THE
BEHAVIOR OF
THIN-FILM SUPERCONDUCTING-PROXIMITY-EFFECT
SANDWICHES
IN HIGH MAGNETIC
FIELDS
by
J.
WILLIAM
GALLAGHER
University
Creighton
B.S.,
1974)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
August,
Q
Signature
1978
Massachusetts Institute of Technology
of
1979
Author
Department of /PhysicFs,
August
11,
1978
Certified by
TheIis
Accepted
Supervisor
by
rhairman, Departmental Committee
on Graduate Students
2
1973
NOV 2
1973
OF
LIBRARIES
2
THE
BEHAVIOR OF THIN-FILM SUPERCONDUCTING-PROXIMITY-EFFECT
SANDWICHES IN HIGH MAGNETIC FIELDS
by
GALLAGHER
WILLIAM J.
on August
Submitted to the Department of Physics
11, 1978, in partial fulfillment of the requirements
for
Degree
the
of Doctor
of Philosophy
ABSTRACT
study the behavior
tunneling Hamiltonian model to
We use a
two-metal
proximity-effect
superconducting
thin
of
the
develop
We
fields.
sandwiches in high parallel magnetic
formalism
in
manner
a
valid
for
all
temperatures
and
field
the
zero-temperature
at
explicitly
calculate
the
potential,
pair
the
states,
of
dependence of the density
free
energy
magnetization,
the
density,
and
the
The
the sandwich.
of
sides
on both
their
in
splitting
field-induced
a
spin-susceptibility
display
sandwiches
The
of states.
spin-densities
metal
superconducting
stronger
effect from
proximity
sandwich keeps
in the
the
the
spin-split
its
after
even
superconducting
metal
weaker
The
level.
Fermi
the
crossed
have
of states
densities
by
accompanied
is
states
of
spin-densities
crossing of the
pair
the
of
degradation
field-dependent
a
of
onset
the
magnetic
nonzero
a
of
onset
the
by
and
potentials
superconducting
the
values,
field
some
At
susceptibility.
may even exceed the
state susceptibility in some sandwiches
Pauli
susceptibility
state.
normal
the
of
Preliminary
indication of the crossing
tunneling experiments do give an
We
Fermi level.
at the
states
of
the spin-densities
of
comment on these experiments and indicate several directions
optimize the experimental
pursued in order to
which can be
In a final chapter we show how to properly
characteristics.
estimate
the
spin-orbit
scattering
times
in
thin
reasonable agreement between our
superconductors and find a
values.
experimental
the
and
estimates
Thesis supervisor:
Brian B. Schwartz
Head, Theory Group,
MIT Francis
Bitter
National
Magnet Laboratory
Dean,
School
of Science
Brooklyn College of CUNY
3
ACKNOWLEDGEMENTS
I
B.
am
pleased to
for giving me
Schwartz
aspects
have the
an
of superconductivity,
I am
for
me
for
for
many
suggesting the
Brian
subtle
problem
patiently supervising the
his openness with
grateful for
also especially
thank Dr.
appreciation
described in this thesis, and
work.
to
opportunity
and for his interests and abilitiy in aiding my education
in matters beyond those related to this thesis.
many aspects
I
of his
influence will
from
benefitted
remain with me.
interesting
physicists
conversations with
particularly Drs.
many
I hope that
and
at the Magnet
instructive
including
Lab,
Sonia Frota-Pessoa,
Robert Meservey, Paul
Tedrow, and
Demetris Paraskevopoulos.
For other useful and
interesting
discussions I am particularly grateful to two of
and Ronald
Andre Tremblay
student colleagues,
my graduate
Pannatoni.
For instruction
programs
Ann Carol
the
IBM
Thomas
J.
of MIT
Watson
Research
for noting
in
using text
Carol H.
Center
Paul Wang of
this
various
editing
document, I thank Mrs.
Mrs.
I also thank
the figures
with
this
Hohl and particularly
some of
Bostock
and help
for the preparation of
Heights, New York.
with
in
thesis
Thompson of
in
Yorktown
MIT for help
and Dr.
Judith
gramatical errors
and
4
For aid
thesis.
statements in a draft version of this
vague
assistance of Don
in using MIT's computers I acknowledge the
Nelson of MIT's Magnet Lab.
the
support of
the financial
gratefully acknowledge
I
National Science Foundation through a Predoctoral Fellowship
of my graduate studies and through
for the first three years
the Core Grant to the Francis
my
am grateful
I also
final year.
Bitter National Magnet Lab for
Center for the preparation of this
In
back
looking
culminating with
the encouragement
Dr.
J.
Sam
Nebraska, and
School
also
this thesis,
of two of
Cipolla
of
I
Omaha.
Wang who,
in a few days,
are
physics instructors,
in
Omaha,
Creighton Preparatory
am
Finally I
that
to acknowledge
University
Creighton
the encouragement and
study
of
my early
continual encouragement of my parents,
months, for
Research
Watson
would like
Fr. Willard Dressel of
in
text
its
document.
years
over the
International
use of
J.
Thomas
the
at
facilities
processing
the
for
Machines Corportation
Business
to the
grateful
for
and, over the last
support of
will be my wife.
the
13
Martha Liwen
5
CONTENTS
Abstract
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2
Acknowledgements
. . . . . . . . . . . . . . . . . . . . . 3
List of figures.
. . . . . . . . . . . . . . . . . . . . . 9
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.12
Introduction .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.13
.
.13
Concepts in the Theory of Superconductivity.
.21
List of tables
Chapter I:
.
.
.
.
.
An Overview of Superconductivity and the Magnetic
A.
Properties of Superconductors.
.
.
.
.
.
.
.
.
.
B.
Basic
C.
Magnetic Properties of Thin Film Superconductors in
Parallel Magnetic Fields
The Proximity Effect
D.
Chapter
II:
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.30
.
.
.
.
.
.
.
.
.
.
.
.
.
.49
for the Proximity Effect
.54
Theoretical Models
.
A.
Introduction
B.
Theories Based
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.54
on the Gor'kov and the
Ginzburg-Landau Equations.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.57
.
.
.
.
.
.
.
.
.
.
.
.
.63
C.
The Cooper Limit
D.
Bogoliubov-de Gennes Equation Approach to the
Proximity Effect
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.66
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.66
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.67
1.
Introduction
2.
Review
.
.
.
6
McMillan Tunneling Model of Thin Proximity-Effect
E.
.
Sandwiches
. . . .
. .
.
. .
.
. .
. .
Tunneling Model of Paramagnetically Limited
Chapter III:
Formalism
Proximity-Effect Sandwiches:
.
.
.
.
.
. . . . . . . . . . . . .. . .
A.
Introduction
B.
The
C.
Green's
D.
Iteration of the Green's Function Equations.
E.
McMillan Solution to the
Hamiltonian.
.
. .80
.
.
. .80
.82
.87
Motion.
Tunneling Model for
.92
.
.
.
the
-98
. .
. . . . . . . . . . . . . ..
Comparison to the Diagrammatic Expansion of the
Function
Green's
2.
of
their Equations
and
.
-. .
.
. . . . . . . . . . . ..
Functions
Proximity-Effect
1.
.74
.
. ..
.
.
. .
.
. .
. .
.
. .
..
..
Equations for the Renormalization Functions.
Chapter IV:
-98
.
.
.
101
the
Tunneling Model Predictions for
Properties of Paramagnetically Limited Proximity-Effect
Sandwiches.
Introduction:
A.
.
.
.
.
. 107
Calculational Procedures.
.
.
.
.
.
107
.
.
.
.
109
.
-
111
.
.
120
. .
. . . . . . . . . . . . . ..
Renormalization Functions.
.
.
.
.
.
.
.
.
.
.
..
1.
The
2.
The Density
3.
The Magnetization, Susceptibility, and Free
Energy
4.
.
.
of States.
.
.
.
.
.
.
.
.
.
.
.
.
.
Finite Temperature Calculations
.
.
. .
.
.
.
.
.
.
7
B.
A Normal-Superconducting Sandwich.
C.
A Two-Superconductor
Chapter
V:
Sandwiches
.
.
Sandwich......
.
.
.
.
.
124
.
.
.
.
.
14 1
on Thin Proximity-Effect
Experiments
in High Magnetic
Fields.
.
.
.
.
.
.
157
.
.
.
157
.
.
.
159
A.
Introduction
.
.
.
. .
.
.
-
-
-
- .
B.
Tunneling Experiments.
.
.
.
.
.
.
.
.
.
.
Tunneling.
.
.
.
. . . . . . . . . . . 159
Measurements
.
.
. . . . . . . . . . . 165
.
.
Theory
2.
Tunneling
3.
Suggested Future Tunneling Work.
.
. . . . . . . . 176
Measurements of the Magnetic Susceptibility.
Summary
Chapter VI:
.
.
.
.
.
.
.
.
.
.
.
.
--..
.
.
A.
Introduction
B.
Background
.
.
.
.
.
.
.
Superconductors.
Spin-Orbit
.
.
.
.
.
.
.
.
181
183
.. .....
.
.
.
. .
. . . . . . . . ..
on the
.
in
Spin-Orbit Scattering Times
Superconductors
.
. .
.
-
185
.
185
Interaction in
.
.
.
.
.
.
.
. -- 188
.
Matrix Elements from Spin-Orbit and Regular
C.
Impurity
D.
.
1.
C.
D.
of
.
.
.
.
.
.
.
.
195
Scattering Hamiltonian.
.
.
.
.
.
.
195
.
.
.
198
. -
-
- 203
Scattering.
Impurity
.
.
.
.
.
.
.
1.
The
2.
Estimation of Scattering Matrix
Ion Core Screening
Delocalization
and Metallic
.
Elements
Electron
. . . . . . . . . . . .. .
8
E.
Comparison of Estimated Spin-Orbit Scattering
Times
1.
to
Experiment.
Comparison of the
. .
. .
. .
. .
.
. .
Ratio to
.
. 207
the
of the Scattering Potential.
207
Estimating the Contribution of Surface
Scatterers
3.
.
Scattering Time
Ratio of the Square
2.
.
. . . . . . . . . . . . . . . . . . . 211
Conclusion and Suggested Further Experimental
Work
. . . . . . . . . . . . . . . . . . . . . . 214
Chapter VII:
Summary and Conclusion . . . . . . . . . . 217
Some Results from the Bogoliubov-de
Appendix A:
Gennes-Equation Approach
to
the
Proximity
Effect.
.
.
220
Calculation of the Sandwich Superconducting
Appendix B:
Transition Temperature as a Function of Field
Biographical Note.
References
.
.
. .
.
.
.
.
.
229
. . . . . . . . . . . . . . . . . . . 233
.
.
.
.
.
.
.
.
.
.
.
235
9
List of Figures
I.1
BCS density of states
I.2
Film
.
.
.
.
.
. .
.
.
.
.
.
.25
and sandwich geometry.
.
.
.
.
.
.
.
.
.
.
.
.32
I.3
Meissner diamagnetic effect
.
.
.
.
. .
.
.
.
.
.
.33
I.4
Partial Meissner effect in thin films
.
.
.
.
.
.
.34
1.5
Spin-split
.
.
.
.
.
.
.37
I.6
Pauli Paramagnetic limit in thin films .
.
.
.
.
.
.39
1.7
Pair potential as a function of field
= 0.
.
. 40
1.8
Free energy
in
the paramagnetic
limit
at T
= 0.
.
.42
1.9
Free energy
in
the paramagnetic
limit
for T 1
0
.
.44
1.10
Paramagnetic limit phase diagram.
I.11
Metamagnet T-H-Hst
density
of
.
.
states.
.
.
.
.
T
at
.
.
.
.
.
.
.45
.
.
.
.
.
.
.47
II.1
Density of states from the tunneling model.
.
.
.
.78
IV.1
Free energy, Magnetization, and Susceptibility
phase
diagram.
of an isolated film at T =0
IV.2
Density of states for
Density
of
states for
.
.
.
.
.
.
.
.
.
.
.
.
116
.
.
.
.
.
.
.
.
.
.
127
.
.
.
.
.
. .
.
128
Pair potential as a function of field for a
strongly coupled sandwich at T
IV.5
.
a strongly coupled
sandwich at T = 0 .
IV.4
.
a strongly coupled
sandwich at T = 0 .
IV.3
.
.
= 0.
.
.
130
.
Total density of states for the 'n-side
of a strongly coupled sandwich at T = 0
.
.
132
10
IV.6
Total
the s-side
density of states for
of a strongly coupled sandwich at T = 0
IV.7
. . . . . . . . . . . . . . . . . . 135
.
.
.
.
.
.
.
.
.
.
IV.14
.
.
.
.
.
.
.
.
.
.
137
.
.
.
.
139
.
.
.
145
.
.
.
.
.
.
146
Pair potential as a function of field for
a
weakly coupled sandwich at T = 0.
.
.
.
.
148
Density of states for
.
.
.
.
.
.
.
a weakly coupled
.
.
.
.
.
.
.
.
.
.
Magnetization of a weakly coupled sandwich
.
.
.
.
.
Total density of states
.
.
.
..
149
. .
. . .
. . .
for the n-side
.
.
.
.
150
.
.
.
.
151
.
.
.
153
Total density of states for the s-side
Susceptibility of a weakly coupled sandwich
at T = 0.
IV.18
.
.
.
of a weakly coupled sandwich at T = 0
IV.17
.
. . . . . . . . . . . . . . 143
of a weakly coupled sandwich at T = 0
IV.16
.
.
at T = 0.
IV.15
.
.
sandwich at T = 0
IV.13
.
.
Density of states for a weakly coupled
sandwich at T = 0
IV.12
.
.
Density of states for a weakly coupled
sandwich at T = 0
IV.11
.
.
.
.
Free energy of a strongly coupled sandwich
at T = 0.
IV.10
133
Susceptibility of a strongly coupled sandwich
at T = 0.
IV.9
.
Magnetization of a strongly coupled sandwich
at T = 0.
IV.8
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Free energy of a weakly coupled sandwich
at T = 0.
.
.
.
.
.
.
.
.. .
-.
. . . . . . 154
11
V.1
Thermally smeared n-side density of states
for a strongly coupled sandwich
V.2
.
.
.
.
163
sandwich
.
.
.
.
.
.
.
164
. . . . . . . . . . . . . . . . 171
Conductance for tunneling into 37 A Al backed
by
VI.1
.
Conductance for tunneling into 25 A Mg backed
by 40 A of Al
V.4
.
Thermally smeared s-side density of states
for a strongly coupled
V.3
.
15 1 of Cu
. . . . . . . . . . . . . . . . 175
Scattering time ratio
atomic number
as a function of
. . . . . . . . . . . . . . . . 208
12
List of Tables
.
.
.
.
.
.
* .93
IV.1
Proximity effect sandwich parameters.
.
.
.
.
.
* 123
VI.1
Parameters
data.
.
.
.
.
.
.
205
VI.2
Measured
.
. .
213
III.1
Green's
function dictionary
times
.
.
.
.
.
.
.
spin-orbit scattering
predicted
.
.
spectral
from atomic
and
.
.
.
.
.
.
.
.
.
.
.
.
13
INTRODUCTION
CHAPTER I:
A.
AN
OVERVIEW
PROPERTIES OF
OF
the potential
and
magnetic fields
magnetic properties
of
Kamerlingh
interest
Onnes's
and
Meissner
destroy
and
the
perfect diamagnets,
there
was
no
have
it
was
determined
hysteresis
This implied that
London 4
electrodynamics
beyond
afterwards
the
formulated
for superconductors.
flux
applied to
were
in
fact
cooling.
Thus
below its
cooled
a magnetic field
and Gorter and Casimir
theory in
a
in 1933,
reversible thermodynamics
two-fluid thermodynamic
soon
that
Later,
upon
metal was
after
hundred gauss was
that
all flux
when a
be applied to superconductors
out a
realized
superconductors
expelling
of
constant
Shortly
superconductivity.
superconducting transition temperature and
applied.
been a
bewilderment.
(B =0),
by H.
variety
implied by Maxwell's equations
conductors
perfect
fascinating
field of a few
Ochsenfeld 2
change expulsion
the
discovery,
1911
uses of superconductors to
of superconductors
relatively small magnetic
enough to
MAGNETIC
SUPERCONDUCTORS
Kamerlingh Onnesi,
subject
THE
discovery of superconductivity in
Since the
generate
AND
SUPERCONDUCTIVITY
1934.
a
Later,
F.
3
could
worked
and H.
macroscopic
in
1950,
H.
14
Londons
and the other
perfect diamagnetism
function was what led to
wave
of the
a "stiffness"
that somehow
suggested
electromagnetic properties of superconductors.
By
experiments 6 .
superconductors came in
when Cooper 8
to
state
a bound-pair
Finally
free electron gas.
in
phonons,
at
the Fermi
the
to
the
surface and
of the
ground state
50
nearly
1957,
such as
led
by
of the normal
an instability
demonstrated that
interaction,
mediated
interaction
formation of
thus
1956
attractive electron-electron
effective
spectrum of
The key insight into what was happening in
superconductors.
an
giving
were
the excitation
gap in
an energy
evidence for
measurements 7
heat
Specific
shift
isotope
from
in
coming
were
superconductivity
phonons in
to the role of
early 1950's clues as
the
years
after
the discovery of superconductivity, a microscopic theory was
put forth by Bardeen, Cooper, and Schrieffer'.
The
of
genesis
end
signaled the
properties
Abrikosov 1 0
properties.
Ginzburg-Landauli
transition
discovery of more
of the
theory of
to predict
superconductor
superconductor
including
superconductors,
of
in
had
earlier
the second
the existence
which magnetic
in a
theory
the microscopic
regular array
by
no
means
fascinating new
new
(1956)
magnetic
used
the
order superconducting
of a
flux would
second type
of
penetrate the
of quantized
vortices.
15
Abrikosov's startling prediction went
when, stimulated by Goodman's
until the early sixties
years
publicizing 1 2
Abrikosov's
was
it
work,
of Abrikosov's
realized
that
theory could clear up several anomalies that had
properties
magnetic
the
in
observed
been
unnoticed for several
of
superconductors.
is
It
now
that there
well known
two classes
are
of
superconductors distinguished most clearly by their magnetic
properties.
The first type is
characterized by the perfect
Ochsenfeld.
diamagnetism first observed by Meissner and
second is
flux
can
characterized by an
quantized vortices.
superconductor
the
penetrate
Type II
in
600 kilogauss'
some
europium
doped
3
for Pb 1
ternary
detailed microscopic theory
Type
II superconductors
an
in which
array
of
superconductors, in contrast to
type I superconductors, can have very large
up to
state
intermediate
The
MO
1
critical fields,
S(06 and even
molybdenum
sulfides1 4 .
of the upper critical
has been
work out
higher for
A
field of
by Maki's
and
16
Werthamer, Hohenberg, and Halpern
though certain anomalies
between theory and experiment have
become evident in recent
years*.
Also in
*See
the early sixties,
Chapter VI for references.
Brian Josephson 1 7
was led by
16
some considerations of broken symmetry and by meticulous use
some
separated superconductors.
function and
junctions
Nowadays Josephson
fields.
as
potential
are commonly
and
in
elements
storage
and
switching
placed in
and galvanometers
sensitive magnetometers
employed in
intimately connected,
features when
show dramatic
the wave
phase of
Because the
two
in
functions
wave
potential are
the vector
Josephson junctions
magnetic
a thin
result of the difference in
macroscopic
the
between
phases
tunneling
two superconductors separated by
These effects are a
insulator.
the
in
effects
dc
and
ac
startling
characteristics of
show
to predict
newly introduced tunneling Hamiltonian's
of the
computers.
was
Historically there
This
understand.
of
shift
lowered
many
towards
proportional to
to
was the
Knight
the
finally
cleared
difficult
very
As
was
shift
up
by
most
expected
to
was
to be
susceptibility of
of the
combined
problems
refinements
shift debate is
of the Knight
*The history
the spin-orbit
Chapter VI, where
fully in
is analyzed
superconductors
given.
references are
in
Knight
temperature
the
the vanishing Pauli spin
superconducting electron pairs*.
were
be
anomalously nonvanishing
superconductors.
zero
magnetic effects
of
class
proved
that
superconductors
one
systematically,
here
of
discussed more
interaction in
and
detailed
17
experiment and theory.
done
on
aluminum showed
vanishing
combined
Knight
as
the
cause
of the early experiments
that aluminum
shift.
realization
superconductors
finite
Repetition
in
Theoretically
that
(1)
the
temperature
did
is
there
spin-orbit
Pauli
superconductor
approaches
was
susceptibility
lowered and
have a
the
impurities
(2)
to
the
orbital susceptibility does not vanish as the
the
fact
in
remain
Van
Vleck
temperature of
zero.
While presently the Knight shift problem is thought to be
understood,
quantitative
systematic,
theoretical
analyses
of
the
spin-orbit
superconductors have not been done.
importance
of spin-orbit
This
in
scattering
and
experimental
scattering
in
is in spite of the
allowing
for
very
ultrathin
films
high-critical-field superconducting materials.
In
in
recent years
high magnetic
tunneling
fields
have
in
shown directly
experiments
allow
scattering
times
a direct
the reduced
from
density of stateszo.
However,
scattering in
weaker dependence on the
the theoryz1.
determination
spin
of
experiments
atomic number
Zeeman
the
These
spin-orbit
splitting of
the magnitude
these
the
states 1 9 .
the quasiparticle density of
splitting of
spin-orbit
experiments
the
of the observed
shows
a
much
Z than that given by
18
of
between
competition
a
showing
Materials
research.
of
field
fascinating
a
remain
themselves
properties
the
superconductors,
of
properties
magnetic
the
in the understanding
remaining difficulties
Besides these
superconductivity and long range magnetic ordering have been
practical,
Moreover,
recentlyzz.
discovered
are mandatory
high-critical-field superconducting materials
for
being
now
technologies
energy
new
the
of
many
pursuedz 3 .
ultrathin
namely
superconductor or to a
and normal
each
other
over
high
parallel
proximity
to
another
Superconducting metal
normal metal.
thousands
of
proximity effect.
In
of
tens
up to
distances
contact, influence
placed in
metal films, when
Angstroms via an
this
in
placed
regime,
in
superconducting films
fields
magnetic
unexplored
an
in
superconductors
of
properties
magnetic
the
of
study
the
with
deals
thesis
This
effect known as the
proximity effect is extended to
thesis a theory of the
apply to thin films in parallel magnetic fields and dramatic
changes
in
the structure
superconductor-normal
Preliminary
stimulation
experiments,
systematic
metal
however,
enough
to
predictions show
are
are
sandwiches
undertaken
experiments
by these
density of
the
for
not
verify the
yet
as
a
states
predicted.
result
new features.
clean
detailed
of
of
The
enough
and
structure
and
19
indeed can eliminate totally)
scattering
this
on
the spin effects of a parallel
to a superconductor,
magnetic field applied
present so
in
of the
nature
superconductors,
which
features
of
Instead
recently
have
superconductors,
are considered
intuitive ideas about
of the
spin-orbit interaction are
use a simple
from
emerged
from
single
in
We find
in detail.
in
that
the atomic number Z dependence
simple,
interaction in
is not
scattering
experiments
tunneling
at
perplexing
certain
the spin-orbit
spin-polarized
the conventional analysis
theory
to the
spin-orbit scattering
here.
detail
a systematic comparison
compared
quantitatively
the complication of
considered
proximity
The proximity effect experiments
to experiment can be made.
yet be
effects of
the
of thin-film
the properties
sandwiches must be considered before
cannot
(and
drastically
alters
scattering
Because spin-orbit
thesis.
in this
systematic behavior calculated
variance with
vastly at
of the magnitude of the
show further
We
these experiments.
spin-orbit
how to
assumption about the contribution of scattering
displaced
estimate of the
surface
atoms
to
yield
spin-orbit scattering time,
a
and
quantitative
we compare
these estimates to experimental values.
Following an introduction in this
of
superconductivity
which
chapter of the concepts
are important
in
this
work,
20
Chapter II presents an in depth review of theoretical models
for the proximity effect.
of one of the models,
thin sandwiches
In
tunneling Hamiltonian model, to the
a
in high parallel
detailed
a
IV
Chapter
Chapter III provides an extension
fields of
of
picture
zero-temperature behavior of these
interest here.
expected
the
sandwiches is presented.
In Chapter V we give a comparison of preliminary experiments
with
theory.
experiments
We
also
indicate
directions
the
future
should take in attempting to test critically for
the properties discussed in Chapter IV.
improved quantitative
superconducting
films.
conclusion and summary.
picture of
Finally
Chapter VI gives
spin-orbit scattering
Chapter
VII
provides
an
in
a
21
IN THE
BASIC CONCEPTS
B.
baffling
as
to
Schafrothz 6
Hamiltonian
showed
remained
25
based on
Bardeen
,
self-energy,
Furthermore
derived
be
effect cannot
Frohlich
the
from
electron-phonon
the
features
of
myriad
the
starting
that,
which described
Meissner
the
role
superconductivity.
of
characteristic
this
the electron
resembling
nothing
and
FrohlichZ4
modifications of
phonon induced
led
by
theories
play
they
How
superconductivity.
for
explanation
the
in
role
crucial
a
play
recognized that phonons would
fifties it was
In the early
SUPERCONDUCTIVITY
OF
THEORY
interaction,
of
order
in any
perturbation theory.
Also
specific
in the
early
measurements 7
heat
an
energy
superconductors.
excitation
spectrum
of
demonstrated
27
Pippard's
that
for
nonlocal
from
evidence
there was
fifties,
in
gap
the
further
Bardeen
electrodynamics
28
would likely follow from a model containing an energy gap.
Leon Cooper
that,
supplied the
in the presence of
interaction
(such as
electrons in the
ground state
that
key missing
He
showed
any type of effectively attractive
mediated by
vicinity of the Fermi
of the
concept8 .
electron gas
phonons)
surface,
is unstable
between
the
the normal
against the
22
formation
of
consists of
states
states so
and spin
attractive scattering
enjoy the
as to
of opposite momentum
pairs
scattering coherently into other
electrons
and spin
momentum
opposite
bound-pair
The
electrons.
pairs of
bound
interaction.
binding energy
pair
2.A
where
problem are
pairs in the Cooper
The
4
IwD
given
2A
e
WAL~D
is the width of
surface where there is
by a
characterized
by:
N(EF:)V(1)
the Fermi
the energy region above
assumed to be an attractive effective
electron-electron potential of strength V,
2N(E
and
) is the
density of electron states at the Fermi level when the metal
is in
the normal
This result
state*.
function of the potential V and
an analytic
is not
it explained the
failure of
the earlier perturbative approaches.
this concept of
With
Cooper,
and Schrieffer
bound pairs
of
(BCS) 9 were able
electrons Bardeen,
to write down a new
ground state wave function consisting of many bound pairs
electrons
and
to
describe
superconductors in
terms
that N(E )
*Note
orientation ony.
is the
of
the
observed
this ground
density
properties
state and
of states
for one
of
of
single
spin
23
electron and
pair
excitations
found an excitation spectrum
clap
N(E,)e V
VE
where
Fermi
They
state.
with a zero-temperature energy
by:
given
2A 0
ground
this
above
N(E)V
)
is now the width
2p),
surface
which
in
effectively
the
is
there
the
near the
region
of the energy
The
attractive electron-electron interaction of strength V.
quantitity
A , which is nonzero
ordered phase,
referred to
often
is
only in
as
the
low-temperature
the
order
parameter
or pair potential of the superconductor*.
Bardeen,
Cooper,
and Schrieffer
temperature of a superconductor was given by
I
.~
S
.
where k.
-t
e
u.
and
is Boltzman's constant
zero-temperature
energy gap
related
is
_
_
I.3)
N(F)V
is Euler's constant.
'
seen that in the weak coupling
It can be
_
1.13 '(w#e
N(F,)V
transition
that the
found
limit
to
the
(V-
O)
the
transition
temperature by
A0
IC
(1.4)
G.
*Although the order parameter equals
half the energy gap in
the
not
is
this
superconductor,
BCS
a
proximity-effect
the
In
superconductors.
here,
considered
well
as
as
in
magnetic impurities, the density of
the BCS form displayed
shape from
energy
gap,
different
which we
from twice
the
will
order
denote
all
for
case
superconductors
superconductors
with
states has a different
and the
in Figure I.1,
by 21).,
parameter,
2A.
is
generally
24
this is generally useful for NCE )V less than 0.25.
and
plot of the density of states
Figure I.1a is a schematic
states is
BCS
N
theory the
gap.
In the
the Fermi
level,
A measurement
of the
below and above the energy
singular
near
of states
density
density of
that the
It is evident
"semiconductor model."
the
as
known
is
what
in
superconductor
BCS
a
of
is given by:
(E),
IEI.5)
Ns(E) = N(E,) Re
I.1b.
is plotted in Figure
and this
junction,
metal-insulator-superconductor
normal
a
of
conductance
Giaever 2 9 ,
by
as pioneered
the superconductor's
measurement of
direct
provides a
density of
states and
therefore its energy gap.
The
is lower
superconducting state
normal
state
ground
condensation
G
energy
temperature T = 0)
of
the
-
G61
N
free
per
than the
in energy
by
gas
electron
unit volume
a
(at
given
by:
2
S(T= 0)
One
can get
FigureI.1
energies
a
-
"feel"
by noting that
lowered
by
G(T o)
for
-
this
2N(E
aMounts
N rN(E,)
condensation energy
from
occupied states have their
)
of
order
/\
A
type
I
25
BCS DENSITY OF STATES
(0)
0
-EF
(b)
-A
0
ENERGY
Figure 1.1
A
26
the
of
lines
thermodynamic critical field Hcb
(T =0)
simply related
the Fermi level.
was done
here, is
chapter where
is
critical field
density of
and the
order parameter
states at
this
given by:
the thermodynamic
to the
a
-(1.7)
-rT=0)
seen that
can be
It
to
leads
this
and
field
magnetic
the flux
field expels
in a magnetic
superconductor placed
Balancing free energies, such as
section of
the next
done frequently in
of thin BCS superconductors
the behavior
in high magnetic fields is discussed.
The original BCS theory was
a
gauge
particular
wave function
many-body
Once the
number.
assimilated, more
the
using
theory
were
independently,
31
transformation techniques;
aspects
Green's
were
Anderson
useful for
of superconductivity;
function techniques
few lines.
