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MSU lo An Afflrmotivo Action/Equal Opportunity Institution
mm:

A Quantitative Test of Quantum Transport in
Ag Films

By

Paul J. McConville

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1 995

Abstract
A Quantitative Test of Quantum Transport in Ag Films
By
Paul McConville

The consequences of long-range electron phase coherence to low-
temperature electrical transport in metals are now well appreciated. Even in
weakly disordered metals, there are substantial deviations from the predictions of
semiclassical Boltzmann transport theory. Two examples of phenomena arising
from electron interference are weak localization (WL) and universal conductance
ﬂuctuations (UCF). The WL corrections to the average conductivity arise from
the interference between pairs of time-reversed paths that return to the origin.
The interference extends over the length scale L4,, the phase breaking length.
UCF, on the other hand, are sample-speciﬁc ﬂuctuations in the conductance that
occur as a function of impurity conﬁguration, magnetic ﬁeld, and chemical
potential. UCF arise from the interference between all of the possible paths an

electron can take in traversing the sample. UCF are governed by two length
scales, Lth and L4,. Lth is the thermal length and is the distance two electrons

separated by KT in energy remain phase coherent. Since L(p is common to both

quantum interference theories, measurements of Lq, can be used to test the

consistency between the two different theories.
We have measured the magnetoresistance and 1If resistance noise
reduction in a magnetic ﬁeld in the temperature range 1-25K for three Ag

samples. Fits of magnetoresistance to WL theory provide an estimate of the

phase breaking length, Lq,. Fits of the 1/f resistance noise reduction in a
magnetic ﬁeld provide another independent estimate of Lq,. By comparing the

independent estimates of Lq, given by the two quantum interference theories, we

can quantitatively test the consistency between these theories. Our results
indicate excellent agreement above about 10K and statistically signiﬁcant
discrepancies below about 10K. We discuss several possible sources of the

discrepancy.

To the pursuit of knowledge

Acknowledgments

I thank my advisor, Norman Birge, for all his help and guidance over the past several
years. His gentle prodding and eye for details have certainly had a major impact on the work i
have done in physics. He is not only my advisor, but my friend as well.

My fellow labmates also deserve thanks for all their support and many discussions that
have taken place over the years. Kookjin Chun taught me many things about e-beam lithography
as well as Korea. I have beneﬁted from numerous discussions about UCF with Jeong-Sun
Moon...though I still dislike random matrix theory. Dave Hoadley deserves special thanks for help
in gathering data and the analysis present in much of this dissertation. The days of the day-shift
and night-shift, I hope, are over forever. I also thank Diandra Leslie-Pelecky, Matt Miller, and Rod
Lambert for introducing me to the physics of the glass transition.

I thank my guidance committee for their support and reading of this dissertation,
professors R. Brock, P. Duxbury, W. Pratt, and W. Repko. Personally I recommend waiting for
the movie. A special thanks to J. Cowen for his help in securing employment and to all the
professors that helped in one way or another.

I give special thanks to the "shop guys" for their help and humor, not to mention the
morning coffee: Tom Palazzolo, Tom Hudson, and Jim Muns. I also want to thank all the PA staff
for their help, in particular Stephanie and Reza.

Many more physics graduate students are to thank for their support and advice. Too
many to mention, but I will name a few: Catherine Mader, Ninamarie Levinsky, Carl Nelson, Lowell
McCann, and Mike Jaeger. Special thanks go out to the friends that l have lived with over that
past several years, Erik Hendrickson, George Jeffers, and Lilian Hoines. Also my friends in

Science Theatre deserve thanks, in particular Bill Abbett for his many discussions.
V

To my many friends not in physics, I give thanks: Mike Mohar, Mark Jameson, Buck
Worthington, Rob Selden, Heidi Aseltine and Danielle Casavant, just to name a few.

I thank all my friends from the Peanut Barrel and Dagwood's, for their continuing support
and reality checks. In particular at the PB, SEED for his awesome burgers, Joe Bell for his
generosity, Lisa Silsby for her interest and support, and all the other people that have been so
kind to me over the years. At Dagwood's, I want to acknowledge Tommy Hayes, the answer man,
and perhaps the funniest person I know, for his witty comments that always lift my spirits. I want
to thank all the other people who work at Dagwood's for their generosity, including owner Bob
Locca. The friends I have gained from both establishments deserve thanks. In particular, Charlie
and Karen Lauterbach, Tee and Jackie Downes, Bubba, Red, Sid, Ed & Beth, Boot, Van, JD,
Kevin, Steve, Dave, Greg, Lisa, Petey, Barry, Hiney, Rick, Lee, Dale, etc.

My deepest acknowledgments go out to my family. I especially thank my father for his
raw creativity and work ethic, my mother for her caring and gentle nature, and my sisters for their
understanding and appreciation of my many years of school. I thank my grandparents for their
support and all the McConville and Kavanagh clan.

Thanks is not enough to give my good friend Jeff Guerkink, who has always kept my feet
on the ground, and not too far from the barnyard of life. Not only for his ﬁnancial help during my
undergraduate studies while working on his farm, but also his continuing support is appreciated.
My only regret is I won't be farming anytime soon. I also want to thank his family.

I further acknowledge the PA department, NSF, CFMR, Department of Education, and the
College of Natural Science for their ﬁnancial support of this work.

Finally, I want to suggest to my high school counselor, who said I should go to vocational

school because I was not smart enough to make it in college, that maybe he was wrong.

vi

Table of Contents

List of Figures ........................................................................................... x
1. Introduction
1.1 Ohms Law ............................................................................. 1
1.1.1 Sample Size Dependence of Resistance ................... 1
1.2 Conductivity ........................................................................... 2
1.2.1 Drude Model ............................................................... 3
1.2.2 Modern Quantum Theory of Conduction .................... 5
1.2.3 Scattering Rates ......................................................... 7
1.2.4 Discussion of Validity .................................................. 8
1.3 Disordered Electronic Systems ............................................. 9
1.3.1 Localization Length ...................................................... 9
1.3.2 Scaling Theory of Localization ..................................... 10
1.3.3 Landauer Formula ....................................................... 11
1.3.4 Random Walk of Feynman Paths ................................ 12
1.4 Quantum Interference ............................................................ 13
1.4.1 Quantum Phase Coherence ........................................ 14
1.4.2 Weak Localization ........................................................ 15
1.4.3 Universal Conductance Fluctuations ........................... 16
1.5 1lf Noise .................................................................................. 17
1.5.1 Resistance Fluctuations in Time .................................. 18
1.5.2 Local Interference of Electrons .................................... 21
1.5.3 Power Spectrum of Resistance Fluctuations ............... 22

vii

1.6

1.5.4 Defect Dynamics .......................................................... 25

1.5.5 1lf Noise versus Temperature ..................................... 29
Motivation ............................................................................... 29
1.6.1 Enhancement of 1/f Noise at Low Temperatures ........ 30
1.6.2 Consistency Test of Quantum theories ........................ 30

2. Experimental Considerations

2.1

2.2

2.3

Choice of Materials ................................................................ 33
2.1.1 Comparison with Free Electron Parameters ............... 33
2.1.2 Ability to Make High Quality Disordered Films ............ 34
2.1.3 Magnitude of 1lf Noise ................................................ 34
Sample Fabrication ............................................................... 35
2.2.1 Triple-Layer Photolithography ..................................... 35
2.2.2 Single-Layer Projection Lithography ........................... 38
Measurement Circuits ............................................................ 41
2.3.1 1/f Noise Circuit .......................................................... 42
2.3.2 Magnetoresistance Circuit .......................................... 43

3. Weak Localization

3.1.

3.2

4.1

Introduction ............................................................................ 46
3.1.1 Interference of Time-Reversed Paths ......................... 46
3.1.2 Magnetic Field Dependence of WL ............................. 48
3.1.3 Spin-Orbit Scattering .................................................. 49
2-D Weak Localization Theory and Analysis ......................... 50
3.2.1 2-D Weak Localization Theory .................................... 51
3.2.2 Data Analysis .............................................................. 52
4. Universal Conductance Fluctuations
Introduction ............................................................................ 56
4.1.1 Origin of UCF .............................................................. 56

viii

4.1.2 Effect of Temperature on UCF .................................... 60
4.2 UCF Enhanced 1/f Noise ....................................................... 62
4.2.1 Sensitivity of the Conductance to Impurity Movement 62
4.2.2 1lf Noise Crossover Function ..................................... 64
4.3 Data Analysis ........................................................................ 67

5. Comparison and Summary

5.1 Discussion of Results ............................................................ 74

5.1.1 Discussion of Discrepancies ....................................... 74

5.2 Summary ............................................................................... 76

I. Derivation of the Noise Crossover Function ............................... 80
References ................................................................................................ 86

ix

Figure 1.1
Figure 1.2

Figure 1.3
Figure 1.4
Figure 1.5
Figure 1.6
Figure 1.7

Figure 1.8

Figure 1.9

Figure 2.1

Figure 2.2

List of Figures

Chapter 1.
Two different paths 3 particle could take to get from A to B... 14

An example of two paths returning to the origin to interfere... 16

An example of two paths propagating in the sample to
interfere. The top path shows a scattering site movement

that would alter the resulting interference pattern. ................. 17
Circuit to measure noise from a metal sample ....................... 18
Voltage versus time with current passing in the resistor ........ 19

Voltage versus time with no current passing in the resistor 20
Typical set of averaged power spectra with 64 averages ....... 24

(Top) Random telegraph signal; (Middle) Power spectrum
of random telegraph signal; (Bottom) Power spectrum of
many random telegraph signals ............................................. 27

(Top) Normalized 1lf noise versus temperature for two Ag
ﬁlms with sample dimensions shown in the ﬁgure; (Bottom)
Sheet resistance versus temperature for the ﬁlms ................. 32

Chapter 2.

(Top) Side view of the beginning process for triple-layer
photolithography; (Middle) Side view of undercut and
evaporation method for triple-layer photolithography;

(Bottom) Side view of the resulting tapered lead from
triple-layer photolithography ................................................... 37

(Top) Side view of the beginning process for projection
lithography; (Bottom) Side view of projection lithography
after development ................................................................... 39

X

Figure 2.3

Figure 2.4

Figure 2.5

Figure 3.1

Figure 3.2

Figure 3.3

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Top view of sample pattern. The top leads are voltage
leads and the bottom leads are current leads. This
geometry forms the bottom half of a Wheatstone bridge ........ 40

Circuit schematic of 1/f noise measurement ........................... 44
Circuit schematic for magnetoresistance measurement ......... 45
Chapter 3.

Electron scattering paths that illustrate Weak Localization 51

Magnetoconductance and ﬁt to WL theory for two different
temperatures for a 7pm x 53pm x 0.014pm Ag sample. The
data are the dashed lines and the best ﬁt to theory are the
smooth lines. .......................................................................... 54

Magnetoconductance and ﬁt to WL theory for a 5pm x 50pm
x 0.017um Ag sample. The data are the round symbols and
the best ﬁt to theory are the smooth lines. The ﬁt parameters

are shown in Figure 5.1 .......................................................... 55
Chapter 4.
Two electron scattering paths that interfere to give UCF ....... 60

Magnetoﬁngerprint of a 5pm x 50pm x 0.017pm Ag sample
at 4.4K .................................................................................... 69

Correlation function of the magnetoﬁngerprint in Figure 4.2 .. 70

1/f resistance noise versus magnetic ﬁeld for a 5pm x 50pm
x 0.017um Ag sample for four temperatures .......................... 71

1lf resistance noise versus magnetic ﬁeld for a 7pm x 53pm
x 0.014um Ag sample for three temperatures ........................ 72

Contour plot of X2 for the T=1OK data from Figure 4.4 ........... 73

xi

Figure 2.3

Figure 2.4

Figure 2.5

Figure 3.1

Figure 3.2

Figure 3.3

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Top view of sample pattern. The top leads are voltage
leads and the bottom leads are current leads. This
geometry forms the bottom half of a Wheatstone bridge ........ 40

Circuit schematic of 1/f noise measurement ........................... 44
Circuit schematic for magnetoresistance measurement ......... 45
Chapter 3.

