many .,,body effects metals by graham simpson a thesis ... · 2. localised spin fluctuations at...
TRANSCRIPT
/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
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