We will employ
Bogoliubov-Valatin
techniques
3
apply
2 developed
seeing the
Gor'kov
33
and,
canonical
a pseudospin
powerful
utilized
results in a
3
both a generalization 4
and
were
broken symmetry 1 7
to derive the BCS
transformation
in this thesis.
to
quick
30
Bogoliubov
forth:
put
a
of formulating
concise ways
elegant and
theory
of the
ideas
on
particle
not conserve
did
which
quickly
approach
variational
a
fundamental
Valatin
formalism which is
rather clumsily formulated in
Green's
of the
function
27
superconductivity,
scales
importance in
of
diamagnetism characteristic of type
fact perfect
from
follows
equations,
length
A
samples.
As
electromagnetic
constituative
London4
the
I superconductors is in
oriented bulk
only for suitably
"perfect"
The
superconductors.
length
two
are
that there
knows
one
of
theories
macroscopic
earlier
the
From
diamagnetic screening occurs exponentially over a
called the
clean
In
depth.
London penetration
superconductors this is given by:
mC
where mcz
100
Typically
2
is
such
as
superconducting metals,
in clean
prototype Ginzburg-Landau theory
scale
near
theories,
Ginzburg-Landau phase-transition
of length
order parameter
T
the London
electrons in
of superconducting
phase transition, contain
the
the electron, and nS is the
to 500 R.
All
the
%-rvi~e(I.8)
is the rest energy of
number density
theory.
A
the
of the superconducting
over which
of the
spacial variations
For temperatures
are energetically allowed.
transition
that gives
a coherence length
temperature
T
,
length
this
--
characteristically
superconductors
temperatures,
this
where
diverges
as
length
also
it
can
be
(1
has
-
T/T
meaning
thought
of
)
In
.
at
as
low
the
28
At
pairs.
Cooper
the
of
radius
root-mean-square
zero-temperature it is simply related to the energy gap:
(1.9)
tr A,
as
electrodynamics
over
when
lengths
in
Impurities
the
are
clean superconductors
length 3
coherence
5
the
decrease
impurities
temperatures the
coherence
the
order
length
$
Coherence
is
given
104
of
approximately
decreases
and
(
and
t by:
I
The penetration
low
At
in terms of the pure superconductor coherence length
the mean free path
f.
depth and
leng th.
coherence
in
reasonable,
intuitively
seems
As
.
on
show
can
parameter
London penetration
the
both
modify
the
be important
chapter.
next
the
in
proximity-effect sandwiches
order
the
of
variations
consider
we
parameter
This interpretation will
spacial variations.
of
the minimum distance
order
superconducting
the
degree
the
temperatures,
all
be thought of as
length can
which
characterizing
at
Physically,
non-locality.
coherence
length
the
non-local
Pippard's
in
appears
also
length
This
depth, on the
at low
0
other hand, increases
temperatures
is
10)
2(I.
given by:
as
.,
29
Near the transition temperature this length also diverges as
(1 -
T/Tc
'
30
IN
SUPERCONDUCTORS
FILM
OF THIN
PROPERTIES
MAGNETIC
C.
PARALLEL MAGNETIC FIELDS
depth
penetration
the
by
characterized
superconductors,
diamagnetism of
of the
The interplay
A
,
the
and
superconducting wavefunction, which varies on a length scale
coherence length
given by the
properties
of magnetic
without
other
hand,
<2 A
in
,
type
In
array
type
ignored
of
II
and
flux
penetrates
the
regime
considerations
some depth 3
6
.
of
the
On
result
when
the
the diamagnetic
the
respectively,
superconductor
can be
almost
the Zeeman energy associated with
In high fields
is
flux to
intermediate and mixed states
superconductors,
type
spin alignment of
I
in which a flux penetrates
parallel fields,
I and
the
type
quantized vortices.
ultrathin films in
uniformly.
which
II superconductors,
effects which give rise to the
in
not allow
superconductivity.
display a mixed state
a periodic
are
samples to
the
destroying
Superconductors
(
than
less
in bulk
superconductors and
enter
superconductors.
of
depth
penetration
with
results in a rich variety
,
electrons
interest
important in
this
here
becomes
and
regime
are
important.
the
free
This
energy
now described
in
31
of thickness d located in
Consider an infinite flat film
superconductor
illustrated
an
has
planes
between the
According to
in Figure I.2a.
film
a
such
is located
internal
field
*C-dok
%
The
H.
of strength
field
external
parallel applied
a
x=0 and x=d as
Londons' equations
with
spacial
a
distribution given by:
H (x) =
This distribution
thin film
the
bulk.
The
(.12)
(
for
The excluded field for
respectively.
be much reduced from
can be seen to
Gibbs free energy per
(average)
I.4a
Figures I.3a and
is plotted in
thick and thin films,
the
H
that in
unit volume G, (T,H)
of a superconducting film in a field is given by:
H(TT H =0 H =
S
I
(1.13)
the zero field free
where G (T,H=0) is
By
energy density.
performing the integration we find:
GS(r, H)
The
&(Tr)
G
H= )
Ho
(I. T
87r
corresponding Gibbs free energy
(1c14)
IA
density for the film in
the normal state is:
G,(T
The
)
zero-temperature
GT,0
) -1.15)
(
81r
free
energies
are
plotted
in
Figures
32
Y
(a)
x =d
b. X
Ha
Z
Y
( b)
+
~X
X =d n
Z
Figure 1.2
MEISSNER DIAMAGNETI C EFFECT
Hcb
Free
Energy
i
k
2
8 7r
T hick
Film
H
Figure 1.3
Hcb =4r
N(EF)A
0
MEISSNER EFFECT
Hcb,'
Free
H
Energy
2
Hcb
8-fr
/
/
/
/
H
Thin
Film
He > Hcb
Figure 1.4
35
One notes that the thinner the film is,
I.3b and I.4b*.
superconducting state
(d <<
A),
state
the normal
crossing of
by the
field determined
critical
the
the higher is
the excluded field and
is
less
and
thin films
For very
free energies.
the
6 H1 b/d which can be quite
the critical field is
large indeed.
far
So
critical fields
on
the
films.
order of
on
In a magnetic field
bulk
of the
the order
not those
of thin
critical field
a temperature
a normal metal has
susceptibility
Pauli paramagnetic
independent
states.
of elemental superconductors but
the high parallel
spin
the
consider
superconducting
and
fields
permissible for
is
to
neglected
the normal
in
paramagnetism
This
have
we
37
4,
given
by:
(1.16)
X1P'KEF),1
where A49
is the Bohr
other hand,
have a ground
electrons separated
thus
have a
temperature
*For
clarity
Superconductors,
magnetont.
state
consisting
by an energy gap
vanishing
approaches zero
we actually
plot
of spin-paired
from excited states and
spin susceptibility
Pauli
as was
GA
tWe adopt the convention that IA
(T,H)
on the
first
pointed out
+ H 2 /(8r).
is positive.
as
the
by
36
Yosida
38
The
.
susceptibility of BCS superconductors
vanishing spin
can be understood quite simply by looking at what happens to
superconducting
limit.)
2uasiparticle states
up-spin
electrons and
raised
by
down-spin
lowered by
split
A
H.
as shown
Ferrel 3"
Tedrow, and Fulde1
7
aligned with
Figure 1.5.
and it
possibility
The
was
spin
first pointed out by
by Meservey,
was first observed
occurs
in
filled
states below
Fermi level can shift and lower their energy.
remain
locked in Cooper pairs
shows
no
in
because, as the
are no empty down-spin states into
up-spin quasiparticles
increases.
of
.
field is increases, there
change
the field.
spin densities of states are
The vanishing spin susceptibility
which
energy
have their
holes
splitting in the density of states
Fulde and
paramagnetic
and up-spin holes have their energies
The resulting
in
(Such
consisting of superpositions
Similarly quasiparticle states
of down-spin electrons
the
formed from a superposition of
their spin is
,mH when
that
neglected.
the Pauli
be in
said to
superconductors are
so
enough
be
can
diamagnetism
Meissner
thin
is
that
film
to a
field is applied
a parallel
of states a
the density
the
The particles
and the superconducting state
paramagnetic
One can however anticipate
energy
as
the
field
that there should be
-
a a.
........ ~-
:1
:1
1%
I-
.' I
I
--
*00
37
38
some drastic behavior when the
at their singular
of states would cross
the spin densities
where
field reaches H =,61/4
points.
normal
the corresponding
and
Chandrasekhar-Clogston4 0
or
field,
Pauli
the
plot with
limiting
paramagnetic
The Pauli
included.
paramagnetism
state
free
paramagnetic limit
superconductor in the Pauli
energy of a
Gibbs
of the
resulting plot
the
gives
1.6
Figure
is
field
critical
determined by the crossing of the normal and superconducting
condensation
N(E, )A,/2,
energy,
^eA /2,
energy,
equating
by
i.e.
energies,
free
superconducting
the
paramagnetic
the
with
and is:
-4
-
For
a
(1.17)
"7
zero-temperature energy
related by
gap is
transition temperature TC
by
limit, the
weak coupling
in the
BCS superconductor
Eq.
1.4
to the
Lo = 1.76k 8 TC' and this gives a
zero-temperature critical field of
18.4 kilogauss
per degree
Kelvin of the transition temperature.
Sarma4 1
and later Maki
and Tsuneto4 2
generalized the BCS
gap equation to include spin paramagnetism and Figure 1.7 is
a
plot
function
of their
of
field
spin-paired nature
results
at
the
for
zero
temperature.
of the ground
as
order parameter
state, the
Due
to
a
the
applied field
PAULI PARAMAGNETIC LIMIT
Gn
Free
Energy
N(EF
2
H
Very
Thin
Film
Figure 1.6
N(EF) A
Xn
0
AO
=2Bo
40
Order Parameter as a Function of Field
Paramagnetically Limited, T = 0
1.0A(H)
As
U-1
0.5
z
U
00
D.5
0.707
MAGNETIC FIELD
Figure 1.7
1BH
AS
1.0
41
A,/'A,
6 0 /4 8 ,
at
Right
H,= A./f2,
the
normal
are equal
energies
field
supercooling
solution
unphysical
at
and
The
superheating and supercooling points.
also
is
there
connecting
equation
the gap
the
is
Similarly there
/ASH,,= &,/2
of
free
state
transition to
order
occurred.
have
state will
normal
and superconducting
a first
and
reached, specifically at
field is
state
saw
As we
field.
superheating
the
is
above however, before this
P,
with the
a catastrophic decay into the normal state is
This
inevitable.
are
pairs
single Cooper
and aligning both spins
unstable against breaking
field and thus
For fields above
a superconducting solution of the
there is no longer
equation.
gap
pH <4.
when
gap equation
BCS
generalized
from their
obtained
parameter
the order
effect on
has no
free energies
a
an
the
of
unphysical solutions of
the superheating, supercooling, and
the gap equation are given along with the physical curves in
1.8.
Figure
At finite
First
change.
temperature two
the energy
difference between
energy
states.
Second, in
superconductor
features of
gap decreases
the
the
are
the
superconductor acquires
magnetic susceptibility
pairs
thermally
now
normal
in
the
excited
to the magnetic field.
quasiparticles and these can respond
Thus
ground
picture
the free
as does
superconducting and
addition to
there
the above
a
and its free
temperature-dependent
energy is
lowered by
0
0.5
0.707
Fiaure 1.8
1.0
P-BH
43
the
application
of the
magnetic field.
The
features.
normal
zero
reflecting the
function of
a
independence
temperature
of
the
below the
(for temperatures much
susceptibility
Pauli spin
two
temperature is raised above
field remains unaffected as the
absolute
as
free energy
state
plot
these
illustrates
and
temperature
finite
at
field
is a
function of
energy as a
superconducting state free
of the
1.9
Figure
Fermi temperature.)
As
temperature
the
normal
the
eventually
increases,
and
superconducting state free energies meet with the same slope
are
there
and
T = 0.
56
T
supercooling
and
superheating
longer
the temperature
At
fields.
no
,
the
order of
phase transition changes from first to second order.
I.10 is
these
sketch of the field-temperature
a
films
superconducting
and
H-T
to the
for
Frota-Pessoa
an
phase
example,
was
as
spins to order
is
of second order
into a first order line.
diagram
and Schwartz"
applied field, there
for the
and
as well as the first
phase diagram, displays a line
transition points changing
FeCl12
diagram of
order transition lines.
and second
identical
Figure
superheating
the
supercooling field lines are indicated
This
phase
the
4
.
a
found
in
recently
This is
some metamagnets4
out
pointed
When a metamagnet is placed
competition between
in an antiferromagnetic
a
3
,
by
in
tendency
manner and
T >0
8=.88
%~
t=68
t=.044
t=.22
t0
00.5
1.0
Figure 1.9
HBH
A0
45
Paramagnetic Limit
Phase Diagram
1.0PBH
Hs
AO 0.707- Ist He
0.5 -
enormal
B CS
0.5
T
0
t=Tc
Figure I, 1 0
2nd
Hc
1.0
46
spins
of
alignment
transitions which
antiferromagnetic
strong
changes into
a first
order
phase
as the
model
with
neighbor
nearest
next
equally
and
coupling
stronger* ferromagnetic
or
This
order line
Ising
neighbor
nearest
phase.
second
of
An
lowered.
is
temperature
paramagnetic
line
a
in
results
competition
a
into
the
which aides
magnetic field
an applied
effect of
the
and thus the
coupling, reproduces the phase diagram of FeCl
phase diagram in Figure 1.10.
The order parameter in the metamagnet is the magnitude of
between the
the difference
easily pictured and one
It is
imagine
applying
alternates
This
a
staggered
tendency to order
field
magnetic
which
Ht
antiferromagnetic
staggered field is conjugate to the
enhances the
in principle,
can, at least
the
direction with
in
magnetizations.
two sublattice
spins.
order parameter and
antiferromagnetically.
One
then has the more complicated H-Hat-T phase diagram which is
given in Figure
in the Hst
two "wings"
The first order
I.11.
= 0 plane
can be seen to be the
of first order
each terminate in a line
The point
line of transitions
phase
of second
in the Hs* = 0 plane
boundaries.
The
"wings"
order phase transitions.
where
details of
for
reference 43
*See
strong.
how
of
precise definition
meeting line of
the second
the
theory
order line
and for
a
Metamagne t
T-H - Hst
Phase Diagram
H
2nd
Paramagnetic
Tricritical Point
\2nd
Antiferromagnetic
T
Hst
Figure 1.11
48
first order line is
changes into a
point
named this
order line)
(and the first
order lines
also
Critical phenomena
3
He-
4
He mixtures.
particular have been the fruitful focus
flexible
system
work in
with
an
enormous
value.
As
of a fair
will be
points in
amount of
years
recent
experimentally
controllable field conjugate to the
of
occurs in many
tricritical
in general and
and experimental
Griffiths4 5
meet.
a tricritical point and it
metamagnets as well as in liquid
theoretical
where three second
and
accessible
a
and
order parameter would be
described
in
the
next
section, and developed at length in the succeeding chapters,
the proximity
field conjugate
these
effect affords a
to the
flexible way of
superconducting order
applying a
parameter in
Pauli-paramagnetically-limited superconductors
display a tricritical point in their
phase diagrams.
which
49
D.
THE
EFFECT
PROXIMITY
coworkers 4 6 ,
effect was
late
until the
Misener
and
1950's that
the
D.
superconductors.
thin film
on
first done
in
Misener et al.'s experiments
clearly identified.
among the
were
not
it was
A.
by
mid-1930's
the
in
observed
fact
was
superconductors
in
proximity effect
Although the
They observed the disappearance of superconductivity in lead
and
films
tin
top
deposited on
of normal-metal
the films
were thinner than certain
(~104
This effect
the
1).
thin film
"critical" thicknesses
characteristic of
was thought to be
to
not
and
superconductors
when
wires
be
related
to
the
substrate metal.
Later
by
experiments
others4
7
done with
,
to
down
of
earlier experiments
of
thickness
Misener
et al.
films
that tin remained
deposited on insulating substrates, showed
superconducting
thin
about
were
then
50
A.
The
thought to
be in error4 8 .
in the late
Only
observed
and
1950
was
identified
experiments 4 9 , H.
carry a
such.
Meissner showed that
wires when coated with thin
would
as
the proximity
supercurrent
(<, 104
A)
between
In
two
effect finally
a
series
of
superconducting
normal metal coatings
them when
they
were
50
supercurrent only when they
support the
the
than
thinner
demonstrated
deposited
that magnetic normal metals
He observed
placed in contact.
on
coatings.
non-magnetic
superconducting
a
when
that
above a
it were
superconducting unless
corroborating the earlier experiments
A
few years
a
later,
induced
superconductivity
proximity effect was given
demonstrated
the
Moreover
he
coating
is
not
coating would
the
metal,
a normal
were substantially
certain thickness,
of Misener et al.
of
the
via
the
definitive fingerprint
a
in
be
metal
normal
by tunneling experimentsso which
energy gap in
existence of an
the normal
side of a superconductor-normal metal sandwich structure.
An
effects
intuitive understanding
concepts of
superconductivity introduced
basic concept utilized, and
other
the
condense.
state into
which
the
above.
that most characteristic of
that of the
superconductivity, is
of
effects and
observed in proximity effect structures can be given
using the
The
of these
macroscopic wavefunction
Cooper pairs
of
electrons
Interference effects involving this wave function
are responsible for the
and characteristics of
Josephson effects 1
this
7
.
wave function also
The existence
explain the
origin of the proximity effect.
The starting
point of all quantum
mechanics, and indeed
51
of
most
is
physics,
every point in
space
equation
at every point
that it be satisfied at
The requirement
time.
space and
This
equation.
the electron wave functions
is satisfied by
in
the Schrodinger
boundary conditions that
leads to the
result in the appearance of energ y and momentum quantization
in
sandwich
metal
(on
discontinuous*
Furthermore,
significant spacial
is
occur
proximity effect is
mechanics
spacial
of
the
over which
the order
parameter
can
the
coherence
length.
The
of
thus the natural result
a macroscopic wave
variations
A
variations of
order
the
on
the distances
mentioned above,
as
that
is
interface.
the
through
extend
must
superconductivity
means
f unction
wave
macroscopic
continuous
Angstroms).
few
a
of
scale
the
phonons
by
induced
interaction
electron-electron
effective
the
though
even
continuous
be
satisfying
that the wave
is
interface
at the
the Schrodinger equation
function
prerequisite for
a
sandwich),
(s/n
superconductor/normal
of a
case
the
In
systems.
many
on
which can
function
scale
the
of
of the quantum
only
show
superconducting
the
coherence length.
The leaking of superconductivity
the
*To
consequent
be
sure the
influence
the
of
derivatives
into
of
the
the potential in the
continuous when
is discontinuous.
a normal metal and
normal
wave
metal
function
Schrodinger
on
the
are
not
equation
52
result
obvious is the
enhancement or depression of
in
density
the
thicknesses
on the order of
geometrical
resonances 5 4~5
from quasiparticle
the coherence length, there are
3
and
reflections
normal metals
been used to
alloys 5 8
Additionally,
behavior of these
of
promise
7
in
superconducting
proximity
induce superconductivity
in Kondo
as
the
above,
mentioned
fields shows
nature
of
been brought
to
of the
elucidation
the
study of
in high magnetic
sandwiches
aiding in
structure5S-S
Recently the
density of states of n/s sandwiches.
effect has
pair
proximity effect
the
induced in
via structure
resulting
discontinuous
the
off
studying phonon
for
is emerging as a tool
states 5 4
bound
Moreover the
the interface.
potential at
with
sandwiches
clean
In
states.
of
the transition
characteristic structures appear
Additionally,
temperature.
Most
phenomena.
of
a variety
in
superconductor
tricritical points.
The
theoretical techniques
which have
bear on proximity effect sandwich
of
above features
the
is
differ
scrutiny.
under
chapter a review of the theoretical models
chapter
following
models
in
we
detail
then begin
in
proximity-effect sandwiches
reader
interested in
is referred to Ref.
other
59 and
order
to
In
the
next
of
60 where the
the
In the
of
these
understand
thin
one
magnetic fields.
in high
aspects
which
is given.
explore
to
to
according
proximity
The
effect
proximity effect has
53
been reviewed through 1976.
54
II:
CHAPTER
A.
INTRODUCTION
I
Chapter
films
The
were discussed, including
density
properties
these
characterizes
proximity-effect well
quasiparticles.
This
of
chapter
become clear,
is
which
theory
a theory
to
superconducting
of the
the
the
of
aspects
generalize such
reviews
be
other
and
a
use
zero-temperature
and to
of
proximity-effect
these
to
will
we
thesis
tunneling theory developed by
context a
superconducting
thin
proximity
various
the aim of introducing
effect theories with
and putting in
McMillan
6 1
,
which,
theory for
the most appropriate
purposes.
Actually, as
of
this
the spin-paramagnetism
account for
as will
need
will
We
sandwiches.
of
spin-splitting
this
on
zero-temperature
parallel magnetic
the Zeeman spin-splitting
In
states.
of
concentrating
our
properties
magnetic
behavior of
discussion of the
sandwiches in high
thin proximity-effect
fields.
superconductivity
of
concepts
the
introduced
be needed for our
that will
the
EFFECT
PROXIMITY
MODELS FOR THE
THEORETICAL
effects
mentioned in Chapter I,
which
fall under
the
there
category
are
of
a number
proximity
55
appropriate to use depends
on the Gor'kov equations,
principle rigorous, but the solution of
theory
is
been used near
with Ginzburg-Landau type theories.
only been
calculate
used to
Even in this
the sandwich
A second
proximity-effect sandwiches.
Cooper and discussed in Section C
in thin films.
Gennes equations
the Bogoliubov-de
layer,
is
reasonable
quite
potential.
to
The
are observed
complicated than that
but does
imposed
in most
(Section
for
when
the
encompass
due to a simple
of the
Gennes-equation theories.
for
D)
discussing
the
has
those
sandwich interface
effective electron-electron attraction,
is
based on the solution
(Section E)
McMillan theory
phenomenological origin,
which
proximity effect
self-consistency for
achieve
of
properties
a
the
superconductors, but it is
geometrical resonance effects in
difficult
transition
of this chapter, gives an
theory,
Still another
regime, it
theory, developed by
intuitive and reasonable description of the
double
only
to make a connection
detailed
the
not
and
temperature,
of
has
the sandwich transition temperature, where it
is possible to linearize the theory and
has
of the
the equations
date this theory
To
quite formidable.
in
are
B),
Section
first (in
discussed
be
will
which
on which effect one is interested
The theories based
in isolating.
is
effect
proximity
the
of
theory
Which
effects.
BCS
pair
rather
a
effects
is
more
discontinuity in the
a restriction which
Cooper-limit and
Bogoliubov-de
It is this tunneling theory which
56
will emerge as
the
low
sandwiches.
the most suitable for
temperature
properties
simply characterizing
of
proximity-effect
57
BASED ON
THEORIES
B.
THE
GOR'KOV AND
THE
GINZBURG-LANDAU EQUATIONS
As
is
particularly
and
in physics
the case
always
in
solid
state physics, one must begin by developing a model which is
to be
simple enough
still
the
contains
of
proceeds
One
discussing.
basis
the effects
then
where necessary in such a way so
the
which
interpretable and
solvable and
by
is
one
interested
in
approximations
making
as to treat the essence of
problem correctly.
that introduced
interaction
it
(and the
Coulomb
overcomes)
are
electron-electron
presumed
their
to
be
momentum
surrounding
the
vectors
Fermi
and
repulsion
generalization
of
Schrieffer
only
within
surface.
when
a
of the
.
an
effective
certain
BCS
narrow
case,
is
have
both electrons
In the
All
electrons
interaction
This
V.
9
electron-phonon
into
lumped
interaction
attractive
a
retarded
the
of
a slight
Cooper,
by Bardeen,
complications
which
which is
with
start
effect
the proximity
Hamiltonian
simplified
the
of
theories
All
region
V
was
58
presumed to have
no spatial dependence*.
proximity-effect
sandwich
it
is
In the
presumed
case of a
that
the
potential has a position dependence reflecting the
attractive
interactions
bilayer.
Thus
the
in the
Hamiltonian
two metals
1'J
for
a
BCS
different
which form
the
proximity-effect
sandwich takes the following form:
where U(x)
is
the one-electron potential and
V(x)
is given
by
VS
-dGS <
X
<0
(II.2)
VV,
where
the
s-n
interface
illustrated in Figure
of the
phonons is
I.2b.
local and
interaction
changes
Realistically
the phonon
atomic layers.
is chosen
to
It is assumed
be
at
at
interaction
as
that the effect
that the nature of
abruptly
x = 0
the
changes
the
phonon
interface.
over a
few
This is a small distance on the scale of the
momentum space as
being a
*Since V
is actually defined in
nonzero constant only
near the Fermi surface,
its Fourier
transform
into
real
space technically
results
in
some
oscillatory
behavior.
(See
Ref. 62
for
a
precise
definition of the spatial form of the BCS potential.)
Here
we
will
ignore
this
fine point
and
speak
as
if
the
potential in real space is a constant.
59
step-function change
superconducting coherence length so the
indicated
spatial
above is
in
variation
of
effective
the
choice for
reasonable
fact a
the
electron-electron
interaction.
The subscripts n and s above
refer to a normal metal and
a superconductor, respectively.
The n-metal is not a normal
a "weaker" superconductor than the
but rather
metal per
se,
s-metal.
Thus the
itself,
superconductor
n-metal may be a
or it may not be.
most direct
The
properties
Eq.
II.1
proceeding to
and write
Green's
(x'
;t')>
F4 (Hxx';v-')e
Gor'kov 3
follow
to
is
regular
=
indicate thermal
the next chapter.
function
GI
the
averages
)
anomalous
(X',')>.
and T.is the
which is
Green's
The
for
=
(,x';z-*')
function
brackets
here
Matsubara imaginary
function is related
local, self-consistent pair potential
basic equation of the
of motion
defined more precisely in
The anomalous Green's
F itself is an integral function of
the
the
finite
to
(generalized
equations
<Trtt(x,
3
3
down the
and
time ordering operator 6
is
calculate
of a system described by the Hamiltonian given in
temperature)
to the
of
way
A
(x)
theory, is
A&(x) by
so Eq.
II.3, which
in fact a nonlinear
60
integral equation.
extremely difficult.
addressed
by
situation which might be
The simplest
the only situation
considered, and
is
formulation,
this
date been
which has to
to
look
nontrivial
for
transition
solutions of this equation in the vicinity of the
temperature.
=
AJ)
K(xx')
(II.4)
K
metals
of
the properties
depends only on
clean
In
be linearized to
AX
X
state.
normal
the
cX)
the kernel
where
11.3 can
In this case Eq.
is
11.3
Eq.
of
solution
self-consistent
fully
A
simply
is
the
6
electron-hole pair propagator 4 given by
c'
(W
.
.., ..
&.
IV
111.16 in
is a Matsubara frequency and is defined by Eq.
chapter),
the next
because
rigorously
treat
clean limit is
though the
6
limit
this
in
s
difficult to
the
spatial
dependence of the kernel is difficult to handle.
a calculation
Even
proximity-effect
transition temperature
of the
sandwich is
difficult in
many different approximations have
of these
successful
a diffusion
Guyon6
6
.
is
the
Recently Silvert'" has
calculation
been used
by de
a
and
The most
been employed.
dirty-limit
approximation has
this model
of
in
which
Gennes and
numerically solved 11.3 in
61
various
to compare the
have some standard with which
an attempt to
of
none
that
finds
He
approximations.
the
approximations seem to be reliable.
of results one
of the type
We have given a brief account
gets from trying to solve directly the Gor'kov equations for
a
For our
proximity-effect sandwich.
there
sandwiches,
is
to
little
transition temperature of the film
yields.
this
from
gained
just the
which is all this
rigor of a direct
is clear that the
It
be
the
proximity
thin
of
more detail than
are interested in
We
approach.
properties
the
of
characterization
purposes here,
theory
attack on
the full Gor'kov equations will have to be sacrificed before
of the detailed properties
any account
of proximity-effect
sandwiches can be given.
It should be mentioned that Ginzburg-Landau theories
of course,
in
also
the
temperature 6
7
.
In
regime
in more
Gor'kov approach
boundary conditions which the
Ginzburg-Landau wave function must
are interested
transition
the
near
fact, results from the
are useful in determining the
we
proximity effect
useful for describing the
temperature
are,
than
satisfy.
Again however,
this limited
temperature
regime.
Actually
even
the
calculation
of
the
transition
62
for thin film sandwiches
temperature
is
This
indicates.
discussion
above
the
is more difficult than
because
the
superconducting properties of a number of metals change when
thin films.
they are in the form of
much increased transition temperature
is observed to have a
other
0
K
vs.
1.17
degradation
a
displays
hand,
(2.5
film
is a thin
when it
Aluminum, for example,
0
transition
its
of
temperature when it is made into a thin film,
on the
Tin,
K).
and this could
be caused by a number of effects 68 .
In
section of
the next
opposite extreme
here.
We shall
offered by
Cooper lead
treatment
of this
of this
the
indicated
of a very thin proximity-effect
see that
sandwich.
we describe
complicated calculation
of the
We examine the limit
understanding
this chapter,
to a
limit.
simple considerations
qualitative and
The
limit, however,
first
quantitative
simplicity of
Cooper's
precludes anything
but
rather unexciting properties for proximity-effect sandwiches
described by this theory.
63
C.
THE COOPER LIMIT
In
superconductivity
bilayer 6 9 .
in
pictured the
He
that
rate
length of
to the
both
superconductor,
states and the
=
0)
having
the
(V
metal
of a normal
simple case
metals
[
where
s-slabs
d
and
dS
are the
respectively.
This
the energy gap as well as
sandwich using
the simple
of
density
argued that the
is given by
IN(EF )V
4J~
-s N(1E0)V,
~EF
(
the
to a
in proximity
same effective mass, Cooper
effective BCS coupling parameter
for
Thus
same
the
proportion
in
side.
in each
time spent
of
average
two sides
the
electron-electron interaction of
at such a
two metals
an
only
feel
they
structure to
such a
in
electrons
be scuttling back and forth between the
rapid
metal
superconductor-normal
thin
a
theory of
microscopic
Cooper presented a simple
1961
(11.6)
the
thicknesses of
formula can be
n-
used to
and
the
calculate
the transition temperature of the
BCS expressions,
Eqns. 1.2
and
1.3.
Others 7 0 have generalized this
obvious
manner
nonvanishing
Fermi-level
to apply
BCS
for
two
interaction
averaging procedure in an
metals
and
both of which
which
(normal-state) densities of
a
differing
have
states,
have
N
(EF ), i
64
The result is:
n or s.
[N (E"V",,1
F
NhEF)
V
+ N
JA
averaging
with the simple
however, some problems
There are,
(EFM
s
procedure indicated here.
Derivations of this result, as well as generalizations to
temperatures
Debye
differing
Gor'kov
formalism described
possible
to use
fact,
In
derive Cooper's
next section to
in the
above.
