented later use the full

summation. We can now make use of equations (B.19 - B.22) and

insert the self energy terms (4.47 - 4.49) to obtain :7

t • 1"2_

Iptovoi 2. Q. "t• C"11

cos 4) + sm 4 sv4".4)

(4.52)

where the effective Dingle temperature x is given by

ilka X c. T/2- (lid) S(S+0 — 2 4S2) +40i( LOo \2.1%-#040. • tli((ttiO1)02- 4 1

A ti L (4.53)

The spin dependent term in the imaginary part of '5 contributes

a g-shift in the spin splitting factor (cosi+) given by

[l ciXI<St) 1 % 2/4814 JITir Arau0ka-rft+tov.-1

AN I (4.54) keTkr

The difference in scattering for up and down spin conduction

electrons gives rise to unequal amplitude contributions and

consequently the interference between the two signals can no

longer be complete. This is reflected mathematically by the

presence of the term in the amplitude. We find

aFC c *tit. 2-

si>

Lair pc0 487)1 + ate (kau

. (4.55)

RS

The fourth contribution from the spin independent term in the

imaginary part of , provides a negligibly small change in the

frequency of the oscillations.

We will now investigate the behaviour of the amplitude

in terms of the expressions (4.52 - 4.55), with particular

reference to the experiments on ZnMn and CuCr. The ZnMn

experiments were performed on third zone needle orbits of very

small effective mass, and so there is no possibility of a spin

splitting zero. We can set sh= 0, and by virtue of the small effective mass, lit4: 1 and cosh,' 1. 1. Hence the amplitude is

completely characterised by the Dingle temperature (4.53). We

notice that this result is of a completely different form from

the expressions given by Hedgcock and Muir (96) and Paton and

Muir (98) who obtained results with temperature independent

Dingle temperatures and additive temperature dependent

correction factors. The present simplified expression is much

closer to Nagasawa's (99), since both are effectively high

temperature approximations - if the field dependent terms in

equation (4.53) are dropped the result is essentially..the same

as that of Nagasawa. The result also confirms that the approach

used by Paton, Hedgcock and Muir (101) is valid for this

particular case. The Dingle temperature inferred by them using

the magnetoresistance is very similar to equation (4.53). As

Paton et. al. point out, the effects of the field are felt

mainly in the Brillouin function type behaviour in the second

term rather than in the logarithm. We note that, since the

effective mass is small and hence the Landau energy level

spacing large, the full summation over frequencies should be

carried out. The summation has the effect of making the Dingle

temperature x differ from the self energy evaluated at the

energy (ITV), equation (4.53). In fact x turns out to be a

weaker function of field than the self energy as is shown in

figure 4.2. The consequent amplitude plots in figure 4.3 show

qualitatively the experimental behaviour seen by Hedgcock and

Muir (96) and Paton, Hedgcock and Muir (101), shown in figure

4.1. The amplitude plots at 1°K and 4°K cross over at a field

of about 4kG. At high fields, the Dingle temperature (4.53)

becomes independent of temperature and hence the amplitude

c 9

0 10 Zo 30 40 so 14 ( KG)

9<4

2

10 2.0 ZO 60

HOAG)

Figure 4.2

Vak0-0 cc

and the Dingle temperature x obtained from a calculation of

the amplitude using data appropriate to the experiments of

Paton et. al. (101): Spl = 0.1, S = 3/2, Tk = 0.2°K and ‘IN

J = 0.3eV.

The conduction electron self energy (equation (4.53))

zoo

0.4 0.5 0.4, 1 /14 (k cc)

Figure 4.3

The logarithm of the amplitude using the Kondo self energy

plotted against the reciprocal of the field with data

appropriate for znMn. The 1°K and 4°K plots cross at a

field of about 4kG.

curves converge to the same infinite field intercept. The

constants used in calculating the amplitude in ZnMn are those

used by Paton et. al. (101).

Coleridge and Templeton (40) interpreted their

anomolous amplitudes in CuCr in terms of a spin splitting zero

modified by an exchange energy. The present analysis is in

agreement with this interpretation, but with some imnortant

changes in detail; also we provide an explanation for the

incomplete zeros observed under some conditions.

