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1/ F NOISE AND QUANTUM TRANSPORT IN THE
LOW SPINfORBIT SCATTERING LIMIT

presented by

Jeong-Sun Moon

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1/F NOISE AND QUANTUM TRANSPORT IN THE
LOW SPIN-ORBIT SCATTERING LIMIT

By
J eong-Sun Moon

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy
and Center for Fundamental Materials Research

1995

ABSTRACT
l/F NOISE AND QUANTUM TRANSPORT IN THE
LOW SPIN-ORBIT SCATTERING LIMIT
By

J eong-Sun Moon

Recent advances in microfabrication technology have enabled the
experimental investigation of quantum interference effects on electron transport in
disordered systems. At low temperature, the electrons diffuse coherently for a
distance which can become much longer than the elastic mean free path. The
resulting quantmn interference among diffusive paths gives rise to universal
conductance fluctuations(U CF), in which the conductance fluctuates chaotically as a
function of a control parameter by a universal amplitude SG~e2/h at zero temperature,
independent of the magnitude of the conductance itself. Recent theoretical
developments provide a connection between UCF and random matrix theory(RMT),
pioneered by Wigner and Dyson. It is shown that the conductance is related to the
eigenvalues of the transmission matrix and the relative amplitude of the conductance
fluctuations is governed by the symmetry of the transfer matrix.

This thesis describes low temperature measurements of l/f noise on a quench-
condensed quasi-1D Li wire, to study UCF in a low spin-orbit and spin-flip scattering
system, which has the feature of maximum symmetry in the absence of a magnetic
field. We observed two distinct reductions by factors of 2 in the noise versus
magnetic field: the first from breaking time-reversal symmetry and the second due to
lifting the Zeeman degeneracy. We measure and calculate the complete crossover

function for both reductions, and find good agreement over the range 0 to 9 T. Our

results show that the magnetic field scale for the Zeeman crossover is determined by
the sample temperature, rather than by the Thouless energy.

We also studied the effect of the spin-flip scattering on conductance
fluctuations in the regime where the magnetic spin-flip scattering rate is comparable
to the electron phase breaking rate. We found that the magnitude of conductance
fluctuations is affected in a dramatic way: l/f noise is reduced by factor of 2 at low
magnetic field and increases dramatically at high field. The results are interpreted
that the magnetic field induces a transition from a low-field state where the impurity
spins are free to flip and destroy the phase coherence to a high-field state where spin-

flip scattering is frozen and recover the universal conductance fluctuations.

To my dearest wife Kyung-ah and son sol
and my parents

iv

ACKNOWLEDGMENTS

I am grateful to Professor Norman 0. Birge, my thesis advisor, for his insight,
encouragement and efforts to provide me with a complete Ph.D. training. He always
taught me deeper understanding of physics and instrumentation. I also thank
Professor Brage Golding who helped design this research project and provided us
with lots of useful information and microfabrication facilities. I have benefitted a
great deal fiom Professor Mark Dykman and S. D. Mahanti. I also thank Professor H.
Smith and V. Zelevinsky for being guidance committee members.

I thank the support from other members of Birge's group who care about my
research: Dr. Kookjin Chun, Dr. Paul McConville, David Hoadley and Matt Miller.
Dr. Chun, we shared a wonderful time together while discussing lithography and life
in E. Lansing, and even while finding a leak in the dilution refrigerator. Dr.
McConville, he taught me the complicated UCF theory with simple cartoon-type
pictures. Mr. Hoadley, he is behind me but is smart so that I have learned from him.
Mr. Miller, he always has nice jokes to make me laugh and I won't forget the old leak
detector he completely fixed. I especially thank A. Engebretson and M. Jaeger for
helpful discussion on the microfabrication. I enjoyed the friendship with other
Korean students, and especially thank Sangil Hyun.

Most of all, I would like to thank the support from my parents who always
show me what is valuable in my life, and my dear wife who gives me a smile with
love while we are traveling against the wind. My happiness is rooted and grows with
the love fi'om my family.

Once more I like to thank my advisor for his warm care about my outside life.