Bogoliubov-de Gennes
the
given
have been
for
the
using
the
cases
(in both
,
only)
temperature
transition
71
states and
densities of
differing Fermi-level
metals with
it is
formalism outlined
Eq. 11.6*.
result,
We discuss this more in the next section of this chapter
in
Appendix
that the
this
A of
are
We
thesis.
simple averaging procedure
to
able
properties
all the
when there is
a difference in
two metals.
The reason
when
originally
of
Eq.
indicated in
these
thin
and
demonstrate
for the effective potential in proximity sandwiches
characterize
also
11.7
does not
sandwiches
the Fermi wave vectors of the
for this,
discussing his
hypothesized
as Cooper
formula,
Eq.
11.6,
is
that
for the
have recently done so
*Entin-Wolman and Bar-Sagi 7 '
metals
to
11.6
Eq.
and generalized
transition temperature
with
finite mean
free
satisfy all
do not
however,
suspect.
are
results
their
Appendix
A.
Their
paths.
the
See
Green's
functions,
so
boundary conditions
and
section
next
the
65
differing
Fermi
wave
will
vectors
in
result
partial
transmission and partial reflection of electrons incident on
the
The
interface.
electrons
on
attraction of
each
side
will
be
describe
different
all
feel
and
properties
a
thin oxide
transmission
the
entire
also
and reflection
the
extent feel
sample.
cannot
(Other
at the interface, such
would
and
formula
simple
Cooper's
barrier,
extent
to a greater
complications which might be present
as
lesser
the
the properties on each side
Thus
of
a
to
the other side and
that of "their own" side.
means that
transmission
reduced
result
affect
in
partial
the averaging
in
similar deleterious ways.)
The Cooper
of
limit is useful for
but we
proximity-effect theories,
such a check
may be misguided.
not sound and the result, Eq.
appropriate,
is too
checking certain results
The
11.6,
trivial, for
have also
basis of
the
theory is
in the limit where it is
the Cooper limit
much more use to us in this thesis.
indicated
to be of
66
D.
GENNES
BOGOLIUBOV-DE
PROXIMITY
THE
TO
EQUATION APPROACH
EFFECT
1. Introduction
one
based
on
This
superconductor.
nonhomogeneous
a
for
equations
(BdG)
Gennes 34
Bogoliubov-de
the
solving
above, is
approach outlined
the Gor'kov
closely related to
quite
actually
proximity effect,
the
to
An approach
approach has been fruitful for elucidating the nature of the
geometrical
resonances
full self-consistency in the
potential
with a
to
this
solve
simple
manner
familiar
Equivalently,
a
functions allows
functions for
Green's
these
from
simple
It is only
matching
function
This
pair
allows
separate
a
wave
the
finite-layer Green's
metals and
then
so as
double layer
57
to use
to
electrostatics,
functions
in
mechanics.
of
dependence
spatial
Green's function of the complete
techniques
quantum
elementary
Green's
trial
a
equations for a nonhomogeneous
familiar from
separate layer
for
dependence.
one to construct the
each of the
theorem,
equation
spatial
explicit solution of the BdG
superconductor using wave
achieve
gap function, one would have to
of the Gor'kov equations.
solve the equivalent
feasible however
To
structures.
in proximity
to get
17 2
.
match
the
67
simple closed form expressions for the
If one is to have
of the pair potential
Green's functions, the initial choice
must
a
one of
remain
the achievement
practice prohibits
is the
potential, which
the pair
approach.
Nevertheless
successful
in elucidating
interference
Furthermore
sandwiches.
main deficiency
of
the
reflections
there are simple
quite
geometrical
destructive
and
constructive
of this
been
has
nature
quasiparticle
of
in
self-consistency for
of
approach
the
by
caused
resonances
the
This
functional form.
simple
clean
in
geometries,
as the thin-film Cooper-limit geometry, where
such
it is possible
to achieve self-consistency.
2.
Review
of this
The use
Gennes
and Saint-James 5 4
states
in a
normal
1963 when
de
total density
of
a superconductor
by
back to
approach dates
calculated the
metal backed
by
solving the wave matching problem:
(E7~
E~VQ.(0E)VE)
EFt.
+ AxVX)
A4~)3
(11.8)
68
electron-like and
v(x) are respectively the
where u(x) and
the hole-like amplitudes of the quasiparticle wave function,
A
where
is
(x)
A,
taken as
0<x<d,,
and where the wave
at the
x=
functions are required to vanish
states of
They found
d, surface.
for
as zero
-ds<x<O and
for
energy less
than A, localized or "bound" in the n-side of the barrier as
a consequence
at the
interface.
not clearly observed until about
These bound states were
years later 7
ten
3
resonant state had
by a
the
McMillan
unbacked superconductors
53
energy gap
when
been
had
tunneling into a
51
'sz
into superconductors
resonance,
Another
superconductorsz.
involving two reflections
this one
the energy
been observed by Tomasch above
weaker
of
another type
long after
much enhanced, by tunneling
and then,
backed
This was
.
by tunneling into
gap, first
above
pair potential barrier
the
of reflections off
off
interface,
n-s
the
observed by
Rowell
normal metal backed
and
by a
superconductor.
McMillan
other
in the
and Anderson7 4
oscillations
theory of
the
for the
Later
McMillan 7
Tomasch effect
5
these
a discontinuity
based on scattering off
pair potential.
complete theory
developed
developed
using an
similar to that of de Gennes and Saint-James.
a more
approach
69
density of states a
calculated the local
as the thickness of the
This
barrier.
result from one
description of
distance d,
taken
well the
Tomasch
reflection off
the n-s
not capable of providing a
but the calculation is
interface
and
superconducting slab, away from the
calculation characterized
oscillations which
metals
semi-infinite
two
considered
McMillan
first of
(the
reflections
the higher-order
which McMillan had discussed in the earlier calculation with
Anderson 7 ").
Recently
solved
the
Arnold
57
Bogoliubov-de
geometry assuming a
and the s-side
Bar-Sagi 7 2
and Entin-Wohlman and
equations
Gennes
both the n-
constant pair potential in
of a sandwich structure.
finite
a
in
have
Entin-Wohlman and
Bar-Sagi discuss the limit where both n and s are thin which
is a case in which full self-consistency of the
be
constant
achieved with
metal*.
backed
geometry
claims one
a
pair potentials
discusses the case
Arnold
by
gap
semi-infinite
used in
some recent
can achieve
BCS pair potentials
solution can
of a thin
superconductor,
experiments 7 6 .
self-consistency with
in each side of the
metal.
for
each
normal layer
which
He
is
a
likewise
constant gap
This claim
as the
in the calculation
to be some errors
*There appear
conditions
boundary
the
does not satisfy
Green's function
that it is stated to satisfy.
70
on
based
is
superconductor over
s
on
calculating
interferences over
to
coherence
the
length,
parameter need
to be
been done
on
potential looks
is interested
variations
taken into
account.
forms),
to treat
has
more
(though only of
parameter
though calculations
for the normal-superconductor
7
Bar-Sagi 7
possible
it is
order
the
of
the
of
the order
spatial
realistic variations of the order
prescribed functional
transition
sandwich
of the resonances, which are
lengths
demonstrated that
recently
when
neglected
be
However, if one
from the interface.
in studying the precise nature
due
Spatial variations
on what the pair
temperature which depends
like far
semi-infinite
the
as
such
quantities
assumed
indeed
can
scale
length
this
is perturbed,
layer.
thickness of the superconducting
the
potential
the
to
compared
small
is
which the pair
into
distance
the
that
belief
a
have not
sandwiches described
here.
As
above, Entin-Wohlman
eluded to
used a theory based
from
solutions
on
of
normal-superconductor
potential in s).
expression, Eq.
films
which are
difference
in
construction of the
sandwich
(with
a
in all
electron-electron
a
for
constant
pair
the validity of Cooper's
for the transition temperature of
identical
have
Green's function
equations
BdG
the
They demonstrate
11.6,
and Bar-Sagi 7 z
thin
respects
except for
interactions
in the
a
two
71
metals.
function at
with an
author shows
Appendix A this
In
potential.
average pair
the density of states,
a BCS behavior
shows
zero-temperature similarly
the Green's
that
Cooper limit
Thus in the
by the
example, is characterized
for
same average interaction as is the transition temperature.
result is rather uninteresting.
This average interaction
It
despite the
that is probed in
interaction is
layer
in
really what is
is
in Eq.
Hamiltonian
indeed
be
that
from
interaction
surface
may
layer
this
the
is
of
a tunneling experiment7 8
If
bulk.
measured,
It
different
a
have
unimportant.
11.8
This is
the local density of states
fact that it is
a surface layer
and
tunneling experiments.
observed in
are routinely
states
densities of
bulk total
characteristics reflecting
tunneling
good
why
explain
however,
may,
an
BCS
average
then this
different
appears
that
incapable of
giving
any
the
more
complicated structure than that which can be described by an
average
BCS
interaction
thin
in
proximity-effect
sandwiches.
In Appendix
BdG derivation
A we sketch a further
of the Cooper limit
generalization of the
for the case
also differing in their Fermi wave vectors.
this case one does obtain
density of
states than
of metals
We show that in
more complicated structure in the
that due
to one
average BCS
pair
72
physical mechanism
The
potential.
original
in his
fact indicated by Cooper
occurrence was in
this
for
responsible
6
discussion of the "Cooper limit"
9
vector leads to reflections and
reduced transmission of
the
that the
retain
density
two metals
will to
The
characteristics.
their own
in
states
of
the naive
pictured in
straight transmission
each
metal
complicated than that which would be
because
other
coupled.
considered
this
of
added
in deriving Eq.
the Cooper limit
reflection
to metals differing in
and its
be
then
more
BCS
the structure
properties of
still
intimately
this chapter, the
not
was
mechanism
11.7, the naive generalization of
Fermi-level densities of states.
this effect
are
metals
above in Section C of
As noted
occurrence
the two
the
to an average
present in each metal will be reflected in the
the
extent
structure in
would expect however, that
One
interaction.
the
Cooper limit.
a greater
should
due
all
instead of
the interface,
electrons encountering
This means
A differing Fermi wave
.
their normal-state
In fact, the occurrence of
significance have
not been
generally
appreciated.
Structure
similar
to
structure
this
the
same normal
properties but
barrier separating the two metals.
by
a
been noted by Bar-Sagi
discontinuous Fermi wave vector, has
and Entin-Wohlman7 9 for a proximity
caused
sandwich of metals with
with
a regular
potential
Physically this is not a
73
surprising result since
would
also lead
similar to
vector.
to
a potential
Experimentally the
interface cannot
and reduced
reflections
that attributed
between the
to a
transmission
difference in
characteristic
be very well
two metals
Fermi wave
parameters of the
indeed,
controlled and,
are
even difficult to characterize. Nevertheless it is certainly
such a potential
plausible that
but the most perfect metal-metal
barrier is present
at all
interfaces.
The effects of differing Fermi wave vectors in the metals
comprising
effects
of
calculated
a potential barrier
using
The calculations
model
the BdG theory
are very long
which captured
the
between the metals,
outlined in
of the
essence
for
McMillan 61
proximity-effect
about
the
next
two
chapters
for
quite useful.
Appendix
A.
and
Such a
given
by
is introduced
chapter and developed in
discussing
sandwiches in high parallel magnetic fields.
*See
was
This model
briefly in the next section of this
a simpler
reflection
sandwiches
ten years ago.
can be
this section.
and tedious* and
reduced transmission effects would be
model
as well as the
a thin proximity-effect sandwich,
proximity-effect
74
OF
MODEL
TUNNELING
MCMILLAN
E.
PROXIMITY-EFFECT
THIN
SANDWICHES
out that any non-BCS-like
vectors
effect).
between the
of
barrier at the
6
approximately by McMillan 1, which
The
of such a potential barrier.
for
simple model
a
solved
and
introduced
incorporates the effects
and its results are
model
A detailed discussion is
only sketched here.
similar
a
in
result
outline
first
sandwiches,
Fermi wave
to differing
which
section we
In this
proximity-effect
metals,
two
the
(or
two metals
reduced
and
the reflections
sandwich with a potential
transmission of a
interface
of thin proximity-effect
features
be traced to
could
structures
was pointed
this chapter it
the previous section of
In
given in the
next chapter where the model is generalized to apply to thin
proximity-effect bilayers
Pauli
in high
paramagnetic effects
McMillan
that
noted
proximity-effect sandwich
by a potential
of
This
if
consists
was
one metal
a more
8
Falicov, and Phillipsi
the
important.
of
two
to another
familiar
that
imagines
one
barrier, the situation is
tunneling from
barrier.
are
parallel fields where the
metals
a
separated
identical to that
through an
problem
and
oxide
Cohen,
had introduced a Hamiltonian, called
tunneling Hamiltonian,
which had
proven
to be
quite
75
useful in
the calculation
junctions.
also
McMillan
approximately
tunneling
which
can
In
the
and
of wave
each
i
n
.
higher order
in
coupling of the metals
in
tunneling problem,
with
considered by
electrode
or
s)
is
Expressed in
annihilation
first-order
vector k and
Cohen, Falikov,
(signified
described
terms of
operators a
here
by its
the
own
BCS
creation operators
for electrons
z spin component
by
in states
o=+ localized
in
side
these Hamiltonians are given by:
Ej ~~
where
E;
-
~ZVaO'O
~
~ Q1 ~&L 0
is the energy
relative to the
the attractive
HT
the
of the higher orders
adequately
tunneling problem
Hamiltonian H
i,
a
if
theory.
superscript
a
sandwiches
Better treatment
treated
Phillips1 8 ,
and
would
compared to the regular
be
perturbation
this Hamiltonian
treated to
to the more intimate
problem as
such
describe proximity
perturbation theory.
this
characteristics of
proposed that
Hamiltonian were
is needed due
of the
of the single-particle
Fermi energy and V
BCS interaction.
which couples
the two sides
The
gives
state
1k x>
the magnitude of
tunneling Hamiltonian
is defined to
be*
*The
meaning
of
the matrix
elements
appearing
equation is discussed in the next chapter.
in
this
76
.....
For
St
VA
known
in
result
the
that
oxide
to
proportional
is
current
the well
obtained
Phillips
and
Cohen, Falicov,
Hr.
real
a
of
calculate the current to first
barrier, it is sufficient to
order
(11. 10)
characteristics
of the
calculation
the
ai
t
a
convolution of the densities of states of the two metals.
In
the sandwich,
other.
It
is
therefore
reflecting
metal.
functions
for the
conditions
which
potentials in n
be
thin enough
sides,
two
be
must
satisfied
The n-
and s.
that the
of this
the Green's
self-consistency
by
and s-films
in
BCS
the
pair
are presumed to
spatial dependence
solution
the other
processes into
there are
is
the other metal,
potentials within each metal can be neglected.
nature
*The
chapter.
order
the sum
on
depends
equations coupling
addition to the
In
metal
Each
function of
virtual tunneling
the
first
beyond
go
function which
states of the Green's
over all
to
solution*.
perturbation
a Green's
described by
necessary
the
McMillan postulated a "self-consistent
perturbation theory.
second-order"
has on
effects each
in the
but rather
just
two metals comprising
current between the
interested in the
not
is
one
proximity effect,
the
treating
of
these
This
discussed in
BCS
is true
the
next
77
for
films in which the total thickness is much less than the
coherence lengths.
Because
next chapter,
8=
.8Aand
the
a
the
n-side has
one of
is a gap
side density of states because
of
potential
of
0.11AS(in units
A,=
bulk
a
of
gap
energy
There
results
densities
of the
pair
self-consistent*
has
superconductor).
in the
sides of a proximity sandwich.
of both
zero-temperature
the
of
sketch only
I1.1 is a plot
Figure
of electronic states
s-side
give a
we here
McMillan obtained.
The
in great depth
this theory is discussed
induced in
s-metal
the normal
in the normal side
electrons
can tunnel back and forth to the s-side where they enjoy the
interaction.
Note
of states
is
s-side.
the
This is
also
the
electron-electron
attractive
effectively
s-side's
that the energy
gap
as it is
same for the n-side
a reflection of the fact
coupling between
the
two metals,
in the density
the
that,
the
value
of
the
potential, the states have
s-side's
because of
no
states are
longer localized on either side of the barrier.
below
for the
At energies
self-consistent
pair
a higher density on the n-side
of
specified
various quantities
of the
precise meaning
*The
reader
the
now
For
chapter.
next
the
in
here is clarified
"standard"
some
mean
to
this
interpret
just
should
The
metal.
a normal
coupled to
superconducting junction
calculating the
McMillan parameters used in
values of the
0.5 and T1/As= 0.2.
are: 1/As
curves in Figure II.1
Nn(E)
Nn(EF)
Ns(E)
Ns(EF)
-2
2
0
-4
ENERGY
Figure 11.1
eV)
\As
79
s-side's
the
of
energy
the
Above
sandwich.
the
self-consistent pair potential the states on the s-side have
of states on the n-side
appreciable density and the density
rapidly diminishes to its
1
transmission due
on the
barrier
to
lead
Physically,
These
tunneling
to the
mention should be made of
to introduce
Zuckermann 8 0
back and
forth
freely moving electrons
must
rates
for
decay
to the McMillan model, some
extensions of the model.
attractive simplicity,
and study
and
Machida 8
added phonon
i
have
Because
been possible
it has
various complications.
impurities in the n-metal and
have
which can
other metal.
In closing this introduction
of the model's
scuttling
the characteristic
below
energies
limit.
Cooper
the
which contribute
to be freely
between the two metals.
have
of
modifications
really be considered
his
the potential
presence of
to the
only electrons
the
within
anticipated above, that restrictions
He finds, as
.
the bulk normal state.
Cooper limit
the
discussed
McMillan also
model
value in
considered
Kaiser and
magnetic
Chaikin, Arnold, and HansmaS6
self-energy effects
to
the model.
As
mentioned above, in the next chapter we generalize the model
to include the quasiparticle spin-paramagnetism.
80
CHAPTER
INTRODUCTION
constant except
for a
proven quite
useful
approach has
in
of
density
the
approximations
states
the
scattering off
for
a
to
leads
in
reasonable
of
but rather
interesting features
also
makes
Bogoliubov-de
and
trivial
more realistic
wave
in the
the
Gennes
at
vector
regular potential barrier at the
It
namely
and
from
the
simplest
only
that the
is
in the
however,
uninteresting
more
slightly
such as
in the Hamiltonian,
Fermi
the
Employing the
superconductivity,
for
structure
interference
resonant
to
the
two metals in the bilayer
responsible
complicated details
change
studying
for
a thin-film bilayer,
Inclusion
result.
due
This
step discontinuity.
discontinuity.
difference between the
interaction
the
solving
Gennes equations for an assumed pair-potential
Bogoliubov-de
which is
on
is based
approach
One
effect.
proximity
to understand
been used
which have
theoretical approaches
various
the
reviewed
we
chapter
preceding
the
In
the
PARAMAGNETICALLY LIMITED
FORMALISM
PROXIMITY-EFFECT SANDWICHES:
A.
OF
TUNNELING MODEL
III:
an abrupt
interface
or
a
interface, results in more
properties of
theoretical
formalism quite
these sandwiches.
calculations
formidable.
the
in
It
is
81
there
apparent that
presence of
Such a
the two metals.
potential barrier between
a regular
in the
a difference
or from the
normal properties of the two metals
which
simpler model
a
effects resulting from
the
reproduces
for
a need
is
for studying all
model is preferable in the thin-film regime
but the most detailed properties.
It
which
was
detailed discussion
generalizing
it
paramagnetism of
applied
chapter
In this
to
of this
the results
the
model while
include
of this model.
The next
of calculations of some
next, we
of
at the
of
give
we
a
same time
the
quasiparticles in
chapter
In this
simple
A
final section
effects
the
the superconducting
magnetic field.
basic formalism
in the
and
a model
introduced
features.
these
was given
the model
II.
McMillan'i
exactly
incorporates
overview of
Chapter
out that
pointed
develop
spin
an
the
chapter presents
interesting properties
of paramagnetically limited proximity-effect sandwiches
THE
B.
82
HAMILTONIAN
in the
As sketched
effects
complicated
the
occurring
at
interface with a tunneling Hamiltonian, H.
so that the
sandwich's
the
,
given by:
111.1
-.
S
H, =(
the sandwich, HTOT'
total Hamiltonian of
the
into account all
takes
is that it
McMillan tunneling model
essence of
chapter, the
last
can be
written as the sum of three separate components:
The matrix
from a
element T-
Ik'T>
state
n-side.
HsV
H
=
HTO
H
gives
(111.2)
<
the amplitude
to a
on the s-side
dictates
Time reversal symmetry
amplitude to the amplitude for
this
1-kl> on the n-side to the
definition
of
HT
superconductors n
sufficiently
state
incorporates
and s will
dirty that there
state
for tunneling
the equivalence of
tunneling from the state
tunneling
this
eventually be presumed
carry
we will place
no such restriction
the analysis as far
to be
is enough momentum scattering
the other side.
wave vector dependence of these
The
equivalence.
into all states on
moment, however,
The
|-k'l> on the s-side.
so that all states on each side have equal amplitudes
for
on the
fkt>
T/
sT
For the
on the
matrix elements and we will
as possible without imposing this
83
restriction.
The
potential
barrier
interface.
The
transmission of the electrons
Above
number of physical mechanism.
Fermi wave
metal-metal
potentials at
of
interfaces
well defined
very
not
bilayers are
the
speaking,
Generally
in the
mechanism
Another
presence of impurity scattering
interface.
a
out that one
difference
a
metals.
the two
vectors of
might be the
the
we pointed
would do this is
other mechanism which
from
arise
interface, could
on the
are incident
rise to,
barrier gives
effects which this
a
the
at
encounter
electrons
which
namely reflection and reduced
which
directly represents
most
Hamiltonian
tunneling
or
well
characterized.
We shall
in the calculations of this
rates
f
matrix element appears
see that this tunneling
=fV(2)
theory in the form of scattering
for scattering
from side
i to
other
side.
These are given by
-
(
ZSN
.1-T
where A is the
which
have
12A
2't'4
h
thicknesses
N
(III.3)
5(F)
(III-4)
the interface between the two metals
area of
densities of states
s
p
d,
(E )
and
normal-state
These scattering
Fermi-level
rates
can be
84
related
to the barrier
transmission
coefficient
C0
by
4 B; A;
where vF.
is
the
Fermi velocity
of metal
i and
B.
is
a
dimensionless function of the ratio of the mean free path in
metal i to
If we
the
thickness of metal i and is
take the
parameters
r
ratio of
we
to
of order unity.
find that
these two
are related by:
-1-.NE
ar.
(111.6)
Ns,:)
leaving only one free parameter in the theory.
have the
dimensions of energy
energy
units
in
of the
bulk,
and we will
Each of
scale of
as
constant.
the coherence length
that its BCS
11.3,
can be
(See
magnetic properties
Chapter
I.)
Thus
Figure 1.4
taken as
be taken
applied
When
field.
as
spatially
the scale
discussion of
superconductors given
field
constant
magnetic
also
of
significant Meissner
and the
magnetic
superconductor can
on the
field effects, we
is no
of thin-film
the
4S
pair potential
two metals are thin enough on
penetration depth that there
diamagnetism.
gap
to be thin enough
For treating the magnetic
assume that the
the
Eq.
energy
(the s-metal).
the metals is assumed
defined by
usually measure
zero-temperature
of the stronger superconductor
F;
The rates
flux
within
and equal
penetrates
the
in
each
to the
the
85
superconductor it
the field on the magnetic moment
the Bohr
is
where
from the three
quasiparticle is
of the
the order of
s-side BCS
Hamiltonians
The n-
generalized from Eq. II.9
+MIS
and
are:
A
H-)O
(111.7)
ii
\/
in this equation
The symbols
formed
vector
a
fields because it can be on
pairing energy.
H(e
is
z direction as illustrate in Figure
field alignment energy
The
of the electron,
= -A,
(r
effects of
We take the magnetic field
Pauli matrices.
important in high
the
1
magneton and
H to be directed in the
I.2b.
consider the
is necessary to
have been defined
in Chapter
II.
In all but the lowest atomic number
include the
necessary to
which
moderate
acts to
magnetic
field on
include spin-orbit
since at
to
allow
effect.
effects of
the
superconductors it is
spin-orbit scattering
of
the
We do
not
paramagnetic effects
the superconducting
state.
scattering in the theory
developed here
the present time experiments are not refined enough
extraction of
the
In
Chapter
VI,
any
information
however,
we
about
include
this
some
consideration of the magnitude of the spin-orbit interaction
in
thin
dominant.
metals
where
Spin-orbit
surface
spin-orbit
effects
have
been
scattering
is
observed
in
86
paramagnetically
should
be
limited
included
for a
thin
film
complete
superconductors
description
proximity-effect sandwiches in parallel fields.
of
and
thin
87
C.
GREEN'S FUNCTIONS AND THEIR EQUATIONS OF MOTION
We
use matrix
matrix space.
identity
and
i = 0 to 3,
Green's functions
This space is spanned
the
Pauli
with the
matrices
written as,
3
The matrix Green's
and
space
,
or
i
,
two-by-two
= 0 to 3.
0
/
-
6
=
-;
0
0
functions will
down-spin electron
anomalous
by the products of the
Green's
off-diagonal
thus
in
These products
include both the up-
functions
Green's
important
in
the
presence of
penetrates
the
superconductor.
the
The Green's function is defined
given by:
kTr
C r)
as
well as
functions
The explicit inclusion
(t)
,
0~
0
o
o
0 0C0
nonzero elements.
operators
spin space
for example,
,
1
a four - by-four
in
identity and the Pauli matrices
the two-by-two Nambus 2
are
in
G
among
of the
magnetic
the
its
spins
field
is
which
in terms of vector field
88
where T
Matsubara, 6 2 ' 6 3 i.e.,
formalism of
finite temperature
in the
imaginary time
is the
evolution of
the "time"
the field operators is described by
The
-Vr-r I
-
;tVro 7
1 ( II. 8)
finite temperature Green's function is then defined as:
)r
imaginary time
(Operators with a more positive
ordered further to the left with
denote
brackets
proximity
For
states.
calculating
the
Hamiltonian
or
coupled
by
Josephson
3
Eq.
the
energy
the
so Eq.
current
tunneling
flowing
the barrier to
useful
are
with
for
tunneling
the
between
Hamiltonian,
not
(q,q';t:),
and G
of
is
defines new
111.9
functions
associated
occupied
it
sandwiches,
from one side
Green's
Such
other.
i',
this ordering.)
thermally
over
G S(R,k';T )
Green's functions,
which describe propagation
the
effect
i be equal to
necessary that
types of
an average
argument are
a sign alteration for each
interchange of operators requi-red to produce
The
(111.9)
operator.
ordering
time
imaginary
the
is
TI
where
c
metals
the
such
as
the
current at finite temperatures.
111.8 implies that
the imaginary time derivatives of
89
the field operators are given by:
T2
7(III.10)
1U
We use this to differentiate the Green's functions to obtain
For the s side Green's function,
their equations of motion.
we
find
3
A8s
+
(III.11t)
'I"
where the matrix T{.. is defined as
-0
O
%i:~o
0
-T- .. 0<m
0 0
a
0-T.
of the
the standard BCS-Gor'kov factorization
We have used
two particle Green's
function resulting from the interaction
term in the Hamiltonian using the definition
LZ7
t
-0, 'te)
As should be obvious but is worth emphasizing,
in
111.13
is of
sandwich evolving
full
the
field
according to H
of the
H t.
self-consistent pair
Thus the
also depends
on the
properties of
coupling between the two
sides.
the averaging
operators for
and not the
uncoupled metals,
the operators
(III.13)
the
average of
which evolve
potential on
the n-side
n-s
with
the s-side
and on
the
90
define
We
imaginary time dependence
the
of
representation
series
the Fourier
in the standard way by:
t
Got
(3(III.14)
where the
sum is over all
j and the frequencies (),'
integers
are given by
(=
|4/'
(3=1/k
with
T
(k
is
16)
'TL(III.
temperature) guaranteeing
have
the
in
Fermi statistics.
the
is
and T
will
Green's functions
that the
periodicity
1/(PAi)
constant
Boltzman's
?
appropriate
which is
for
Using these we get
v3( &a.~
3
o3)
.06 Li~r Z'r Z(A4.ivJ
A
,->
t
(III.17a)
P
The quantity in the square brackets in the first term on
left hand side
function of
superconductor,
GS (k,iw)].
s-side
inverse
of the
an unperturbed,
bulk,
of this equation is just the
superconducting Green's
the
-/
equation as
SS,
55t.p.I)1
-a)17b)
We
can rewrite
this
91
In
G
a similar manner one
which
finds
describes propagation
for the
Green's function
from the
s-side into
the
n-side:
/(III.18)
WIr./
There are
111.18
G
also analogous
the
for
expressions
Green's
(k,k';76v) as well as
to
G
functions
to Eq.
Eqns.
111.13 for
111.17
and
(k,k';/W)
and
the pair potential
on the n-side.
A
The equations
A sAfs
G
,and
G
for the
and the
pair potentials,
functions, G
31 -
A K60
G
,
two self-consistency equations for the
A rhand
tunneling theory of
four Green's
Ar, are the basic equations of the
the proximity effect.
In
the next two
sections of this chapter we
discuss the analytic solution of
these equations
as this
discuss
the
equations.
in so
direct
In
far
iteration
the
McMillan 6 "
in
the case
The next chapter
the
section,
We
Green's
we
first
function
discuss
an
equations first introduced by
where there
then gives
method of solution.
of
following
approximate solution of these
is possible.
numerical
is no
magnetic field.
results of McMillan's
92
ITERATION OF
D.
we introduce
section
on
a
describes
expression for
form
a closed
propagation of
the
perspective
The
Green's function
and
H'
respectively, are
i (i=n
with superscript
subscript
The
referring
referring
to
respectively,
to
or i=s,
respectively) and
each side,
the
of
Finally
propagation
from
are
final
and
excitation
metal
denoted by
'),
wave
functions which
to
the
other,
directed double
and
a
)',
vectors,
by
described
are
with
again
two subscripts, k
the Green's
one
lines
lines
III.1.)
(k,k';/
G
H
with a
(See Table
momentum.
double
and s,
directed
by single
the
give
Hamiltonians
denoted
initial
the
BCS
the
by
but this time with
propagator.