Equation (4.54) differs in detail from Coleridge and

Templeton's result by a factor of 2, 07) instead of the

saturated value S and the logarithmic term. Consequently the

exchange energy has some field and temper gE2relolHe in Coleridge

and Templeton's expression. At T = 1°K and a field of 38kG the

exchange energy is approximately 5/2 larger than that given by

Coleridge and Templeton. Using the experimental value of this

exchange energy we thus infer a value of J somewhat smaller than

5.5eV calculated by them, which is rather high compared with

resistivity. Figure 4.4 shows the amplitude close to:a spin

splitting minimum at two temperatures. We see that the minimum

moves to a lower field value when the temperature is raised in

agreement with experiment.

The depth of the minimum is determined by t which

is given by equation (4.55). Again, the most important

variation is contained in <S1) . At high fields (S2)

saturates and t is inversely proportional to field; that is,

the minimum sharpens as the field rises. The effect of raising

the temperature is to reduce ‹Sz) and increase the size of the

logarithm, thus causing the minimum to deepen. The field

property is seen experimentally, but Coleridge and Templeton

state that the effect of raising the temperature is to make-

the minimum less well defined. The present theory cannot account

for this behaviour.

We see that the theory presented in this chapter can

account systematically for the experimental results on ZnMn

and Cr and incorporates some of the earlier explanations of

1.02

20

30

40 1-4 (kG)

Figure 4.4

The dependence of the dHvA amplitude due to Kondo scattering on temperature and magnetic field. The values of the parameters Xe/N = 0.1, S = 3/2, Tk = 1oK and J = O. 63eV are realistic for CuCr.

103

special cases. It can he used to predict the dHvA behaviour in

other alloys so long as the fields and temperatures are such as

to keep the system in the regime of perturbation theory. It

will probably be more interesting, however, to try to predict

and understand the behaviour as the low temperature'spin

compensated state is approached.

5. Recent Developments

Since the work described in this chapter was

completed there have been several publications which are

directly relevant to the problem. Bloomfield, Hecht and

Sievert (104) have attempted to obtain a full solution to

Nagaoka's equations in an external magnetic field. They

decouple the equations of motion in the same way as us and then

go on to try to force the problem into a simple Hilbert form -

a technique used successfully by Bloomfield and Hamann (81) in

the absence of a field. By some lengthy analysis they convert

the problem into a set of simultaneous non-linear equations for

two "t-matrices" evaluated at a finite number of points along

the imaginary axis. The equations are solved numerically, but

take a great deal of computer time. Their inability to obtain

an analytic solution confirms our experience.

They apply the results of the computation to several

properties - the magnetisation, magnetoresistance and the

spatial dependence of the conduction electron polarisation.

The magnetoresistance results are only in qualitative agreement

with experiments on CuFe, but this discrepancy is presumably

largely due to their neglect of potential scattering. They

find that the reduction of the local moment at temperatures

below the Kondo temperature is due to a strong spin correlation

between conduction and impurity electrons rather than due to a

compensating electronic cloud. Also they conclude that the

conduction electron contribution to the excess susceptibility

is very small. These results are in conflict with the work of

Heeger (84) based on the variational method of Appelbaum and

Kondo (89). However, some of the most recent experimental work

on Fe lends support to Bloomfield et. al. (105, 106).

104-

A particularly interesting consequence of this work

in the'present context is the energy dependenc'e of the

imaginary part of the conduction electron t-matrix for T<K Tk.

The logarithmic dip in the t-matrix at the Fermi energy in zero

field becomes shifted and fills up as the field is increased.

This behaviour is just what is to be expected from a simple

treatment where the field is introduced into the logarithm, as

in equation (4.33). Besides this shift, Bloomfield et. al.'s

result has some relatively unimportant structure close to the

Fermi energy. This encourages one to believe that a

perturbational treatment could be useful at low temperatures,

particularly if a more realistic behaviour for <St) than the

Brillouin function is used.

A theoretical attempt to explain the dHvA data in

CuCr and ZnMn has been presented by Miwa, Ando and Shiba (107).