Electron scattering paths that illustrate Weak Localization 51

Magnetoconductance and ﬁt to WL theory for two different
temperatures for a 7pm x 53pm x 0.014um Ag sample. The
data are the dashed lines and the best ﬁt to theory are the
smooth lines. .......................................................................... 54

Magnetoconductance and ﬁt to WL theory for 3 5pm x 50pm
x 0.017um Ag sample. The data are the round symbols and
the best ﬁt to theory are the smooth lines. The ﬁt parameters

are shown in Figure 5.1 .......................................................... 55
Chapter 4.
Two electron scattering paths that interfere to give UCF ....... 6O

Magnetoﬁngerprint of a 5pm x 50pm x 0.017pm Ag sample
at 4.4K .................................................................................... 69

Correlation function of the magnetoﬁngerprint in Figure 4.2 .. 70

1/f resistance noise versus magnetic ﬁeld for a 5pm x 50pm
x 0.017pm Ag sample for four temperatures .......................... 71

1If resistance noise versus magnetic ﬁeld for a 7pm x 53pm
x 0.014pm Ag sample for three temperatures ........................ 72

Contour plot of x2 for the T=1OK data from Figure 4.4 ........... 73

xi

Chapter 5.

Figure 5.1 The results of comparing WL and UCF theories for a sum
x 50pm x 0.017pm Ag sample ................................................ 78

Figure 5.2 The results of comparing WL and UCF theories for two Ag

ﬁlms. Sample 1 is 7pm x 53pm x 0.014pm and sample 2
is 18pm x 162nm x 0.014um.

xii

Chapter 1. Introduction

1.1 Ohms Law

One of the successes of Condensed Matter Physics in this century has
been the qualitative and sometimes quantitative explanation of electrical
resistance in metals. This resistance is usually measured by injecting a known
current through the sample while measuring the voltage induced across the

sample. From Ohm's law, the resistance is calculated:

RzV/I
1.1

where the resistance R, in units of Ohms(Q) is determined by the ratio of the
voltage V, in units of Volts(V) with the current I, in units of Amps(l). In the
semiclassical limit, all the sample dimensions are contained within R as well as
the speciﬁc material parameters. In the present work, Ohm's law will always be

obeyed.

1.1.1 Sample Size Dependence of Resistance

We now wish to discuss the dependence of resistance in the
semiclassical limit on sample dimensions and speciﬁc material parameters. The
resistance is deﬁned in terms of either the resistivity (p) or the conductivity (0')
which are characteristic of the particular metal one is measuring. The deﬁning

relation is:

_ Wt th
1.2

where L is the length, W is the width, and t is the thickness of the sample. If the
length is doubled, the resistance doubles, which is exactly like adding resistors in
series. If the width or thickness is doubled, the resistance is halved, like adding
resistors in parallel. It will turn out later when we discuss universal conductance
ﬂuctuations (UCF), this relation will no longer be strictly valid because of

quantum interference effects of the conduction electrons [Lee et al.,1987].

1.2 Conductivity

Thus far we have not given an explanation of exactly where resistance
comes from. We will attempt to do so in this section. Normally what is

calculated is the conductivity (0'). Ohm's law takes the form:

.7sz
1.3

d ‘

where J is the current density(l [W t) and E is the electric ﬁeld(V I L) in the

sample.

1.2.1 Drude Model

The ﬁrst model of electrical conductivity relevant to this work is the Drude
or free electron model. This model has the basic assumption [Ashcroft and
Mennin, 1976] that the electrons in the sample can be treated as non-interacting
gas particles with a charge equal to an electron. All dynamics can be calculated
using Newton's laws and the probability of a particle suffering a collision in a time
interval dt is dt l 'r, where r is the relaxation time exactly like an ideal gas.

The current density, which is the number of electrons on average passing

by a point in the sample, can be written as:

.7 = —ne(\7>

1.4

where n is the average number of electrons per unit volume, 6 is the charge of
an electron, and <17) is the drift velocity of the electrons. If we apply an electric

ﬁeld to the sample then from Newton's second law:

d (m1?)

=F=—eE,
dl‘

1.5

where if we integrate up to time t we get:

mi? = mfio — eEt.

 

1.6
Assuming that <m170> = 0, we ﬁnd:
(m1?) = <—eEt> = -eEt .
1.7
Thus (1.3), (1.4), and (1.7) simply we ﬁnd:
2 _.
_. —. _. ne 13E
J =oE = —ne<v> = .
m
1.8
Now the conductivity is identiﬁed to be:
116213
0' = ———,
m
1.9

which is the Drude conductivity. This simple model is excellent at describing the
conductivity of metals once we take into account the quantum mechanical nature

of electrons, which we will do next. It will turn out later when we discuss weak

5

localization, the above equation will have quantum corrections due to quantum

interference of conduction electrons.

1.2.2 Modern Quantum Theory of Conduction

It is clear from the assumptions of the Drude model that it is highly
oversimpliﬁed. A surprising result is attained when the quantum mechanical
nature of the electrons (Free Electron Gas) and electron-electron interactions
(Fermi Liquid Theory) are taken into account. The form of the conductivity does
not change, only the interpretation of what carries the charge changes [Kittel,
1986; Ashcroft and Merrnin, 1976]. The following models add one more piece of
information that was not present in the above discussion, which is that the free
electrons are the valence electrons of the neutral atom. It is assumed that the
core electrons are bound tightly to the nucleus and thus don't effect the valence
or free electrons.

We begin by discussing the free electron gas. Similarly to the Drude
model, we start with a gas of non-interacting electrons except we add the
periodicity of the ions (nucleus with core electrons). This time we solve the
Schrodinger wave equation to get the eigenfunctions and eigenenergies. The
electrons being fermions have to be put in the gas following the Pauli exclusion
principle with each energy level doubly degenerate because of the spin of the
electron. Several important features come out of this model, including the Fermi
energy Ef (energy of the last electron put in the gas), Fermi wavelength if
(characteristic wavelength of the electron at the Fermi energy), and density of
states at the Fermi energy D(Ef) (number of states per unit energy at the Fermi
energy). Another important feature of this model is that the electron waves

extend throughout the entire system and are referred to as Bloch waves. Thus

6

the electrons don't scatter off the ions as one might think when they move
through the system. Because of the Pauli exclusion principal, only electrons
within kBT of the Fermi energy can be involved in the conduction process.

The connection to conductivity is made by replacing the momentum of an

electron mv in the Drude model by the characteristic momentum of the electron

wave at the Fermi energy 71k f = h / Af in the quantum free electron gas

model. Following the above arguments in the Drude model, the conductivity is:

1.10

which is exactly what was derived from the Drude model.

We can go further with this idea of free electrons and put into the
Schrodinger equation the electron-electron interactions (Fermi Liquid Theory).
When this is done, the qualitative picture of conduction is not changed [Ashcroft
and Mermin, 1976]. Instead of electrons carrying the charge, quasi-particles
which are excitations in the quantum ferrni liquid, do the work. There is a one to
one correspondence of quasi-particles to electrons because of the translational
invariance of the system [Altshuler and Lee,1988]. For Ag, these quasi-particles
have the same characteristics as an electron, hence Ag is well described by the

free electron gas described above.

1.2.3 Scattering Rates

We now discuss what causes the electron wave to be scattered. As
stated above, the electrons don't scatter off the ions when they have perfect
periodicity, but when there is a deviation from this perfect periodicity, the
electrons scatter off the imperfections. These imperfections come in two forms.
First, if there is a different ion (impurity) occupying a spot or an ion missing or
misplaced (vacancy or defect), this will destroy the perfect periodicity of the
system. Secondly, due to thermal vibrations of the ions (phonons), there is a
distortion of the perfect periodicity. The other source of scattering is from the
electrons themselves (electron-electron). It is these scattering mechanisms that
ultimately lead to the resistance in metals.

According to Matthiessen's rule [Kittel, 1986; Ashcroft and Mermin, 1976],

the total scattering rate can be written:

1 1 l 1
_:____.+____+___

I total t impurity I phonon
1.1 1

assuming the relaxation processes are independent. This total relaxation time,
12 total is used in the conductivity expression derived earlier. These scattering

rates all have different temperature dependencies but combine as above to give
the total temperature dependence of the resistance of a metal. The impurity
scattering is, to ﬁrst order, temperature independent, so this is the dominant

process at low temperatures as will be seen later in this work. The phonon and

8

e-e scattering rates have the temperature dependence associated with the
resistance of a metal. For a more complete discussion on the temperature

dependencies of these rates see Ziman [Ziman, 1972].

1.2.4 Discussion of Validity

The purpose of the present work is to quantitatively understand the
quantum interference corrections to the above models. To further this
understanding, we must ﬁrst point out where the above models are expected to
break down. Since it was emphasized above that we started with a perfectly
periodic system, the question is how far can we deviate from that perfectly
periodic system before the models are totally incorrect. Certainly if we added
disorder until nearly every ion strongly scatters the electron, the above theory
would break down. It turns out that as long as the mean free path of the electron

hk

(distance between scattering events I = vf'rmm, = 713mm,) is much

larger than the Fermi wavelength (£>>}.f), and the electron when scattered loses
all its memory (Its new direction and speed are completely uncorrelated with its
initial direction and speed), the above description is correct [Ashcroft and
Mermin, 1976]. We will see later that it is these assumptions that break down at
low temperatures and give the quantum interference corrections that are the

topic of this thesis.

9
1.3 Disordered Electronic Systems

We now discuss electronic conduction in a system where an increasing
amount of disorder is added. It will turn out that the qualitative picture discussed
above will break down and new concepts will be required to understand this
regime. What is needed is a new approach to understanding conductivity. The
early work of Anderson [Anderson, 1958] in which he considers the electronic
states in a random potential is a starting place in understanding how disorder

effects the conductivity .

1.3.1 Localization Length

The discussion above of Bloch waves revealed that the electronic states
in a clean metal extend throughout the sample. As disorder is added, these
waves qualitatively change such that when the disorder has reached some
critical limit, they will be localized. This is called Anderson localization and the
critical limit is known as the metal-insulator transition. If the states are localized,
we no longer have a metal but rather an insulator at zero temperature due to
quantum interference effects. For an inﬁnite sample with disorder, the states
don't extend throughout the entire sample but rather extend over a distance
called the localization length, 5 [Lee and Ramakrishnan, 1985]. As long as your
sample is smaller then a, then you will have a metal. If your sample is much
larger then g, then you will have an insulator. Next, we will investigate the
crossover from a metal to an insulator due to only quantum interference effects
at zero temperature. Speciﬁcally, we will consider how the electron energy

levels play a role in this crossover.

10
1.3.2 Scaling Theory of Localization

The next important step in understanding conduction in disordered metals
is largely due to Thouless and his coworkers [Edwards and Thouless, 1972;
Thouless, 1974]. We begin by looking at the zero temperature eigenenergies of
a d-dimensional hypercube of length L and volume Ld. The eigenenergies come
from diagonalizing the Hamiltonian for a single particle interacting with the
random potential that is the source of the disorder. Suppose that w, the average
spacing between energy levels, is (NoLd)'1, where No is the density of states.
Next, we double the length of the hypercube so that we now have volume (2L)d.
The eigenstates of the (2L)d sample will be a linear combination of the Ld
eigenstates with the amount of admixture depending on the overlap integral and
energy denominator. Thouless observed that, the overlap integral is of order the
energy difference between having periodic boundary conditions on the

hypercube versus aperiodic boundary conditions. This energy difference, AE
can also be understood from the uncertainty principle as h / T D, where 1: D is

the time it takes for the particle to traverse the hypercube of length L.
Furthermore, Thouless reasoned that the conductance G=1IR was related
to the ratio AE/w by a fundamental overall constant equal to (e2/h), i.e. G=(e2/h)(
AEIw)P, where p=2 in one dimension [Lee and Ramakrishnan, 1985]. Thus, if
the eigenstates are localized, the energy difference AE will be small, and the
sample will be an insulator due to quantum effects. Also, if the states are
extended, the energy difference will be large, and the sample will be a metal.
This follows from noting that the localized states remain in their individual
hypercubes and are insensitive to the boundary conditions while the extended

states spread out over several hypercubes up to their localization length g.