Gas k,k';;w-),
metals n
directed
by
superscript i,
for the isolated
functions for
full Green's
represented
solution.
described
G (k;iW)
and
G (k;iW.)
system.
the
of
approximate
to McMillan's
to
function which
Green's
the
this section, however,
The considerations presented in
some
are unable
We
normal modes
the
of
diagrammatic equations,
and re-sum the resulting infinite series.
find
In
G.
representation
diagrammatic
We iterate these
these equations.
function
Green's
the
which itself depends on
11.18,
described by Eq.
this
depends
for Gss
Equation 111.17
Gs
FUNCTION EQUATIONS
THE GREEN'S
this
describe
such
as
lines, again
93
G0 (k Ito*
J 1%
S
7w
It
G
(k,k';oI/)
S.7t
Table III.1:
with two
Green's function diagrammatic dictionary.
momentum subscripts,
superscripts
referring
to
the
initial and final electrons.
Eq.
111.12
all
Table
this
sides
The
of
time also
the
with two
sandwich of
the
The transfer matrix defined by
is indicated explicitly
summations.
Green's
but
diagrams
in the
introduced
functions are summarized in
diagrams
for
the
as are
various
the dictionary given in
III.1
Multiplying
Eq. 111.17 on the
left
by Go(k;-a)),
we get
94
for the full s-side propagator:
S
S
(III.19)
111.18 for the
Similarly,
Eq.
the n-side
can be rewritten:
kk r
propagator from the
"f
k
(111.20)
find
Plugging this into the preceding equation, we
S
s-side to
Se
A
%I
(1II.21)
which is an
s-side of
the
integral* equation for
the
Green's function on
the bare n-
the sandwich and depends only on
and s-propagators.
manipulating Eq. 111.21
By
form involving a sum over
over full s-side
an equivalent
full n-side propagators
propagators.
Setting p =k',
instead of
multiplying
.
*
A
,
by Tt,
we can obtain
and summing over k',
we get an
integral equation
for the summation over the s-side propagator in Eqns.
and
A
111.21
I_
A
111.20
5
*In our calculation
of values.
k
k is
allowed to range
r
over a continuum
95
Putting this equation back into itself, we find
S
^
4-
P
-
A
S
el
*'~i
~Js
can
One
this
continue
.b,,Jr
(111.23)
scheme
indefinitely.
111.21 for the
s-side Green's
iteration
Plugging this result into Eq.
function, we get
AS
S.
4Ac:
?1
±
VI.'
4.
Te
Tx
~'I
do
j
/..&
-
-
k
e
jA~'
This
-I
-+Pt
r
A% A
~
may be re-summed to give
S
A
S
(111.25)
hII
which is the desired rendering of
involves
sums
over
all the
Eq.
111.21.
intermediate
This equation
n-side
propagators
96
an
physical interpretation for these equations in terms
obvious
of
There is
propagators.
(explicitly) over s-side
and none
contributions
from
of
crossings
more
and
more
the
interface.
is a complicated
Equation 111.25
can
little headway
be made
some approximations.
and
111.24
one might
bear
strong
in
The infinite
series in
resemblances
to
drastic simplifying
rather
of the
momentum dependence
consider this
will not
tunneling
means of
illustrative purposes below)
series
possible
matrix
solution
only
placed on
are
and
closed form
obtain a
assumptions
111.23
Eqns.
geometric
This is
these series.
expression for
introducing
without
it
solving
is possible to
imagine it
equation, and
integral
elements.
when
the
We
than for
(other
or the exact solution
of this
equation.
McMillan61
gave an approximate solution for the tunneling
He did
model.
not proceed along
the
lines developed here,
but we shall comment in the next section on the relationship
proposed
of his
to Eqns.
We
111.21
solution to
and
approximation
contributions to the
above
series,
in
particular
111.25.
give an expression for
drastic
the
that
series
the Green's
all
function under the
off-diagonal
111.24 can be
(in
momentum)
neglected.
This
97
will
in the sandwich because the
coupling between the two metals
series for
the
terms in
the
we find that Eq.
that
such
with phases
By neglecting
averages to zero.
S
in
order
the higher
to
sums contributing
come
contributions
be expected that
Furthermore, it might
T.
zeroeth and
vanishes in
the off-diagonal terms
first orders in
for weak
approximation
be a reasonable
contributions might
off-diagonal
the
Neglecting
solution.
McMillan
the
compared to
are
equations
when these
be useful
their
sum
diagonal terms,
these off
111.25 becomes
~
9
7
5
When iterated it becomes
(II.
111.25 can be seen to be
and fourth
that the wave vectors in the second
propagators in the
that
this
importance
tunneling
Eq.
tends
to
approximation
of the
back
and
the higher
order
forth between
deficiency.
right-hand
at the same time,
summed independently in
McMillan solution compensates for
of this
term on the
third
side of Eq. 111.27 are summed over
these are
that in Eq.
series and
between this
The biggest difference
.27)
111.25.
wheread
This means
underestimate
contributions
the
two
of
metals.
the
the
The
the most significant part
98
E.
THE
TO
SOLUTION
MCMILLAN
MODEL
FOR
THE
Expansion
of
the
TUNNELING
PROXIMITY-EFFECT
to
Comparison
1.
Diagrammatic
the
Green's Function
the Green's
McMillan proposed an approximate solution for
function of the
to
permits
of
by Ansatz
was given
the
pair
a coupled
61
.
Dyson's equations
propagators
of
down.
form which
in a
section.
and
the n-
wave vector arguments
written
"solution"
diagrammatic expression given in
McMillan assumed that
his Anzatz solution,
In
This
be rendered
but can
comparison with the
preceding
form of solutions
Hamiltonian 111.2 in the
s-sides
done done when
as was
However,
diagonal
were
McMillan replaced
Eq.
the full
in
their
111.26 was
the final
bare
propagator in this equation by the full propagator giving
T+
A0
S1
111.28S)
Tez(III.29)
<
trk
These
can each indeed be seen
to be in the
form of a Dyson's
99
equation.
(p,iO.)
Defining a matrix self-energy
-
S
I
&
we
such that
-J
)
(j
(111.30)
see that*
A
(111. 31)
k
m.2,
Eop
A diagrammatic expansion of
yields:
+
+)
+
5
(terms of order
T6 )
+
14
(T8 )
+
42
(III.32)
+
In
comparing
10
)
CTi
with Eqns.
this
111.24
and
111.27,
there are
two things to note:
(1)
There are
111.32 than
more higher
many
111.27.
(2)
In some of these higher
in
the
term
last
*McMillan's original equations
T)
T
=
T't
whereas
correct expression is T
are however correct.
order terms
1
TI
given by
in Eq.
.
111.12
111.32,
the
explicitly,
are such that (with
from Eqns.
=
given
Eq.
terms in
"diagonal" approximation
in the
Eq.
notably
order
and
III.
Tg=
31,
the
McMillan's final results
100
what is
similar to
in the roughly approximate
the s-propagator
these diagrams, however,
In
is
111.24 and
exact Eq.
done in the
different from what is done
111.27.
This
independently.
summed
are
n-propagators
intervening between two n-side sums is restricted to be
bare
the
the
the
original
The
electron.
not itself summed over as
includes the
propagator
not
does
representing
Specifically
exact
many
a
in
It
order)
The
111.24.
Although this
it should be.
most
important contribution, it
there
versus n-1
intermediate
those from
contributions.
of terms
is a plethora
sums.
restricted
are
there
T,
n/2
only one
for
but
in 111.27 also
higher
in
summations
in
propagators
the
these
which one would expect
Thus this
solution
should not
approximation.
McMillan Ansatz is
best justified
between the two metals is
weak, in which
is clear that the
coupling
case it is not important to
the higher order terms.
identical
is
in
in each
when the
propagator
order
n-th
the
(as
bad
intervening
of
restricted summations are
be
as
combinations
111.32
the greatest
wave vector
the
everything.
in
summations
term
include
higher orders in T,
Going to
same
s-propagator with
to order
Ansatz is in
T 2 .)
know the exact contributions of
(Indeed Eqns. 111.24
The
ultimate test
the comparison of the results
and 111.32 are
for
the
McMillan
derived from it
101
to
experiment.
sandwiches
There
are
(specifically,
classes
thin,
9
Hamiltonian
interface,
.
This,
does
are
the
McMillan Ansatz to
the
thin
2.
Equations
The
bases
for
solution
side of
the
for
the
These
important
a
in
that
the
of
the
physics
generalization
the magnetic field
111.28
this
of
the
effects for
work.
and
111.29
can be expressed
of two renormalization matrices
ZI(iW )
for the
) for the pair
on each
mass renormalization
potential
renormalization
are defined in terms of the self-energies by:
$ a H'
WZ
where we now
belief
Renormalization Functions
of Eqns.
(iW
and
effect
coupled sandwiches)
the
using
describe
the barrier,
matrix.
states
with
sandwiches considered
compactly in terms
matrix
along
contain
weakly
proximity
gives a reasonably quantitative
for which the McMillan model
description
of
(1II.33)
assume a constant tunneling
contributing
to
the
sum,
amplitude for all
namely
T-..,
rk
renormalization matrices have the forms
= T.
The
A(
0)
0
0
7/M.
7A.- =
the
superconductor,
electron
i
A+
coupling the down-spin
H.
0
0
As
is
equations
Putting the
find that
the
34b)
the same
cases is
BCS
a
up-spin
the
couple
electron and the up-spin
as
those
hole.
just + ^H.
The
If
renormalization
define the
of one scalar function eachO.
self-energy given
inverse Green's function given in
we
(III.
for
case
the
are
hole
this shift, we can
matrices in terms
34a)
0
which
paramagnetic energy shift for these
we allow for
(III.
0
the down-spin
and
102
Id )
0
0
tO-)
where
.2
C)
0
A
0
in Eq.
Eq.
111.17
Green's function itself is
renormalization functions
111.33 into
the
and inverting,
related to the
by
the
and
up-spin
the
of
uncoupling
"This
occur when
does not
renormalization equations
scattering is included.
down-spin
spin-orbit
103
O0
~:
~tz
0
O
6)-aj
( Zi"
0
0
zir/)_4__
evaluated explicitly
[i'11-
in Eq.
diagram
the
a sum
by performing
states on this Green's
function.
We find,
Ti-60
(7I7
over the
is
n-side
for example*,
(111.36)
alp
t
Z.
Lz
rate
111.31
Z
S.
where we
-
(III. 35)
given by
self-energy
0
0
-iwfi'
X->f- 0
(z
decay
0
#
z</
The
^'' -
5
have used
i.
Eq.
v&'I
)
III.3
Finally we can
iwo
(III.37)
1- 0
which defines
use
Eq.
the tunneling
111.33 to relate the
contour in the
were defined by closing a
*The square roots
to have positive
so they must be chosen
upper half plane,
imaginary parts.
104
renormalization functions
s-side
on the n-side
to those
(111.38)
Y_
equations
From the
n-side Green's
the
(111.39)
/
(
corresponding to
and
111.30
111.31
find identical
function, we
for
equations
with the sub- and superscripts n and s interchanged.
Equations
must
111.38 and 111.39
satisfied
be
necessary that
and their n-side counterparts
the BCS
pair potentials
equations
satisfy self-consistently their
111.13.
Using the
111.35, we
VL
functions in
/~)'
Z &/(kjk=)
0+.
This
type of
~~
k}
jt
handled
in the
after converting
Manner
in Ref.
a contour
.40)
(111. 41)
sum occurs frequently
described
the sum to
(III
-
finite-temperature Green's function formalism
Eq.
become
find that the self-consistency equations
e?
where
in these
used
definition in Eq.
renormalized Green's
VZo4'
,L1 ',(H)
h6,
is
it
addition
In
simultaneously.
and
84.
in the
it may be
We
find,
integral, deforming
105
the contour, and then integrating, that
ApH) =Vi
N5(EF
Re
hhf
where
is
:
quti
o.42)
the solution ofE the implicit equation
Z.
--
(111.43)
as
and where we have defined the renormalized gap
(111.44)
In the case where
:
to
find it convenient to make this
In the next section we
of variables.
change
form allows
present
The
gap equation
to the
superconductor
the
a degradation
At
temperatures.
curve
field
Pauli
this equation
factors in
to
in
of
paramagnetic
the
calculated the
(for
the
Fermi
The
limit.
curly bracket)
at
pair potential
zero temperature
BCS
isolated
an
terms in the
(the
of
who first
drawn
be
to
direct connection
a
of Sarma4 1 ,
versus
parameter
order
lead
from
integration variable
changing the
equation by
E~s.
McMillan's self-consistency
a form identical to
be cast in
111.42, can
Eq.
H = 0, the gap equation,
BCS
finite
case)
factor in the bracket gives unity unless the field energy
greater
allowed
than the pair potential.
to
be
smaller
that
the
the
is
When the pair potential is
gap
energy
we
get
the
106
unphysical
solution
connecting
the
at
=
,H
proximity effect sandwiches,
the physical curves
influence at
energy starts
term in the
It
chapter.
the next
in
see
The
causing
the first
the gap
to exceed
energy
bracket to suddenly go from one to zero.
Eq. 111.42,
reflected in
because
renormalization
for
final equations
in these
acquired a field
sandwich
the term in brackets influences
the field
non-trivial way.
be
In these
Figure 1.7.
in
the
zero-temperature always begins when
is only
potentials,
and
at zero-temperature as well as at finite
we shall
as
temperature
Os /2
equation
gap
aH =,&,
at
point
superheating
supercooling point
dependent
field
the
of
that the
BCS potentials
dependence and this field
of
functions
the two
of
implicit
the
on
BCS pair
magnetic field enters in a
The self-consistent
the properties
the
the
have
dependence will
sides of
the
of
the
dependence
self-consistent
pair
analogs, and
Eqns.
potentials.
Equations 111.38, 111.39,
111.42 are
their n-side
the fundamental equations of the tunneling model.
Their solution,
which must be
the most trivial cases,
done numerically in
is discussed
all but
in the next chapter.
107
OF
A.
PROXIMITY-EFFECT SANDWICHES
PARAMAGNETICALLY LIMITED
INTRODUCTION:
superconductor
of
solution
of the
of two
pair
field.
a
the
Green's
the sandwich
can be
potentials,
equation,
functions
found
we
functions on the two sides are
of
this
that
for
the
Given
the
model
Hamiltonian
absence of
original
McMillan's
of
for
functions
expressed in
Zi(E)
renormalization functions,
Green's
the
Dyson's
BCS
comprising
each metal
When
the
effect in
proximity
terms
tunneling
the
limited
paramagnetically
lines
the
along
a solution of the
presented
we
Pauli
a
for
model
tunneling
values
CALCULATIONAL PROCEDURES
preceding chapter,
In the
PROPERTIES
FOR THE
PREDICTIONS
TUNNELING MODEL
CHAPTER IV:
form were
and
put
(E).
into
renormalization
the
interrelated by
the following
equations:
Y
z
-
(
(IV.1)
(IV.
2)
108
E)
1zt
-=
il(IV.
Zn
3)
4t)
(IV. 4)
The BCS
pair potentials in
the two metals
themselves must
satisfy the self-consistency equations
4z(E) E
0
0 tE)Z
(IV.5)
and are thus functions of
In
both field and temperature.
this chapter we present
these
equations
and
discuss
the
a
results of
number
proximity effect sandwiches which can
solutions.
We calculate
density of
states of the
three main
potentials
in
self-consistent pair
n
and
(3)
of
these
be derived from these
quantities:
s;
(2)
the
calculational procedures.
this
In
the
field-dependent
of Eqns.
IV.1
the magnetization, susceptibility, and
In
(1)
be obtained
potentials which satisfy Eq.
sandwiches.
of
to IV.4 for given values of the
which depend implicitly on solutions
and
properties
sandwiches which can
from solutions of Eqns. IV.1
pair
of
solutions to
section
we
IV.5 and
to
IV.4;
free energy
outline
Sections B and
our
C of this
109
a
sandwich and
normal-superconducting
for a
calculations
zero-temperature
of
results
the
present
we
chapter
two -
superconductor
The numerical method of self-consistently
calculating the
coupled)
proximity-effect
(weakly
sandwich.
Renormalization Functions
The
1.
using Eqns.
renormalization functions
Af
and
A
The
which are
properties:
be
below
ftL
Eqns.
IV.1
(5 and
E\j).
solutions
to
the
energy in
gap
renormalization
of the
,
as well
can
be
well above
as energies
solved by
At
t(A
E)
Initial values of
values are
used in
IV.4 to calculate
The iterated
are
chosen to
are chosen to be
the right-hand
the next
_a
straightforward
a
A'and initial values of Z1 (E)
These
unity.
the
the
Above this energy they are complex.
iteration procedure.
be equal to
BSC pair potentials
associated with
to IV.4
IV.1
Eqns.
following
which
equations are real.
energies
have the
A!(and
assumed
spectrum)
excitation
IV.4
to
which is
out to
turns
IV.1
to IV.4.
At energies below a certain energy fl,
of the
a function
IV.1
in solving Eqns.
used
Eqns.
solutions to
pair potentials
initial values for the
We assume
follows:
as
IV.5 is
to
IV.1
sides
of
iterate values of
values of these
quantities
again put into the right hand side of these equations
are
giving
110
To get solutions in
iteration procedure does not converge.
with
the
of E
initial value
out
turns
In practice it
down tol.f.
energy region where
that the
differential
to be obtained from the
which is large enough for solutions
iteration procedure
are differentiated
resulting
integrated from an
E are
in
equations
to IV.4
The
energy.
to
respect
IV.1
Eqns.
regime,
this energy
this
slightly above -01,
For energies
converges.
until it
The procedure is repeated
A%)and ZjE).
of
estimate
a better
procedure does
iteration
not converge is very narrow and it is possible to accurately
interpolate
above
far enough
those solutions
-f2t to have
interpolation scheme when we calculate
interpolate
interpolation
potentials
We
however,
could,
(We
regime.
in
scheme
use
the
of
the
pair
as well.)
that
require
self-consistent. It
the
values
above
is necessary
functions, when put into Eqns
assumed
whole energy
justifiably
calculation
the
not
do
we
over the
solve the equations
but
the
In calculating
however,
potentials,
pair
this simple
the density of states
and the magnetization as outlined below.
self-consistent
a converging
We use
iteration procedure.
solution from the
and
just below
between solutions at energies
for the
quadratically convergent
pair
solution
fully
renormalization
that the
IV.5, reproduce
potentials
be
the initially
Anh(H). We
Newton-like method
use
a
to numerically
111
solve
2.
these nonlinear self-consistency equations.
The Density of States
well known that the analytic
It is
temperature Green's
are such that the
functions
electronic states* N(E),
of finite
properties
density of
defined by 8 s
N'CE) = 3
(where the
by
E-
)
sum is
over all states (k o>
we
function
spin-up
are
and we
electrons
electrons
respectively.
HN(E)
with
from
For example,
matrix
a four-by-four
can separately
NI(E)
related to
The usual relationship is
function.
dealing
with energies denoted
representation which is
has a spectral
that of the Green's
Here
civ 6)
and
G
the density
calculate
the
(k,kjiW)
Green's
density
and
of
spin-down
of
G
(k,kii)
for N(E) we have
each side
of states for
defining a "bulk" density
*We are
of states in
equal to the local density
which is actually
on the
films are so thin
This is because the
each side.
superconductors
lengths in
characteristic
of the
scales
vary
not
do
states
of
densities
local
their
that
appreciably across the films.
112
7--
of
terms
in
expressed
functions
Green's
The
(IV. 8)
the
renormalization functions given in Eq.
111.35 can be used in
the right hand side of this equation.
We
Et
2(IV.9)
N (E)VEt)Ree
--
_(__
N'(EF)
- <tE)
.[ZtE)E±
before, MT(E
where, as
find
of one
of states
is the density
)
spin orientation of side i at the Fermi energy when it is in
the normal state.
3.
The Magnetization, Susceptibility, and Free Energy
magnetization
The
M
on each
side of
the sandwich
is
defined by
The
thermal
Green's
average here may be
functions
manipulated to
and
expressed in terms
resulting
the
be an integral
expression
over the density
of the
can
be
of states.
The result is
MC,
VO
E
Thus once we have calculated
[N(EHEI
the
+
(IV.10b)
density of states from the
113
Green's
function we can easily calculate the magnetization.
the magnetization with respect to
From the derivative of
the
magnetic
we
field,
get
the
spin
paramagnetic
susceptibility
M
(IV. 11)
In
performing
account has
pair
the differentiation
to be taken of
potential
for
a
superconductor,
the implicit dependence
on H as well
as the explicit
of the
dependence of
the magnetization on H.
If
Gibbs
we work
at constant temperature,
the change
in the
free energy density as we vary the field is given by
Ot H
_
where B is
--
(IV.12)
-
the magnetic induction of side
i.
The magnetic
induction is related to the magnetization by
13
(IV.13)
-
Thus the magnetization is related to the free energy density
by
SH
If one is
o
MH(IV.14)
'rr
interested in the free
between the normal and the
energy
density difference
superconducting states,
the
first
114
term in
B cancels
and we
have
(IV.
_____
the magnetization, it would
Since we can easily calculate
be
integrate
to
able
be
to
convenient
At H
integration.
the
sandwiches,
discuss
the corresponding results
Pauli paramagnetically
we
can
point of
effect
constant
for example).
susceptibility
magnetization, and
energy,
proximity
coupling
conventional
the
integration' 2 ,
We first
for these
0
=
one
this in itself would be a formidable calculation
doing
(when
so is
doing
states at the starting
superconducting
normal and
and
difference between the
know the free energy
to
first needs
difficulty with
The
magnetization.
IV.15
Eq.
using our knowledge of
calculate the free energy difference
the
15)
use knowledge
of an
limited superconductor.
of the
form of
for the free
We
isolated
see how
the proximity-effect
phase diagram gleaned from this simpler example to avoid the
difficult initial
the
between
calculation of
normal
and
the free energy difference
superconducting
states
of
the
sandwich.
In
isolated
Chapter I
Pauli
we
discussed
paramagnetically
the
phase diagram
limited
of
an
superconductor.
115
(See
Fig.
1.7.)
we
see
field,
From the
that
and
supercooling
at
free
energy as
low
temperatures
of the
end point
supercooling curve.
In Figure
first and second
derivatives
These
IV.1 we
are
as
an
that of
the
display again
the
these curves and
of
derivatives
the
give
well
curve to
superheating
zero-temperature free energies
include the
there
of
gap equation which connected
unphysical solution of the BCS
the
as
curves
superheating
a function
of
we
these
magnetization
also
curves.
and
the
susceptibility, respectively.
The susceptibility of the normal
state is the well known
constant Pauli susceptibility
84
and
this
results
linear normal
in the
state magnetization
and magnetization reflecting
a vanishing susceptibility
the fact that the up- and
down-spin electrons in the Cooper
are locked together and the paramagnetic
pairs
great
enough to
break
state susceptibility
the
<IV.16)
A superconductor at zero-temperature
shown in Figure IV.1.
has
N'(E,)
solid
horizontal
indicated those
state
lines.
curves
The
these
pairs.
superconducting
and magnetization are thus
lines
in
Figure
portions of the normal
that can be
vertical
The
energy is not
IV.1.
in
the
We
have
and superconducting
physically realized by
jump
plotted as
solid
line
the solid
for
the
116
Free
Energy
Gs
0
0.5
0.707
1.0
BH
/PBM
X Pauli A
normal
0
0.5
0-
/
N
/
)
/
/
/
/
-N
(b)
\
/
-"supercond.
0
0
I
I
0.5 0.707
0
I
1.0
L
BH
A
k
x
X Pauli
4-
-Q
normal
I
0*
- -
0
supercond.
0.5 0.707
Figure IV. 1
(c)
1.0
*p H
B
117
magnetization and the susceptibility occurs at
free energies
the
of the normal and
superconducting states
lines
the extensions of these
We indicate by dashed
cross.
beyond the
solutions
for the superconducting
field H
for the normal curve.
of the unphysical
where the
energy gap
a
has
the
to zero.
function of
as the
The
is greatest
and
It
supercooling point
field for
the
an isolated
energy and
gap
all the
Pauli
Fig. 1.7)
of
magnetization
to zero as the energy
spin aligning
of the
(See the plot
superconductor given in
point
superheating
2 H
net magnetization.
value at the
energy gap goes
unphysical state goes
large
has a
This solution
negative slope.
the normal state
as a
connects the
less than the spin-alignment energy
paramagnetically limited
At
solution which
have broken pairs and a
magnetization
equal to
curve to the supercooling
supercooling points.
energy gap which is
so it does
superheating
dashed line the susceptibility and
We also indicate by a
superheating and
to the
field
critical
field H
free energy
is the field at which
This
critical field.
the first order
/,r2,
AH
this
is then as
electrons
become spin paired.
Many features of the
the
susceptibility
paramagnetically limited
free energy, the magnetization, and
curves
of
an
isolated
superconductor will be seen
Pauli
to be
118
for paramagnetically
the corresponding curves
reflected in
there
In particular
effect sandwiches.
limited proximity
will be a supercooling field at which the pair potentials of
the
solution goes
Thus
same
energy.
these two solutions have the
starting
point for
use
thus
can
We
the magnetization
Then, by
the
by
density
When we
free
field, the
be equal
to that of the
integrating
the difference
energy of the unphysical state will
superconducting solution
between the
integration, and
superheating
to the
superconducting state.
as
the unphysical state.
between the normal state and
integrated out
this point
in free energy
using IV.15 calculate the difference
have
unphysical
state solution.
into the normal
continuously
at the supercooling point
free
point the
this
At
zero.
to
sandwich go
and the
normal state
solution, we can calculate the difference in the free energy
density
between the
state.
This
proceeding, but
minimum
of
is
normal state
a
albeit
it gives
the
additional
and the
rather
superconducting
calculations
of what we know
beyond
from these solutions and
of
with a
free energy densities
self-consistent solution for the Green's function.
full use
way
roundabout
the
It makes
what we
119
the qualitative shape of the free energy curve*.
know about
This
calculational procedure yields the difference in the
density between the normal
free energy
solutions
on
each
side
transitions are determined
We shall
sandwich.
free energy density
is
equal
to
normal-state
the
by the total free
for each side
gi
free
energy of the
normalized
calculating a
actually be
the sandwich which
of
divided
density
energy
phase
The
sandwich.
the
of
and superconducting
by
the
Fermi-level density of states,
(IV.17)
N (E,)
The total free energy of the sandwich is then
+
Sd N
d )S
pK
Thus
proper weighting of
the
determined
parameters
ratio.
the next
by the
that
same
can be
(See Eq. III.6.)
section it
ratio
the free energy
of the
determined from
For
CIV.18)
densities is
proximity
coupling
the specific
heat
the sandwiches we consider in
turns out that
the critical
field is
of the
an explicit calculation
also have avoided
*Note we
and
normal
the
between
difference
energy
free
the
with
associated
is
that
states
superconducting
tunneling Hamiltonian.
120
almost entirely determined by the free energy differences of
the s-side.
pair
This is because near the transitions
potentials,
differences,
and
therefore
are small
in
the
n-side
comparison
the n-side
free
with those
energy
of
the
s-side.
4.
Finite Temperature Calculations
All
of
the
temperature
above
sandwiches.
numerical results
temperature
formulas
We
more
zero-temperature ones.
point of isolated
We
finite
present
Finite
difficult than
They
would
be
the
most
phase diagrams
the region near the tricritical
paramagnetically limited superconductors.
choose not
concentrate on
for
sandwiches.
certain details of the
of these sandwiches such as
sandwiches.
no
valid
however, only
for zero-temperature
useful for exploring
Here we
shall,
calculations are
corresponding
are
to explore
these
the detailed properties
shall see that these
features
but instead
of zero-temperature
properties
themselves
are quite interesting.
Finite temperature
calculations
vicinity of a second-order phase
parameter,
zero.
in this
case the
In this regime,
become
simpler in
transition where
BCS pair
the
the order
potential, goes
to
the renormalization equations become
121
possible to make some headway analytically.
it is
linear and
We discuss some analytic results in Appendix B.
procedures we will
We have now outlined the calculational
sandwiches
results
of
order parameters,
states, the
of
calculations
numerical
the
present
sections we
two
next
In the
.
an
proximity-effect
limited
paramagnetically
Pauli
isolated
results for
corresponding
indicated the
and have
use
densities
the
of
magnetizations, free
and the
and susceptibilities of two representative
energy densities,
First, in Section
paramagnetically limited superconductors.
B,
we consider a sandwich consisting of a superconductor and
a
much
superconducting
weaker
so
the
of
superconductivity
n-side
consider
superconductor
sandwich
a
coupled
these films
remains
(n-side) and this
n-metal
film).
take
the
the weaker
The
parameters of
(i.e. an
the
strong
but
still
proximity
loss
of the
superconductor
transition can be compared to
weaker superconductor
a
case, there
In this
to be weak.
superconductivity of
isolated thin
We
transition associated with the
a phase
"inherent"
of
weaker,
another
superconductor.
moderately strong,
coupling of
to
In Section C we
consists
which
its
by
swamped
are
proximity-effect-induced superconductivity.
then
"inherent"
the
of
vestiges
all
that
strong
be rather strong,
these two films to
proximity coupling of
the
take
We
"n-metal."
that of an
isolated thin
two sandwiches
we
122
study are
given in Table IV.1.
123
Table IV.1:
Proximity-effect sandwich parameters
Sandwich in:
As
Section
F
X
(E )v
Section
1
1
0.0011
0.32
152.
N'(E )V
B
C
6.46
0.08
0.27
0.175
0.39
0.46
0.02
0.184
0.04
'i- is the energy gap of metal 1 in the absence of
the proximity effect.
124
A NORMAL-SUPERCONDUCTING SANDWICH
B.
next we
this section and the
In
proximity-effect
sandwiches
zero-temperature.
In
magnetization,
the
parallel
in
we consider
this section
spin
the
thin
of
susceptibility
the
and
results of
for
model
pair potential,
the
energy,
free
the
states,
of
densities
tunneling
the
using
calculations
present the
at
fields
a sandwich
The
which is effectively a normal-superconducting sandwich.
much weaker superconductor
"n-metal" used is actually a
superconductivity
proximity-effect-induced
the
at
look
the
two-superconductor
a
of
properties
overwhelms
In the next section we
its weak inherent superconductivity.
will
and
sandwich.
The
are
parameters of the sandwich we discuss
in
given
first column
the
superconductor, when
isolated and not
zero-temperature, zero-field energy gap
the
unit of
calling
a
The n-metal,
energy.
"normal
superconductor which
zero-field energy
gap
metal"
of
widths
associated
sufficiently
strong
4
that
the
in proximity,
has a
AS
which
a bulk,
=0.0011,65.
with
s-side
is
metal,
would have
The
IV.1.
Table
of
the
in this section
which we
take as
we are
loosely
weaker
a
actually
zero-temperature,
The
tunneling
tunneling
model
proximity-effect
decay
are
induced
125
overwhelms its weak inherent
superconductivity in the n-side
superconductivity.