This work is very similar in approach to ours and was done

concurrently. They calculate the conduction electron self

energy in a magnetic field within perturbation theory and apply

this to the dHvA effect by means of an identical equation to

our equation (2.20). In doing so, they treat the effects of

the real and imaginary parts separately to deal with the

different aspects of the problem. The Dingle temperature that

they obtain contains some magnetic field dependence through the

logarithmic factor, but does not have the more important

dependence through the impurity spin, seen in our equation (4.53).

It is not Clear from their paper whether Miwa et. al. find that

their Dingle temperature accounts for the crossing of the

logarithmic amplitude plots.

They also give a result for the shift in the spin

splitting zero identical to ours and remark that the position

of the zero is enhanced over the value without the Kondo effect.

However, by treating the real and imaginary parts of the self

energy separately, they do not consider the effects of the spin

dependent scattering on the depth of the spin splitting zero.

Fenton (108) has criticised Miwa et. al., and by

implication the present work, on the enhancement of the position

of the spin splitting zero. He claims that the Kondo effect

105

must work to lower the field value of the zero and that this

effect would anyway be very small at the temperatures considered

in CuCr. We cannot agree with this criticism since firstly it

is well known that the inclusion of successively higher orders

of perturbation theory always reduces the value of J required

to fit experimental data, which is the result found by both

Miwa et. al. and ourselves. Secondly, we find that the

logarithmic term, whilst slowly varying, does provide an

appreciable enhancement under the experimental conditions in

CuCr.

Templeton, Coleridge and Scott (109) have reported

preliminary measurements of spin splitting minima in other copper

based alloys; CuFe where the Kondo temperature is well above the

experimental range, and CuMn where Tx is very low. In both

cases, there are significant shifts but the minima are rather

shallow. Until more detailed experiments are performed there

does not seem any chance of interpreting these results

successfully.

As we have said in the previous section the present

description has been shown to be adequate to describe the dHvA

experiments reported so far. Further, the work of Bloomfield

et.al. (104) has given some theoretical backing to the

approximations made in the analysis. There would appear to be

two important lines on which the theory could be developed.

Firstly, either by using the present treatment or by using a

more rigorous approach to the low temperature properties, an

attempt could be made to describe the effects of the break up of

the spin compensated state with increase in field or temperature'

on dHvA amplitude, particularly on the position and depth of the

spin splitting minimum. Secondly, one might try to account for

the variation of the dHvA results over different parts of the

Fermi surface using a Lifschitz and Kosevich type expression for

the amplitude and a model for the wave vector dependence of the

exchange energy J.

9.06

CONCLUSION

The information on the geometrical features of Fermi

surfaces obtained from the measurements of the period of the

oscillations has been interpreted on a sound theoretical basis

since the work of Onsager (6) and Lifschitz and Kosevich (7).

On the other hand, there has been little theoretical work on the

amplitude of the dHvA effect apart from Dingle's

phenomenological theory (12) of the effects of electron

scattering. In this thesis we have followed Luttinger's

approach (21) to the thermodynamic potential and applied it to

the case of electron-phonon interactions. In doing so we have

corroborated Fowler and Prange's result (31) and established a

general formulation which is valid under certain well defined

conditions. We find that, within perturbation theory (infinite

order if necessary) the free electron energy can be replaced

by the free electron energy plus the full self energy within

an integral, so long as the oscillatory part of the self energy

can be neglected. This term will always be small if the

cyclotron energy is small compared with the Fermi energy. The

area where these conditions do not hold has not been

investigated. It would be interesting to predict what happens

at very high fields or more practically, what happens when

perturbation theory is invalid such as the low temperature

bound state found in the Kondo effect.

Further we are able to see under what conditions the

phenomenological approach holds and the measured Dingle

temperature can be equated to the relaxation time of the

conduction electrons. These are found to hold if the

relaxation time is energy independent or if it is a smooth

function of energy and the temperature is low enough to keep

excitations close to the Fermi surface. When these conditions

do not apply the formulation shows how the dHvA effect samples

the excitations which will necessarily be different from other

experimental probes. In most cases the magnetic field does not

directly affect the excitation but just defines the temperature

range and hence the energies of the electrons over which the

signal can be detected.