This work is supported by the Center for Fundamental Materials Research at
Michigan State University and the National Science Foundation under Grant # DMR-
9321850.

vi

TABLE OF CONTENTS

LIST OF TABLES
LIST OF FIGURES
INTRODUCTION
References
THEORY OF QUANTUM TRANSPORT
A. Diffusive Quantum Transport in Metal
B. Landauer's Conductance
C. Universal Conductance Fluctuations (U CF)
C.1 Random Matrix Approach
C.2 Random Hamiltonian Approach
C.3 Diagrammatic Approach
CA The Amplitude of UCF
D. Quantitative Analysis of UCF
D.1 Field correlation, Variance and UCF noise
D.2 UCF Crossover Function at weak Magnetic Field
D.3 UCF Crossover Function at strong Magnetic Field
E. Weak-Localization and low-field Magnetoresistance
References
EXPERIMENTAL TECHNIQUES
A. Quench-Condensation
B. Electron-beam Lithography
3.] Resolution and Undercut Profile

3.2 Sub-micron Metal Stencil

vii

ix

10
10
10
11
12
15
17
18
20
20
22
28
32
34
41
41
43
44
46

C. Optical Lithography
D. 1/fNoise Measurement
C.1 Dual-Phase Technique
C.2 Digital Lock-in Amplifier
References
ZEEMAN EFFECT on UCF
A. Introduction
B. Experiment
C. Magnetoresistance Measurement
D. I/f Noise Measurement
D.1 Quantitative Analysis of GOE-GUE Crossover
D.2 Quantitative Analysis of Zeeman Crossover
E. Earlier work
References
SPIN-FLIP SCATTERING and UCF
A. Introduction
B. Experiment
C Weak localization
D l/f noise and UCF
D.1 Orbital effect on UCF
D.2 Spin effect on UCF
E. Summary
References
CONCLUSIONS
APPENDIX

1/f noise

viii

52
55
55
56
59
74
74
74
75
76
76
77
78
80
9O
9O
91
91
93
93
96
97
99
108
l 10
1 10

LIST OF TABLES

Table 1.1
The relative amplitude of universal conductance fluctuations in various regimes

determined by an applied magnetic field and spin—orbit scattering rate.

Table 3.1
The procedure for trilayer electron-beam lithography to make sub-micron metal

stencil.

Table 3.2

The procedure for bilayer optical lithography.

Table 3.3

The characteristics of samples are summarized.

Table 3.4
An assembly code used for real-time signal processing in the implementation of a
dual-channel digital lock-in amplifier.

Table 5.1

The parameters of metallic lithium Samples studied for quantum transport.

ix

LIST OF FIGURES

Chapter 2
Figure 2.1

(a) Evaluation of the one-dimensional (ID) field correlation by Beenakker and von
Houton.

(b) Evaluation of the 1D variance and l/f noise crossover function in the particle-
particle channel.

Figure 2.2

Evaluation of the 1D 1/f noise crossover as a fimction of magnetic field in the
particle-particle channel at different values of L¢.

Figure 2.3

(a) Evaluation of the 1D variance crossover in the particle-hole channel.

(b) Evaluation of the 1D l/f noise crossover in the particle-hole channel.

Figure 2.4

Evaluation of the 1D noise crossover function in the particle-hole channel for the low-
temperature and hi gh-temperature limit.

Figure 2.5

Evaluation of the particle-hole channel noise crossover function in 2D with different

spin-orbit scattering rate.

Chapter 3
Figure 3.1
(a). Schematic set-up for the quench-condensation with a cryo—evaporator inside the

vacuum-can.

(b). Schematic view of the substrate with a metal stencil for in-sr'tu measurement.
Figure 3.2

Plot of line width versus line dose obtained by electron-beam lithography.
Figure 3.3

Scanning electron microscope (SEM) photograph of the bilayer electron-beam resist
pattern just after it is developed.

Figure 3.4

(a). SEM photograph of the undercut profiles and line resolutions obtained in the
~400 nm thick bilayer system.

(b). SEM photo which shows the ~80 nm deep undercut profile of the bilayer resist
exposed by the line dose.

Figure 3.5

SEM photo which shows the ~lOO nm deep undercut profile of the bilayer resist
exposed by area dose, 225 uC/cmz.

Figure 3.6

Schematic procedure for trilayer electron beam lithography.

Figure 3.7

(a). SEM photograph of the top view of a sub-micron metal stencil

(b) SEM photograph of the side view of a sub-micron metal stencil on top of the
copolymer resist.