In Figure II.1
for
prediction
plotted the tunneling model
we
the zero-field density of states for both the n-
and later
the curves in this figure
figures indicates which
semiconductor model.
in the
considered occupied
states are
in the n-side
also the energy
gap in the s-side density
half
of
the
amplitude in
energy states
longer localized
are no
the sandwich
but
sandwich,
be seen
s-side density of
states
finite
that the
lowest
n-side.
in the
are primarily localized
in either
a
have
they
rather
It can
each side.
of states, which
that in the n-side because the
must in fact be identical to
in
This is
density of states.
of 2 Jl0=2(0.34)
states
energy gap
a substantial proximity-effect induced
There is
the
shading under
The
a proximity-effect sandwich.
s-sides of
and
value
is greatest near the
The
of its
self-consistent pair potential which, in the presence of the
proximity effect, is reduced to 0.81 As.
this energy reflecting the
of states decreases rapidly near
fact
states
there is
that suddenly
on the
states
s-side.
resembles, except
Abrikosov-Gor'kov8 6
by Fulde
and Maki0
for
density
depaired superconductor.
7
that
a much
larger amplitude
rounded shape
The
The n-side density
the
of
of
the density of
two hump
states
(Indeed, it has
for
feature,
of
been
proximity effects are
a
the
partially
pointed out
in certain
126
to the
to
identical
limits
depairing
other
of
effects
perturbations.)
As
a parallel field is
and down-spin quasiparticle states are
top
the
of
portion
of states
H =
plot,
this
energy gap
£Q0
and is not great
The number and
this field strength
cross
pair
the Fermi level.
state Cooper
field case.
of
field
is
There
zero-temperature.
and
As
pairs
a result,
potentials have not been altered by
a
of
application
(half)
enough to cause the up-
from the zero
the self-consistent pair
As
the zero-field
the nature of the ground
remains unchanged
these
The applied field for
states to cross at
down-spin densities of
the
than
less
In
corresponding densities
electrons.
0.2gry is
In
down-spin
n-side and the s-side.
we give the the
for the up-spin
the
give
we
figure
densities of states for both the
the bottom portion
The
shifted by ±dAf.
split as shown in Figure IV.2.
spin density of states is
the
the up-
applied the energies of
also no
this
magnitude
net magnetization
and the spin susceptibility
at
at
is zero at
field values.
the field
at
increases enough
the Fermi
states
have
quasiparticles with
with the field.
level
(Fig.
energies
to cause
IV.3),
some of
exceeding
both of their magnetic
In this
the states
that
to
the ground
of
two
moments aligned
case, as the shading in Figure
IV.3
N(E)
N (E F)
2
sn
N1(E)
N(E F
0
0
-2
ENERG
Figure IV. 2
2
2
N (E)
N(EF)
2
Nt(E)
N(EF)
0
ENERGY
Figure IV. 3
2
( eV
-)
129
field
exceeds _1l./,us,
start
zero-temperature
superconductor
at
in Fig.
IV.4)
(indicated by the dashed line
the critical field
from zero through
as the
potential
pair
a constant
would have
s-metal
limited
paramagnetically
isolated
an
of
behavior
The
reduced.
be
to
IV.4)
in Figure
plotted
lines
self-consistent
in the
is reflected
(the solid
pair potentials
breaking of pairs once the
This
the spin alignment energy.
due to
at lower energy
down-spin which are now
and occupy
their spins
electrons flip
of the up-spin
some
indicates,
field varies
up to the-
all the way
superheating field.
As
the field
decrease.
continue to
sandwich when
Pauli
the free
energy of
the
and the
potentials
pair
the
and
increases
depairing
resulting
the splitting
increases further,
of the
the critical field
We reach
energy of
the
sandwich in
the
equals the
sandwich
normal state.
The
proximity-effect sandwich then undergoes a sudden transition
into the
reflection
normal state.
of the
zero-temperature
is first
The transition
of the
first-order nature
transition
of
an
paramagnetically limited superconductor.
with
an isolated
there is,
paramagnetically limited
associated with the
proximity effect
equations for the
sandwich, a
order, a
field-driven
isolated
Pauli
In direct analogy
superconductor,
first order transition of the
superheating solution
pair potentials as well
of the
as an unphysical
130
1.0
ph
P
~A(H)
S.
/
As
LU
In~
0.5
n/
-/
0
ph
0~r
0
0
hsc hc hsh
MAGNETIC FIELD
Figure IV. 4
1.0
h = P-BH
131
normal state
supercooling extension of the
Figures IV.5 and
In
The total
respectively.
these
density
the
sandwich,
the peaks
in the
another once
cross
appears once
the
which
of states
the
a superconductor's
zero-bias
n-side.
states to cross
states
of
These
Fermi level.
peaks
region
in
The
states
density of
a
to cause
large enough
are
the field-induced splitting is
at
The most
peak
spin densities
in the
energy gap
measured by
curves is the large zero-bias
large peak results from
This
Fermi level.
identical to
at zero-temperature.
and down-spin densities of
the up-
of
of states is
induced splitting is
magnetic field
them to
density of
s-sides
density
sides
dramatic feature of these
the n-side
plot the total
characteristics which would be
the conductance
tunneling into
IV.6 we
and
n-
the
for
states
the
does
but it
1
for the isolated film at finite temperatures" .
occur
in
This
corresponding
temperature,
zero
superconductor at
limited
superheating
paramagnetically
Pauli
isolated
an
of
parameter
order
dotted
for the
not occur
evolution did
the
by
the unphysical curve.
solution goes continuously into
continuous
These
be seen that the
It can
in Figure IV.4.
lines
solution.
indicated
are
gap equations
the
of
solutions
the
to
point
superheating
the
connecting
solution
the addition of
just
above the
added to
one
sufficient for
zero-energy
and,
at the
region of
correspondingly
normal-oxide-superconductor
132
Tunneling to Normal Side
4
2
=0.0
U)
O
(I)
N(E)0
>.N(EFI
-
/.BH
As
z
LU
0.2
O
4
_J
0
2
-=0.4
OL
-2
ENERG Y
rigure IV.5
2
0
\
eV
s I
Tunneling to Superconducting Side
4
133
2
H
s
0-
(fO
=0.0
L- N(E) 2
0~ N(EF)
= 0.2
-2-
H
0-2
Figure IV. 6
0
ENERGY
eV
$ '
=0.4
2
134
tunneling experiment, are usually
The dramatic peak structure in
should
be
readily
featureless flat regions.
the n-side density of states
observable.
In Figure IV.7 we plot the magnetization on both sides of
this
h =
sandwich
MOHg/Ag.
as
We
a
function
of
indicate
a
dimensionless
with
solid
superconducting state magnetizations of the
sandwich
and
also
magnetization.
the
corresponding
Dotted lines indicate
superheating
comparison
the
and
the
two sides of the
state
the magnetizations
solutions,
magnetization
s-metal paramagnetically limited
lines
linear normal
the unphysical
the corresponding
field
of
and
a
of
for
isolated
thin film is given
as the
dashed line.
As
we
indicated above,
magnetization
reflecting
of
the
of
states
both sides
cross
is
an onset of
the superconducting
breaking of
electrons can align
Once the field
there
of the
Cooper
at
pairs
-Wco
the point at
the Fermi
level,
sandwich starts
the
h
so
their magnetic moments with
exceeds
at
some
state
a nonzero
=/tv
that the
the field.
which the density
magnetization
to grow.
The
rate
on
of
growth of the n-side magnetization is very rapid and exceeds
that of the linear Pauli
The growth
magnetization in the normal state.
of the magnetization
increased further until
continues as the
the critical field of
field is
the sandwich
135
a-
m
0.5
z
0
N
ZJ
Or
0
a0o hhsc
hChhsh
hc
As
MAGNETIC FIELD
Figure IV.7
h=A 'BH
I
136
is reached at
which point the sandwich makes
a first order
transition into the normal state.
The susceptibility, the
is
plotted
in
physically
Figure
derivative of the magnetization,
IV.8.
The
solid
are
the
realizable curves.
The
magnetizations of
the
lines
superheating solution,
the supercooling
solution, and
the
unphysical
connecting
superheating
and
solution
supercooling points
are indicated
fields
the
which allow
susceptibility,
the
n-
the
with dashed
breaking of
pairs
and
susceptibilities
s-side
similar in shape to the respective
states
above
identical if
decreasing as
the energy
gap.
n-
and
(These
the self-consistent
unusual
the field increased.)
In the next chapter
behavior
in
the
and
a
For
nonzero
are
s-side density
would
in
fact
pair potentials
Note that
field dependent susceptibility actually
normal state.
lines.
of
be
were not
the n-side
exceeds that in the
we comment on
susceptibility
how this
might
be
experimentally verified.
A
zero-temperature
superconductor
case in
is
which this has
partially depaired
In the
an
nonzero susceptibility
unusual occurrence.
been predicted
occur for
The only
to occur is
state postulated by Fulde
Fulde-Ferrel state,
momentum states
the depairing
localized
in
a
pure
other
in the
and Ferrel 3 9 .
is predicted
in certain
to
regions of
137
xS
2
I
F0
0
-1
D
co
-2L
0
hsc he h sh
M AGNETIC FIELD h=A
Figure IV.S
I
Ss
138
As one might guess even a small amount of
the Fermi sphere.
in
defined regions
well
prohibit the
scattering would
impurity
momentum
been
Fulde-Ferrel state has never
the
sandwiches, the depaired states are
and
indeed
In the
such
the
case of
proximity-effect
these
localized in real space,
the possibility
of observing
reasonable.
For example,
primarily on
the n-side,
such a state
appears to be quite
the n-side Knight shift should
space, and
observed.
in
state
depaired
partially
existence of
show
a sharp growth as we go
into the partially depaired state*.
curves
We integrate the magnetization
to get the
Figure
densities which are
Gibbs free energy
IV.9.
The
realizable solutions
lines indicate
and the normal state
the unphysical
the
indicate
lines
solid
as described above
plotted in
physically
solution, dotted
extrusions of
these curves.
The dashed curve is the corresponding free energy density of
a isolated paramagnetically limited
for comparison.
The n-
and s-side
the superconducting state
cause the
Thereafter
free energies are flat in
until the field is great enough to
up- and down-spin
the free
s-metal and is included
densities of states
energies densities
start to
to cross.
decrease
reflecting the response of the increasing number of depaired
next chapter we
*In the
systematically.
discuss
possible
experiments
more
139
O
normal
state
n
s
LU
LL
LUJ
LL
0
I
I
1
S2
h
h h
A
sc
MAGNETIC FIELD
Figure IV. 9
C
sh
h
=PBH
As
140
electrons
sandwich free
and s-side free energy densities*.
IV.18) of the n-
sandwich, it
of
density
susceptibility
occur
level.
most interesting
that the
field needed
exceed the
spin-split density of states to
We indicated in Chapter
cross
we
have
seen
that
the
superconductors.
is
still
at the Fermi
transitions
In
effect
proximity
zero-temperature transition with increasing
phase transition
to
I that the proximity effect
should also influence the nature of the phase
paramagnetically limited
n-side
These effects
of the n-side of the sandwich.
values which
features
larger-than-normal-state
the
and
states
in the
which occurs
zero-bias peak
for field
cause the
this proximity
the predicted behavior of
is clear
the huge
were
the total
weighted sum
given by the
energies which is
In summarizing
of
by the crossing
determined
entire sandwich is
(Eq.
The critical field of the
to the magnetic field.
first order.
in
this example
rounds
field,
In fact
but
the
the
the most
interesting features of the phase transitions occur not when
the entire sandwich
superconductor
loses
goes normal but rather
its
inherent
when the weaker
superconductivity
but
induced
the proximity-effect
weakness of
of the
*Because
very
occurs
crossing
the
the n-side,
superconductivity of
free
superconducting
and
the n-side normal
close to where
the
in
difference
the
Indeed
cross.
densities
energy
discerned
be
cannot
s-sides
and
nthe
crossing fields of
on a plot of this scale.
141
The inherent superconductivity of
effect.
the proximity
of
superconducting because
remains
the sandwich discussed in
this section was too weak compared to the proximity coupling
of its own superconductivity.
strength to show any vestiges
In the next section we study a sandwich which consists of
comparably
two
strong
are
which
superconductors
weakly
coupled by the proximity effect.
C.
A TWO-SUPERCONDUCTOR SANDWICH
In
this
in
interested
we are
section
transition and the influence
zero-temperature phase
the
studying
of the
proximity effect from a stronger superconductor on the phase
transition
with
superconductor.
Chapter I
and
I.10
expect,
We
the
that
increase the critical field
loses
its
"inherent"
this section
moderately
strong
s-metal with only
we
the
of
a
weaker
analogy
and a metamagnet
in
drawn
(see Figs.
act
effect will
proximity
to
where the weaker superconductor
superconductivity
superconducting only because
In
from
between this system
I.11),
field
increasing
and
left
is
of the proximity effect.
consider
superconductor.
is
n-metal which
an
We
couple
a weak proximity effect.
it
to
a
the
The parameters
142
given in column 2 of Table
for the sandwich we consider are
IV.1.
Again we choose an s-metal
have a zero-temperature, zero-field energy
table
1
widths
coupling decay
0.32,A,
gap of
in the
are given
magnitude smaller than these pair
and are an order of
potentials.
we
The n-metal when isolated would
take as our unit of energy.
The proximity
which
6&
potential
pair
zero-field
zero-temperature,
which when isolated has a
This makes the proximity coupling rather weak.
plot the zero-field, zero-temperature
In Figure IV.10 we
density of states for this
As should be expected
sandwich.
for the case
of weak coupling, each metal has
states which
is only slightly
a density of
its
perturbed from
BCS-like
The most significant alteration from a BCS shape
bulk form.
which now has
occurs in the stronger superconductor
a small
but non-zero density of states below its bulk gap energy all
the
the
to
down
way
the
of
gap
energy
weaker
This is again a reflection of the fact that
superconductor.
the states are no longer completely localized in either side
superconductor, but this
plot.
anomaly in
a slight
states near the gap energy
n-side density of
this
is also
There
sandwich.
of the
of the s-side
cannot be discerned on the scale of
and
zero-temperature
At
the
field,
the
self-consistent pair potential of the s-side is reduced from
its bulk
value of
increased from
A,
0.32&S
to 0.96 AS,
to
0.344s.
and that of
The
the n-side
zero-field
(half)
143
Nn (E)
N(E)
N, (E) 4
Ns(EF
-2
0
ENERGY
Figure IV.10
( As)
eV
144
0.364A3
sides, 14, is
on both
energy gap
just
is
and this
slightly greater than the pair potential on the n-side.
As the
the
splitting of
Zeeman
parallel or antiparallel alignment
occurs due to the either
There should be
spin with the field.
of a quasiparticle's
again
states
densities of
spin
the
(Figure IV.11),
is applied
field
magnetic
no reduction in the pair potentials until we reach the field
ASH
=1J6,
where
the first-order critical field
However such a field exceeds
an
of
superconductor.
its
isolated
form by
proximity effect,
the weak
"inherent" superconductivity
the
lost.
The n-side is then
its
proximity
perturbed from
is only
the n-metal
H /,
h'
first-order critical field
reach a
of
Since
n-metal
limited
paramagnetically
isolated
states cross.
densities of
the spin-split
of
we first
=0.33 where
n-side metal
the
is
left superconducting only because
still
to the
superconducting
strongly
s-metal.
In Figure IV.12 we plot the
density of states at a field
above this first order transition
the zero-field density
Fermi level.
It can
of states would have
be
seen
that
single-spin density of states has
through
this
self-consistent
phase
but below the field where
transition.
n-side pair
the
crossed at the
energy gap
in the
been much reduced on going
is
This
potential
as
true
well.
of
This
the
is
145
N (E) 4
N(EF)
0
Nt(E) 4
N(EF)
o0
-2
0
ENERGY
Figure IV.11
1
AS
2
N (E)
N (EF)
Nt (E)
N (EF)
0
-2
0
E NERGY
Figure IV.12
1
( eV
)
147
plotted in
a discontinuity
shows
IV.12),
the phase transition (Fig.
have gone through
there
quasiparticles.
of spin-aligned
number
a substantial
at hG.
plot after we
of states
under the density
From the shading
are
and
Figure IV.13
in the magnetization on the
There is also a sudden increase
n-side which is plotted in Figure IV.14.
IV.14
and
Figures IV.13
the
also show
corresponding
behavior of the s-side pair potential and magnetization near
h
discontinuities at h'C .
This
is
a
reflection
The jumps
the
of
are rather
that
fact
show
also
these
that
seen
be
can
It
.
the
jump
small though.
states
with
pair potential energy of
energies below the superconducting
the s-side are predominantly localized on the n-side because
of the weak coupling.
are plots of the total density of
Figures IV.15 and IV.16
states of the
field
where
n-side
n-side and the s-side
values below
and
the n-side loses
total
spin-splitting.
just
its
respectively for these
above the
transition at
inherent superconductivity.
h
The
clearly
the
This splitting takes a rather curious
shape
density
of
states
shows
e
near zero-energy
The s-side,
at fields above
on the other hand,
the transition
field he.
also shows the spin-splitting
clearly but the features near zero-bias are small.
148
1.0
-J
------------------h
ph
A(S(H)
.
A
s
LUJ
-0.5
ph
0
s
nP
r:1
0
hscIhcIhshI hsc2
MAGNETIC FIELD
Figure IV.13
hc2
hsh
2
h=
I
BH
U)
<
L
0.5
0
normal
state
IA
NLU
0
0
----------I I
I
hschc hsh
h2sc
MAGNETIC FIELD
Figure IV.114
h2
c
h =
/-BH
AS
Tunneling to Normal Side
150
8
4
Cl)
LU
H
LL
0
H
O
8
N(E)
N(EF)
4
--
4
0
-2
0
0
ENERGY
Figure IV.15
2
2
( eV
151
Tunneling to Superconducting Side
4
LU
F0
Co
I-
0
CO
z
4
N(E)
LU N(EF
O
4
0'L
-2
0
ENERGY
Figure IV.16
2
eV)
\As
152
Pauli
paramagnetically
superconductor
limiting
field labeled as he
in the order
B
above.
This
stronger
a transition
parameter and magnetization
is
to the
IV.14, is similar
the normal-superconductor
transition of
the
transition, which occurs at a
in Figures IV.13 and
plots given
of
then have a flat density
Both sides
This second phase
of states.
field
and the whole sandwich undergoes
into the normal state.
Section
we encounter the
field increases still further
When the
in
film discussed
because
the
"inherent"
superconductivity of the weaker superconductor, the n-metal,
is
destroyed by
the
field
and
the
is only
n-side
left
superconducting because of the proximity effect.
Finally in
respectively.
densities,
realizable
portions of
s-metals.
behavior
the free
In the
dashed lines
fact that both
the
above,
the dotted
solution,
unphysical
corresponding
As
of these curves
portions
the free energy curve
the
Figure
and
susceptibility
paramagnetic
of
and
the
we
the curves
of
the
of the
to get
free
energy
plot the
line gives
the
physical
physical
lines.
and
the unphysical
unphysical
n-
we indicate
solutions.
is
of
the
curve
of isolated
curve
In
the behavior
dashed
transitions are first order
the existence
IV.13
with solid
energy densities
susceptibility
differentiate and
we
magnetizations given in
integrate the
the
and IV.18
Figures IV.17
and
with
The
reflected in
superheating, supercooling,
and
153
(n
0-
2
n
statfe
.-~2
is
0
n ss
s
- =-
n
0_
LLJ
-2
0
hChc hsh
MAGNETIC FIELD
I
h2
h/-BH
A
Figure IV, 17
s
I
H0
0
LUi
z
LUJ
LUJ
LUJ
O0
2
hschC
hhhhshh sc
M AGNETIC FIELD
Figure IV.18
c2
h
sh
= BH
155
unphysical excursion
he
are
There
.
a
to the
of
number
h.
various
of the
crossings
and
in
divergences
the
unphysical curves and these
susceptibilities of the
traced
energy near both
of the free
nearly
can be
singular
density of state curves.
weakly
coupled
existence
of
two
remnants
the
of
order when
the second
In
effect.
the proximity
by
transition
the normal state.
is
loss
of
leaving only
induced
superconductivity
the
field
from the
first transition results
sandwich goes into
first
as
superconductivity of the n-metal
the inherent
proximity
predicts
sandwich
transitions
phase
such a
tunneling model for
proximity-effect
The
increased.
the
that the
we have seen
Thus
weak
a
the
entire
transitions are
Both
coupling of
the films
is
weak.
for
evidence
is experimental
There
state
the
as
raised"8 .
is
I,
order.
of
(instead
it would
As mentioned
be
in
the
interesting
experimental and theoretical exploration
and field dependence of the order
the
sandwich into the normal
the
In this temperature case, however,
second
Chapter
temperature
of
and then the phase
inherent superconductivity of the n-side
transition of the proximity effect
the loss
field)
is
the transition
introduction
to have
a
in
detailed
of the temperature
parameter of the n-side
in
156
transition should change from first
terms
in
interpreted
order,
all*.
and finally to no transition at
of an exploration
this
temperature
increasing
With
superconductivity.
inherent
its
loses
n-side
the
where
region
the
to second order,
This change
of the
could be
"wings" of
the
tricritical phase diagram given in Figure I.11.
We
survey of
given a
have now
expected for Pauli-paramagnetically
we
describe
shall
some tunneling
undertaken in an attempt to
in the
spin-split density
experiments were
suggest
some
experimental
experiments
besides
experiments
were
that
see
these
conclusive, and we
shall
of states.
We
shall
tunneling
also
shall
lead
which might
modifications
We
In the next chapter
observe the structure predicted
promising but not
results.
limited superconductors
model.
based on the McMillan tunneling
of properties
the type
to
discuss some
experiments
clearer
other
which might
be
attempted.
*In fact in the sandwich considered in Section B illustrates
considered in this section
the no-transition case and that
Both of these
illustrates the first-order-transition case.
things very
did not vary
zero temperature and we
were at
Nevertheless it is clear one could use the
systematically.
model presented here and to do such a systematic study, and
get results
to indicate one would
these calculations seem
the winged
in terms of
indeed be interpreted
which could
phase diagram.
157
CHAPTER V:
IN
A.
EXPERIMENTS ON
THIN PROXIMITY-EFFECT SANDWICHES
HIGH MAGNETIC FIELDS
INTRODUCTION
In
the
McMillan's
the
preceding
tunneling
behavior of
chapter
model
for
we
the
used
an
proximity
thin proximity-effect
study
sandwiches in
high
We saw that such
predicted
and striking
have unusual
their densities
of states and
magnetic susceptibilities.
IV
of a
sandwich
coupled
to
in their
sandwiches are
characteristics
in
magnetizations and
Furthermore the study in Chapter
consisting
another
of
effect to
parallel magnetic fields.
to
extension
weaker
of a
strong
superconductor
superconductor
as
well
as
preliminary considerations given in Chapter I indicated
that
these
some
two-superconductor
sandwiches
interesting phase transitions as
transition
of
states
and
into the
in the
at
the
normal state.
reason for undertaking
more detail
sandwiches would thus be
to look both at these
interesting in order
have
the field is increased.
Experiments on proximity-effect
densities
should
nature
There
such experiments.
next chapter,
features in the
of
is
the
one
As we
spin-polarized
phase
further
discuss in
tunneling
158
from isolated superconductors
spin-orbit scattering
effect it may be possible
By
mixing
spin
Furthermore it
spin-orbit scattering times
to measure
in
proximity
extend the measurable range of
to
metals which are not superconducting
of
with moderate
using the
spin-orbit interaction.
the strength of the
would be possible
superconductors
interactions.
strength spin-orbit
The
superconductors.
times in these
limited however to
method is
affords a way of measuring the
these
in
by analyzing the degree
metals
when
have
they
a
proximity-effect-induced superconductivity.
In
this
chapter
preliminary tunneling
Drs.
their
R.
Meservey, D.
unique
we
describe
experiments
tunneling
National Magnet
Francis Bitter
comment
which were
Paraskevopoulos,
high-field
and
undertaken by
Tedrow using
and P.
facilities
Laboratory.
some
on
We
at
MIT's
shall also
present suggestions for future tunneling experiments and
experiments
to
measure
the
field-dependent
susceptibility of proximity-effect sandwiches.
for
magnetic
159
EXPERIMENTS
B.
TUNNELING
1.
Theory of Tunneling
most
the
of
One
across
an
oxide
the
is
superconductivity
metal
below,
directly
is
of the
quasiparticle states
tunneling between
The
we
density
the
to
shall
of
superconductor, and
Josephson
(which we
shall not
two superconductors
reflects the coherent nature of the superconducting
discuss)
ground
normal
structure, as
related
another
and
a
of
of
conductance
the
of
superconductor
conductance
metal-oxide-superconducting
outline
a
probes
experimental
measurement
placed between
The
metal.
detailed
state.
current-voltage characteristics
between two metals
tunneling
are easily
Hamiltonian
we
of
interpreted1 8
have
been
oxide
an
barrier
using
the same
to
describe
using
160
two
between the
coupling
metals
enough that a potential difference
weak
to be
supposed
is
the
case however,
this
In
sandwiches*.
proximity-effect
between the metals can be
maintained and that the tunneling Hamiltonian can be treated
order perturbation
from metal
1 to metal 2 is then given by
A E IT NJE) f(E)N2(E+e\/)I-f(E *eV)J
I
where
current flowing
The
theory.
in lowest
difference
applied
that
is
V
N(E) is the density of states in metal i,
and A
function,
is
a
constant
probability
of
(proportional to
states in metal
electrons
ITI2)
metal
energy in metal 2),
N (E+eV) [
+ eV
-
f(E) is
1, N,(E)f(E),
from
the number of
2 at energy E
metals,
the Fermi
This
product of the number
tunneling
times
resulting
of proportionality.
equation results from integrating the
of occupied initial states in
the
the two
of
chemical potential
in the
is
eV
voltage,
(V.1)
these
.
states
available empty
(relative to
f(E+eV)
times the
the Fermi
Summing this
and
the reverse current, we find
AI4
ITIl N,(E) NI(E t eV)[f (E) -4f2(E +e V)]
(V.2)
a nonequilibrium
through the barrier is
*A current flowing
treats
theory
Hamiltonian
tunneling
This
situation.
nor the perturbation
neither the nonequilibrium statistics
has
Feuchtwang 78
Recently
rigorously.
case
this
in
of tunneling for barriers
developed a more rigorous theory
perturbation
nonequilibrium
a
using
metals
normal
in
to
theory
this
extended
has
Arnold 8 9
and
theory,
in
theories are
these
of
The results
superconductors.
accord with the results of the arguments reproduced here.
161
that
Assuming
the
considering
the
matrix
case
where
element
metal
is
T
constant
and
metal,
this
normal
1 is a
becomes
7JT12 N\)(E,,) {'4
( E eV)
Or
where
is
G
the
corresponding
are in the
both metals
of
Eq.
(V.3)
V.3
with
we find
voltage and
to
respect
given by the derivative
is
The conductance
(JE~eV)LF&)-A 4-eV)3
f(E eV)1
(V.4)
when
constant conductance
zero temperature
At
normal state.
just
this is
,
VI(
N,(e Ivl)I
(V.5)
NZ(EF
superconducting density
Eq.
the derivative
smearing results from
This
temperature
average of the density of
V.3 gives a thermally smeared
states.
At finite
of states.
the
yields directly
zero-temperature conductance
Thus the
of the
Fermi function which is a bell shaped function with width of
order k T.
We have not solved
IV.5, for
rough
idea
the self-consistency equations,
finite temperature.
of
temperature looks
what
the
like by
We
can,
conductance
performing
the
Eqns.
however, obtain
at
finite
a
(low)
thermal smearing
162
calculations carried out
resulted from the zero-temperature
Chapter
in
IV.
considered there
of Table IV.1,
V.2 for
For
the
normal-superconducting
results given in Figures V.1
we find the
at
bias
zero
spin splitting of the density of
conductance.
This
is
the
conductance
at low
bias is
would be obtained from a
As
the field
(times the
states
is reflected in the
visible
an
the
the thermal
in the
glancing angle from
readily noted
In
in
the splitting
curves from a
s-side.)
the
a
the field is applied the
to mask
below, the splitting can be more
curves of
result from
density of states but
sufficient
(By viewing
would
than
is clearly
splitting
thermally smeared
smearing here
field
and
sides are predicted to have
As
smeared BCS density of states.
n-side.
1
the normal and superconducting sides respectively.
conductance
s-side
sandwich
and having the parameters given in column
In zero magnetic field, both
more
which
states
density of
the
over
V.3
Eq.
in
indicated
in the
applied field,
again larger
than that
lower
the
which
spin-split BCS density of states*.
Bohr magneton) increases beyond the
more
show
curves
conductance
spin-split
*Generally,
the
by
explained
be
than can
zero bias
conductance near
spin-orbit
and
depairing
orbital
both
of
inclusion
It is not implausible to suppose that there is
scattering.
some proximity-like perturbation near the oxidized surfaces
bias
low
enhanced
this
causes
which
metals
the
of
tunneling
the
that
recall
should
One
conductance.
conductance actually reflects the local densities of states
near the oxide surfaces.
n -side
C
0.48
<
0.45
00.3
0.2
0
BH
s
C
-o O
02
-2
Voltage
Figure V. 1
eV2
s- side
0(
:D
0.4
0
00.
-
0
00
.3
0.2
- -B
0
Voltage
Figure V.2
--
H
I
2
eV
165
oero-field induced
densities
of
energy gap,
Jl0,
states cross at the
which
resulted in
indeed
reflected in
the normal
the up-
and down-spin
Fermi level and
metal density
the conductance
for
the
peak
of states
tunneling
is
into the
n-side of the sandwich.
At
peak
fields great
in the
enough to
n-side density
smaller peak in the s-side
IV.5
and
however,
peak.
IV.6.)
In
the thermal
For
of
the
large zero-energy
states, there
was
also
a
density of states.
(See Figures
this finite-temperature
conductance,
smearing is
At lower temperatures
this s-side
cause
enough
to obscure
this
there would indeed be a peak in
conductance at zero bias.
suitably chosen
proximity coupling
parameters,
the
s-side peak may be pronounced enough to show up even after a
significant amount of
account.
The
always be
energy
thermal smearing has
low bias
peak on
greater than that on
states which
been
the n-side,
the
are involved
taken into
however, will
s-side because
have larger
the low
probability
densities on the n-side.
2.