11

1.3.3 Landauer Formula

The above description can be extended to ﬁnite temperatures as long as
the electron maintains phase coherence throughout the sample. This means
that the electron only undergoes elastic collisions. For small samples at low
temperatures, this can be achieved quite readily and has produced a new area
of physics called mesoscopic physics. Since the electron doesn't exchange
energy in the sample, that means all the energy dissipation occurs in the voltage
and current leads. This allows us to think about conductivity in a different way.
One can use the Landauer formula to calculate the conductance of a sample
from the transmission matrix computed for the sample region [Landauer, 1957;
See Stone and Szafer, 1988 for a review]. Once the transmission matrix is

computed, then the conductance is given by:

e2
G = Z— T 7116607?) .

1.12
where t is the N x N transmission matrix connecting the incident ﬂux in the N
channels in the leads on one side, to the outgoing ﬂux in the N channels on the
other side.

One of the consequences of the Landauer formula is that there are no
perfect metal conductors. For perfect transmission, the conductance of a sample
is N e2/h, where N is the total number of channels . Another consequence of the
Landuaer formula is that if two equivalent conductors are connected in series,

maintaining phase coherence throughout the system, the resulting conductance

12

is not G/2 but rather has some higher order correction terms that take into

account the multiple scattering processes.

1.3.4 Random Walk of Feynman Paths

Electronic conduction in disordered systems leads to some different ideas
that have not been discussed. The ﬁrst is the idea that the electron moves
diffusively through the sample. If you want to know how far an electron has
moved in a time t greater than the time between collisions, you need to use the

concept of a random walk, which leads to:

L=xlDt

1.13

where D = —Ver is the diffusion constant and [e is the mean free path of the

electron. The second modiﬁcation comes from the lack of ability of the electrons
to screen the random potential. This leads to an enhanced electron-electron
interaction that the Fermi liquid theory is incapable of handling [Lee and
Ramakrishnan, 1985]. Though this second modiﬁcation is large for disordered
samples, it does not effect the low magnetic ﬁeld corrections to the conductivity
which are the focus of this work.

At the heart of this work is the formulation of quantum mechanics that
Feynman put forth, which states that a particle will take all possible paths to

traverse from one point to another. This idea, coupled with having to square the

13

probability amplitude to get probabilities, leads to all the quantum interference
effects that will be discussed in this work. These concepts will be considered in

more detail below.

1.4 Quantum Interference

As discussed above, the essential physics behind this work comes from
quantum interference effects. To appreciate this consider two different paths as
shown in ﬁgure 1.1. Using the formalism of quantum mechanics, we ﬁnd for the

probability to get from A to B:

= A]2 + A22 +2A1A2 cos(¢1—¢2)

1.14

|Alei¢1 + Azei‘l’z '2

where the (I) 's represent the phase of the particular path. Depending on the

phase differences between the paths, one can get a result which differs

signiﬁcantly from the classical case which is A12 + A22 .

14

 

2

Figure 1.1 Two different possible paths a particle could take to get from A to

B.

1.4.1 Quantum Phase Coherence

The above example elucidates the importance of phase in quantum
mechanics. If we were to phase average the result, we would be left with only
the classical limit, where we compute the probability by squaring the individual
probability amplitudes and summing. The phase (I) is usually a sum of the wave

vector dot postion vector and a time dependent term as shown below:

at it"? ’1?)
e :8

1.15

where hk is the momentum, F is the postion, and E is the energy of the
particle while t is the time it takes to traverse the path. As long as the phases
are well deﬁned, we will have quantum interference effects. We will see below

that the quantum interference of conduction electrons is very sensitive to phase.

15

1.4.2 Weak Localization

The ﬁrst quantum correction we would like to discuss is weak localization
(VVL). WL deals with the subset of Feynman paths that return to the origin as
shown in Figure 1.2. These paths are a time-reversal of each other and thus
have identical phases. If a single path has probability amplitude A, then their
probability to return to the origin is 4A2, not 2A2, which is the classical result.
This enhancement of returning to the origin weakly localizes the electron at the
origin, hence the name weak localization theory.

This enhancement can be destroyed with a magnetic ﬁeld perpendicular
to the path trajectories because the paths pick up an additional phase factor from

‘

the vector potential A that is produced by the magnetic ﬁeld. This additional

phase is given by the semiclassical approximation (I) B = (ie/ 71)] A 0 d] .

Thus the phase difference between the two paths is 2(1) B- This is the origin of

the low magnetic ﬁeld magnetoresistance of WL theory.

At ﬁnite temperatures, the distance an electron travels before losing

phase coherence, L4, , must be greater than the path length in order for there to
be interference effects. For paths greater than L4, , the phases average to yield

the classical result that has no magnetic ﬁeld contribution. L4) is a ﬁtting

parameter in WL theory that we can estimate from ﬁtting the theory to
magnetoresistance measurements. In chapter 3, we will discuss WL in more

detail.

 

‘ in

/ * 2--.. r, h I

/ \ \\ ‘

' * ‘1*\ ‘
/ /
\“v .7732

\A r 7!, ‘ ‘

,4

 

 

Figure 1.2 An example of two paths returning to the origin to interfere.
1.4.3 Universal Conductance Fluctuations

The second quantum interference theory that we would like to discuss is
UCF. UCF is a result of different paths interfering together as shown in Figure
1.3. The resulting interference pattern of the conduction electrons is similar to a
speckle pattern of a laser shown through a disordered medium with bright and
dark spots resulting from constructive and destructive interference, respectively.
Because UCF involves two different paths, the resulting interference pattern is
highly sensitive to the location of the scattering sites. This sensitivity leads to
UCF enhanced 1/f resistance noise. This 1lf noise enhancement can be
reduced by 50% in a magnetic ﬁeld thus allowing us to use it as a tool in testing

UCF theory.
At ﬁnite temperatures, two lengths play a role in UCF, L4, and Lth- L4,
is the distance an electron maintains phase coherence as described in WL

theory. and Lth is the distance two electrons separated by energy KT maintain

phase coherence. L4, is a ﬁtting parameter in the reduction of UCF enhanced

17

1/f resistance noise, allowing us to estimate this distance and compare it to the

WL result. UCF theory will be discussed at length in chapter 4.

 

 

 

Figure 1.3 An example of two paths propagating in the sample to interfere.
The top path shows a scattering site movement that would alter the resulting

interference pattern.

1.5 1lf Noise

One of the major focal points of this work is the measurement of
resistance ﬂuctuations in small Ag ﬁlms. A substantial amount of work has been
done in understanding the dynamics of these ﬂuctuations in metals [\Neissman,
1988; Dutta and Horn, 1981]. We will discuss the current understanding of these

ﬂuctuations in this section.

18
1.5.1 Resistance Fluctuations in Time

When you look closely at the output voltage of a metal resistor with
current passing through it, you will notice ﬂuctuations or noise. This noise comes
from three sources, the Johnson or thermal noise, circuit or ampliﬁer noise, and
1/f or excess noise. A typical schematic of a resistance measurement is shown
in ﬁgure 1.4. A battery drives current through the ballast resistor, which is much
larger then the sample resistor, to produce a constant current through the
sample. The output voltage of the sample is increased with an ampliﬁer to
produce a voltage versus time trace as shown in ﬁgure 1.5. To see the effect of
1/f noise, open the circuit so current no longer passes through the sample. A
typical output voltage versus time is shown in ﬁgure 1.6 for this situation. The
trace in ﬁgure 1.5 has the thermal, circuit, and 1lf noise where as the trace in
ﬁgure 1.6 has only the thermal and circuit noise. Qualitatively, ﬁgure 1.5 has
more wiggles and a slowly varying component as compared to ﬁgure 1.6. This is

what 1/f noise looks like in the time domain.

 

Ballast Resistor

 

 

Battery

I. .

Sample Amp. OUtpUt

I.

Figure 1.4 Circuit to measure noise from a metal sample.

 

 

 

19

 

0.5 _

0.4 f
0.3 _'
0.2 '
0.1 If

(Volts)

a) 0.0

Voltag

 

-0.1 :
~0.2 :
-o.3 t
-o.4 _~

L J l L i l I r l I l l l A I I l

 

 

-0.5
0

Figure 1.5

2 4 6 8 10 12 14 16 18
Time (Seconds)

Voltage versus time with current passing in the resistor.

20

20

 

.0 .0 9 .0
N w A 01
, .

.0
A

-0.1
-0.2 _'
-0.3 .—
-0.4 f

Voltage (Volts)
O
O

 

 

 

-0.5 I I r I l I L AL I I i l i_ l l l I I
0 2 4 6 8 10 12 14 16 18

Time (Seconds)

Figure 1.6 Voltage versus time with no current passing in the resistor.

20

21

The thermal or Johnson noise is a result of the electrons in the sample
being in equilibrium [Kittel, 1980]. They bounce off each other producing voltage
noise independent of the measuring current. Thennodynamically, this thermal
noise is well understood and can be explained from the ﬂuctuation-dissipation
theorem [Callen and Walton, 1951]. The thermal noise is proportional to
temperature and resistance and independent of sample volume or any other
speciﬁc material parameters.

The circuit noise is produced by the ampliﬁers that also have voltage
noise, but since they are outside the cryostat, it is temperature independent. At
higher sample temperatures (above about 20K in this work) this is a small
contribution because the Johnson noise of the sample dominates, but at lower
sample temperatures the circuit noise becomes dominant.

The 1/f voltage noise is quite different than the thermal and circuit noise.
It scales with the current like Ohm's law allowing one to infer that it comes from
resistance ﬂuctuations. These resistance ﬂuctuations scale like 1Nolume so that
they average out in a large sample. This noise is a result of scattering sites
changing in the sample either by moving or rotating to alter the cross-section
. scattering of the impurities [Weissman, 1988]. There are two limits of-interest
here, local interference, which is the topic of the next section, and long-range
interference known as UCF enhanced noise. The UCF enhanced noise will be

discussed later in chapter 4.

1.5.2 Local Interference of Electrons

The local interference model calculates the change of resistance due to a
scattering site (usually a defect) that has changed its cross-section, hence its

electronic scattering rate. An important note is the cross-sectional scattering can

22

only be changed within a few fermi wavelengths of the defect [Pelz and Clarke,
1987]. No multiple scattering effects are taken into account. A defect in
cooperation with its nearest neighbors can locally change the resistance leading
to an overall resistance change. From this model it's clear why the resistance

ﬂuctuations average to zero in a large sample. For N independent scatters,

the ﬂuctuations, assuming Gaussian statistics, would be proportional to m

tending to zero as N —-) 00.

1.5.3 Power Spectrum of Resistance Fluctuations

We now discuss the frequency dependence of the thermal, circuit, and 1lf

noise. The power spectrum is deﬁned:

51/0”) = 4:(<V(I>V(0)> - <V>2)008(21tfv >d—r oc lFT<6V<r>>I2

1.16

where F T (5 V(t)) is the Fourier transform of the voltage ﬂuctuations and f is

the frequency. For thermal and circuit noise, the power spectrum is independent
of frequency where as for 1/f noise, the power spectrum is roughly inversely
proportional to frequency, hence the name 1lf noise.

The thermal noise as stated above depends on only the sample
resistance and temperature. The power spectrum of the thermal noise is [Callen

and Welton, 1951]:

23

1.17

where k3 is the Boltzman constant. Its form is independent of frequency as long
as kBT >> h f, which is always true in the present work.

The circuit noise coming from the ampliﬁers is basically the result of
thermal noise and 1/f noise in the ampliﬁer components. The truth is if we made
our measurements using the circuit in ﬁgure 1.4, there would be 1/f background
noise coming from the ampliﬁer. To eliminate this we use an alternating current
method of measuring the resistance ﬂuctuations that will be discussed in detail in
section 2.3.2. Using the AC technique renders the circuit noise independent of
frequency allowing us to treat it the same as thermal noise. The sum of thermal
and circuit noise is referred to as background noise.

The 1If noise is analyzed in terms of normalized resistance ﬂuctuations so
we need to subtract the background noise and scale out the current as shown

below:

SR (f) = [SV(1)(f) - SV(I=O)(f)] / 12

1.18

where SV( I)( f ) is the 1lf and background noise; SV( I=O)( f ) is the

background noise and I is the drive current. Following the above analysis, the

power spectrum of the resistance ﬂuctuations is inversely proportional to

frequency ,SR (f) 0C 1/ f . In the next section we discuss the reason for this

frequency dependence in metals.