107

In the case of electron-phonon interactions it was

found experimentally (33) that the cyclotron mass in mercury

showed no temperature dependence over a wide temperature range

despite a low energy phonon mode and strong coupling. This is

contrary to cyclotron resonance experiments (30) in the same

temperature range. The present formulation shows that this

result is consistent with the standard electron-phonon. theory

and is due to a cancellation of the temperature effects in the

real and imaginary parts of the self energy. One important

consequence of this result is that the Dingle temperature is.

not a true measure of the scattering rate in the case of the

electron-phonon interaction.

The dHvA effect has also been shown to be affected by

spin fluctuations and to a first approximation, the cyclotron

mass is enhanced in the same way as the specific heat and the

argument of the spin splitting factor is enhanced by the same

amount as the spin susceptibility. The detailed behaviour is

analysed for localised spin fluctuations in nearly magnetic

dilute alloys within the RPA. A treatment of the effects of

temperature and magnetic fields on band spin fluctuations might

stimulate dHvA experiments on a series of nearly magnetic

alloys such as Ni Rh or Pd - Ni.

In the case of Kondo alloys, the magnetic field plays

a direct role in altering the nature of the problem and the

dHvA effect can be a most useful tool in investigating this

behaviour. In particular the observed shifts in the spin

splitting zeroes in copper base alloys (109) would seem to hold

a great deal of information. kparticularly interesting

experiment and tough theoretical problem is the study of the

magnetic field induced break up of the low temperature condensed

state; in this case the present formulation might be suspect.

Perhaps the most important recent development in the

study of electron scattering on the Fermi surface is the

inversion of dHvA Dingle temperature data to provide maps.of the

conduction electron relaxation time in momentum space. This

development should stimulate theoretical calculations of the

log

anisotropic behaviour of the self energy-for different

scattering mechanisms. The most obvious case for study, as

there is not plenty of experimental data, would be the Kondo

scattering in copper based alloys.