Tunneling Measurements
A
number
of
considerations
restrict
the
choice
of
166
materials
which
boundary between the metals must
the
low solubilities
The metals chosen can have
in order that
the superconductor to low atomic
beryllium90 ,
alloys 9 1 ,
and gallium' 2 .
in which
spin splitting of
case
(In the
only
limits the choice of
vanadium-titanium
or
These are the only
superconductors
been clearly
the densities has
V-Ti alloys,
the
and
of V
the
number metal atoms such as
vanadium
aluminumzo,
splitting has
this
In practice
states.
densities of
splitting of
spin
an observable
be
there still
in
which does
preparation
amount of spin-orbit scattering
only a small
observed.
junctions,
be sharp and well defined.
of sample
and a method
not allow much atomic diffusion.
the
in
must choose metals with
This means one
one another
behavior discussed
make good proximity effect
To
thesis.
proximity-effect
construct
exhibit the
that should
sandwiches
this
used to
be
may
the
To avoid
very recently been observed.)
the depairing which accompanies the Meissner diamagnetism in
thicker films,
If the
thin as possible.
may
form
disconnected
electrically
it
always
necessary to use
it is
is,
to
use
islands of
metal
Finally,
metals
temperatures as possible so that
be carried out
with
are as
rather
than
it is desirable,
as
high
an
as
transition
the experiments can easily
at a reduced temperature
smearing effects are minimized.
which
are too thin, however, they
films
continuous film.
films
T/T. where thermal
167
in V and its
The spin splitting
large enough to make the effects hard to
scattering in Ga is
The initial candidates for experiments were Al and
observe.
films
Be
the
warming
Upon
a transition
Be has
substrate,
0
K.
their
lose
and
transition
high
relatively
The
superconductivity.
8.7
around
temperature
anneal
glass
cooled
helium
liquid
onto a
deposited
When
Be.
spin-orbit
the
because
observed and
been
recently
very
only
Ti has
alloys with
temperature of thin film Be makes it a desirable material to
use,
designed for low-temperature tunneling
film in an apparatus
experiments in high
fields and the
magnetic
make it a difficult material with which
temperature
transition
film),
has
the
on
Aluminum,
(2.4
0
K
used
experiments,
extensively
and
the
proved
were
for
undertaken
contained in this thesis.
as
to choose a metal with
a
for
the
of
the
result
For the n-metal, it
low atomic number
little spin-orbit scattering
the up- and down-spin states
in the n-side also.
and copper were chosen for these
It
tunneling
choice
simplest
low
thin
a
as
spin-polarized
calculations
there will be
relatively
to deposit and handle.
which
so that
a
when deposited
experiments
is also necessary
toxicity of Be
to work.
has
other hand,
but it is relatively easy
been
such a
cryogenically depositing
the necessity of
but
to mix
Magnesium
initial experiments.
168
The
films used were
made by vacuum
peak in the
the predicted zero-bias
in the
were made
with Al
on the normal
side of
had shown that
50
thickness for
a single
electrically
continuous
f
first
the
was
tunnel
sandwich.
junctions
Experience
about the lower
Al film
if it
film.
sandwiches were made with both the n-
limit of
is still
to be
the
Therefore
Since
is largest
conductance
sandwich, the
n-side of the
evaporation.
an
initial
and the s-films having
thickness on this order.
First
a 500
%
Al
film was
evaporated onto
Next the Mg was
substrate and allowed to oxidize.
as
a
cross
thicknesses
those
top
strip
of
of the
Al
and
the Mg
this
was
would subsequently
thickness
of
the Al superconducting
deposited
thickness.
itself has
in the form of
bulk Al).
The proximity effect
this
transition
substrate
by
could only be cooled to
so
Al.
to
(versus
much
about 0.4
25 R and
of the
the
actual
than the
of
1.175
2.5*K
0
K for
metal lowered
as
0
The
Some
temperature
from the Mg
as
deposited
film is less
a thin film
temperature
f
X.
oxidize
a transition
when it is
by
40
50 A to 37
layer
Aluminum
covered
layer varied from
top Al layer from
a sapphire
K
0.8
0
K.
(using
The
3
He at
low pressure), so there was considerable
thermal smearing of
the density of states
characteristics.
in the conductance
169
The
40
a
1.7 0 K.
the
evident, but
states was
of
density
of the
splitting
spin
field the
parallel
in a
placed
When
of
transition temperature
zero field
and had
of Mg
about
50 R of Al deposited on
initial sandwich consisted of
sandwich was too thick to neglect the orbital effects due to
the
state long before
into the normal
for
enough
spin-split peaks
the
sample returned
The
field.
of the magnetic
screening
splitting was
this
conductance
the
in
large
to
cross.
The transition
caused
and becomes more and more
of states with increasing field
significant in thicker films.
of
a
series of
reflecting boundaries.
taken
to
spheres
equal to the eigenvalue of
the eigenvalues
those
of the
spin
of this
operator
operator, both
the
densities of states will be spread out.
the
occupied states have larger
the angular
a magnetic field
will be raised or lowered
the energies of these eigenstates
Since
specularly
sphere can be
z component of
In the presence of
momentum operator La.
by an amount
have
which
The eigenstates of the
be eigenstates of the
To
imagine that the film
explain the broadening qualitatively,
consists
apparently
This effect causes
largely by the orbital depairing.
a broadening of the density
was
normal state
the
into
the operator
AL*.
are unrelated
upFor
and
to
down-spin
larger spheres,
orbital quaptum numbers and
170
but the effect of the
spheres,
films are really not
Thin
the orbital depairing increases.
diamagnetic screening will be
similar to that in this simple sphere picture.
Paraskevopoulos evaporated thinner
Meservey, Tedrow, and
this deleterious
films to try to mitigate
The
characteristics they observed for
side of a 25
normal state
from the
state from
2.50K to
into the superconducting
increases
further
and there
is an appreciable
3.2 Tesla
there is
a
be
must
In a field
about 2.30K.
reflection
of
the
bias.
at zero
crossing
actual
At
bias which
at zero
a substantial peak
As
is evident
more spin-splitting
conductance
of 2.0
is evident.
spin splitting of the conductance
Tesla, the
the bias
in Figure V.3.
a rather broad transition
a midpoint at
2.20K with
tunneling into the Mg
Al sandwich are shown
t Mg-40A
junction displayed
This
thickness effect.
of
the
spin-split densities of state at the Fermi energy.
When the
the
sandwich
the conductance
curve is
field
returns
increased
is
to
3.4
further to
state and
the normal
Tesla;
flat.
Due
to
densities
temperature
conductance
the
significant
of states
in
orbital
this sample,
smearing, there is not
characteristics
to
of
the
as to
the
broadening
as
well
sufficientt detail in the
justify
quantitatively fit the characteristics.
an
attempt
to
However there are a
171
Tunneling into 25A Mg on 40A Al
U,
:D
0
0
:
0
0
0
-l
0
Voltage (mV)
Figure V.3
I
172
things to
couple of
Figure V.3 to the
n-side characteristic of
self-consistent
the
fields,
lower
qualitative calculation
values.
peaks
at
for
the field
included
field
at these
strengths
This is a result of
There is
a greater
as
compared
zero-temperature field dependence of the pair potential.
must
include
also
the
caused by the
potentials
It is clear that the
degradation
dependent
theor'etical
reproduce the
the lack
calculation
meaningful.
does
We
pair
the
proper inclusion of the temperature
of
is
the
pair
in
the
experimental curves.
data
of
to
orbital depairing.
of a sample which
experimental
degradation
added
and
self-consistent pair
temperature
finite
at
potentials
included.
increasing field of the
decrease with
In
the theoretical curves show
of
most
which should be
the
appropriate
decreases to about 0.3 mV at H = 3.2 T.
two effects
in
the experimental peak remains
^ +( A5 + AH) whereas
0.4 mV
At
have apparently
At higher field values,
nearly
at
Figure V.1.
the theoretical curve.
we
is
smearing than
more thermal
smeared density
evident
clearly
splitting is
experimental curve though not in
our
in
given
pair potential
experimental
zero-temperature
the
from
calculated
states
of
comparing the
note when
potentials
right
At this
direction
For a quantitative
such
a
the
to
time however,
gives distinct features
not make
in
in the
comparison
too
investigation of the theory
173
a
more distinctive
achieved
be
be
could
with thinner
diamagnetism, or
with lower
showing less
films
This
orbital
and
temperature measurements,
sandwiches
measurements on
systematic
is required.
higher-transition-temperature
with
superconductors,
more
curve
experimental
with
varying
did
attempt
proximity-effect coupling parameters.
diamagnetism which
went to
They
curves.
film.
orbital
the
25 A Mg
These sandwiches
with
1.50K,
a
again
splitting
The
conductance.
the
broadening
a 35
i
Al
transition to
a broad
the transition ranging from 2.30K
about
midpoint at
was
is
films backed by
again showed
the superconducting state,
to
an attempt to
even thinner film in such
measurements on an
lower
Tedrow
and
Paraskevopoulos,
Meservey,
clearly
evident
diamagnetism
was also
very
lowered
1.850K.
The
spin
in
the
apparent.
Although these sandwiches had a lower transition temperature
than
A
the 25
given in
Mg-40 R Al
Fig. V.3,
film
they had
characteristics
whose
a much higher
critical field.
Unfortunately, at 0.40K, the critical field was
the
3.8 Tesla maximum field
in which
performed.
the measurements
states cross.
greater than
of the superconducting solenoid
on this
At this maximum field,
reached the point
were
where the up- and
particular sample
the
were
sandwich had not yet
down-spin densities of
174
Paraskevopoulos,
across
conductance
%
to the
that
illustrates
transition
differ
sandwiches
proximity-effect
This
K.
is
Mg-Al films
sharp
The
n-sides.
thicker
had
which
above
discussed
0
for the
broad transitions
the
into
transition
a temperature of 2.19
superconducting state at
in contrast
These films displayed
of Cu.
zero-field
sharp
rather
a
%
Al film backed with 15
the
this for a
They did
s-side of a proximity-effect sandwich.
37
on
constructed
oxide barrier
an
field-dependent
at the
Tedrow looked
and
Meservey,
measurements,
n-side
to these
addition
In
these
thinner
greatly
from
Cu
the
Mg
sandwiches discussed earlier.
H = 0.22
Figure
in
displayed
T does not
side of the s-side
zero-bias
reached.
this
s-side
maximum
If
before
just
this were
sandwich both
of this
conductance
whereas one
sandwich which is
sandwich
its
One
the
fields and
the appearance
before the
field driven
shape of
of the
also displays
critical
tunneling into
the
large
transition into
at
a bump on
There is
tunneling Hamiltonian model.
well characterized by the
notes the
curve
s-side density of
look like any of the
in Chapter IV.
are
junction
conductance
the low energy side for a
is expected on
also
The
V.4.
states curves displayed
the high energy
this
obtained for
conductance curves
The
field
the normal
the normal
is
side of
conductance at
zero-bias
a
low
maximum
state
Tunneling into 37A Al on 15 A Cu
4-
_0
C:
-o0
Ll
C)
0
-1
Figure V.4
0
Voltage
(mV)
176
explain
model can
for s-side tunneling.
is indeed doubtful that
however, that it
the McMillan
so thin on an
in this film is
The copper
even doubtful
that the
pair potential is
Cu film is
that it is not possible to draw any
validity of the
the assumption of
It is
justified.
one.
a continuous
of this
metallurgical characterization
atomic scale,
step-function shape
tunneling model about the
of the self-consistent
to the
the
tunneling into
is
features the tunneling
these are not
s-side and
this
However
understood.
could be
junction is
The
so poor
conclusions with respect
tunneling model from the conductance
characteristics for tunneling into this sandwich.
In summarizing these results,
measurements
show
did
tunneling model.
we
can say that the n-side
agreement
reasonable
It would
have been
with
very interesting
the
if
s-side measurements had been made on sandwiches identical to
ones used
the
section we
for the
discuss other
n-side measurements.
In the
next
suggestions for future experimental
work.
3.
Suggested Future
The
results
Tunneling Work
presented
here
of
initial
experiments
177
and suggestions
result of the calculations
undertaken as a
in this thesis are promising in some respects and perplexing
in
clear
It is
some others.
In
required.
subsection
this
and
additional
to theory,
quantitative comparison of experiment
experiments are
a systematic
that for
we
indicate
promising for future experimental
what directions look most
work.
thermal smearing of the density of
states which occurs in a
the
either decreasing
the conductance
temperature at which
by increasing the transition
is measured or
the proximity effect sandwich.
orbital diamagnetic broadening
the films
is desirable to make
each
them
can be done by
This
conductance versus voltage measurement.
have
to minimize the
it would be advantageous
Most obviously
to reduce the
of the density of
states it
as thin as possible and still
continuous
be
superconductors
well
pair potentials.
constant
by spatially
characterized
Additionally,
temperature of
comment on each of these three suggestions
We
in turn.
Lower Temperatures
a
By using
3
dilution refrigerator
He-4He
samples, instead of cooling with liquid
it
would
be
possible
temperature by a factor of
temperature
tunneling
to
100
easily
3
the
He at low pressure,
reduce
or more.
to cool
the
sandwich
The high-field low
facility at the Magnet
Laboratory
is
178
going to ultra low temperature would
or
0
K)
0.026 0 K)
bulk
(with a
Be
One is restricted to
condensed
Beryllium,
temperatures, was,
0.84 0 K)
temperature
transition
of
Superconductors
spin
until very
advantage for
an additional
Low temperature
sandwich
transition
1 ow deposition temperatures has
proximity effect
experiments.
in
proximity-effect
two
layers
to a sharper interface
between the
by
most of the
a sharp interf ace is assumed
The high transition temperature
another advantage
only candidate
interdiffusion of
This leads
and such
He
deposition mi nimizes the
the
between
liquid
at
recently, the
the
observed.
be
splitting can
maintained
and
Going to such
temperature.
theories.
using materials with low spin-orbit
increasing
for
material
sandwiches.
the
so that
scattering
films
of about
might be the weaker superconductor.
Higher Transition Temperature
atoms
stronger superconductor
(with a bulk transition temperature of
and Zn
annealed
be the
thin film transition temperature
(with a isolated
2.5
would
Aluminum
Chapter IV.
of
Section C
in
discussed
geometry
superconductor
the
allow
superconductor-stronger
weaker
the
of
realization
experiment, the
would
temperature
of
decades
two
additional
a number
be useful for
For this
this one.
besides
of experiments
capability of
the additional
facility, and
already a unique
for its use. Aluminum
of Be gives
can be used
still
for the
179
n-metal
and
Be the s-metal,
realization of
resulting in a
the realization of the two-superconductor sandwich discussed
in Chapter IV Section C.
spin-splitting
recently
Very
states has been very clearly observed
density of
some advantages
and offer
have
40K respectively
30K and
of about
transition temperatures
in Vanadium
These films
films.
alloy
in Vanadium-Titanium
and
superconducting
the
of
Additionally,
over the use of Al.
V forms compounds with other low atomic-number elements such
Al,
as
Ga, and
and
Si,
and temperatures.
critical fields
spin splitting of
the
comparisons as well as
have very
in these compounds
quantitative
for
high
observe the
Attempts to
density of states
useful
very
be
would
compounds
these
single
film
for possible uses in proximity-effect
structures.
Thinner Films
Going to thinner films
the
advantages of
acts to
from
from 40 A
to 35
to an increase of
3.4 T to
occurred
reducing the
smear the density of
thickness
Mg leads
in the proximity sandwiches offers
states.
the
diamagnetic
which
A reduction of the Al
A in an Al sandwich
with 25
A of
the 0.40K sandwich critical field
well above 3.8 T. This
despite
temperature in
orbital
fact that
the thinner film
the
higher critical field
sandwich
was lowered from
transition
2.40K
to
180
1.85
0
proximity-effect
for
thickness
this
the n-
gained from studying both
to be
sandwiches.
and the
s-sides
of
(or else identical) sandwiches.
the same
In minimizing
to make
it is
London
the
because
proportional
to
the
n-side
the
s-side thinner
higher
The
superconducting
electrons.
s-metal results
in a smaller
of
root
is
the
simply
inversely
is
depth
penetration
square
This
thinner.
orbital
of the
effects
the deleterious
diamagnetism it is more important to make the
than
just
of information
be a significant amount
Further there would
work
continuity and
for electrical
minimum thickness
determine the
to
do experiments
be worthwhile to
it would
above
films appears possible, and
Preparing even thinner
K.
density
density
in
of
the
London penetration depth on the
s-side.
Summarizing these suggestions
we can say that there are
improve
a
for
number
future tunneling work,
of directions to take to
the tunneling characteristics which were obtained on
the first
experimental samples.
Better resolution
in the
tunneling characteristics and more systematic investigations
of the
influence of
should
make
possible
various parameters
a
quantitative
of the
sandwiches
comparison
between
181
theory and experiment.
OF THE MAGNETIC SUSCEPTIBILITY
MEASUREMENTS
C.
of
were also predicted to have some
proximity effect sandwiches
Both sides
unusual features.
strikingly
magnetic properties
that the
Chapter IV
saw in
We
strongly
have a
field dependent susceptibility and the n-side susceptibility
which is several times larger
actually can approach a value
the corresponding
than
should
sandwich
S2UID
a
with
The spin
magnetometer.
on the
information obtained
nearly
as
sensitive
very
normal
susceptibility of each side
separately
with
the separate
in
for the
susceptibility
as
that
would be very
in the n-side
One
sandwich susceptibility
nuclei
detailed
experiment*, but it
susceptibility
or
measured
be
can
measurements
not
the total
be able to measure
directly
state susceptibility.
normal
given
Knight
films.
metal
of the
shift
The
measurement is
by
interesting
a
tunneling
to observe a
of a proximity-effect sandwich
potentials is small
dependence of the pair
*When the field
at a
susceptibility
the
temperature,
zero
at
are
and we
the
to
proportional
just
is
IV.11
Egn.
by
field H as given
measurement
a
Thus
pH.
of
energy
an
at
states
density of
point in the
at a field H gives one
of the susceptibility
at
measurement
tunneling
a
whereas
curve,
states
of
density
curve.
whole
the
yields
strength
field
the same
182
the
in
sandwich is
the
when
actually greater
is
which
as
the normal state
superconducting state than when it is in
predicted in Figure IV.8.
The
measurement
a
response
proximity-effect
with
the
this
layers of proximity effect
difficulty, one can deposit many
sandwiches
overcome
To
itself.
sandwich
of
response
the
overwhelms
which
a
proximity sandwich itself, may
much greater volume than the
give
having
support,
This
films.
for the thin
the
to have a
is necessary
sandwich may be difficult because it
rigid support
of
susceptibility
total
of the
insulating
nonmagnetic,
intervening thin
layers.
arise when measuring the Knight
This difficulty does not
shift.
In
a
by the
a sample
nuclei in
when they
angular momentum eigenstate to another.
levels
involved are
given by
the values
character near the nuclear cores)
frequency.
the
and
When the
nucleus is
in
this
of the
turn
by
one
magnetic
The
(with s-like
affects the local field at
influences
electrons are polarized,
changed
the
The energies of the
conduction electrons
spin paramagnetism of the
nuclei
go from
the nuclear magnetic moments.
field at the nuclei times
the
measures
of the resonant absorption of
frequency in a magnetic field
energy
one
experiment
Knight shift
an amount
the
resonant
the
field at
proportional to
the
183
separately measure the shifts of the
the nnot
and
s-side nuclei,
obstruct the
has
resonant frequencies of
the presence of the substrate does
side of the
sandwich, one
each side
on the magnetization of
a handle
resonant
measuring the
By
measurement.
the nuclei on each
frequency of
separately.
to field
the magnetization with respect
By differentiating
can
one
Because
magnetization.
the
to
proportional
is also
resonant frequency
The shift of the
magnetization.
for the electrons
one can obtain the susceptibility
on each
side of the proximity-effect sandwich.
Both of
be
interesting
see
to
if the
Knight shift beyond the normal
As
the
for
feasible
these measurements appear
choice
optimal
considerations apply
for Knight
predicted
and it would
increase of
the
state value can be observed.
materials,
of
same
the
as were
shift experiments
given above for tunneling experiments.
D.
SUMMARY
In
summary it appears
and susceptibility
sandwiches
are
that both
measurements on
feasible
and
tunneling measurements
these proximity
interesting.
effect
Preliminary
tunneling experiments were encouraging though it appears
sample can be
optimized considerably by suitable
the
choice of
184
the materials and
the
thicknesses of the films
the measurements
at which
temperature
and by lowering
were taken.
A
systematic study of the conductance into both the n-side and
the s-side
of otherwise identical
valuable.
It
interface
between the
sandwiches would be quite
two
the
oxidize slightly
possible to
should be
metals and
in
exert
this way
additional control over the strength of the proximity-effect
coupling
the
in
quantitatively
sandwich.
investigate
Chapters III and IV.
The
basic
features
these
If better
theory
needed
to
given
in
was
resolved and more systematic
experimental data becomes available, it will be necessary to
add
the
spin-orbit
effects
of
interaction
the
of
orbital
the
and
diamagnetism
electrons
for
a
precise
quantitative comparison between theory and experiment.
can be done in a straight forward manner.
the
This
185
A.
SUPERCONDUCTORS
IN
TIMES
SPIN-ORBIT SCATTERING
CHAPTER VI:
INTRODUCTION
limited
paramagnetically
sufficient
to
extent
the
In
properties.
sandwiches to
proximity-effect
a
predict
we
zero-bias
properties,
the
predicted
conductance
for
tunneling into
the
a sandwich.
We suggested a number
experimental
situation so
to
as
peak in
these
higher
field
weaker superconductor
usefulness
increase the
and we
the
about
indeed produce
do
characteristics
parallel fields,
some
it
addition effects
thin
states.
the
films
in
sandwiches
theory.
One
is present to a
effect is
high
of
the
small extent in
to broaden the
the density
in
necessary to consider
the effect of spin-orbit
spin-splitting of
cause a mixing
the
and which acts
Another is
these
will no doubt be
orbital diamagnetism which
these
of
If these
information
detailed
more
of
also suggested
a some other experiments which might be attempted.
experiments
of
of ways of modifying the
theory,
the results for comparison to
the
is
of
for one
from tunneling experiments
some evidence
that there
saw
a
interesting
of
number
chapter
preceding
for
model
tunneling
the
developed
have
We
density of
scattering on
states.
This
will
of the up- down-spin states and a decrease in
186
the extent of the splitting.
Adding these effects to the theory so far developed is an
which the experiments are
we
of such an
on
the theory here,
to
understood
poorly
some
Pauli
isolated
in
spin-orbit scattering
the
of
aspects
pursued and the success
concentrate
instead
will
extent to
add these effects
Rather than
effort.
depends on the
a usefulness which
exercise with
paramagnetically limited superconductors.
We
simply estimating the magnitude
develop arguments for
of the spin-orbit scattering time from impurities, including
interpretation of these experiments,
the
in
factors
With
elements.
and
factors
agreement
becomes
a
trace this
many
even so,
and,
neglected
of
between
We then
are
scattering matrix elements,
estimate
numerical
fortuitous
experiment93
estimation of
to the
of
as
We
how
spin-reverse
the
spin-orbit
and a simple hypothesis
many displaced surface atoms act
matrix
magnitude.
orders of
contributing
use our
the
of
theory and
an incorrect
discrepancy to
numerical
scattering
apparently
an
discrepancy of several
surface atoms
scattering.
inclusion
the screening,
recently noted
this screening has been
spin-orbit
proper
an
conventional
the
some missing
there are
estimated
the
In
electrons.
core
its
by
impurity
charge of
the nuclear
screening of
of the
the effect
"impurities,"
of how
to compare
187
theoretically
predicted
experimentally
spin-orbit
magnitude agreement which is as
expected
in the theory
order
of
arguments can be
this chapter (Section B)
aim of putting
this interaction into context.
C and D,
we give arguments which
proper numerical factors
in the estimate of the
in Sections
Following this,
lead to the
we review
spin-orbit interaction
of superconductivity with the
the current understanding of
spin-orbit scattering times
arising from impurity scattering
We further include the effects of the screening of
centers.
of the
nuclear charge
electrons.
scattering
the
Tedrow to our
the data
data to an improper estimate of
contributing to
results
of
the
number
in reasonable
of
and
the
We
trace
the
tabulated
the number of surface
the spin-orbit scattering
counting
and experiment.
theory and
our improved
(Section E)
by Meservey
improved theoretical estimate.
discrepancy between
potentials
tabulated
core
by its
center
In the final section of this chapter
then compare
proper
an
good as our
the inclusion of the
the history of
we
to
to give.
In the next section of
the
times
find
We
ones.
determined
scattering
and show
surface
atoms
that a
scattering
agreement between
theory
188
IN
INTERACTION
SPIN-ORBIT
THE
ON
BACKGROUND
B.
SUPERCONDUCTORS
role of the spin-orbit
Understanding the
interaction in
superconductors has proven to be a long and evasive process.
developed,
Yosida 3 8 calculated
and found
it to be
reflecting
conducting
ground
Sn,9'
which
was
and A19
scattering
all
out
impurities
Ferrell"8
Anderson
and
a theory
excitation
in
Knight shift
spin
spin-orbit
that
and Abrikosov
found
indeed
and
99
but
Hg,
the
Al
a
random
non-vanishing
by a spin-flip
lifetime,
spin
Ta,.
non-vanishing Knight shift in
results
remained
a
mystery.
Al results might be explained
suggested that the
surfaces
and Gor'kovI 0 0
scattering from
of spin-orbit
by spin-flip scattering from
at the
super-
nonvanishing
a
suggested
This presumably explained the
Matthias 1 01
a nonvanishing
reflect
susceptibility characterized
and
the
at surfaces was responsible for this nonvanishing
susceptibility
Sn
the
in
gap
of
function of
Knight shift experiments' 4
showed
to
thought
susceptibility.
worked
and the
state
7
pair nature
bound
the
On the other hand,
Hg,9 5
spin susceptibility
an exponentially vanishing
temperature
spectrum.
Pauli
the
was
superconductivity
of
theory
BCS
the
after
Soon
paramagnetic
of the Al samples,
later proved to be unnecessary.
oxygen impurities
though this explanation
189
Ferrell 1
0 2
to the Van Vleck37
non-vanishing susceptibility might be due
be present in addition to
orbital paramagnetism which should
the
is
and
states
Appel 1
superconducting.
Sn,
normal Hg,
and
It
.
in
this contribution
4 estimated
order needed
the
of
small
too
much
was
but
3
becomes
metal
a
when
Sn data,
bands 1 0
non-s
for
Al and found it was
and
the Hg
to explain
0
to high
spin) electrons
important
unaffected
remain
should
paramagnetism arises
orbital
(both
of
transitions
from virtual
energy
This
paramagnetism.
spin
measured
the
that
suggestion
the
made
to
that of Al.
explain
Later
the
Knight shift
Al
repeated 1
experiments were
106
the
be vanishing as
was found to
Os
the
and
temperature
approached zero in agreement with the original expectations.
standing
long
The
Anderson 1
07
resolved by
finally
it, was
has called
Knockabout,"
Shift
Knight
"Great
as
this
combination of refined theory and experiment.
Spin-orbit
scattering
superconductors.
interaction,
the
In
in high
also important
is
absence
the
difference
between
the
of
the
superconducting state susceptibilities results
field
where
the
magnetic
superconducting and normal
energy
states
difference
equals
the
field
spin-orbit
normal
and
in a critical
between
the
superconducting
190
18.4 kilogauss
paramagnetically limiting field of
or Pauli
per
0
is the Chandrasekhar-Clogston4
(This
condensation energy.
the spin
into
the superconducting
from impurities
is only spin-orbit scattering
paramagnetism, which resolved the
The orbital
metals,
in transition
discrepancies
the
in
superconductors.
of high field
theory of the critical field
the spin
scattering
impurity spin-orbit
and
paramagnetism
Knight shift
in
not important
is
it
which effects
only include
Thus one need
this susceptibility.
Furthermore,
critical field.
paramagnetically limiting
the
alone determines
it
state,
transition
upon the
susceptibility changes
only
Since
transition temperature.)
of the
degree Kelvin
determining the critical field.
Hohenberg
16
lifetime
ls,
spin-orbit
.
Maki 1"
by
superconductors
characterizes
in
scattering
is
spin-orbit scattering
dramatic, as in Pb
0
Mo5
this
smaller, but even a
where
of say
a
the effect of
10 percent
120 kilogauss
also.
the effects
limiting field is exceeded by over
less extreme cases,
case
II
and
spin-orbit
of
the
When
the
effects
the
of
all
large,
S6 ,
the
parameter,
of type
Helfand,
Werthamer,
and
single
A
theory
in the
included
have been
impurities
from
scattering
spin-orbit
and
paramagnetism
Pauli
quite
can be
the Pauli paramagnetically
a factor
spin-orbit
change
material, is
of three
13
.
scattering
In
is
in the critical field
extremely significant
191
we
In this paper
means of effecting such a change.
lead to a
the
reexamine
carefully
will
an
would
spin-orbit scattering
role of
of the
understanding
that
hope
might
One
applications.
practical
for
of
theory
impurity
spin-orbit with this in mind.
Gor'kov1 0 0 provides the
Abrikosov and
including
As their
microscopic
theory of superconductivity.
was in the
effects of spin-orbit scatterers
origin
interaction.
spin-orbit
Z
the
on
commented
parenthetically
structure
and
constant
(m Z) 2 , where
of
the
of
the
the fine
is
c.
of
number
atomic
is the
Z
the
in
Since
impurity.
as
scale
to
ratio
the
merely
to regular sc attering
spin-orbit scattering matrix elements
matrix elements
no t in the
and
dependence
expected
They
the
interest
and Gor'kov
Abrikosov
scattering,
of this
in
impurities
of
way of
most natural
from
scattering
spin-orbit
technique
Green's-function
impurity-averaging
The
impurity-averaging
the
t echnique
relevant scattering diagrams involve two scattering from the
same impurity
ratio
of
potential,
transport
scattering time,
Experimenters
indeed
the first
field
data
from
TC5,
scattering
time,
that
as
scales
set out to
thin
'r,,,
08
films
,
to
spin-orbit
(<C.Z)".
measure this
experiments1
into a
translates
comment
their
Z
dependence and
using parallel
agreed
well
critical
with
the
192
result.
this
a much slower dependence on Z than
T,
to
normalized
is
T,
when
of Z4
corroborate
did not
These experimenters found
that
experiments by
and critical field
Tedrow 2 o
and
Meservey
Later
constant of unity.
dependence with a proportionality
spin-polarized tunneling
(04Z)4
an
in fact
finding
prediction,
Abrikosov-Gor'kov
the
transport
scattering time.