24

 

  
  
   

 

 

 

4 _ I I ll ' ' ' l ' l ' VTV'IYITI'VI l I ll ‘ T ' I ' ' ‘l"'l"'l'T'l I l l I _~

3»? :3

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I A background .

104 : i

: ' 1If+ background :

1; 4 __ slope- -0.98 p

E 3 :— 3‘

N L J

a _— .J

o . .
.8 5r A ‘ ‘

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Z10 : :

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10.6>4111 . I linlrl.1tl.l.I.I.L LILI . 1.1.1.1.l.1.l.1.1.llllil

10-1 2 3 4 5 6 100 2 3 4 5 6 101

Frequency (Hz)

Figure 1.7 Typical set of averaged power spectra with 64 averages.

25

In general, several (more then 60) power spectra are taken and averaged
together to make a data set. A typical data set of 1/f noise plus background
noise, background noise, and 1If noise is shown in ﬁgure 1.7. As can be seen,
the 1lf noise has a slope of z -1 over the entire measurement frequency.

A 1/f spectrum has some interesting features, including scale invariance
and increase of signal at lower frequencies. The scale invariance implies that if
one made a recording of resistance versus time and played it back on a tape
player, it would sound the same no matter what speed it was played back at
[Weissman, 1988]. The increase of signal at lower frequencies is a feature we
often take advantage of, however it's not cheap. Every reduction of a factor of
two in frequency costs us double in data taking time. At our lowest temperatures

in this work, it takes a minimum of 4 hours to acquire a single data set.

1.5.4 Defect Dynamics

We now investigate the reason resistance ﬂuctuations are inversely
proportional to frequency. Recall from section 1.5.1, the resistance ﬂuctuations
arise from defects moving in the sample. For our purpose we can think of a
defect having two different locations in the sample that then lead to two different
resistances. A possible resistance versus time for this model is shown at the top
in ﬁgure 1.8 and is known as the random telegraph signal. If we compute the

power spectrum of the random telegraph, we ﬁnd a Lorentzian which is:

26

 

 

12
S (f) 0C
R
1 + (21:11:)2
1.19
where T is a characteristic time of the defect given by [Machlup, 1954]:

1 1 l

— = — + .

I I up I down
1.20

13 up is the time the defect spends in the upper resistance position and 1: down is

the time the defect spends in the lower resistance position. The Lorentzian gives
a ﬂat power spectrum for low frequencies and a 1/f2 for high frequencies as
shown qualitatively at the middle in ﬁgure 1.8.

To get the 1/f spectrum, we need to add many random telegraph signals
together [Dutta and Horn, 1981] as shown qualitatively at the bottom in ﬁgure

1.8. The noise spectrum will be given by:

D(‘t )t I
1+ (21tfr )2

 

SR(f)°C.I

1.21

27

 

 

 

 

 

 

a
E
u

 

 

 

 

 
 

 

 

 

9
c2: 1/f2
8
3
o
D—l
Frequency
5
s
8 l/f
o.
m
3'5 " .
i \\\ \""
o -.
a ;
Frequency

Figure 1.8 (Top) Random telegraph signal;(Middle) Power spectrum of
random telegraph signal;(Bottom) Power spectrum of many random telegraph

signals.

27

 

 

 

 

 

 

 

 

E

 

 

 

 

 

 

  

 

 

 

E
.3.
8
v8" 1/f2
33
B
o
:1.
Frequency
8 "
é
8 l/f
c.
m
B
i \\\ \"
O '.
9.. 5:
Frequency

Figure1.8 (Top) Random telegraph signal;(Middle) Power spectrum of

random telegraph signal;(Bottom) Power spectrum of many random telegraph

signals.

28

where D(‘t) is the distribution function of the characteristic time constants of
the moving defects. If D(ln(‘r )) is constant, which is equivalent to
D(17 ) 0C 1/1: , then the resulting pewer spectrum will be 1/f [Dutta and Horn,

1981]. There are two temperature limits of importance here. First, the high
temperature limit where the defects are thermally activated, and second, the low
temperature limit where the defects move via quantum mechanical tunneling.

We now consider temperatures where the defect is thermally activated. In

08’” , where SE is the activation barrier

this regime, ‘L' is given by 1? =17
energy of the defect to move from one spot to the other and 1: 0 is the attempt
time. The condition of D(ln('c )) being a constant is equivalent to D(8E) being

ﬂat or slowly varying as compared to KT. For disordered metals with a broad
distribution of defect barrier heights, this condition is satisﬁed.

The low temperature argument is similar except that the characteristic

1: 1/2
Vo

barrier width d, and where V0 is given by V0 = h2/2md2. Assuming that V is

tunneling time constants are given by 1? = 1: De for a barrier height V,

broadly distributed on the scale of V0, then again the this result leads to 1/f noise
[Phillips, 1971; Feng, 1991]. Thus in the entire temperature range of this work,

we expect a nearly 1/f frequency dependence in the resistance noise.

29
1.5.5 1lf Noise versus Temperature

From the above argument, we expect the 1/f noise to decrease with
decreasing temperature because the defects begin to freeze out and become
less active. Figure 1.9 shows the normalized 1lf noise versus temperature for
two Ag ﬁlms as well as the resistance versus temperature for these ﬁlms. From
room temperature to about 35K, the noise drops very rapidly with temperature as
expected. Below 35K, the noise begins to increase! It is important to point out
that the increase in noise is not due to extra defects moving or speeding up but
is from increased sensitivity of the resistance to a defect's movement. This is the
crossover from local interference noise to UCF enhanced noise. It is this low
temperature region that we focus on in chapter 4 and is a major portion of the
present work. It is worth pointing out that the resistance versus temperature

shows no unusual behavior.

1 .6 Motivation

The purpose of the present work is to show that UCF enhanced noise can
be seen in a ordinary metal such as Ag and, more importantly, to test
quantitatively the consistency of two quantum theories, weak localization and
UCF. Weak localization will be discussed in chapter 3 and UCF will be
discussed in chapter 4. The result of the comparison between the two theories

will be discussed in chapter 5.

30
1.6.1 Enhancement of 1If Noise at Low Temperatures

Although the increase in 1/f noise at low temperatures has been seen in
many materials (including Ag) [Beutler et al., 1987; Meisenheimer et al., 1987;
Meisenheimer and Giordano, 1989; Birge et al., 1989 and 1990; Garfunkel et al.,
1988; Alers et al., 1989], the connection to UCF was only deﬁnitively tested in Bi
[Birge et al., 1989 and 1990] by its reduction in a magnetic ﬁeld. Figure 1.9
shows beyond a shadow of a doubt that in A9 below 35K there is an increase in
1/f noise as the temperature is lowered. In chapter 4 we will show that its
dependence on magnetic ﬁeld is very well described by UCF theory. Now that
we have established UCF enhanced 1If noise, we will use it as a tool to extract
quantitative information by measuring its magnetic ﬁeld dependence and ﬁtting

that dependence to UCF theory.

1.6.2 Consistency Test of Quantum Theories

The primary purpose of this work is to test the consistency between weak
localization and UCF. This was attempted in Bi but because of signiﬁcant
differences between the bulk and thin ﬁlm properties, any quantitative
comparison was very difﬁcult [Birge et al., 1989 and 1990]. Thin ﬁlm Ag on the
other hand is very well described by its bulk values and free electron theory
making comparisons much easier. The consistency test is accomplished by

comparing a common ﬁtting parameter in the two theories. The common ﬁtting
parameter is the phase breaking length L(p , which is the distance over which a

conduction electron remains phase coherent.

31

In weak localization, L,p is obtained by ﬁtting magnetoresistance data (R

Vs B) to weak localization theory. In UCF, L“, is obtained by ﬁtting the

reduction of 1/f noise in a magnetic ﬁeld (1/f noise Vs B) to UCF theory. Since

the two measurements are completely different, this should yield two
independent estimates of L(p. In chapter 5 we will discuss the results that
indicate at the lower temperatures a discrepancy between the inferred values of

L(" and at higher temperatures excellent agreement.

32

 

 

 

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100 2 3451012 345102 2 345
Temperature (K)

Figure 1.9 (Top) Normalized 1/f noise versus temperature for two Ag ﬁlms with
sample dimensions shown in the ﬁgure; (Bottom) Sheet resistance versus

temperature for the ﬁlms.

Chapter 2. Experimental Considerations
2.1 Choice of Materials

We chose to study Ag ﬁlms for several reasons, including similarity to bulk
free electron values, ability to make high quality ﬁlms, and magnitude of 1If
noise. Because we are attempting to do a quantitative comparison between
Weak Localization (VVL) and UCF, it is necessary to have a reliable estimate of
material parameters so that the mean free path of an electron can be calculated
and used in WL and UCF theories. The capability to measure 1If noise is a must

so that UCF theory can be used as a tool.

2.1.1 Comparison with Free Electron Parameters

Both WL and UCF are weakly dependent on the mean free path of an
electron. To calculate this, it is advantageous to use the free electron result
derived in section 1.2.2. In order for this to be valid, we want the Fermi liquid
theory to match as closely as possible the free electron result. For Ag, this is the
case [see Kittel 1986] because Ag has a nearly spherical Fermi surface. Since
in the present work our ﬁlms are not more then ten times the bulk resistivity
value, we believe that free electron theory is valid. Also several groups have
already used free electron theory to analyze WL theory in even more disordered
Ag ﬁlms [Bergmann, 1982; Kawaguti and Fujimori, 1983] without difﬁculty. The
accuracy needed for the material parameters such as the diffusion constant, is
roughly a factor of 3 before signiﬁcant changes occur in the WL and UCF

comparison [McConville and Birge, 1993].

33

34
2.1.2 Ability to Make High Quality Disordered Films

High quality A9 ﬁlms can be easily prepared by thermal evaporation on a
cold Si or Si02 substrate. Thermal evaporation is a prOcedure for evaporating
metal onto a substrate by heating the metal in a vacuum until it vaporizes. This
evaporation technique allows one to make very thin continuous ﬁlms with a high
degree of disorder. Typically the substrates are cooled between 100K - 200K
before the start of the evaporation but warm up as the evaporation is begun.
The evaporation is done at a rate of 30 - 50 Als to ensure purity and disorder in
the ﬁlms. Electrically continuous Ag ﬁlms have been achieved at thicknesses
below 75 A using this method, however the thinnest Ag ﬁlm used in the present

work was 140 A.

2.1.3 Magnitude of 1If Noise

Since by far the greatest experimental challenge of this work is the
measurement of 1/f noise, a material must be chosen that has signiﬁcant 1ft
noise. Ag ﬁlms, as compared to Au or Cu, have this quality at room temperature
[Scoﬁeld and Mantese, 1985]. Unfortunately, not only does the 1/f noise drop
with temperature, but also the maximum drive current must be reduced because
the ﬁlm loses its ability to dissipate the joule heating of the electrons with
temperature. Nevertheless, measurements can be performed down to 1K in A9,
which is the lowest temperature in this work, with a great deal of effort and time.
This ability to measure 1lf noise in Ag allows us to make the comparison

between WL and UCF that we would like to perform.

2.2 Sample Fabrication

35

Several reasons, including the 1lvolume dependence of 1/f noise and
joule heating, lead to the decision to make small Ag ﬁlms usually a few microns
(micrometers) in width and several microns long. Since this size sample is only
visible under a microscope, several methods must be combined to pattern the
sample as well as connect electrical leads. The leads are formed using a
process called triple-layer photolithography and the sample is patterned using

projection photolithography. These two processes are discussed in detail below.

2.2.1 Triple-Layer Photolithography

Triple-layer photolithography is a ﬁve step process that allows us to form
thick Au leads that later will electrically connect the sample to the outside world.
This multi-layer process is needed because we would like to produce thick leads
(several sample thicknesses) to reduce lead resistance. In order to make
electrical connection later to a much thinner sample, we need to taper the leads.
Not only do we require tapered leads, but also we would like to bring the leads
as close as possible to the sample area while on the other end of the lead have
them big enough to attach wires. These are the reasons a multi-layer process is
chosen.