109

REFERENCES

1. W.J. de Haas & P.M. van Alphen : Leiden Comm. 203d, 212a (1930).

2. R. Peierls : Z. Phys., 81, (1933).

3. L.D. Landau : Appendix to D. Shoenberg, Proc. Roy. Soc., A170, 341, (1939).

4. D. Shoenberg : Proc. Roy. Soc., A170, 341, (1939).

5. D. Shoenberg : Phil. Trans. Roy. Soc., A245, 1, (1952).

6. L. Onsager : Phil. Mag., 43, 1006, (1952).

7. I.M. Lifschitz & A.M. Kosevich Sov. Phys. J.E.T.P., 2, 636, (1955).

8. E.M.Gunnerson : Phil. Trans. Roy. Soc., A240, 340, (1957).

9. D. Shoenberg : in "Prog. in Low Temp. Phys. II", ed. C.J. Gorter (North Holland, Amsterdam), (1957).

10. D. Shoenberg : Phil. Trans. Roy. Soc., A255, 85, (1962).

11. A.P. Cracknell : Adv. Phys., 18, 681 (1969) and 20, 1, (1971).

12. R.B. Dingle : Proc. Roy. Soc., A211, 517, (1952).

13. M. Springford Adv. Phys., 20, 493 (1971).

14. S.J. Williamson, S. Foner & R.A. Smith : Phys. Rev., 136, A1065, (1964).

15. Y.A. Bychkov : Sov. Phys. J.E.T.P., 12, 977, (1961).

16. A.D. Brailsford : Phys. Rev., 149, 456, (1966).

17. E. Mann : Phys. Kondens. Materie, 12, 210, (1971).

18. N.F. Mott & H. Jones : "Theory of Metals and Alloys" (Clarendon Press, Oxford, 1936), p. 262.

7

19. J.M. Luttinger & J.C. Ward : Phys. Rev., 118, 1417, (1960).

20. J.M. Luttinger : Phys. Rev., 119, 1153, (1960).

21. J.M. Luttinger : Phys. Rev., 121, 1251, (1960).

22. S. Nakajima & M. Watabe : Prog. Theor. Phys., 29, 341, (1963) and 30, 772, (1963).

23. H. Scher & T. Holstein : Phys. Rev., 148, 598, (1966).

24. P. Goy & G. Weisbuch : Phys. Rev. Letters, 25, 225, (1970).

25. G. Grimyall : Phys. Kondens. Materie, 9, 283, (1969).

26. W. McMillan & J. Rowell : in "Superconductivity" ed. R.D. Parks, (Marcel Dekker, 1969).

27. J.J. Sabo, Jr. Phys. Rev., Bl, 1325, (1970).

28. P. Goy Physics Letters, 31A, 584, (1970).

29. R.G. Poulsen & W.R. Datars : Solid State Comm., 8, 1969, (1970).

30. D. Chen & J.J. Sabo, Jr. : (unpublished).

31. M. Fowler & R.E. Prange : Physics, 1, 315, (1965).

32. J.W. Wilkins & J. Woo : Physics Letters, 17, 89, (1965).

33. C.J. Palin : PhD Dissertation, Cambridge University, 1970 (unpublished).

34. A.A. Abrikosov, L. Gor'kov & 1. Dzyaloshinskii : "Quantum Field Theoretical Methods in Statistical Physics" (Pergammon Press, Oxford, 1965).

35. G.M. Eliashberg : Sov. Phys. J.E.T.P., 16, 780, (1963).

36. A.B. Migdal : Soy. Phys. J.E.T.P., 7, 996, (1958).

111

37. R.E. Peierls : "Quantum Theory of Solids" (Oxford University Press, London, 1955).

38. • E. Titchmarsh : "Introduction to the theory of Fourier Integrals" (Oxford University Press, London, 1948).

39. D.L. Randles : PhD Dissertation, Cambridge University 1971, (unpublished).

40. P.T. Coleridge & I.M. Templeton : Phys. Rev. Letters, 24, 108, (1970).

41. S. Hornfeldt, J.B. Ketterson & L.R. Windmiller : Phys. Rev. Letters, 23, 1292, (1969).

42. J. Bardeen & D. Pines : Phys. Rev., 99, 1140, (1955).

43. J.J. Quinn : in "The Fermi Surface" eds. W.A. Harrison & N.B. Webb, (J. Wylie, New York, 1960).

44. S. Engelsberg & J.R. Schrieffer : Phys. Rev., 131, 993, (1963).

45. D.J. Scalapino, J.R. Schrieffer & J.W. Wilkins : Phys. Rev., 148, 263, (1966).

46. W.L. McMillan & J.M. Rowell : Phys. Rev. Letters, 14, 108, (1965).

47. S. Doniach & S. Engelsberg : Phys. Rev. Letters, 17, 750, (1966).

48. N. Berk & J.R. Schrieffer : Phys. Rev. Letters, 17, 433, (1966).

49. J. Hubbard : Proc. Roy. Soc., 276A, 238, (1963).

50. R.R. Oder : Bull. Am. Phys. Soc., 14, 321, (1969).

51. J. Friedel : Can. J. Phys., 34, 1190, (1956).

52. P.W. Anderson : Phys. Rev., 124, 41, (1961).

53. A.D. Caplin, C.L. Foiles & J. Penfold : J. Appl. Phys., 39, 842, (1968).

54. N. Rivier, M. Sunjic & M.J. Zuckermann : Phys. Letters, 28A, 492, (1969).

112.

55. N. Rivier & M.J. Zuckermann : Phys. Rev. Letters, 21, 904, (1968).

56. A.A. Abrikosov Physics, 2, 5, (1965).

57. J.R. Schrieffer : J. Appl. Phys., 38, 1143, (1967).

58. A.D. Caplin & C. Rizzuto : Phys. Rev. Letters, 21, 746, (1968).

59. D.R. Hamann : Phys. Rev., 186, 549, (1969).

60. J.R. Schrieffer & P.A. Wolff : Phys. Rev., 149, (1966).

61. H.P. Meyers, L. Wallden & A. Karlson : Phil. Mag., 18, 725, (1968).

62. J.R. Schrieffer & D.C. Mattis : Phys. Rev., 140, A1412, (1965).

63. R. Aoki & T. Ohtsuka : J. Phys. Soc. Japan, 26, 651, (1969).

64. C. Hargitai & G. Corradi : Solid State Comm., 7, 1535, (1969).

65. B.E. Paton & M.J. Zuckermann : J. Phys. F., 1, 125, (1971).

66. R. Tournier & A. Blandin : Phys. Rev. Letters, 24, 397, (1970).

67. L. Dworin : Phys. Rev. Letters, 26, 1372, (1971).

68. A.D. Caplin : Phys. Letters, 26A, 46, (1967).

69. M.B. Maple, J.G. Huber, B.R. Coles & A.C. Lawson : J. Low Temp. Phys., 3, 137, (1970).

70. B.E. Paton : Can. J. Phys., 49, 1813, (1971).

71. A.P. Klein & A.J. Heeger : Phys. Rev., 144, 458, (1966).

71 14. D.N. Zubarev

Soy. Phys. USP, 3, 320, (1960).