Very recently Meservey and Tedrow
data assuming
time.
impurity scattering
scale,
log-log
of
role
plays the
time
thin films
that in
many
over
93
have
the boundary
scattering
(i.e.
transport)
the regular
agreement, on
now find
They
reanalyzed their
Here
of magnitude.
orders
a
we
consider reconsider the calculation of the scattering matrix
elements which enter into the Abrikosov-Gor'kov theory.
drastic
screening
of the
inclusion
of
reductions
ruins
and
potentials
Meservey and Tedrow.
surface
scatterers.
contribution
this
brings
cores results
ion
apparent
agreement
the
noted
effective concentration
reestimate
data
into
by
agreement to an
We trace the lack of
Our
in
scattering
the
of
strength
the
of the
improper estimation
of the
Our
of
the
of the
surface
order-of-magnitude
agreement with the corrected theory.
In
this chapter
theory which have
we examine
a number
of points
never been clearly stated:
in the
Exactly what
193
is
predominantly from actual impurities
which have a different
the
Given the
the scattering
correct identity of
spin-orbit scattering to experiment?
future
the
impurity
What directions
should
take?
experiments
Because
what
center,
theory of
compare the
should one
How
potentials?
the scattering matrix elements
are reasonable estimates for
involved?
lattice
the host
as
number
same atomic
centers with
resulting in scattering
imperfections
from
Is it
atomic number than the ions of the host lattice?
lattice
Is it
impurity spin-orbit scattering?
the origin of the
behind
arguments
Abrikosov-Gor'kov
the
atomic-number dependence prediction have never been formally
an
the
constant.
dependence
Z
the Z 4
effect
electrons
atomic
to
give not
proportionality
the
also
but
matrix
prediction attributed
The arguments are readily extended to include the
effect of core electron screening
the
of
magnitude
Furthermore the arguments
Gor'kov.
Abrikosov and
only
to
leads
elements, which
the
on
based
argument,
includes
Section C,
given, the next section of this chapter,
of
delocalized
the
compared
to their
of the nuclear charge and
nature
metallic
the
counterparts.
atomic
effects are considered in Section D
of
These
where they are shown to
lead to a much more erratic and overall slower dependence of
T / TsO than simply Z 4 .
prediction with
In the Section E we confront the new
the experimental data.
We
show
that when
194
the
correction is
screening
and experiment
between theory
and
Tedrow
becomes
magnitude.
We
normalize
the
argue
normalizing
boundary
a
that
spin-orbit
scattering time.
of
properly
We
the
scattering
which
discrepancy
Meservey
scattering
was
noted
of
many
and
time
and
by
Meservey
orders
of
Tedrow incorrectly
time
to
then describe our new
spin-orbit
agreement
the
included,
scattering
demonstrate
the
and
boundary
simpler way
times
an
magnitude agreement between theory and experiment.
to
order
the
of
195
ELEMENTS
MATRIX
C.
IMPURITY
SCATTERING
clarify exactly
1.
The
what
Schrodinger
the
are.
"impurities"
The Impurity Scattering Hamiltonian
nonrelativistic limit of
electron in
We also
matrix elements.
Hamiltonian and then estimate its
attempt to
impurity scattering
down the
we write
this section
In
AND REGULAR
FROM SPIN-ORBIT
an external scalar
equation with
an an
the Dirac equation
for
an
potential V results
in the
potential
U given
equivalent
by 1 09
V
The
the
first term,V(r), is
second
the
from
resulting
Eq.
IV.1
interaction
magnetic
of
field
one
the
gets
limit
Other
terms
the
Dirac
of
spin-independent corrections
Except
of importance to us.
is what
interaction.
nonrelativistic
in small
equation result
are not
the regular scattering potential and
the spin-orbit
is
VI. 1)
A
for a factor
classically
electron's
resulting from
magnetic
a Lorentz
by
of two,
considering
moment
and
with
transformation
the
the
of
196
the electric field through which the electron is moving.
seen by a single electron moving
The total potential V
through an
array of
potentials
like Eq.
lattice ions
VI.1 centered on each of the ion sites:
- R)
In a
a sum of
is then given by
(VI.2)
may
V
potential
real solid the total
be separated
into a periodic part and an impurity part:
V
d
VostbI
Vto.
The first
over
sum is
R
.v-R)(VI.3)
+
over all lattice
all impurity sites.
a periodic host lattice
impurity
This equation defines the
potential as the difference between
potential.
second
the
vectors and
the total potential and
This difference
clearly
contains negative host ion potentials at vacancy sites, host
ion potentials
potentials
potentials.
destroy
at
at interstitials, and
dislocations
(We are
the
impurity
assumption to cause
better justification
host ion
"real"
impurity
as
well
assuming that the dislocations
the long range order so
periodic potential
from
as
a series of
that the subtraction of the
eliminates most
potential.
We
of the
do
total potential
not
procedure
can be
this point we also implicitly assume that the same
can account for surfaces and
grain boundaries.)
this
expect
serious difficulties and expect
for this
do not
that a
given.
At
procedure
The reason
197
for
separating
periodic
the
potential
potential
in
and
impurity
the
different physical effects.
Bloch states,
the
manner
is
that
potential
The impurity
and effective
the
lead
The periodic potential
effective masses,
usual way.
this
to
leads to
g-factors
potential causes
in
scattering
between Bloch states.
With
this picture in mind
What is the
it
dominant source
predominantly
from that
"real"
one can answer
the question:
of the impurity scattering?
impurities
of the host metal?
with potentials
Or is it
with the same potential as the host?
be
that unless
apparent
impurity
doping,
it
there
is
the
that
impurity
are
lattice.
In
same
as
the
amount
host
of
a
series
potentials
particular the "impurity"
given by that of the host lattice.
in
of
the
of
lattice
"impurity" scattering.
Hamiltonian consists
the
from
from
It should
a substantial
is
scattering
potentials which dominates the
different
predominantly
"impurities"
Is
Thus
potentials
periodic
atomic number
Z is
198
impurity
treatment of
takes
conventionally
uses these
the
scattering
spin-orbit
regular
the
and
in metals
one
If
one
states.
impurity matrix
the
it is
clear that
impurity
scattering
VI.1,
the potential in Eq.
elements from
the Green's
In
electron basis
free
matrix
potential
states.
states and calculates
basis
Elements
Matrix
impurity
basis
must chose our
elements, we
function
the
can estimate
Before we
both
of Scattering
Estimation
2.
matrix elements will be proportional to the atomic number of
the
center.
scattering
down by a factor
that is
element
spin-orbit matrix
a
to
This leads
-5
(V
the
number.
atomic
ratio
transport
ten orders
and,
failure
cores,
time,
where
in
as
find
and
can be
of
independent
This
traced to
the spin-orbit
that
of magnitude
result
the
the
is
fact
the
spin-orbit
longer than the
atomic
host
the
type
impurity
of
fact, independent
well.
estimate
result to
this
square
times
scattering
concentration
The
We
of scattering
scattering time is
number
independent of
regular scattering matrix element,
from the
and
nonsense.
clearly
that near
the
ion
strongest, the
interaction is
free electron wave functions are a poor approximation to the
real wave
basis
functions
in
underestimates
the solid.
the
Clearly
electron
a
free
density
electron
near
the
199
positively charged core and gives too
rg .
large a value of
wave function accurately it is
To estimate this part of the
more reasonable to go to the opposite extreme and start from
a tight-binding basis.
r In
ket)
wave
tight-binding
our normalized
take
We
functions
as:
k*
J
re
where n
of
number
atoms
function, and the sum over R is
taking the
we are
that
all integrals
wave
functions from
q)
,
the solid
in
index,
Ow, is a spin
band index,
is the
is
an
the
is
atomic
wave
over all ion core sites.
limit, we
extreme tigh t-binding
except thos e
the same
N
involving
atomic
assume
and
potentials
site vanish.
As
Using
these basis states the second qu antized form of the impurity
Hamiltonian in Eq. VI.3 is:
Y%'k**7-
(V
C
where
The
1K
o
(VI.4)
-t-
er'Kh
spin-orbit part of
this
matrix element
can
be written
so
terms of
the corresponding atomic matrix
elements La.
4 t'as:
in
200
I
'
so
Li
EquationVI.4
(VI.5a)
14,40C,
of exactly
is
the
point of the Abrikosov-Gor'kov theory
the
basis
and
atomic matrix elements,
use
wave
their
of
1/N should
of a unit atomic cell so that:
/
50
S "(VI.5b)
is
identical in
to
form
Abrikosov-Gor'kov theory, we
of the
defined
Abrikosov and
splittings.
normalization
be replaced by the volume v
VI.4
are in a
well
functions, in which case the normalization factor
Since Eq.
the
starting
point
their results
can use
directly with our interpretation of the matrix elements.
obtain transport and
spin-flip
scattering
times
Ir
I;
N(Yj)
NI
M
F
where
n
density
represent
is
in
the same as those that appear in the
continuum
a
are
the matrix elements
calculation of atomic spin-orbit
Gor'kov
free electrons
for
Here however we
presence of random impurities.
tight-binding
the starting
same form as
the "impurity"
of states
averages
at
of the
level,
appropriate
given by
(VI.6)
A(VI.7)
concentration,
the Fermi
We
N(E
)
and the
scattering
is
the
brackets
matrix
201
time can be measured in
resistivity experiments whereas the
be measured in spin-polarized
spin-flip scattering time can
tunnelingz 0
measurements of the
determined from
theoretical estimates
our
of these
can be
temperature dependence
in thin films.
field
parallel critical
of the
In addition both times
experiments.
92
transport scattering
The
the Fermi surface.
elements over
times
We compare
experimental
to
values in Section E*.
We
leads
conclude this section by showing how a simple argument
to the
dependence
((Z)4
to Abrikosov and Gor'kov.
We
of
't/'Cso
p electron
relevant
ones.
charge Z,
the
VI.6
and VI.7.
and
spin-orbit matrix elements are the
p electrons
For
attributed
use hydrogenic wave functions
to estimate the matrix elements in Eqns.
note that the
which is
(radial part of the)
in
an
atom with
nuclear
spin-orbit matrix element
is l I
L
(VI.8)
3
the principle quantum number and ao
where n is
radius.
e&
On
the other
hand,
the
is the
Bohr
of
the
matrix element
*The experiments are done on thin films and, as Meservey and
determined
resistivity
the
point out,
correctly
Tedrow
unknown
the
eliminate
not properly
time does
scattering
the
normalization of
The proper
impurity concentration.
regime
scattering
surface
in the
"impurity" concentration
is discussed more in Section E.
202
regular Coulomb potential is:
Using these
and taking
-
-(VI.9)
a.o
v1
the ratio
of Eq.
VI.7 to
VI.6 we
find:
---
where
n
is
the
in question.
principle quantum number
This result,
(3~
(VI.10)
340M
%,
5S
of
the
with the coefficient
i.
effectively set to unity has
to be the
been interpreted
Abrikosov-Gor'kov prediction for the Z dependence
This
result
modifications
is
simplistic
at
section.
best.
of core electron screening
charge and of delocalization of
next
p electrons
We
of
't/T
consider
the
out of the nuclear
conduction electrons in the
203
D.
ELECTRON
CORE SCREENING AND METALLIC
ION
DELOCALIZATION
this
In
section we
ion
of
effect
their atomic counterparts.
of
"pulling out"
the
interested
this
make
in
order
complication.
for
splittings
neglected
and
(In
and
primarily
are
we
we will
fact a later approximation
numbers
atomic energy
levels
acts
ignore
will
from
spin-orbit
this
to cancel
delocalization effect.)
has
Yafet
effects
of magnitude
less
in
spin-orbit
the
paper we
In this
effective quantum
the
resulting
where
core
the ion
strongest.
interaction is
be seen to result from
This can
the electron orbitals
density near
electron
% from
25 to 40
about
reduces spin-orbit matrix elements by
typically
solids
in
delocalization
electron
The
charge.
nuclear
of the
screening out
core
the
by taking into account
naive Abrikosov-Gor'kov Z dependence
the
correct
data to
use atomic
will
quoted
most of the simple metals.
state
valence
atomic
One
A5 0 for
splittings
can fit the expressions for
each row of the periodic table with
)
where
,,P
is
the orbital
M
atomic number
. 11)
(here
. =
1),
o
is
the
204
The values
number.
we simply take
this paper
for
p
effective quantum number
The
the approximations
2 and consistent with
is always near
in
are
White 1 1 2
by
those given
given in
are
here
quantum numbers
Rb.
Na, and
in Li,
effective quantum
and neff so determined
agreement with
reasonable
electrons
r
effective
The
VI.1.
Table
of
the
nefi is
and
core screening parameter,
be its
this to
in
Thus we
value.
take:
4
y_=_.__.
(VI.
matrix element in Eq.
for the value of the spin-orbit
have dropped
(Note we
the
which enters into the
IV.11
in the
(I +
final
(
+
to
1/2)
in Eq.
factor
(j + 3/2)
IV.6.
splitting
into the calculation
case but does not enter
atomic
1/2)
12)
in the solid.)
The
regular scattering matrix elements
by atomic
hydrogenic matrix elements
from
are
approximated
screened nuclei.
In the tight-binding approximation we get:
(V.
'e~(~-O)
(If
we had
used
scattering part,
different from Eq.
a free
electron
our numerical
VI.13.)
basis
for the
results would
We then
regular
not be
obtain for our
13)
much
ratio of
transport to spin-orbit scattering time:
I
TSO 36
ot'(2-
tr)
soY
Ve;
'(VI.14)
205
Spectral Data
Parameters from Atomic
Table VI.1:
Lb)
(C .)
Z
________________
n
ne~f
'I
I-U
I
nef f
3
7
1. 92
2
2 84
1 6
11
15
7. 69
3
2 76
2
29
33
23. 45
4
2 05
47
51
41. 25
5
1 68
79
83
68. 8
6
1 93
from spin-orbit splitting
(a)
Estimated
(b)
Taken from Ref.
112.
12
2.29
tabulated
in
Ref.
111.
206
This
result
chapter.
is
discussed
in the
next
section
of
this
207
E.
COMPARISON OF
TO
TIMES
ESTIMATED
EXPERIMENT
1.
Comparison of the
the
Square
A plot
of
the
of Eq.
Meservey and
field
Bhatnagar, 1
0 8
92
Time
Scattering
(H
)
given in Figure VI.1
data
own
their
Knight shift
conduction electron
normal metals).
1 1 5
of
1 7
93
of
Crow,
Ratio
of
data, 1 1 3
(KS)
Meservey
11
along with
the
parallel
Strongin,
spin-polarized
and
(SPT)
tunneling
4
and
data
from
(CESR) experiments
spin resonance
-1
the
Potential
Scattering
VI.14 is
Ratio to
Tedrow's recent compilation
critical
data,ZO
SCATTERING
SPIN-ORBIT
and
Tedrow
have
(in
normalized
the spin-orbit scattering time to the surface collision time
T
given
by
L.
L
(VI.15)
F
where
L is
velocity.
the
thickness
The data
has a
of the sample and v,
scatter
of orders of magnitude, but falls
which
has
been
interpreted
to
that
ranges
roughly on
be
the
is the
over
Fermi
a couple
the line
(K Z)4
Abrikosov-Gor'kov
prediction.
Our result, Eq.
VI.14, can be seen to display a much more
208
0
10
-2
10
-4
10
T
so
-6
10
-8
10
-\0
10
2
Figure VI.1
5
10
20
50
Atomic Number z
100
209
erratic behavior with
increasing atomic number.
This
reflection of the closing of atomic shells and the
is a
resulting
abrupt
jumps in the spin-orbit scattering potentials seen by
outer
electrons.
The
striking
feature
however is not this erratic behavior
of
this
figure
but rather the size of
the discrepancy between theory and experiment when screening
is included
It
and the
is unusual
theoretical
magnitude
numerical coefficients
an experimental
to call
prediction
which
extends
are estimated.
verification of
seven
orders
a
of
an error in favor of a theory which disagrees with
the data by
as
be emphasized,
magnitude.
It should
the considerations
presented
much as six orders of
however, that
above are quite simple and in accord with what is well known
about the spin-orbit interaction in other
branches of atomic
and solid state physics.
If we
accept the validity of the theoretical estimates of
the scattering times,
the other
in the normalization
of the
the surface scattering
possible source of error is
spin-orbit scattering
times.
Indeed, it is
times
to
here where we
find the source of the disagreement between our more careful
theoretical estimates and the data in Figure VI.1.
When the
manner of
experimental data is
Figure VI.1,
compared to
theory
in the
it is assumed that all of the regular
and the spin-orbit scattering
is
governed by the scattering
210
off
the ion core potentials
is
further assumed
results in a
surface
that these
proportional
and inversely
of impurity scatterers
that
correct
is
it
is
spin-orbit scattering
scattering.
regular potential
assume
to
governed
However, we believe
squares
of
VI.14.
We
as given in Eq.
the corresponding matrix elements
This
is independent of
ratio of the
to the
scatter
bulk metals.
scattering time ratio which
the presumably unknown concentration
argue
impurities
impurities in
strength as
with the same
It
of displaced surface atoms.
that
the
surface
spin-orbit
by potential
it is incorrect to say that
scattering has
anything to
do with
the
boundary scattering time.
The
difference can be understood as follows.
electron incident on an interface.
Independent of what the
nature of the scattering potentials is at
electron
backwards.
a
has
unit
Spin-orbit
occur, however.
The
same contribution as
probability
scattering
Consider an
of
does
therinterface, the
reflected
being
not
necessarily
displaced surface atoms will
make the
always to spin-orbit scattering whereas
the boundary scattering occurs
independent
the impurity scattering potentials.
of the nature of
211
2.
Estimating the Contribution of Surface Scatterers
what
now
Consider
potential is
scattering
"impurity"
estimate the
to
used
regular
the
when
doing
is
one
Equation
concentration when surface scattering is dominant.
VI.15 gives the surface scattering time as
the time it takes
an electron to travel at the Fermi velocity from one
side of
hits the
surface
the other.
the metal to
be reflected in some
it will certainly
we say
When an electron
that the scattering
such a
probability for
the necessary amount
necessarily overestimate
of the number of
When this overestimate
is then
Eq.
used in
of
VI.9,
we will
VI.6
potentials
of surface
the number
Eq.
we use
Then if
much
get something
concentration of scattering
calculate the
would give
and we integrate the
atoms
scattering, we
unit probability.
less than
only to
surface is due
at the
scattering off displaced surface
However if
manner.
to
which
scattering, we
surface scatterers.
surface
scatterers
spin-orbit
estimate a
scattering time which is much smaller than it should be.
The correct way of estimating the effective concentration
of impurity scattering centers when the scattering
displaced surface atoms
of
encounters
relative to
a
is merely
electron will
the times
to estimate
with
have
it encounters
an
is due
to
the fraction
surface
ion core
atoms
from the
212
lattice*.
In tunneling experiments
the tunneling electrons
move primarily perpendicular to the
plane of oxide barrier,
and the fractional number of encounters
will have
with displaced
merely given by
sample
surface scattering
units
a film
of
which is
will
the
lattice
50 atomic
tunneling
electrons
"impurity"
scattering potential about 2 *
proper
be
under
"concentration" of impurity
then given by
constant.
layers
the
the fractional
(0.02
in the example).
Table
we
theoretically.
We
calculated
of
a
The
scattering potentials is
compare
spin-orbit scattering rates with
the
of the time.
per unit volume
number of encounters with
VI.2
For
thick,
influence
the number of host ion cores
times
In
potentials is
1 atom (say) divided by the thickness of the
measured in
example, in
a tunneling electron
the
experimentally
surface
measured
scattering rates predicted
the
theoretical
rates
by
assuming the fractional concentration of scatters was indeed
given
by
measured
1
in
over the
lattice
matrix elements to
spin-orbit
experimentally
given
film
constants.
took
our
be those
splittings.
We
tabulated by
(This
was
the
thickness
spin-orbit
Yafet from atomic
source
of
our
*Note this method of estimating
the surface contribution to
the "impurity" spin-orbit scattering time gives an estimate
which
is
independent
of the
magnitude
of
the
regular
scattering potential of displaced surface ion cores.
w
w
Table VI.2:
Element
Atomic
No.
Measured and Predicted Spin-Orbit Scattering Times
Method
Size
Measured
(meV)
Al
11
13
Ref.
so
(1)
Na
Predicted
(meV)
CESR
6000
(sphere)
4.6 -10-6
1.2 -10-6
115
CESR
700
(sphere)
3.4-10-5
1.1-10-5
115
SPT
50
(film)
0.063
0.010
H
50
(film)
0.15
0.010.
108
20
CESR
16500
(foil)
2 -10-4
3 -10-5
116
CESR
8000
(foil)
5.2 -10-4
6 -10-5
116
100
(film)
0.9
0.126
92
Ga
31
SPT
Sn
50
H
38
(film)
100.
26.
108
H
63
(film)
50.
16.
108
5.8
113
7.0
3.3
113
(elpsd)
5.8
2.5
113
(elpsd)
3.6
1.7
113
KS
170
(elpsd)
12.4
KS
300
(elpsd)
KS
400
KS
580
214
18
plotted by
the data
only points obtained from
Tedrow in Figure VI.1
Meservey and
Fermi level
get the
to
selected from
We
states.
density of
Kittel
data from
specific heat
used
We
above.)
given
constants
screening
empirical
thin samples in which surface scattering is dominant.
Furthermore
spin-orbit
predicted
our
one
It can be
estimate that
atom effectively contributes as an impurity scatterer on
reasonable
displaced atoms
to
that
assume
have
elements which
an
it is probably
In fact
there
is
are
five
or
so
surface scattering
to assume that there is
only
atom which contributes.
Conclusion and Suggested Further
We
the electrons
which contribute to the
on each transversal than it is
one displaced
by
factor of five.
underestimate by a
more
sample
of the
transversal
each
3.
that our
to mean
quantitatively interpreted
rates
scattering
encouraging result.
actually an
This is
values.
measured
of five below the measured
consistently fall about a factor
values.
the
with
of magnitude
order
in
agree
we estimate
all of the times
first of all that
We note
estimated
enter
spin-orbit scattering
the
into
Experimental Work
spin-orbit
the
scattering
Abrikosov-Gor kov
in superconductors.
We
matrix
theory of
included all
215
numerical coefficients as
of the
have
We
displaced
surface atoms
from
in superconductors and have achieved
experimental
agreement with
good
estimate
to
scattering
spin-orbit
contribution of
the
how
indicated
further
properly
reasonably
previously been
interaction than had
spin-orbit scattering
supposed.
a much smaller
These considerations lead to
the ion core.
screening of
inner shells on
by the electrons in the
charge
the nuclear
well as the
determinations
of spin-orbit scattering times.
is the
The recognition that surface spin-orbit scattering
spin-orbit scattering times
dominant contribution to the
spin-polarized tunneling experiments was
by
and
Meservey
or
whether
not
mechanism
is to
thickness,
but
scattering
this is
indeed
do experiments
to
spin-polarized
scale with
vary
the
tunneling
scattering
series of
on a
films.
The
would
be
demonstrate
a factor of
possible,
and
because
two variation of
the size dependence.
would
It
significantly
sizes
experiments
this
surface
thickness.
requirement that the orbital diamagnetism be small.
appear that
different
if the spin-orbit
see
the sample
film
verifying
for
dominant
the
identical
otherwise
only given recently
test
crucial
mechanism will be verified
scattering times
difficult
A
Tedrow.
in
be
of
is
in
the
It does
sample thickness
sufficient
to
216
One should
predicting
samples
be able
spin-orbit
that
are
to
scattering
intentionally
concentration of impurity.
all
of
the information
spin-orbit
scattering
Preliminary doping
Tedrow, and Bruno
by
doped
working
with
a
of
with
known
case one knows directly
that
is
needed to
time
according
estimate
to
Eq.
VI.7.
were never
Such a direct quantitative
scattering times
the
by Meservey,
a few years ago but the results
predicted spin-orbit
interesting.
In this
times
our method
experiments were attempted
analyzed quantitatively.
the
verify directly
would be
test of
quite
217
We
fields.
magnetic
in
richer
than the properties
which
properties
either metal
of
each other via the proximity effect include
states
the spin-density.of
films
influence
the splitting of
the
to the extent that
peaks in
the sandwich
side of
states on the
densities of
the spin
New
film alone.
two
the
only appear when
coupled
considerably
are
limit
Pauli paramagnetic
the
films
two
of
properties
the
that
saw
parallel
in
sandwiches
proximity-effect
of thin
behavior
to predict the
thesis we used a tunneling.model
this
In
CONCLUSION
AND
SUMMARY
CHAPTER VII:
which is the weaker superconductor can actually cross at the
Fermi level.
enough
great
states to
Once the spin splitting caused by the field is
the
for
cross,
of
energies
the
range,
coupling between
there
will be
density of states
peaks
For
this strength
This
split densities
can
result
a certain
peak in
the total
the addition of
of coupling,
the
rapidly
of states to cross
in
the separated
the down-spin densities of states.
superconducting sandwich grows
the
of
falls within
a sharp zero-bias
in the up-spin and
sides
a net magnetization.
acquires
the films
reflecting
down-spin
BCS pair potentials start
the sandwich, the self-consistent
If
and
depairing in both
there is some
to decrease, and the sandwich
up-
some
a
magnetization of
once
at
the
field
the
causes
the Fermi level.
superconducting-state
spin
218
of the
susceptibility which exceeds the Pauli susceptibility
normal state.
the results of the
Preliminary experiments stimulated by
showed a clear peak in
calculations described in this thesis
just below the sandwich
the zero-bias conductance at fields
the normal metal side of
critical fields for tunneling into
a
was not
experiments
the conductance
structure in
a
experiments might
we
Furthermore
measurements would
can
be it
will
complications
caused
by
severely
We
smearing.
tunneling
the
be modified
to increase
the resolution.
suggested
that
some
susceptibility
be quite interesting.
experiments
If these
ways
data
of the observed
characteristics was
feasible
of
number
amount of
orbital diamagnetic
limited by thermal and by
suggested
the resolution
limited and
taken was quite
the
possible because
the
of
analysis
quantitative
A
sandwich.
proximity
be
are as fruitful as
it appears
they
to
the theory
the
add
necessary to
of orbital
spin-orbit
deparing and
of the
off
scattering
spin-mixing
impurities
and
surfaces.
Although we did not include
our
theory
of
the
this spin-orbit scattering
magnetic
field
proximity-effect sandwiches, we gave
behavior
some estimates
of
in
the
for the
219
impurities.
from
had
surface atoms
we
found
experiment.
tested
films
the number of
td the
measured spin-orbit
When we corrected for both of these errors
reasonable
of
for
surface
matrix
scattering
by
elements
can
of
our
measuring
the
The
be measured
Quantitative
these lines would be quite interesting.
and
times can be
correctness
scattering.
of this
doped with impurities.
scattering
the
the
theory
between
agreement
Our explanation of the
thickness dependence
the spin-orbit
The analysis
rates.
also previously misestimated
experimentally
normalization
nuclear
of an impurity ion drastically
which contribute
scattering time.
the
screening of
predicted spin-scattering
modifies the
data
that
saw
We
core electrons
charge by the
of the
matrix elements
dependence of the scattering
atomic number
strength of
directly
experiments
in
along
220
APPENDIX
A:
SOME
RESULTS
FROM THE
GENES-EQUATION APPROACH TO THE
In
Chapter
methods of
treating
out that it
Gennes 3
7
is
In
the
proximity
the
to
in each half
for
a
appendix we
equations
various
sandwich with
the
pair
a
by others 5 7,
of the sandwich be equal.
case where
equal,
we
of the
show that the
with an
energy gaps of
metals
which
only
of a sandwich.)
has
a
We
Our
72
Fermi wave
is
(2)
the metals on the
things:
of the
is an
(1)
For the
are
two metals
of
states
states are
average of
derivations7 1,
discussed
vector
is
then use the derived
(Other
In
result
because we do
The densities of
the two metals.
have
sharp
function for
zero-temperature densities
energy gap
limit
Cooper
temperature
the
wave vectors
two metals are equal.
BCS-like
the
to demonstrate two
the Fermi
a
potential
thin sandwich.
not require that the Fermi wave vectors of
Green's function
pointed
in very thin sandwiches
construct the Green's
describing
We
sandwich.
a generalization of that given
two sides
theoretical
solve the Bogoliubov-de
thin
assume that
of the
EFFECT
effect sandwiches.
This is possible because
this
BdG
discussed
equations
justified
constant
PROXIMITY
is possible to exactly
(BdG)
interface.
it
II we
BOGOLIUBOV-DE
the
the
72
of
transition
the limit where one of
which
very
much exceeds
221
equivalent to
sandwich are
of the
outside boundaries
at the
Green's functions evaluated
the other, the
that of
Thus
Green's functions of the corresponding isolated films.
in
of
average
no
show
and
differently
sides
two
the
limit,
this
behave
sandwich
the
thus
We
properties.
the
demonstrate that the difference in the Fermi wave vector acts
to
invalidate
of
averaging
the
otherwise occur in
properties would
superconducting
the
the Cooper
limit.
In
Chapter II, this effect of a difference in Fermi wave vector
was
in
justify
to
used
Hamiltonian model.
the
part
use of
tunneling
the
in the
In the tunneling model, the films
of their own properties when
sandwich each also retain some
the coupling via the tunneling Hamiltonian is weak.
the some
It should be mentioned that
.results
derived here,
function for a
sandwich comprised of metals
Fermi wave
vectors, are quite
published.
Here we
results
fully.
the double
as
such
of the intermediate
by no means evaluate
Green's
with differing
general and have
In particular, we
potentials are the
layer
never been
and exploit these
assume that the given pair
self-consistent ones
and do
not discuss
the achievement of this self-consistency.
As
in Chapter II,
superconducting
x = -ds
and
metal with
x = 0, and
a sandwich consisting
we study
a
pair
a weaker
potential
AS
of a
between
superconductor with
pair
222
potential
A,
x = d,.
between x = 0 and
We
rewrite the BdG
equations given in Eq. II.8 as
E-4xx]
3A(,X)
1)*4)
(A. 1)
where
)~2Wx
We
k
(A.2)
the
Fourier transformed
have
=(O,k
,k
1AX
i
The Green's
C (
,
A.1
)=
x E,
)X(X
')
are appropriate for
conditions which
x = -ds
and
to the
satisfies
function also satisfies the
at
to
be
Green's function corresponding
wave functions given by Eq.