Figure 2.1 shows a side view schematic of the process, which is begun by
spinning a photo sensitive polymer known as photoresist onto your substrate.
The photoresist has the property that when exposed to UV light, bonds break,
which leave the photoresist soluble in a developer. Thus where light is exposed,
the photoresist can be removed leaving the unexposed photoresist in place.
Once the bottom layer of photoresist has been spun and baked in an oven to

remove the solvent, it is exposed everywhere. The next step involves thermally

36

evaporating a thin buffer metal layer, usually Cr or Al, at a thickness of 100 - 300
A. This will separate the top layer of photoresist from the bottom layer. Next, the
top layer of photoresist is spun onto the metal and baked. Now a mask that has
the desired lead pattern is used to expose only the regions of interest as shown
in the top of ﬁgure 2.1. (The mask, a quartz slide with a Cr or iron oxide coating,
is previously patterned using a ﬁlm negative as a mask and single-layer
photolithography to pattern the slide before chemically etching the metal.) Once
the lead pattern is exposed, it is developed to remove the top layer of
photoresist, which then leaves a pattern on the metal buffer layer. The metal
buffer layer is then chemically etched leaving the bottom exposed photoresist.
Now with careful timing, the bottom layer of photoresist is developed to produce
a large undercut as shown in the middle of ﬁgure 2.1. This large undercut is
critical for the next step which is metalization of the leads.

The leads are thermally evaporated at an approximate angle of 45° from
perpendicular while the substrate is spun at roughly 300 RPM. Typically a thin
layer of Cr (50A) is evaporated ﬁrst to assure adhesion of the Au to the
substrate. The Au is typically 1500 - 2000 A thick in the center and tapers nicely
to the edge. Once the evaporation is complete, the unwanted Cr/Au is
eliminated by removing the remaining photoresist with acetone. This leaves our

tapered leads as shown in the bottom of ﬁgure 2.1.

37

4/

UV light
unexposed photoresist
Cr or Al

4’ exposed photoresist

4— Si substrate

x 4 Cr/Au

 

 

 

Cr/Au tapered leads

 

 

Figure 2.1 (Top) Side view of the beginning process for triple-layer
photolithography; (Middle) Side view of undercut and evaporation method for
triple-layer photolithography; (Bottom) Side view of the resulting tapered lead
from triple-layer photolithography.

38
2.2.2 Single-Layer Projection Lithography

Projection lithography using a single layer of photoresist is the process by
which we pattern the sample and align it with the previous tapered leads. Figure
2.2 shows a side view schematic of this process. A single layer of photoresist is
exposed using a microscope instead of a mask because this allows us to
demagnify the pattern to the size we want and also leaves a very slight undercut.

The process is begun by spinning and baking photoresist onto the
substrate with leads. After cooling the substrate back to room temperature, the
photoresist is exposed using a microscope. The pattern is projected from a ﬁlm
negative that acts as our mask. The mask is in the path of the light source of the
microscope so that it is projected down onto the substrate allowing the lenses of
the microscope to demagnify the mask as well as project its pattern. After
exposing the photoresist, it is put in developer to remove the exposed
photoresist. This leaves a proﬁle shown in the bottom of ﬁgure 2.2. A top view
of a typical sample is shown in ﬁgure 2.3. Our samples have ﬁve leads because
they become half of a Wheatstone bridge as will be discussed in the next
section. The ﬁnal step before metalization is to clean the sample areas. This is
accomplished using a reactive ion etch (RIE), which uses an oxygen plasma to
remove any organic residue on the sample site.

Once the samples have been de-scummed, they are loaded into a thermal
evaporator for metalization. The metalization is done with the procedure outlined
in section 2.1.2. After completion of the evaporation, the samples are allowed to
warm up to room temperature before breaking vacuum. The samples are then
put in acetone to remove the unwanted photoresist and inspected under the

microscope for viability.

39

Light from microscope

A/

4— photoresist

4— Si substrate

 

 

Figure 2.2 (Top) Side view of the beginning process for projection lithography.

(Bottom) Side view of projection lithography after development.

40

 

Figure 2.3 Top view of sample pattern. The top leads are voltage leads and
the bottom leads are current leads. This sample geometry forms the bottom half

of a Wheatstone bridge.

41
2.3 Measurement Circuits

In this section we discuss the circuits associated with the two
measurements that are required to test WL and UCF. Recall WL theory
explained the magnetoresistance of disordered samples while UCF explained
the reduction of 1/f noise in a magnetic ﬁeld. Both measurements are performed
on the same sample with only the techniques differing. The samples have a ﬁve
lead geometry as shown in ﬁgure 2.3 so that they can be measured as the
bottom half of a Wheatstone bridge for 1If noise and, with the center current lead
open, measured as a four terminal device for magnetoresistance.

The samples are set in a vacuum can of a He4 pumped cryostat that has
a base temperature of 1 K. The samples are in thermal contact with a Cu plate
that has therrnometry and heaters to regulate the speciﬁc temperature. Typically
the temperature can be controlled to better than 0.1%. The lead wires from the
sample are also heat sunk to the Cu plate to eliminate any temperature
ﬂuctuations that occur above the Cu plate. The change in circuits is all done
outside the cryostat thus leaving the samples undisturbed between
measurements.

The magnetic ﬁeld is produced by a superconducting 5 Tesla magnet that
sits in the surrounding He4 bath of the cryostat. The ﬁeld is produced from
current through the magnet, which is measured with a precision resistor and
voltmeter. From the magnet speciﬁcations, we can calculate the ﬁeld from a

given current.

42
2.3.1 1If Noise Circuit

Figure 2.4 is a schematic of the circuit used to measure 1/f noise. The
use of an AC Wheatstone bridge has several advantages [Schoﬁeld, 1987] over
the circuit shown in ﬁgure 1.2. First, since the bridge is run at frequencies of
about 500 Hz, the 1lf noise of the ampliﬁers are not a problem. Second,
because the samples form the bottom half of a Wheatstone bridge, the
measurement of resistance ﬂuctuations is very sensitive to resistance differences
while being less sensitive to overall changes to the resistance from the
temperature or magnetic ﬁeld. Another advantage of using an AC drive is that
we can use a cooled transformer to amplify the signal before the low noise
ampliﬁer. This helps reduce the overall noise background because a cooled
transformer can have less voltage noise then commercially available ampliﬁers.

The circuit uses an AC voltage drive with ballast resistors to produce a
stable AC current through the samples. The ballast resistors are variable so that
the voltage difference between the samples can be nulled out. In parallel with
the ballast resistors are capacitors to adjust any phase differences between the
two arms of the bridge. The signal coming from the samples is then ampliﬁed
with a transformer sitting in liquid nitrogen at 77K before being ampliﬁed again
with a low noise ampliﬁer. The signal is demodulated with a digital lock-in
amplifer into the in-phase channel and out-of-phase channel. The in-phase
channel has both the voltage noise from the background plus the noise coming
from resistance ﬂuctuations in the samples. The out-of-phase channel has only
the background noise because in an AC circuit, a pure resistance has no phase
shift.

The resulting in-phase and out-of-phase channels are run through an anti-

alaising low pass ﬁlter before being digitally sampled. The digitally sampled

43

signal would look like ﬁgures 1.3 and 1.4. They are transferred to a 286
computer where the power spectrum is calculated and stored for further analysis.
Often the data stored on the 286 are transferred via a parallel connection to a

486 for data analysis so that the 286 can resume taking data.

2.3.2 Magnetoresistance Circuit

Figure 2.5 shows a schematic of the magnetoresistance measurement
circuit. The AC voltage drive and the ballast resistor produce a stable current
passing through the sample that produces a voltage from the sample. The AC
drive also drives a variable ratio transformer. By differentially (A-B) putting the
two signals into a lock-in ampliﬁer, one can null the output of the lock-in ampliﬁer
while at zero magnetic ﬁeld. Since only the sample resistance changes with
magnetic ﬁeld, the measured signal from the lock-in ampliﬁer reﬂects the change

in resistance of the sample with magnetic ﬁeld.

44

 

 

 

 

6) ._r

 

 

 

 

 

 

 

2E? Ef/ W

31w

 

 

 

 

 

 

 

 

 

1
Low Noise
Amp.
Ag Samples
‘ X
Lock-in 8-pole A/D ‘
Amp Y ﬁlter Convert.
IEEE-488

 

 

 

 

 

 

 

 

 

 

 

I parallel
, 286 486

Figure 2.4 Circuit schematic of 1/f noise measurement.

 

45

 

 

g’vBallast Resistor

Ratio
Transformer

 

 

 

 

 

Lock-in
" Ampliﬁer

 

 

 

Figure 2.5 Circuit schematic for magnetoresistance measurement.

Chapter 3. Weak Localization
3.1 Introduction

The ﬁrst quantum theory we would like to discuss in more detail is weak
localization (WL). This is the theory that describes the low ﬁeld
magnetoresistance of a disordered metal. It is important to note that low ﬁeld
refers to magnetic ﬁelds in which the semiclassical cyclotron radius of an

electron is much greater then the mean free path of the electron

mv
(1e << rc = 7;). For magnetic ﬁelds greater then this, one expects
e

semiclassical magnetoresistance [Ashcroft and Mermin, 1976], which is not the
focus of this work. For a more exhaustive review of WL see Bergmann, 1984 or

for a more experimental review of the subject see Alt'shuler et al., 1987.

3.1.1 Interference of Time-Reversed Paths

As discussed in chapter 1, WL is a consequence of quantum interference
from a subset of electron paths that return to the origin. These paths are a time-
reversal of one another, thus they have identical phases in the absence of a
magnetic ﬁeld. Figure 3.1 shows a set of electron scattering paths that give rise
to WL. It is important to note that the paths are from the same electron
wavefunction so that their energies are identical. The difference is only the
direction of scattering as they move in the sample. As pointed out earlier in
chapter 1, there is a factor of two enhancement to ﬁnd the electron at the origin,

compared to the classical result. Assuming a single path has a probability

46

47

amplitude equal to A, then the probability to be at the origin is 4A2, not 2A2.
Because the electrons have a higher probability to be found at the origin, they
have a lower probability to propagate through the sample. This leads to an
increased resistance or decreased conductance.

These paths will interfere constructively as long as they are phase
coherent. Since they are a time-reversal of each other, this means that they
must undergo only elastic scattering events to maintain phase coherence. If
they scatter inelastically, say from a phonon, then the time dependent term in the
phase will quickly dephase the two paths and the correction will average to zero.

This distance before dephasing, L4, is the average distance an electron can

traverse the sample and stay phase coherent. This is all that needs to be taken
into account in WL theory when temperature is added to the theory. L4 is the
only temperature dependent correction in WL and since it's a ﬁt parameter used
in ﬁtting the magnetoresistance, the temperature does not explicitly enter in
when ﬁtting magnetoresistance data.

Another interesting point about WL is that if we were to rearrange all the
elastic scattering sites, we would ﬁnd no difference in the magnitude of the WL
correction. WL is an impurity averaged result, which is in stark contrast to UCF
that will be discussed in the next chapter. Since WL is an impurity averaged
result, we can calculate the correction for a square of any size assuming it's

much larger than L4. For a rectangular sample, the correction is just the number

of squares times the result for one square.
The quasi-dimensionality of the sample is set by L4 while the true

dimensionality is still set by the Fermi wavelength. Thus a quasi-2D sample has
a length and width greater then L4 but a thickness less then L4. In the present

work, we will only be concerned with quasi-20 samples. The thicknesses of the

48

samples used in this work are less then 200A while the shortest L4 analyzed is

greater then 750A.
In quasi-20 samples, there is a logarithmic correction to the conductance

from WL theory given by [Bergmann, 1984] :

2 3L2
AGE, = —e—ln —i

7th 1,,2

3.1
where AGE is the conductance correction for a square. Since L4 increases

with decreasing temperature, this leads to a resistance versus temperature
correction. Unfortunately, there is a much greater correction to the temperature
dependence of the resistance from the enhanced e-e interaction, making the WL
resistance versus temperature contribution very hard to interpret. Fortunately,
the behavior in a low magnetic ﬁeld between WL and the enhanced e-e is quite
different. While WL is very sensitive to low magnetic ﬁelds, the e-e interaction is
essentially ﬁeld independent until very high ﬁelds. It is this difference that allows

us to make a quantitative inference of L4 from ﬁtting magnetoresistance data to

WL theory.