73. M. Levine, T.V. Ramakrishnan & R.A. Wiener : Phys. Rev. Letters, 20, 1370, (1968).

113

74. J. Kondo : Frog. Theo. Phys., 32, 37, (1964).

75. C. Zener : Phys. Rev., 81, 440, (1951).

76. T. Kasuya : • Prog. Theo. Phys., 16, 45, (1956).

77. K. Yosida : Phys. Rev., 107, 396, (1957).

78. J. Bardeen, L.N. Cooper & J.R. Schrieffer : Phys. Rev., 108, 1175, (1957).

79. Y. Nagaoka : Phys. Rev., 138, A1112, (1965).

80. D.R. Hamann : Phys. Rev., 158, 570, (1967).

81. P.E. Bloomfield & D.R. Hamann : Phys. Rev., 164, 856, (1967).

82. D.S. Falk & M. Fowler : Phys. Rev., 158, 567, (1967).

83. J. Zittartz & E. Muller-Hartmann : Z. Physik, 212, 380, (1968).

84. A.J. Heeger : in "Solid State Physics" (F. Seitz, D. Turnbull and H. Ehrenreich, eds.), Academic Press, New York (1969).

85. H. Suhl : Phys. Rev., 138, A515, (1965).

86. H. Suhl : Physics, 2, 39, (1965).

87. J. Kondo : in "Solid State Physics " (F. Seitz, D. Turnbull and H. Ehrenreich, eds.), Academic Press, New York (1969).

88. S.D. Silverstein & C.B. Duke : Phys. Rev., 161, 456, (1967).

89. J. Appelbaum & J. Kondo : Phys. Rev. Letters, 19, 906, (1967).

90. R. Behringer : J. Phys. Chem. Solids, 2, 209, (1957).

91. M.S. Fullenbaum & D.S. Falk : Phys. Rev., 157, 452, (1967).

92. M.T. Weal-Monod & R.A. Wiener : Phys. Rev., 170, 552, (1968).

114

93. R.A. Wiener & M.T. Beal-Monod : Phys. Rev., B2, 2675, (1970).

94. R. More & H. Suhl : Phys. Rev. Letters, 20, 500, (1968).

95. G. Simpson (Unpublished 1971).

96. F.T. Hedgcock & W.B. Muir : Phys. Rev., 129, 2045, (1963).

97. C.A. Dominicali : Phys. Rev., 117, 984, (1960).

98. B.E. Paton & W.B. Muir : Phys. Rev. Letters, 20, 732, (1968).

99. H. Nagasawa : Solid State Commun., 7, 259, (1969).

100. G. Holt & A. Myers : Physik Kondens. Materie, 9, 23, (1969).

101. B.E. Paton, F.T. Hedgcock & W.B. Muit : Phys. Rev., B2, 4549, (1970).

102. M.D. Daybell & W.A. Steyert : Phys. Rev. Letters, 20, 195, (1968).

103. P.T. Coleridge & I.M. Templeton : Phys. Letters, 27A, 344, (1968).

104. P.E. Bloomfield, R. Hecht & P.R. Sievert : Phys. Rev., B2, 3714, (1970).

105. A. Narath, K. Brog's, W. Jones Jnr. : (Unpublished 1970).

106. C. Stassis & C.G. Shull : 15th Annual Conference on Magnetism and Magnetic Materials, J. Appl. Phys., 41, 1146, (1969).

107. H. Miwa, T. Ando & H. Shiba : Solid State Commun., 8, 2161, (1970).

108. E.W. Fenton,: Solid State Commun. 10, 185, (1972).

109. I.M. Templeton, P.T. Coleridge & G.B. Scott : Bull. Am. Phys. Soc., 15, 800, (1970).