Figure 3.8

Schematic diagram of the sample geometry.

Figure 3.9

Schematic set-up of the noise measurement using the Motorola Digital Signal
Processing(DSP) boards and personal computer.

Figure 3.10

Block diagram for the digital signal processing of the input signal.

xi

Figure 3.11

Transfer functions of the three Chevychev FIR digital filters used in the
implementation of a digital lock-in amplifier.

Figure 3.12

Recovery of the small 1/f noise signal out of the large background.

Chapter 4

Figure 4.1

Sample resistance versus temperature obtained from one-dimensional metallic Li
wire.

Figure 4.2

1/f noise power spectrum and background Johnson noise obtained simultaneously by
two-phase digital lock-in amplifier.

Figure 4.3

Magnetoresistance data at temperature, T = 1.6 K, 2.8 K, 4.0 K and 5.6 K. The solid
line is the fit to the quasi-1D weak-localization theory with zero spin-orbit scattering.
Figure 4.4

The values of x2 in the fit of the magnetoresistance data at 1.6 K with respect to
different spin-orbit scattering rate.

Figure 4.5

The electron phase breaking length versus temperature, obtained from the weak-
localization fits of low-field magnetoresistance.

Figure 4.6

The resistance noise power spectra, SR(f), versus frequency at 1.6 K with fits to a
power law, at three values of magnetic field, B = O T, 1.0 T and 8.8 T.

Figure 4.7

xii

Conductance noise power, Sg(O.1 Hz), as a function of magnetic field at 1.6 K and
4.2 K. The data are normalized by the noise power at zero field. The solid line is the
theoretical expression for the noise crossover function.

Figure 4.8
Contribution of % UCF to the total noise versus Lyle.

Chapter 5
Figure 5.1
(a) Magnetoresistance of quasi-1D Li sample #1 (W = 0.11 pm, RC, = 5.7 (2) taken at
T = 2 K. The solid line is the fit to the quasi-1D weak localization theory.
(b) Magnetoresistance of 2D Li sample #3 (W = 205 um, R0 = 7.8 0) taken at T =
3.5 K. The solid line is the fit to the 2D weak localization theory.
Figure 5.2
The electron phase breaking length versus temperature, obtained from the weak-
localization fits of low field magnetoresistance (a) for quasi-1D wires and (b) for 2D
films.
Figure 5.3
Conductance noise power as a function of the perpendicular magnetic field below 1T,
normalized by its zero-field value (a) for sample #1 and (b) for sample #2.
Figure 5.4
Values of the phase breaking length, obtained from the fits to the data of noise vs
magnetic field in figure 5.3, and obtained from the magnetoresistance data in figure
5.2.
Figure 5.5
Noise power as a function of magnetic field up to 9T from sample #1 at temperatures
1.6 K and 4.2 K. (The data are normalized by the noise power at zero-field.)
Figure 5.6

xiii

Noise power as a function of magnetic field up to 9T from sample #2 at temperatures
1.6 K, 4.2 K and 10 K. (The data are normalized by the noise power at zero-field.)
Figure 5.7

A fit to the noise power as a function of magnetic field from sample #1 at

temperatures 1.6 K and 4.2 K.

Appendix
Figure A
(a) Temporal representation of Johnson noise.

(b) Temporal representation of 1/f noise.

xiv

Chapter 1

INTRODUCTION

A. Classical Drude conductivity and Quantum correction.

How can we understand the low-temperature electronic conduction in a metal such
as gold? Until recently it was generally accepted that at sufficiently low temperature,
the conductivity of a metal is dominated by the residual scattering by impurities and

is given by the classical Drude formula,

 

(H)

where 1: is the elastic scattering time, related to the elastic mean free path [e = war.
This model predicts that at low temperature, the resistance of a metal will reach a
constant value determined by the concentration of static impurities.

Extensive theoretical and experimental studies showed that the effect of disorder
on the electron transport is more dramatic than the classical transport estimated.[1] If
the disorder is strong then electron wavefunctions can become localized, leading to a
metal-insulator transition. Even in the weak disorder limit -- i.e. the good metallic
regime, the deviation from Eq. (1-1) is significant. The resistance of thin films and
wires increases above the residual value and continues to increase as the temperature
is decreased.