3.1.2 Magnetic Field Dependence of WL

As discussed above, it's the unique low magnetic ﬁeld dependence of WL
that allows us to make quantitative inferences of L4. The ﬁeld dependence is a

result of the magnetic ﬁeld separating the identical phases of the time-reversed

paths. Recall from chapter 1, the phase from a magnetic ﬁeld perpendicular to

49

the paths is given by (1)3 = (i8 / 701A 0 d] . Since the paths each gain this

phase but with opposite sign, the total phase difference is 24) B- The maximum
length a path can be and still contribute to WL is L4 without a magnetic ﬁeld. As

L4 increases, the correction increases because there are more paths that

constructively interfere. The magnetic ﬁeld effectively reduces that number by

shortening the length of the paths that can still interfere constructively. The new
maximum length the paths can be, L B = \l h / 42B , shortens with increasing

magnetic ﬁeld. As the number of paths reduce, the decreased conductance is
reversed, thus we get an increase of conductance with increasing magnetic ﬁeld.
The bottom of Figure 3.2 shows this effect. Next, when we take into account the
spin of the electron, it will turn out that much of the above argument is correct
except when spin-orbit coupling is large. When spin-orbit coupling is large, there
will be a sign difference in the correction leading to what is termed weak anti-

localization.

3.1.3 Spin-Orbit Scattering

Spin-orbit scattering is a consequence of the electron, which has spin 1/2,
moving past a nonuniform electric ﬁeld produced by a defect. This nonuniform
electric ﬁeld when viewed in the rest frame of the electron appears as a magnetic
ﬁeld pulse that rotates the spin of the electron [Alt'shuler et al., 1987]. Thus as
the electron traverses the sample, seeing many different defects, its spin is
constantly being rotated in spin space. The distance over which the electron's
spin wavefunction rotates by 1: is called the spin-orbit distance, L50. This
distance, unlike L4», is temperature independent [Bergmann, 1984], which

explains why Ag is in the weak spin-orbit limit at high temperatures (above about

50

10 K, L4, < L30) and the strong spin-orbit limit at low temperatures (below about
10K, L4, > L30).

When we consider the difference between time-reversed paths, we must
now include the spin portion of the electron's wavefunction. Since the paths are
a time-reversal of each other, the only difference in spin space is the direction of
rotation as the electrons move past the defects. For spin 10 particles, a 4n
rotation is required to transform the spin matrix into itself. A 21: rotation produces
a minus sign in the spin matrix. This minus sign leads to destructive interference
of the time-reversed paths. After doing a statistical average over the possible
spin states from the time-reversed paths, it can be shown in the strong spin-orbit
limit [Bergmann, 1982] that the probability to return to the origin is 1/2 that of the
classical limit. This 1/2 probability to be at the origin means the electron is more
likely to propagate through the sample leading to an increase of conductance,
thus the term weak anti—localization. The above discussion related to the WL
correction is nearly the same except for a minus sign. This is shown for low
magnetic ﬁelds in the top of Figure 3.2., as the magnetic ﬁeld is increased, the

conductance decreases.

3.2 2-D Weak Localization Theory and Analysis

In this section we discuss the WL ﬁtting function and ﬁtting procedures in
determining the phase breaking length L4 and the spin-orbit distance L30. The
ﬁtting function is derived using diagrammatic techniques in many references [see
for example Bergmann, 1984 or Alt'shuler, 1987] and thus will only be quoted

here. The ﬁtting procedures are standard so they will only be discussed brieﬂy.

 

5
7‘

Ti

V :4.‘ \\4 4

is:

 

 

 

<—L——>

Figure 3.1 Electron scattering paths that illustrate weak localization.

3.2.1 2-D Weak Localization Theory

The data are analyzed in the form of conductance corrections as shown in
equation 3.2 below:
AG(B) a G(B) — G(B = 0)
3.2
where G(B) is the conductance of the sample at magnetic ﬁeld equal to B. From
Bergmann, 1984 the WL conductance correction for a square as a function of B

is:

2 B
1th 2 B 2 2 B 2 2 B
3.3

where:

 

 

BO:BeI+BS0; BI=B¢+-4—BSO;
3.4
h 72 3h
B = ; BSD = i e = —
‘l’ 4eL3p 4eL§0 ’ 4e]:
3.5

and:

1 Bx
‘1” -2- + —B— is the digamma function. The digamma function ‘P(x) for large x

is equal to ln(x), which is the case for small magnetic ﬁelds. Thus putting it all

together, we ﬁnd for our ﬁtting function:

1e2 B3
AG(B) = NU 56D —5‘n_hln B IBZ
¢ 0

 

3.6

where NEI is the number of squares in the sample, i.e. LengthNVldth. This

function is used to ﬁt the magnetoconductance data with only L4 and L30 as

ﬁtting parameters as will be discussed below.

3.2.2 Data Analysis

From equation 3.6, there are three parameters needed to ﬁt the data. The

mean free path, I3 is determined from the residual conductivity using the free

53

electron result derived in chapter 1. The other two are adjustable to ﬁnd the best
ﬁt. Since Lso is temperature independent [Bergmann, 1984], its value is

determined at low temperatures with a two parameter ﬁt and then ﬁxed at higher
temperatures. Thus at higher temperatures, the only adjustable parameter is L4.

The ﬁts are carried out to at least ten times the largest ﬁeld scale (either B4 or

Bso) as shown by the arrows in ﬁgure 3.3. Restricting the ﬁeld scale guarantees
that no other corrections need to be accounted for such as the semiclassical
magnetoresistance.

The best ﬁt is determined by the Levenberg-Marquardt method [Press et
al, 1990], which is a nonlinear least squares ﬁtting algorithm. This ﬁnds the
minimum x2 in parameter space and allows us to make a rough estimate of the
uncertainty of the ﬁtting parameters. Since we don't know the statistical errors of
our data, the actual value of x2 is meaningless. This makes estimating the
errors in the true statistical sense impossible. We can however make the simple
assumption that our ﬁts are good in the statistical sense and assign a reasonable
value of X2 to our ﬁt, which is the number of points in our data set [Bevington,
1969]. Will this assumption, we can estimate our uncertainties. The

uncertainties of L4,, using this approach, are shown as error bars in Figures 5.1

and 5.2 for the WL data. The error bars are typically less then the symbol size.

54

 

  

1.0E-5 T=1K, Lso=.25pm, L¢=.93pm

 

6.4E-6 ‘

2.3E-6 ‘

AGl(e2/1rh)

-1.8E-6 l

-5.9E-6 ’

 

 

   

L J.

 

_1 .055 . . . - . . . . . . . . .
-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

Magnetic Field (Tesla)

 

105-5” T=25K, Lso=.25l.l.ITI. L4=.075l1m

    

6.4E-6 ’

2.3E—6

AGI(e2/1rh)

-1.8E-6 ‘

-5.9E-6 ‘

 

 

 

405-5 -42- ‘ ‘ ‘ :
-0.4 03 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

Magnetic Field (Tesla)

Figure 3.2 Magnetoconductance and ﬁt to WL theory for two different

temperatures for a 7pm x 53pm x 0.014um Ag sample. The data are the dashed

lines and the best ﬁt to theory are the smooth lines.

‘ Fat—2‘2??- a

 

Tlllllll‘l'l lTlllTll'Hl lTlTlllf'l‘l lllllTU‘l'l lilllill

C ' -

_L _

AG/Goo (G00 = e2/1rh)
, .
I1. 3
a: '4.
7s 7:
‘—
l I i

 

 

 

_ J

L-llllllllriLLlIIILIIIIIIIJIIIIIIIIJII iillllllIIIl lldllllld

10'4 10'3 10'2 10'1 10°
B(Tesla)

Figure 3.3 Magnetoconductance and ﬁt to WL theory for a 5pm x 50pm x
0.017um Ag sample. The data are the round symbols and the best ﬁt to theory

are the smooth lines. The ﬁt parameters are shown in Figure 5.1.

Chapter 4. UCF
4.1 Introduction

Universal conductance ﬂuctuations (UCF) theory is the second quantum
interference theory that we would like to discuss. This is the theory that
quantitatively describes the reduction of the UCF enhanced 1lf resistance noise
in low magnetic ﬁelds. Again, as in WL, low magnetic ﬁelds refer to ﬁelds where
the semiclassical cyclotron radius of the electron is much larger then the mean
free path. UCF theory began by looking for the theoretical explanation of
aperiodic magnetoconductance ﬂuctuations of small metal wires [Webb, et al.,
1985] and rings [Umbach, et al., 1984]. Since the observation of the aperiodic
magnetoconductance ﬂuctuations, an extensive amount of theoretical work has
been done in understanding their origin [Lee and Stone, 1985; Al'tshuler, 1985;
Al'tshuler and Khmel'nitskii, 1985; Lee, et al., 1987]. Below we will give a brief
explanation of UCF but for a more complete description of the subject, see Lee,

Stone, Fukuyama [Lee, et al., 1987].

4.1.1 Origin of UCF

The main feature of UCF theory is the explanation of the conductance
variance between disordered samples with the same number of impurities. In
fact, the average difference between the conductance of these samples is of
order e2lh, independent of average conductance and sample size at T=0 [Lee,
et al., 1987]. This is the reason that the conductance ﬂuctuations are considered

universal, hence the name universal conductance ﬂuctuations.

56

57

To begin, we would like to explain the variance between samples with the

same number of impurities. The variance of the conductance is deﬁned:

(AGf = ((G— <G>)2)

where G is the average over impurity locations of the sample conductance
<

and G is the sample-speciﬁc conductance. According to UCF theory:

2 2
(AG)2 .-= A 5—
h

4.2
where A is a constant of order unity. To understand the universal nature of the
variance, it is helpful to think in terms of transmission coefﬁcients, which
determine the conductance from the Landauer formula [Landauer, 1957].
Because at T=0, the electrons are phase coherent over the entire sample, they
scatter elastically from the defects and interfere in some complicated matter to
produce a speckle pattern in analogy with a speckle pattern produced by a laser
shone through a disordered system. The many Feynman paths that repeatedly
intersect each other producing the speckle pattern of the electron wavefunctions,
lead to highly correlated transmission coefﬁcients for different paths [Stone,
1989]. In particular, the statistical correlation between paths that start and end
far from each other in the sample but form closed loops in the middle, lead to the
universal value of the conductance variance. Figure 4.1 shows a subset of
electron scattering paths that gives rise to UCF. There are two different types of

paths that are important. Paths that have time-reversal symmetry form the

cooperon and paths that lack time-reversal symmetry form the diffuson.

58

This universal value can be understood by using the Landauer formula
with the concept of effective channels, where effective channels are statistically
independent of one another. These channels add to give the total conductance

of the sample as:

4.3

where Ni are the statistically independent channels. Since the interference due

to the phase of the electron wavefunctions is bounded by constructive and
destructive effects, the total number of statistically independent channels
changes by 1, on average, which leads to the universal conductance variance
that is of order (e2/h)2. This result is independent of the number of effective
channels, hence independent of overall conductance and sample size.

Since a rearrangement of the sample impurities totally rescrambles the
phases of the different paths, a special hypothesis, known as the ergodic
hypothesis [Lee, et al., 1987], was put forth that states that anything that
rescrambles the phases of the electron wavefunctions should yield the same
result as changing the impurity locations. In particular, changing the Fermi
energy of the electron or changing the magnetic ﬁeld in the sample, should
rescramble the phases of the wavefunctions to give UCF type ﬂuctuations in the
conductance. These changes alter the speckle pattern in some complicated way
that yields an effectively rescrambled impurity conﬁguration. The Fermi energy
sets the characteristic wavelength of the electron wavefunction so its change
leads to a change in the wavelength that then changes the entire speckle

pattern. The magnetic ﬁeld changes the phase of the electron wavefunctions in

59

the familiar way, (p B = (ie/ 71)! Z 0 d7 , thereby rescrambling the speckle

pattern.

In a metal, changing the Fermi energy is impractical, however, applying a
magnetic ﬁeld can be done quite easily. Usually this conductance change or
resistance change versus magnetic ﬁeld is called the magnetoﬁngerprint of the
sample. Its description would require knowledge of the impurity conﬁguration,
which is not experimentally practical. Figure 4.2 shows two magnetoﬁngerprints
taken at 4K for a small Ag sample. Note the ﬂuctuations are not time dependent
but rather magnetic ﬁeld dependent as can be seen by comparing the ramp up
data to the ramp down data.