-
dependence
-(A.3)
The zero-temperature
insulator
Y to
) and have defined
and z
y
a
free
(A.4)
following boundary
an interface
surface
with an
at
x= dn,
0G(
(A.5)
respectively
kx) X',E)
---G,(C
-,
=
xx,
E
X=-Ss
=
I=$
E)=
Following and
Green's
theorem
Feuchtwang? 8
to construct the
and
(A.6)
Arnold
Green's
57
, we
function
can
use
for the
223
entire sandwich from the Green 's
sandwich
the
Green's
the
of
and s-sides
n-
of
absence
side.
the other
These
v l
i x,x',E)
and G0 (
denoted G" (k it ,x,x',E)
functions,
the
for
in the
functions for each side of
respectively,
sandwich,
satisfy
-E
(x)]
along with
)=
0k X Xj )=
)
-
x, x',
(
the boundary condition A.5
x,x
<O
X),
c
>
(A.8)
G:,
for
(A.7)
A.6
for
,
G
and additionally, at x = 0,
0
II
Cx
X
x=o
x'450
x':0
X1o
G"(kt)x;e
(A.
0x
10)
single films,
functions for the
of these Green's
In terms
(A.9)
78
the Green's function for the sandwich is given by
G(x,x',E)
(A .11)
G(0(X)xX'E)
we k
-rP
Y. ,- E) =
X, X'E) -
G
X)X' < 0
G(0oX.[GE),(A.)1
XoE)
(0,0, E) + G(S
C
whoyk
It
isolated
is
possible to construct
0,
xxx',
V1
(,
1
A(.12
o
the Green's function for the
metals from the eigenfunctions
of the BdG equations
224
for each isolated film.
5 (XX, Er)
CT
The results are
5 7
-ts I
E
c~rK"
J "IO ft
+n mGe(K; Xy)(#-$(K" (yg+d)
Als
Kv3
4+
(A. 13
)
E f.
AS~
Vet.
±
A V,l
*
(A. 14)
where
K
(v.
.,L
(A. 15)
A,it) %
(A. 16 )
a
'IL
=
Putting these
find for G
I
Ouf~I
(-ds , -d
[S(!
-
<,(x:
A 3+ 1A,)
-2( I - X1) I ( is
G
for
and G"
into
Eq.
A. 11
(1X) (Es. -S
- X SAo(Kdt,--
'K
)
(
-I
CA1(KV-k-)os
4)
+
cod<kf4, -d
(KAs+ 104) - XKi LA(K-
4A.) -x cZ(K- KN'd )3
~,- Es S&) £L*1(i(2' kf)dwJ
(A. 17)
where
'D
-x') (EKEs-,SS
we
, E)
-(E04IX)4 5 + I)
-Ads I
+2
expressions
J-
3o/ (Ks - K!)sc.
-
(K'-g.")k
-; 2 f.O*(k!s As,+ K+"oA k)- X coq(0d,- j<4"dm)j [eOD(
k x2
_IC, +
..
K4.
S +
E
,~
J
A sJ
">ts
j
$
jL~~) S
sC
225
now evaluate
We
identical
differing
Making
the
this for
Fermi wave
vectors and
with
two metals
limits of
metals with
vastly
A.17
in the
two
Fermi wave vectors.
of Eq.
a partial fraction expansion
limit where the
two films comprising
the
sandwich are thin
and have the same Fermi wave vector, we find
k
AS
/s
+A18)
where
tege*+B
.
± + Tr
___
The
local density
imaginary part of
of states
__
(___
at
(A.20)
_(
-d
x
given by
is
this when integrated over k
BCS density of states with an average
(A.19)
.
the
We find a
gap given by
Ais+
If we interchange the n- and
for the
n-side density
s-sides we get the same result
of states
as this
result for
the
226
s-side density of
where
thin sandwich
and s-side
vectors, the n-
the same BCS shape with the
the case
take
We now
densities of states both display
same average gap.
We first assume X-
-1.
Then
A.17 becomes
V
)-
GAs
9
(
+0)C4Pt Ods
which is equivalent to the
s-side film,
not
greatly
metals have
the
where
different Fermi wave vectors.
Eq.
Fermi wave
the same
metals have
both
limit for a
Thus in the Cooper
states*.
be
different
Eq.
A.13.
Fermi
wave
Green's function for an isolated
at X =-d .
evaluated
surprising because
( A .21)
cAt(K.E )I
-)
in
vectors,
the
limit of
there
is
This should
such
almost
vastly
total
states directly
n-side density of
not evaluate the
*We do
in an identical
at X = d
from the local density of states
the s-side density of states.
manner to our calculation of
A.6 imposed on
the boundary condition Eq.
This is because
leads to a vanishing density
the Green's function at X = d
8
indicated how one is to
has
Feuchtwang7
of states there.
more
any
avoid
have
we
but
case,
this
in
proceed
calculations by this interchange trick.
227
reflection at the interface*.
In
this
same
limit of
vastly
differing
Fermi
wave
vectors, we evaluate the equivalent Green's function for the
n-side by
X -+ 1.
interchanging n- and s-
and taking the
limit as
We find
K%(t-X)
si,(-A , E=)
(6,+1)-1
C-t-f(1)-to"" k+'d()
k
(A.22)
This results in a density of states equivalent to that of an
which can be
isolated n-metal
all
the
s-side
counterparts.
obtained from Eq.
quantities
The
replaced
difference
by
between
A.13 with
their
the
n-side
and
tangent
cotangent functions can be traced to an effective difference
the interface between a medium
in the boundary condition at
in which
waves propagate with
a high
velocity (v,= fkF/m)
and one in which they propagate with a low velocity.
This difference
easily pictured
which occurs
is physically
of the partial reflection
at an
interface between
light rope held in tension.
rope are
reflected inverted,
condition requiring the
to one
equivalent
more
and transmission
a heavy
rope and
a
Waves incident from the lighter
as if
there were
amplitude of the wave
a physically illuminating
give
*Below we
explains the near total reflection.
a boundary
to vanish at
analogy
which
228
the interface
are reflected uninverted,
heavier rope
of
the wave
with
were a
as if there
vanishing derivative for the
boundary condition requiring a
amplitude
from the
Waves incident
between the ropes.
at
to position
respect
the
interface.
We have now demonstrated that
one effect of a difference
in Fermi wave vectors between metals in the sandwich is that
each
side
in
Each side
identity.
when
will
the
tunneling
qualitatively then,
results which
in
differing
obvious
Hamiltonian
manner)
tunneling
of
more
used.
perfect interface
This is,
of
own
its own identity
of
least
At
Hamiltonian should
the tunneling
their superconducting
the
is
its
of
more
retain
also retains some
describe a
Fermi wave vectors.
description
some sense
give
between metals
properties
their
and
course, in addition to the
Hamiltonian gives
complicated
containing real tunneling barriers.
(in
interfaces
a
more
perhaps
229
CALCULATION OF THE SANDWICH SUPERCONDUCTING
APPENDIX B:
TRANSITION
TEMPERATURE
1.9),
these
4
(E)
order
transition field
to the
r
+4+
) A
when
made
A
form given in
0
IV.1
111.41
B.1
and
solve
values of
second
we find
&rift(B.
to
the
(a
discussing
1)
of
case
a
simplification
the
transition
by setting
(B.2)
= 2N
from Eq.
Eq.
these
to IV.4,
in the McMillan model'is 80)
=
Now putting
IV.1
sandwich
normal-superconducting
temperature
order.
temperature.
simplified
have
,pk ,
relating the
an equation
to derive
conventionally
is second
and use
A;(E),
for the pair potential
we
sandwich into
parameter
order
of the
Proceeding to linearize Eqns.
where
field
and
Chapter I
(see
it is possible to expand equations
in powers
in IV.5
FIELD
proximity
with increasing
This being the case,
through IV.4
the
transition of
the
normal state
the
A FUNCTION OF
high temperatures
At sufficiently
Figure
AS
into the gap equation
performing
the
in the
sum over states,
230
we find
+
,
SQ
p)
*
(B. 3)
.J
Making use of the relationship
(2 e I(.
N(E,) V,
where
'
Tc,
(I
is Euler's constant (
temperature of
bulk transition
is the
.57722) and Tc
'
the superconductor,
and of
the relationship
IT
2eW)
(B.5)
04 W;4C t
where
we
are
the frequencies appropriate
critical temperature, T
(3, c
T
4s
,
Eq.
W
for
the sandwich
B.3 can be rendered
/~(+
ce
6'
e
(B.6)
These sums can be expressed in terms of the
+(X)=
by using the property 11 9
- - -
++
Finally we
digamma function
get
)
-
(B.7)
231
T
~ ~±~
)
.!-AA4
(B.8)
2,10
-
cancel at
where we have made the approximation that the sums
their upper limits.
function of temperature.
field as a
critical
the
that
temperature
We might
order critical field" which we have
all
at
increasing
with
and we have no experiments with which to compare
a quantitative estimate.
exists
know qualitatively
We
decreases
field
critical
of the
calculations
no numerical
We present
temperatures
just calculated actually
the
whether
of
regardless
first order
the normal state is
sandwich's transition into
"second
mention that the
or second.
In the case of a first order transition, such as
illustrated
in
Figs.
calculated
is
the
parameter
IV.20
goes
equation to the
Eq.
T = 0 plots
calculate
H5
makes the
difficult.
where
the
supercooling
comparison of
We
Eq.
In
this
back
can go
field
function formalism.
I
order
the
singularity of
appropriate transformations
zero-temperature Green's
we find:
field
actually
have
what we
Actually the
zero,
to
IV.13, make the
directly
and IV.4,
supercooling
to zero.
goes
as T
1.7
to
there,
and
using
the
this case
I
E-)
/= He
2.
+
'S
( B .+9
C
WD
(B.9)
232
is the
where &s
T=O,
H=O order
parameter of
one of weak coupling.
absence of
6s/(2A.)
so
a proximity
we can
see
As
isolated
in writing the last line
s-metal and the approximation made
is
the
discussed in Chapter I,
effect, the
that the
expected and reduces this field.
in the
supercooling field
proximity effect
acts
is
as
233
BIOGRAPHICAL NOTE
I
was
There
I grew
Grade
School,
Creighton
the
school's
chapter
of
the
to
summers
I
1972
was
1974
employed
to work
accelerator
to excite
of
physics
and
while
Dr. Sam
Department
May
a
on
Sciences
and
X-rays
was
Senior
as
I was
J.
for
of
a corecipient
support the
to
courses
instrumentation
for
and I
of
1976
superconductivity
I
spent
electron
utilizing MIT's 400 Mev Bates
summer
Physics
small
a
In
analysis.
cum laude from Creighton as a
as a graduate student in physics.
took
During
the
using
element
the
at Creighton,
in
at
aimed
of
Students.
Cipolla
trace
served
President
studying
the
College
started
under
Dr.
three years
first
graduate studies, and in September of
I
I
and
of
Arts
Predoctoral Fellowship from the National
Science Foundation
MIT,
School,
Award.
I was awarded a
at
High
Physics
project
1974 I graduated summa
major
year
of
Catholic
Lourdes
University,
Creighton
the Society
by
of
Our Lady
school
Omaha, Nebraska.
in
Preparatory
At
1972-1973
1953,
24,
attending
up,
University.
Creighton
during
on September
born
of
my
1974 I enrolled at MIT
During my first two years
a few
months working
scattering
experiments
Electron Accelorator.
working
Brian B.
in
the
Schwartz
on
In the
theory
at
of
MIT's
234
Francis
Bitter National
Magnet
Laboratory and
eventually
this became my thesis work.
In
February of
position
Public
as assistant
of
Affairs
1978-79 Chairman to
of
1978 I
IBM's
Thomas
completed, I
theoretical
assistant to
American
J. Watson
will commence a
one
half-time
of Panel
Society.
Physical
Research
Richard L.
Center
in
graduate studies
second half-time
as a postdoctoral
superconductivity.
in a
Chairman
whom I report is Dr.
In September, my
Heights, NY.
IBM, this
the
started working
researcher
on
The
Garwin
Yorktown
now being
position at
in the
area of
235
REFERENCES
124c
122b,
120b,
Leiden Comm.
1.
Kamerlingh Onnes,
H.
(1911).
2.
W. Meissner and R.
787 (1933).
3.
C.J. Gorter and H.B.G. Casimir,
Z.
(1934);
35, 963
Z.
Phys.
(1934).
4.
Proc. Roy. Soc. A149, (1935);
F. London and H. London,
following
the
also
see
(1935);
341
2,
Physica
reference.
5.
F.
London, Superfluids,
6.
E.
Maxwell,
Ochsenfeld, Naturwissenschaften 21,
78,
B. Serin, W.H. Wright, and
487.
7.
W.S.
Wexler,
8.
L.N.
9.
J.
10.
Rev.
Phys.
Phys.
Cooper,
Bardeen,
104,
Rev.
and
C.B. Satterthwaite,
(1954).
1189
and A.
(1956).
Phys.
J.R.Schrieffer,
Rev.
(1957).
108,
1175
A.A.
Abrikosov,
V.L.
1442
Cooper,
L.N.
Ginzburg
Zh.
i Teor.
Eksp.
in Soviet Phys.
English trans.
11.
96,
1950.
C.A. Reynolds,
477 (1950);
L.B.Nesbitt, Phys. Rev. 78,
Goodman,
B.B.
Corak,
New York, Wiley,
Vol. I,
Rev.
Phys.
Physica 1, 306 (1934);
539
15,
techn. Phys.
and
Fiz.
JETP
Landau, Zh.
L.D.
5,
32, 1442 (1957);
1174 (1957).
i
Eksp.
Teor.
Fiz.
20, 1064 (1950), English translation in Men of Physics:
1965, p.
I, Oxford, Pergamon Press,
L.D. Landau, Vol.
138.
12.
B.
also
(1962); see
63
S2UIDS
Applications:
S.
Research and
Goodman, IBM Journal of
Foner, eds.,
New York,
13.
McNiff,
S. Foner, E.J.
49A, 269 (1974).
14.
Decroux,
Fisher, M.
0.
C:
Phys.
J.
Sergent,
(1975).
15.
K.
Maki, Phys.
Rev.
Development 6,
Superconductor
Pippard in
A.B.
Schwartz and
and Machines, B.B.
148,
1977,
Plenum,
and E.J.
p.
Alexander,
1.
Phys.
Lett
M.
Chevrel, and
Roth, R.
S.
L474
8,
Phys.
State
Solid
362
(1966).
236
16.
Hohenberg, Phys.
E. Helfand, and P.C.
N.R. Werthamer,
Werthammer, p.
R.
N.
also
see
(1966);
295
Rev. 147,
817, in
p.
Hohenberg,
P.C.
and
Fetter
A.L.
and
321,
Dekker,
York,
New
ed.,
Parks,
R.D.
Superconductivity,
1969.
17.
B.D.
Josephson, Physics
B.D.
Josephson,
Lett.
Rev. of
Mod.
1, 251
M.H. Cohen, L.M. Falicov,
Lett. 8, 316 (1932).
19.
R. Meservey, P.M. Tedrow, and P.
25, 1270 (1970).
20.
P.M. Tedrow,
R. Meservey,
4224 (1975).
B11,
21.
Meservey and
R.
(1976).
22.
M.B.
Maple,
therein;
Appl.
Phys. Rev.
Lett
Phys.
Rev.
R.C. Bruno,
and
(1976),
references
and
W.A. Fertig, D.C. Johnston, L.E. DeLong,
McCallum,
M.B.
Matthias,
B.T.
and
Maple,
(1977); F. Acker, B.T.
Lett. 38, 987
Rinderer, J. Low. Temp. Phys. 30, 133
Phys.
Physics Today
24.
H.
Frohlich, Phys.
25.
J.
Bardeen, Phys.
26.
Helv. Phys.
R. Schafroth,
M.
Nuovo Cimento 9, 291 (1951).
27.
J. Bardeen, in Handbuch der Physik, Vol. XV,
ed., Berlin, Springer, 1956, p. 274.
28.
Pippard, Proc.
A.B.
(1953).
29.
I. Giaever,
30.
N.N. Bogoliubov, Nuovo Cimento 7, 794 (1958);
English trans.
58 (1958);
Fiz. 34,
i Teo.
Phys. JETP 7, 41 (1958).
31.
J.G. Valatin, Nuovo Cimento 7,
32.
P.W. Anderson, Phys. Rev.
Phys.
Rev.
845
587
80,
Royal
Rev. Lett.
112,
Rev.
7,
p.
(1950).
(1950).
Acta
Soc.
5,
No.
30,
B.B. Schwartz and S. Forter,
34, (July, 1977).
79,
R.W.
Matthias, and L.
(1978).
23.
Rev.
131
58A,
Lett.
Phys.
179
9,
also
Rev.
Fulde,
Tedrow,
Phys.
see
(1962);
251 (1974).
Phillips, Phys.
and J.C.
18.
P.M.
46,
Phys.
147
843
1900
(1951);
241, 645,
S.
Flugge,
(London) A2 16,
547,
(1960).
(1958).
(1958).
Zh. Eksp.
in Soviet
237
33.
34.
Zh. Exsp.
i Teo.
Gor'kov,
English trans. in Soviet Physics
de
P.G.
Gennes,
For
for
a
Superconductivity,
67ff,
36.
For
37.
Metals
and
Alloys,
discussion see,
and quantitative
to
Tinkham,
Introduction
New
York,
review of high
McGraw-Hill,
1975,
superconductivity
field
in Phys. 22,
see P. Fulde, Adv.
pp.
667
in thin
(1973).
Principles of the Theory of
Cambridge University Press,
example J.M. Ziman,
2nd ed., Cambridge,
See for
Solids,
1972,
(1958).
111ff.
a
films
of
(1958);
735
505
chapter 5.
1966,
more detailed
example,
M.
34,
JETP 7,
Superconductivity
New York, Benjamin,
35.
Fiz.
L.P.
p.330 ff.
38.
Y.
Yosida, Phys.
39.
P.
Fulde
Rev.
110,
Ferrel,
and R.A.
769
(1958).
Phys.
Rev.
135,
A550
(1964);
see also A.I. Larkin and
Y.H. Orchinnikov, Zh. Eksp. i
Teor. Fiz. 47, 1136 (1964); Soviet Physics JETP 20, 762
(1965).
40.
B.
S.
Chandrasekhar, Appl.
Clogston, Phys.
Rev. Lett.
Sarma, J. Phys.
Phys. Lett
9,
266
1, 7
of Solids, 24,
Chem.
(1962);
A.M.
(1962).
(1963).
1029
41.
G.
42.
K. Maki and T.
945 (1964).
43.
transitions
in
recent
reviews
of
the
phase
For
metamagnets see J.M. Kincaid
and E.G.D. Cohen, Physics
N.
and
E.
Stryjewski
and
Reports
22,
57
(1975);
Tsuneto, Prog.
Giordano, Advances
44.
S.
Frota-Pessoa
20,
505
45.
R.B.
46.
E.
F.
Roy.
in
Theor. Phys.
Physics,
and B.B.
26,
487
Schwartz,
(Kyoto),
31,
(1977).
Solid
State
Comm.
(1976).
Griffiths, Phys.
Rev. Lett. 24,
Burton, J.O. Wilhelm,
Soc.
Can.
28,
65
715
and A.D.
(1934);
A.D.
(1970).
Misener, Trans.
Misener
and
J.0.
Wilhelm, 29, 13 (1935); A.D.
Misener, H. Gayson Smith,
(1935); A.D. Misener, Can. J.
and J.0. Wilhelm, 29, 1,
of Research 14, 25 (1936)
47.
A.
48.
L.A.
Shal'nikov, Nature
142,
74
(1938).
Feigin and A.I. Shal'nikov,
Sov. Phys.
Doklady 1,
238
377 (1957); D. Shoenberg, Superconductivity, Cambridge,
Cambridge University Press, 1952, p. 166.
Rev.
(1958); Phys.
109, 686
Phys. Rev.
(1959); Phys. Rev. 117, 672, (1960).
49.
H. Meissner,
Lett. 2, 458
50.
Shapiro, J.L. Miles,
P. Smith, S.
Rev. Lett. 6, 686 (1961).
51.
W.J.
52.
16, 16 (1966); W.
W.J. Tomash, Phys. Rev. Lett.
and T. Wolfram, Phys. Rev. Lett. 16, 352 (1966).
53.
J.M. Rowell and W.L. McMillan, Phys.
(1966).
54.
P.G. de Gennes
(1966).
55.
P.M.
1552,
62A,
Phys.
Phys.
56.
P.M. Chaikin, G. Arnold, and
Phys. 26, 229 (1977).
57.
G.
58.
L. Dumoulin, E.
1086, 1977.
59.
A. Gilabert, Ann. Phys.
60.
P.
61.
Equations 6
W.L. McMillan, Phys. Rev. 175, 537 (1968).
Tr; Eq. 15
of
missing factors
and 10 in this paper are
Eq. 39
in
Q
and
2;
place of the
a 4 in
should have
should be
Tomash, Phys.
Rev.
and D.
Lett.
(1965);
672
15,
Phys.
J. Nicol,
and
Rev.
Saint-James, Phys.
Lett.
Tomash
16,
453
4,
151
Lett.
36,
Rev. Lett.
Phys.
P.K. Hansma,
Chaikin and
(1976); E.L. Wolf and J. Zasadzinsky, Phys. Lett.
Finnemore,
and D.K.
Toplicar
J.R.
165 (1977);
C.K. So,
and
Bermon
(1977); S.
B16, 2072
Rev.
Rev. B17, 4256, (1978).
P.K.
Hansma, J.
Low Temp.
Arnold, to be published.
Guyon, and P. Nedellec,
Nedellec, Ann. Phys.
203
(1977).
2, 253
(1977).
2,
Phys. Rev.
B16,
7.
62.
Dzyaloshinski,
and I.E.
L.P. Gorkov,
A.A. Abrikosov,
Methods of Quantum Field Theory in Statistical Physics,
p.
Dover, 1975,
New York,
R.A. Silverman,
trans. by
283-4.
63.
T. Matsubara, Prog. Theor.
64.
36, 1918 (1959);
Exsp. i Teor. Fiz.
L.P. Gor'kov, Zh.
(1959).
1364
9,
Engl. trans. in Soviet Phys. JETP
Phys.
14,
351
(1955).
239
W.
65.
references
gives
Silvert
(1975);
proximity
the
on
work
earlier
to
439
20,
Silvert, J. Low Temp. Phys.
effect using the Gor'kov equation approach.
P.G.
66.
Gennes
de
E.
and
3,
Lett.
Phys.
Guyon,
168
(1963).
P.
67.
W.
Fulde and
(1967);
A.
Kondens.
Morrman, Phys.
Migliori
and
Mater. 6,
403
Rev.
BS,
Phys.
Ginzberg,
D.M.
N.L.
Smith, and
Blackburn, H.J.T.
(1973); J.A.
5063
M.
H.J. Fink,
(1975);
B11, 1053
Phys. Rev.
Rowell,
J.P.
Laheurte,
J.P.
Gilabert,
A.
Sheikholeslam,
B14, 1052
E. Guyon, Phys. Rev.
Romagnan, J.C, Nioray,
Phys. F
and A.A. Babaei-Brojeuy, J.
(1976); N. Rivier
7, L 111 (1977).
Yu,
M.
68.
M.
A.
Strongin,
Rev
Phys.
Paskin,
996
B14,
(1976).
Cooper,
L.N.
Jorunal 6, 75
70.
P.G.
71.
G.V. Minnigerode, Z.
72.
.
de Gennes, Rev. Mod.
0. Entin-Wohlman
Phys. Rev. B.
Phys.
fur Phys.
and
(1961);
689
6,
Rev. Lett.
Phys.
(1962).
69.
36,
225
(1964).
192,
379
(1966).
J. Bar-Sagi,
IBM
published in
to be
73.
D.
Belin,
M.
Leger,
A.
Klein,
J.
Bellanger,
D.
J.M. Rowell,
459 (1973);
Lett 42A,
Defourneau, Phys.
Phys. Rev. Lett. 30, 167 (1973); J. Vac. Sci. Tech. 10,
702 (1973); S.L. Colucci, W.J. Tomasch, H.J. Lee, Phys.
Rev. Lett. 32, 590 (1974).
74.
W.L. McMillan and P.W.
(1966).
75.
W.L. McMillan, Phys. Rev.
76.
S.
77.
J. Bar-Sagi,
1669 (1976).
78.
T.E. Feuchtwang, Phys. Rev. B10, 4121,
3979 (1975); B13, 517 (1976); B13, 517
79.
J.
(1974);
15, 729
Comm.
State
Bar-Sagi, Solid
J.
Bar-Sagi and 0. Entin- Wohlman, Solid State Comm. 22 29
(1977).
Bermon and C.K. So,
Z. Physik
Anderson, Phys. Rev.
559
175,
Phys.
269,
16,
85
(1968).
Rev. B17,
7
Lett
(1974);
4256
J.
(1978).
Phys.
F 6,
4135(1974); B12,
(1976).
240
Rev. B1,
Phys.
Zuckermann,
and M.J.
229
80.
A.B. Kaiser
(1970).
81.
K.
82.
Y. Nambu, Phys.
83.
V. Ambegaokar and A. Baratoff, Phys. Rev.
(1963); Erratum 11, 104 (1963).
84.
of
Theory
2uantum
Walecka,
J.D.
and
Fetter
A.L.
pp.
1971,
McGraw-Hill,
York,
New
Systems,
Many-Particle
248-50.
85.
R.
86.
0.
S. Skalski,
A1500 (1964).
87.
P.
88.
Romagnan, and E.
A. Gialbert, J.P.
Comm. 9, 1295 (1971).
89.
G.B. Arnold, Phys.
90.
Tedrow and
P.M.
(1976).
91.
P.M. Tedrow and R. Meservey, to be published.
92.
57
51A,
Lett.
Phys.
R. Meservey,
and
Tedrow
P.M.
(1975); R. Meservey, P.M. Tedrow, and R.C. Bruno, Phys.
Rev. B17, 2915 (1978).
93.
R. Meservey and P.M. Tedrow, to be published.
94.
D. E. MacLaughlin, Solid State Physics 31, 1 (1976) has
in
experiments
magnetic resonance
reviewed
recently
superconductors, including Knight shift experiments.
95.
F.
96.
Androes and
G. M.
(1961).
97.
R. H. Hammond
(1964).
98.
R.
99.
P. W.
Machida, J.
Rev.
Weiss,
B17,
3576
106,
W.
and G. M.
Ferrell, Phys.
Anderson, Phys.
208
Rev.
Phys.
136,
(1965).
675
18,
Solid State
Lett.
58A,
237
(1957).
Phys. Rev.
Kelly, Rev. Mod.
Rev. Lett.
Rev.
468
(1978).
Knight,
D.
10,
Lett.
Guyon,
Phys.
Meservey,
R.
(1977).
(1960).
Rev. Lett.
Maki, Phys.
Rev.
27 737
Appendix B.
1972,
MIT,
Betbeder, P.R.
Fulde and K.
A.
648
117,
Rev.,
Boro, Thesis,
Reif, Phys.
Phys.
Low Temp.
3, 262
Lett. 3,
121,
779
36,
185
Phy.
(1959).
325,
(1959).
241
Zh. Eksperim.
Gorkov,
Phys. JETP 15,
Soviet
and L. P.
A. A. Abrikosov
100.
42, 1088, (1962);
i. Teo. Fiz.
752 (1962).
Matthias, unpublished;
see Ref.
14.
101.
B.T.
102.
R. A. Farrell, University of Maryland Technical Report
No. 329 (unpublished).
103.
R. Kubo
(1956).
104.
J.
Appel, Phys.
105.
R.
H.
Hammond
Phys.
Obata, J.
and Y.
Rev
139,
G.
and
547
11,
(1965).
A1536
M.
Japan
Soc.
Rev.
Lett
18,
Kelly,
P.
and M.
Strongin, Phys.
156
(1967).
106.
H. L. Fine, M. Lipsicas,
29A, 366 (1969).
107.
P.
W.
Anderson
New York,
108.
J.
E.
Rev.
109.
110.
in Superconductivity,
Dekker,
1969,
R.
D.
I.
1352.
p.
and A. K.
Crow, M. Strongin,
B9, 135 (1974).
Schiff
McGraw-Hill,
ed.,
Parks,
Bhatnagar,
R.
and
Bethe
A.
H.
example
for
See
ed.
Mechanics, second
Intermediate 2uantum
MA, Benjamin, 1968), p. 377.
L.
Lett.
Mechanics,
Q
2uantum
1968, p. 438.
3rd ed.,
14,
1 (1963).
Yafet, Solid State Physics
Phys.
Jackiw,
,
Reading,
New York,
111.
Y.
112.
H. E. White, Introduction to Atomic Spectra, New York,
McGraw-Hill, 1934, p. 109, 112.
113.
F.
W.A.
Wright,
Lett.
18,
115
Hine s,
and W.D.
Knight, Phys.
Rev.
(1967).
114.
F. Wright, Phys. Rev. 163,
W.D. Knight, Phys. Rev. B4,
Hines
and
115.
V.I. Petinov and A. Yu. Ardashev, Fis. Tverd. Tela
3 (1969); Sov. Phys. Solid State 11, 1 (1969).
11,
116.
Sambles,
J.R.
Stesmans, and J.
(1977).
117.
W.M. Walsh,
G.
420
893
Sharp-Dent,
Witters,
Phys.
L.W. Rupp, and
(1967);
(1971).
W.A.
Cousins,
J.E.
Stat. Sol. (b) 79,
P.H. Schmidt,
Phys.
A.
645
Rev.
242
Schultz and G. Latham, Phys.
Lett. 16, 181 (1966); S.
T. Kato, S. Hiramatsu, and
Rev. Lett. 13, 148 (1965);
(1978); D.
Japan 44, 449
3. Phys. Soc.
H. Hirakawa,
1104
Lett. 36,
Rev.
Schultz, Phys.
and S.
Lubzens
Phys. Ref.
R.T. Schumacher,
K. Wang and
(1976);
S.
Phys. Rev.
correction in
(1973); important
B8, 411 9
Zh.
Petinov
and V.I.
Ya. Gen
B10, 21 29
(1974); M.
Ekxp. T eor
Fiz. 48, 29 (1965); Sov. Phys. JETP 2 1, 19
S tatus
Cousins, Phys.
and J.E.
A.J. Watts
(1965);
Chem.
Phys.
Taupin, J.
C.
(1968);
Solidi
30
105
Solids 28, 41 (1967); K. Saiki, T. Fujita, Y. Shi mizu,
32 , 447
Soc. Japan
Wada, J. Phys.
Sakoh,
and W.
S.
(1972).
118.
Physics,
Solid State
Introduction to
C. Kittel,
ed., New York, John Wiley & Sons, 1976, p. 167.
5th
119.
Functions, Vol.
Higher Trancendental
A. Erdelyi,
New York, McGraw-Hill, 1953, p. 19.
I,