These deviations in disordered metals can be understood in a new view of low-
temperature electron transport. Electrons are quantum mechanical waves and the

conductance is given by the quantum mechanical transmission through scattering

2

centers.[2] This remarkable idea led a number of groups to fabricate small wires and
films and study their conductivity at low temperature. These efforts resulted in the
discovery of novel phenomena -- weak localization and universal conductance
fluctuations. Nowadays the quantum corrections to the classical Drude conductivity

are well established in the low-temperature electron transport of disordered systems.

B. conductance as quantum transmission

The most intuitive explanation of quantum transport is based on Feynman's
formulation of quantum mechanics; a particle propagates between two points via all
possible paths. The picture below shows two typical paths between points A and B.
Each step along the path is determined by randomly located elastic scatterers due to
the disorder and these elastic scatterers determine the electron mean free path, Ie. In
disordered metals, at sufficiently low temperature, the electron transport is in the

diffusive regime. The electrons diffuse coherently over a distance called the phase
breaking length, L¢, which can be much longer than the elastic mean free path.

/\
\/

L4

The elastic collisions with impurities do not destroy the phase information but shifi
the phase by some fixed amount. The different paths arrive at point B with different

phases. The probability amplitude that an electron is transmitted from one side of the

3

sample to the other is proportional to the sum of the amplitudes of Feynman paths

that walk randomly across the sample. The transmission coefficient T is

2
Toc few“

(1

(1-2)

 

 

where or represents spatial path.[3] The cross terms in Eq. (1-2) correspond to the
interference between paths, and lead to significant deviations from the classical

Boltzman diffusion.

C. Weak-localization [4]

Consider the special paths that return to the origin as shown below.

 

There is always a time reversed path (dotted line) corresponding to a closed loop
(solid line). These two paths will be in phase when they arrive at A because they are
scattered by exactly the same elastic scattering centers. The interference between
these two loops is always constructive and will enhance the probability of returning to
the starting point A. The enhanced back-scattering decreases the diffusion constant
and diminishes the conduction. This correction to the classical Drude conductivity is
called “weak localization”, and it depends on the amount of disorder but not on the

details of the impurity configuration.

4

A dramatic effect of weak-localization occurs when a weak magnetic field is
applied, in which case the theory predicts a negative magnetoresistance. This is a
striking prediction because, in classical transport, a magnetic field pushes electrons
transverse to the current direction and increases the resistance, resulting in a positive
magnetoresistance. The classical magnetoresistance can be observed in relatively
high fields. In weak-localization, the presence of a magnetic field perpendicular to a
metallic film gives each Feynman path an additional phase factor %I A ~dl , where the
line integral is along the trajectory. Since the path and its conjugate will pick up
opposite phases in a magnetic field, the applied magnetic field destroys the coherent
backseattering. Therefore, the resistivity will decrease as the magnetic field

increases, leading to a negative magnetoresistance.

D. Universal Conductance Fluctuations[5]

An interesting question is what happens when we consider interference between all
the paths. In small structures, mesoscopic systems, with dimensions on the order of
the phase coherence length, the transmission coefficient has a sample-specific value
that depends on the detailed location of the scattering centers. The transmission
depends sensitively on perturbations such as rearrangement of scatterers, magnetic
field and chemical potential, leading to chaotic fluctuations of conductance. This is
the phenomenum called universal conductance fluctuations. In contrast to weak
localization, turiversal conductance fluctuations (U CF) average to zero in large

samples (the ensemble average).

D1. Universality of fluctuation amplitude.
The amplitude of UCF has a remarkable universality: the conductance of a metallic

sample fluctuates by order eZ/h (z 4E-5 mho), independent of degree of disorder and

5
sample size (L) as long as L < L¢.[6] This universality is a key result in mesoscopic
physics.

The universality of conductance fluctuations can be understood following several
approaches. One approach is based on the microscopic theory using the impurity-
averaged Green’s function technique.[7,13,14] An alternative approach is built on the
basis of Landauer’s definition of conductance -- the conductance is related to the
eigenvalues of the transmission matrix.[8] The eigenvalues obey the level repulsion
property of the appropriate random matrix ensemble introduced by Wigner and
Dyson: Gaussian orthogonal ensemble (GOE) for a system with time-reversal and
spin symmetry, Gaussian unitary ensemble (GUE) when time-reversal symmetry is
broken, and Gaussian sympletic ensemble (GSE) in the limit of strong spin-orbit
scattering.[9] This random matrix approach provides the more fundamental
understanding of the universal conductance fluctuations and gives a simple way to

predict the relative amplitude of the UCF. The theoretical result is :

2
2.1 i ’“_2 -
(5G) 4(h] B (13)

where k is the number of independent eigenvalue sequences, 8 is the eigenvalue
degeneracy, and B =1,2, or 4 for the GOE, GUE or GSE, respectively.