The magnetoﬁngerprint data are analyzed in terms of the magnetic ﬁeld

correlation function deﬁned as:
F(AB) a (AG(B)AG(B + AB))
4.4
where AG(B) = G(B) — (G(B)). Figure 4.3 shows the magnetic ﬁeld

correlation function for the data in Figure 4.2. There are two values of interest in
the magnetic ﬁeld correlation function. First, the value at AB=0 is equal to the
variance of the conductance as a function of magnetic ﬁeld. Note that the
variance is much smaller then (e2/h)2 because the sample is at ﬁnite
temperature. The effect of temperature will be discussed in the next section.
Second, the point at which the correlation function drops to 1/2 its AB=0 value is
the point that sets the scale of magnetic ﬁeld that rescrambles the phase of the

electron wavefunction to effectively give a different impurity conﬁguration.

60

Top path
’* ‘ ~ ‘ ‘ :\ =I‘\\\T*L *
A - — 7 ‘ _ * \‘ B Ly
-> * v —>

/

* I ,.

Bottom path

 

 

<———Lx——>

Figure 4.1 Two electron scattering paths that interfere to give UCF.

4.1.2 Effect of Temperature on UCF

Two temperature dependent lengths effect UCF, L4, and Lth- L4, is the

phase breaking length and sets the quasi-dimensionality of the sample as in WL.
Here again, we will only be concerned with quasi-2D samples. Lth is the thermal
length and is the distance two electrons separated in energy by KT remain

phase coherent. The thermal length is deﬁned:

4.5

where D is the diffusion constant of the electron.
To understand the effect of these two lengths, consider the paths in
Figure 4.1. Let us, instead of talking about distances, talk about time scales.

These time scales are related to the distances by:

wll

like

de

be

tin

lir

61

L=Dt

x x

4.6
because we are in the diffusive limit. The paths begin at point A where they
intersect each other. The time it takes to traverse the separate paths in general

will be different. The resulting interference from the paths at point B will decay
like exp[-(ttop + tbottoml/Tcplr [Bergmann, 1994] where 1:4, is the phase breaking

time. Thus if the time to traverse the paths is much larger then 14,, the effect will

2‘

decay rapidly. On the other hand, if the time difference, top -l‘b0,,0m I

 

between the two separate paths is greater then 71/ K T , the internal coherence

time, the two paths will not be able to interfere with well deﬁned phases
[Bergmann, 1994]. Thus, the quantum interference effect will not be present.
From this discussion, the importance of the two lengths on UCF are seen. We
will limit our discussion to samples where the thermal length is shorter than the
phase breaking length, which is the case for most metals.

The addition of temperature reduces the result of equation 4.2 for the

conductance variance. For a quasi-2D sample of length L2 and width Ly in the
limit Lth < L4, the new result is:

2 2L L2
Var(G)=(AG)2=A 5h— :3“

 

4.7
where A is a constant of order unity which depends weakly on the ratio of Lth/Lq,

[Bergmann, 1994]. From the above equation, we see that UCF average to zero

for an inﬁnitely long sample.

62
4.2 UCF Enhanced 1lf Noise

In this section we would like to discuss the sensitivity on conductance of
defect motion. It is this effect that gives rise to the low temperature increase in
1If resistance noise that is studied in the present work. The quantitative
calculations are done in the context of UCF theory. We will only consider the
case where movement by the impurities changes the conductance by less then

(e2lh), technically termed the unsaturated limit.

4.2.1 Sensitivity of the Conductance to Impurity
Movement

The quantitative calculation of the conductance change due to impurity
reconﬁguration has been considered by two groups. We brieﬂy describe the
approach of Feng, Lee, and Stone, along with Al'tshuler and Spivak. The latter
is used in the theoretical calculation for the noise crossover shown in appendix l.

Feng, Lee, and Stone [Feng, et al., 1986] begin by considering the

conductance change from a single impurity movement deﬁned:

(5G)2 = <(G{r1,...,rN} — G{r1 +5r,...,rN})2>

4.8
where r represents the impurity location and Sr represents the distance moved by
an impurity. They then generalize their result to the case of many impurities
moving to give a description that explains the increase in 1/f resistance noise.

Their approach is divided into two steps. First they calculate the conductance
change in a coherence volume (square of length L4,) due to impurities moving.

63

Then they divide their sample into coherence volumes and classically average

the result from a single coherence volume to obtain the ﬁnal result:
2 2 2
(5G) 0C ns(T)LthL(p

4.9
where nsm is the density of moving defects. Typically nS(T) 0C T,

L3,, 0C1/ T, and L3,, 0C T —p , which leads to an increase in 1/f resistance

noise at low temperature because:

2 (MY
Gz

D(t )t
1 + (21rﬁ)2

 

 

Sx(f)°CR I

4.10

where DC!) is the distribution function of the characteristic time constants of

the moving defects. This increase at low temperature was shown in chapter 1.
Al'tshuler and Spivak [Al'tshuler and Spivak, 1985] take a different
approach by considering the conductance correlation between two different
impurity conﬁgurations. Clearly if the impurity conﬁgurations are signiﬁcantly
different, there is no correlation and the result given in equation 4.7 is obtained
for the difference in conductance. We will limit the discussion to small

conﬁguration changes. They deﬁne the conductance correlation function:
F(V,V') '=-'- (G(V)G(V')> - (G(V)><G(V')>

4.1 1
where V denotes the beginning impurity conﬁguration. They show the effect of

changing the impurity conﬁguration effectively adds an additional dephaslng rate

64

to the previous conﬁguration [Al'tshuler and Spivak, 1985]. This seems logical
because any change in the sample impurity location alters the speckle pattern,
thus reducing correlations in the beginning speckle pattern. We can add this

additional dephasing rate to the phase breaking rate to get:

 

_1_ L i
In In 15f
4.12
where
1 1 (VV')
_=_ 1_ 2
if ‘e <V>
4.13

Thus, for small impurity reconﬁgurations, the additional dephasing rate is small

and the two conﬁgurations remain highly correlated.

4.2.2 1If Noise Crossover Function

In this section we discuss the effect of magnetic ﬁeld on the 1/f resistance
noise. We follow the procedure outlined by Stone [Stone, 1989] for the
quantitative calculation. As mentioned in chapter 1, the 1/f resistance noise
reduces by 50% in a magnetic ﬁeld. This is because at zero ﬁeld, the noise
enhancement is equally given by two types of paths, the cooperon and diffuson.
The cooperon contribution, having time-reversal symmetry, is eliminated by the
magnetic ﬁeld, which breaks this symmetry. Thus, the noise in a large enough

magnetic ﬁeld will be reduced by exactly 1/2. The diffuson is independent of

65

magnetic ﬁeld, thus its contribution is unaltered. This means to calculate the
crossover function, one needs only to calculate the decay of the cooperon with
magnetic ﬁeld.

We begin by deﬁning the crossover function as [Stone, 1989]:

 

v(B) = N(B)
N(B = 0)
4.14
where N(B) is the noise at magnetic ﬁeld equal to B. With this deﬁnition of the
crossover function, the value at B=0 is 1 and reduces to 1/2 at large magnetic

ﬁelds. Next we write N(B) as:
N(B) oc 156(8)]2 cc ([6043) - G(V',B)f)

4.15
where G(V,B) and G(V',B) are the two different conductances that give rise to the
1lf resistance noise. We assume that V and V‘ are part of the same ensembles
with the same average conductance and conductance variance. Then by adding

and subtracting the average conductance to the above equation, we ﬁnd:

[50(3)]2 cc ([6043) — <G> — 60/33) +<G>f>
= <[AG(V,B)]2> + <[AG(V',B)]2> - 2(AG(V,B)AG(V',B))

= 2<[AG(B)]2> —- 2(AG(V,B)AG(V', 3))

4.16
where because we assumed that the variance from the ensembles related to the

impurity conﬁgurations V and V' were the same, we can combine the variances.

66

Now we use the result from Al'tshuler and Spivak [Al'tshuler and Spivak. 1985]

that was discussed above to get:

. 2
AG V,B,-1— AG V',B,—1— 0C AG V,B,—1—+—l—
t4 t4 14 If

4.17

where we have included the phase breaking time. Thus putting it all together for

large 1f, we ﬁnd:

2<[AG(B)]2> — 2(AG(V, B)AG(V', 3))

2 2
=2 [AG[V,I,_L)] - [AG[V,I,_L.-I_]]
14, 124, If

 

 

 

 

 

2

.31 d IAGIV’B’LII
t 1 1:
~12.) .

Thus for the crossover function, we have:

4.18

 

k .
‘.——'n_

 

2
d AG V,B, 1
1 r
d( 11>) (p
v(B)= 2
d AG V,B=O,1
1:

4.19
The problem is now reduced to calculating the variance of the conductance as a
function of magnetic ﬁeld for the cooperon. This is done in Appendix I with the
inclusion of spin-orbit scattering to yield the ﬁtting function for the 1lf resistance

noise versus magnetic ﬁeld.

4.3 Data Analysis

We now discuss the procedure for ﬁtting the 1/f resistance noise versus
magnetic ﬁeld to UCF theory. The ﬁtting function derived in Appendix I is
numerically evaluated using the fourth-order Runge-Kutta with adaptive step size
algorithm for the numerical integration [Press, et al., 1988]. The diffusion
constant D, and thermal length Lth are calculated for the given temperature and
used in the ﬁtting function. The spin-orbit length, Lso, is determined from the ﬁts
of magnetoresistance data to WL theory, which is done because the UCF noise

crossover function depends only weakly on Lso- Thus at low temperatures, we
have only two ﬁtting parameters, L4, and Nzero. Nzero is the zero ﬁeld absolute

68

noise level. At high temperatures, the noise does not drop a full factor of two.
This occurs because at these temperatures, the noise is not 100% UCF type but
rather a combination of local interference and UCF type. The local interference
noise is magnetic ﬁeld independent [Hershﬁeld, 1988; Pelz and Clarke, 1987].
The additional ﬁtting parameter, %UCF, accounts for the lack of the 1/2 reduction

in the noise. The ﬁtting function is then:

Nzero

 

FitFun(B) = [2 x (1-%UCF)+%UCF(1+VC(B))]

4.20
where Vc is the cooperon contribution to the crossover function given in appendix
I.

The best ﬁts are determined by minimizing x2 with respect to the allowed
ﬁtting parameters. This is accomplished by making contour plots of X2 and
manually determining the minimum. This is done because computation of the
ﬁtting function is time consuming. Figures 4.4 and 4.5 show data for the 1/f

resistance noise reduction in a magnetic ﬁeld with best ﬁts to UCF theory. The
values of L4, are shown in Figures 5.1 and 5.2 along with the thermal lengths.

An example of a X2 contour plot is shown in Figure 4.6. The estimate of the
uncertainties on the parameters are determined by projecting onto parameter
space a difference of x2=2.3 for the two parameter ﬁts, and X2=3-5 for the three
parameter ﬁts [Press, et al., 1988]. The uncertainties are shown as error bars in

ﬁgures 5.1 and 5.2.

69

 

 

Rm" 6 10'6

 

 

 

I I I I I l I I I I l I I L I l

0 1 2 3 4 5
B Field (r)

 

Figure 4.2 Magnetoﬁngerprint of a 5pm x 50pm x 0.017pm Ag sample at 4.4K.

70

 

 

 

 

5.0E‘6 I l I I l 1 I I I T
4.0E-6 -
“Ax
.35
3 30156 - ~
:1 ' " O
.2 3.
E
O
S 2.0E—6 - ”5 ~
0 0..
"U
:33 O
L“ 10E6 ~ ' g . _
' ' . . . .
i . O
o
OOEO I l I 1 I l I 1 I I I 1 I I I. J
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Delta B (Tesla)

Figure 4.3 Correlation function of the magnetoﬁngerprint in Figure 4.2.

1.0

71

 

1.1 lllll'l'l'T’TTllllll"'l"'T Tllllll ﬁl'l"'l lllllll'r'l'7'

T=1.0K

 

90.}

Normalized Noise Power

 

 

 

04 1111141411JL1 11111111111A1A1 1111111I1111141 111111—1A11111

10'3 2 3 10'2 2 3 10'1 2 3 10° 2 3
B(Tesla)

 

Figure 4.4 1l‘f resistance noise versus magnetic ﬁeld for 3 5pm x 50um x

0.017um Ag sample for four temperatures.