Experimental establishment of the different ensembles can be shown with
application of magnetic field and spin-orbit scattering. Application of a weak
magnetic field breaks the time-reversal symmetry of the electron orbital motion and a
strong magnetic field lifts the electron spin symmetry due to Zeeman splitting.
Strong spin-orbit scattering breaks the spin-rotation symmetry. The relative
amplitude of fluctuations in various regimes determined by an applied magnetic field

and spin-orbit scattering are summarized in Table 1-1. The characteristic field scales

6

involved in these symmetry breakings are denoted as BC] and Bcz- (The field strength

for Beland B02 will be discussed later)

D2. UCF noise reduction induced by magnetic field.

Experimental verification of the UCF amplitude given by Eq. (1-3) can be
achieved by measuring and comparing (60)2 in the different random matrix
ensembles determined by the magnetic field. Measurement of (6G)2 in a single
sample requires that we vary some parameter to obtain different values of G in the
ensemble. The usual method of measuring the static variation of G versus magnetic
field (the “magnetofingerprint”) is inadequate because we like to obtain (86)2 at fixed
values of magnetic field. Fortunately, in disordered metals, impurities and scattering
centers rearrange themselves spontaneously even at low temperature due to tunneling.
The rearrangement of impurities gives rise to 1/f noise in the electrical resistance.[10]
The noise measurement provides a tool to study (8C7)2 with excellent statistics at fixed
magnetic field; the accuracy of the measurement increases with the measurement
time. A further discussion of l/f noise is given in the appendix.

Experiments pursuing the ratio of the UCF amplitude between the random matrix
ensembles have been reported for several special cases. For the strong spin-orbit
scattering case, Birge et al. studied the l/f noise as a firnction of magnetic field in 2D
films of Bi and clearly observed that UCF noise is reduced to half of the zero field
value with a characteristic field scale determined by one-flux quantum(h/e) through a
phase coherent area.[1 1] This reduction corresponds to the crossover from symplectic
to unitary ensemble in the quantum transport with simultaneously breaking of the
Kramers degeneracy. Measurements in the weak spin-orbit scattering limit, however,
have been less clear. Debray etal measured time-independent conductance
fluctuations in a MOSFET as a function of gate voltage at several values of magnetic
field at 1.3 K.[12] A reduction of the variance, (SOP, by a factor of 4 was observed

7

at ~0.7 kG. The authors attributed this reduction to the Zeeman effect with the
characteristic field scale determined by the Thouless energy, EC = hD/ L¢2. It was
not clear, however, whether Bcz is really determined by the Thouless energy because
the value of the g-factor was not known in their MOSFET. In addition, theoretical
arguments about the energy scale for the Zeeman crossover on UCF have been
contradictory.[l 3,14] A second limitation of the MOSF ET experiment is that the low
field data were in the limit of either GOE or GUE «only one data point near the first
reduction is shown and there were no data showing the crossover between those
ensembles.

We studied universal conductance fluctuations in the low spin-orbit limit by
measuring the l/f resistance noise in a quasi-1D lithium wire fabricated by quench-
condensation at 4.2 K. From the magnetoresistance, we confirm the low spin-
dependent (spin-orbit or spin-flip) scattering in out Li wire; the upper bound of spin-
orbit scattering rate is estimated to be 5~10 times smaller than the phase breaking
rate. At temperatures of 1.6 and 4.2 K, we observe that the noise is reduced twice as a
function of magnetic field. The first drop occurs below 0.01 T and the second
reduction occurs above 1 T. We calculate the theoretical crossover function for both
drops and compare with the experimental data. The experimental data are fully
consistent with the theory for the complete crossover function both for the GOE to
GUE transition[15] and for the splitting of the Zeeman degeneracy[l4]. Our results
show that the characteristic field scale for the Zeeman crossover is determined by the
sample temperature rather than the pre-assumed Thouless energy for the case kBT >>
EC.