72

 

11 v1'111111111'11'111111111'1 1 11 111 .111

 

 

 

 

 

b

B

o

9..

8

'8

Z

'0

0

£1

E

0

Z
0.5 _1_ L i1 "
04 l111A1A1111111l lIllIl lllllllI1 iIJIlJILllllIl 111.1 lllllllIn
10“ 2 3 10'3 2 3 10'2 2 3 10'1 2 3 10° 2

B(Tesla)

Figure 4.5 1/f resistance noise versus magnetic ﬁeld for a 7pm x 53pm x

0.014um Ag sample for three temperatures.

73

 

 

 

 

 

2.0E-7 2.1E-7 2.2E-7 2.3E-7 2.4E—7 2.5E-7

Figure 4.6 Contour plot of X2 for T=10K data from Figure 4.4.

Chapter 5. Comparison and Summary
5.1 Discussion of Results

We now discuss the results of the comparison between the quantum
interference theories, WL and UCF. Recall the basis of the comparison was the
common ﬁtting parameter Lq, Figure 5.1 shows the result for a 5pm x 50pm x
0.017um Ag sample and Figure 5.2 shows the results of two Ag samples,
sample 1, 711m x 53pm x 0.014pm and sample 2, 18pm x 162um x 0.014um.
While there is general overall agreement between WL and UCF, at temperatures
less than about 10K there are statistically signiﬁcant discrepancies in the Lq,'s.
The spin-orbit lengths and thermal lengths are also shown on the Figures for

reference.

5.1.1 Discussion of Discrepancies

There are several possible explanations that would give rise to the low
temperature discrepancies that we would like to rule out, including boundary
conditions, magnetic impurities, and ohmic heating. Also, there is a possible
explanation that we can think of at this time that may account for the differences
at low temperature, which is failures in the theoretical development. We will
discuss each possibility below and give an explanation as to why or why not we
feel we can eliminate them as a possible explanation.

Figure 5.2 shows the result from two different size samples that were
made at the same time. For a given measurement (WL or UCF), there is
excellent agreement between these two samples. However, the discrepancy at

low temperatures between the two measurements is observed in both samples.
74

75

From this result, we conclude that the boundary conditions of the samples do not
play a role in the comparison. We expect this because the samples have a
length and width much greater than L‘p, which means the boundary conditions
should be irrelevant.

Though we have not discussed the effect of magnetic impurities thus far,
they do enter into the two quantum theories differently. However, at the lowest
temperatures where Ag is in the strong spin-orbit scattering limit, we would
expect an opposite result from that observed if magnetic impurities were
involved. We would expect the Lq, inferred from WL to be less then the L,p
inferred from UCF. This is because in UCF, a magnetic scattering event
dephases the wavefunction in the same way as an inelastic scattering process.
However in WL, there is still some correlation in the time reversed paths from
magnetic scattering, which then enters into the theory differently than the phase
breaking process.

Another explanation for the discrepancy is ohmic heating, however, the
same drive current was used in the two measurements, and thus the same
ohmic heating was present. To ﬁrst order, this ohmic heating shows up in a
reduced L,p that is common to both the measurements. We did do a heating
experiment on sample 1 of ﬁgure 5.2. The result indicates that indeed the zero
magnetic ﬁeld 1/f noise measurement is more sensitive to ohmic heating than the
magnetoresistance measurement [Hoadley, et al., 1995]. We then lowered the
drive current and remeasured magnetoresistance and 1If noise versus magnetic
ﬁeld. The result of this was no signiﬁcant change in the inferred Lq,'s but we
must note that the signal-to-noise ratio was much worse, which explains why we
did not measure it with the lower drive in the ﬁrst place. There are several

unresolved issues that we are in the process of resolving: The point that we

76

would like to make is that maybe heating comes into the theories differently, i. e.
not just as a reduced Lq,.

Though we have separated the heating issue from the theoretical
development issue, in reality if heating is the culprit, this points to an inadequacy
in the theory, either WL or UCF. In our opinion, it's most likely to be in the UCF
theory. From looking at the data in Figures 5.1 and 5.2, it appears that the
discrepancy occurs when the phase breaking length is larger then the spin-orbit
length. We have, to the best of our knowledge, put into the theories correctly the
effect of spin-orbit scattering. Furthermore, we included all temperature effects
and sample dimensions. In short, the theory should be correct. We have
included all known theoretical considerations that are relevant to our
experimental situation. If the explanation of the discrepancy is not a theoretical

shortcoming, we simply don't understand it.

5.2 Summary

We have measured magnetoresistance and 1/f noise versus magnetic
ﬁeld for three different Ag samples in the temperature range 1K - 25K and have
ﬁt the data to WL and UCF theories, respectively. By comparing a common ﬁt
parameter, Lq,, we have quantitatively compared the consistency of WL and UCF
theories. At temperatures above about 10K, the comparison yields excellent
consistency. At temperatures below about 10K, the comparison yields
statistically signiﬁcant discrepancies that are not understood at this time. We
have put forth some possible explanations but no concrete proof as yet to the
discrepancies.

The sensitivity to ohmic heating between the measurements is a

unresolved issue that will require additional effort to resolve. Future work to

77

resolve the phase breaking length discrepancy may require measurements of

quasi-1 D samples and/or different materials. ~

78

 

ll‘T‘T‘T'l'l‘l'l'llllliﬁr’l“‘

. Fits of Magnetoresistance to W.L. theory
1 ”‘ I Fits of Noise vs. B to U.C.F. theory ~

1
1111

l
h
(I)
C
11141

T
l

H. (Hm)

0.1 — I

 

 

 

Figure 5.1 The results of comparing WL and UCF theories for a 511m x 50um x
0.017pm Ag sample.

79

 

 

 

 

31 111 I111111'T111111 1 TTT 1,
_ v d

2 ‘ V Sample1(WL)
_§ A Sample2(WL) _
. . V 0 Sample 1 (UCF) .

100 ._ é * I Sample 2 (UCF)
: O . :
7T— ' _
65' _
l: :

O
A 5— —
i ELso ¥ :
V 4—...------------..--..-----..--- _..--..--_-_....:
_I’ :. i :
3 — L
2* 4
~ L111 Q 2
10": §"
1 I 1 I l 1Il 1 lIiorrI'1IlI1Il l l l 1 I 141 I
100 2 3 4 5 6 7 101 2 a
T(K)

Figure 5.2 The results of comparing WL and UCF theories for two Ag

samples. Sample 1 is 71m x 5

APPENDIX

Appendix I.
Derivation of the Noise Crossover Function

In this appendix we outline the derivation of the noise crossover function
that is discussed in chapter 4 and used to ﬁt the 1lf resistance noise reduction in

a magnetic ﬁeld. Previous work by Stone [Stone, 1989] calculated the noise
crossover function in the limits Lso << L,p and Lso >> L,p together with the limits

Lth >> Lq, and Lth << L4,. Unfortunately in Ag over most of the temperature

range in this work, these limits are not valid. Therefore it was necessary to
calculate the noise crossover function for any value of Lso and Lth- We will
outline the procedure below for a quasi-20 sample of length L2 and width Lx.

Recall from chapter 4 that the noise crossover function is deﬁned as:

v(B) = N(B)
N (B = O)

 

l.1
where N(B) is the noise magnitude at magnetic ﬁeld equal to B. Since the UCF
enhanced noise is given by two equal contributions, one from the cooperon and

the other from the diffuson, we can write the crossover function as:

v(B) = %vd + %vcw) = :— + %vc(B)
l.2
because the diffuson has no magnetic ﬁeld dependence. Thus the problem is
reduced to calculating the crossover function of the cooperon. We begin with

equation 4.19 from chapter 4:

80

V(B) = N(B) :
N(B = 0)

 

W 132112111
W 121220.111

L3
and only consider the contribution from the cooperon. We ﬁrst must calculate
the variance of G, [AG]2, to calculate the noise crossover function. The

cooperon contribution to the variance of G is given by [Stone, 1989]:
2 2 , ,0
h 1t _.., 2kBT 2kBT
|.4

[ ]= xcoth(x) —l

where sinhz (x) and FC(AE, B) is the energy-correlation

function for the cooperon given by:

“153)-; __1_,.1R.[+]._3_,.3R.[,;]
n 411M=0| 8 XIII-=0 4|).n,j=,| 8 XIII-=1

|.5

82

The eigenvalues, 71.",1- are given by the cooperon diffusion equation:

2
D 46—2321 +-1—+ij-l—-ié-E- =7.,,Ip
7: 17¢ 3 ‘L' h ’1

SO

l.6
where, j=0 is the contribution from the singlet channel and, j=1 is the contribution
from the triplet channel [Chandrasekar, et al., 1990]. This is the way spin orbit

scattering formally comes into UCF theory. The solution of the cooperon
diffusion equation for the eigenvalues, 71. n, j is:
1 1 4 1 AE

kn.=4-e—BD n+— +—-+—j——i—
’1 h 2 1:4, 3 17w h

 

 

 

 

 

l.7
with degeneracy:
e
ZBLXLZZ.
l.8
Therefore we ﬁnd:
' n=3(h/e) "
°° d(AE) AE WEBZBLL [1 3 ]
N B —— x 2 — ,b +—— n ,b
C( )Oci, ZkBT 2kBT] "2:30 (We) 4f(a"" ) 4f(a” )
I.9

where NC(B) is the contribution to the noise from the cooperon and:

83

 

_ a,,(—3a,,+b2 .
f(amb)-( (a3+b2)3 )

 

2 2
a“ =-SBLZ("+l/2)+—-§iz—=BC2(n+1/2)+C3,;
’ 1t(h/e) 1: L ’

(P

_8BL§(n+1/2) L3, 5 L3,

 

 

 

—— =BC +1/2 +C ;
a“ 1t(h/e)1 «ng 371%; 201 ) 3”
b_AE L:
_ 2 2 ;
KTrr Lth
|.10

where we have deﬁned:

 

 

_2LxLz.

‘ (h/e)’
2

C2: 8L2 ;
1r(h/e)
L2

C = 2 ;
3,3 11:sz

(P

84

 

 

2 2
3 _ L. 5 L.
,t — 2 2 2 2
1r Lq, 35 L30
H1
and the degeneracy of the eigenvalues is:
e
ZBLxLz—zBCl.
h
l.12

Converting the sum of eigenvalues to an integral in the limit B—>O, we ﬁnd:

3(h/e)
“T
81:! B

2‘ 801E f1a...b)+—:,°°f(a...b)]~ 1 dn<BCI)[1f<a.,..b)+§f(a.,.,b)]

n=0 -l/2

 

1—[C,(1b2 + 3C3 ,)] + 3 —[C,(b2 + 3C§,)]
=4 2C2(b2+C3S) J 4 2C2(b2+C§,)4

 

L13

The low magnetic ﬁeld corrections to the above approximation are calculated:

n+l/2

Tdn[BCI[ﬁ-f(a.,..b)+21114.41} 1 dn’1BCI)[§f<a.I,..b>+§f<a..,,,b)]]

—l/2 n—l/Z

 

 

_B 1C,C,(b4-14bZC§,+9C;,) 3C1C2(b4—14b2C32H,+9C§,)
‘ 4 24(b2+C3,) +4 24(b2+C31)
l.14

85

Finally we ﬁnd:

 

 

°° d(AE) AE 1 3
NC(B-_> 0) xi[ ZkBT ZkBT][4f(B’b’Cl’C29C3s)+4f(Bab9C1,C2,C3t)]]
l.15
where:
—Cb2+3C2 4_ 2 2 4

2C2(b2 + C32)4 24(1)2 + C32)4
l.16
As discussed in chapter 4, we add two more parameters to the ﬁtting
function that we use to ﬁt our data, %UCF and Nzero. %UCF allows for the case
at higher temperatures when the noise is not 100% UCF type. Nzero is the zero
ﬁeld absolute value of the noise. Vlﬁth these, along with the derivation above, we

ﬁnd for our ﬁtting function:

Nzero

 

 

FitFun(B)= [2x(1—%UCF)+%UCF[1+ NAB) J]

N.(B = 0)

|.17

which is equation 4.20 in chapter 4.

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