References

[1] For a reviews of this subject, see P. A. Lee and T. V. Rarnakrishnan, Rev. Mod.
Phys. 57, 287 (1985).

[2] R. Landauer, Phil. Mag. 21, 863 (1970).

[3] D. S. Fisher and P. A. Lee, Phys. Rev. B23, 6851 (1981).

[4] see G. Bergmann, Phys. Rep. 107, l (1984).

[5] Mesoscopic Phenomena in solids, edited by B. L. Al’tshuler, P. A. Lee, and R. A.
Webb (North-Holland, New York, 1991).

[6] B. L. Al’tshuler, Pis’ma Zh. Eksp. Teor. Fiz. 41, 530 (1985) [JEPT Lett. 41, 648
(1985)] ; P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55,1622 (1985).

[7] P. A. Lee, A. D. Stone, and H. Fukuyama, Phys. Rev. B 35, 1039 (1987); S. Feng,
Phys. Rev. B 39, 8722 (1989).

[8] Y. Imry, Europhys. Lett. 1, 249 (1986); B. L. Al’tshuler and B. I. Shklovskii, Zh.
Eksp. Teor. Fiz. 91, 220 (1986) [Sov. Phys. - JETP 64, 127 (1986)]; P. A. Mello,
Phys. Rev. Lett. 60,1089 (1988) ; K. A. Muttalib, J.-L. Pichard, and A. D. Stone,
Phys. Rev. Lett. 59, 2475 (1987).

[9] For a review of random matrix theory, see T. A. Brody et al., Rev. Mod. Phys. 53,
385(1981)

[10] S. Feng, P. A. Lee, A. D. Stone, Phys. Rev. Lett. 56, 1960 (1986).

[11]N. O. Birge, B. Golding, and W. H. Haemmerle, Phys. Rev. Lett. 62, 195 (1989).

[12] P. Debray, J.-L. Pichard, J. Vicente, P. N. Tung, Phys. Rev. Lett. 63, 2264
(1989).

[13] A. D. Stone, Phys. Rev. B 39, 10736 (1989).

[14] S. Feng, Phys. Rev. B 39, 8722 (1989).

[15] The quasi-1D noise crossover function between GOE and GUE is derived from
the field correlation, <G(B)G(B+AB)>.

Table 1.1: g is defined as G/(eZ/h).

 

 

 

 

 

 

 

 

13<<13cl 13c1<<13<<13c_2 13>>13c_2
weak spin-orbit (8g)2 5 1 (6g)2 5 1/2 (6g)2 2 1/4
limit 3:1, k=1, s=2 13:2, k=1, s=2 [3:2, k=2,s=1
strong spin—orbit (25g)2 5 1/4 (fig)2 5 1/8 (fig)2 5 1/8
limit [3:4, k=1, s=2 (i=2, k=1, s=l 13:2, k=1, s=l

 

 

10

Chapter 2

THEORY OF QUANTUM TRANSPORT

This chapter describes the theoretical background necessary to understand our
experiments. In the first half, we explain the important concept of diffusive quantum
transport in disordered metals, and its connection to random matrix theory. In the
second half, we summarize the previous quantitative calculations of the amplitude of
UCF and the UCF crossover functions between random matrix ensembles. In
addition, we present our quantitative formulation of the UCF crossover and its

evaluation, which is used in the data analysis.

2A. Diffusive quantum transport in metals

In disordered metals, at sufficiently low temperature, the electrons diffuse
coherently through the disordered medium over a distance called the phase breaking
length, L4,. One can think of the electron motion as a random walk where the step
distance is the elastic mean free path. When the sample size L is much longer than

elastic mean free path, we call the transport "diffirsive".

2B. Landauer's conductancefl]

Landauer considered the electrical conduction of a sample connected to two ideal
probes, and showed that the conductance G is related to the transmission probability
of electrons through the sample. Consider a coherent box of length less than L4, and
of transverse dimensions W. The number of quantized transverse momentum states

within the probes is NC as (Wk,.~)d'l in dimension d.

11

__A

ar

 

The a. is the incident flux from the left and the a, is the transmitted flux through the
scattering medium. The transmission matrix t of the scattering medium can be
defined as a, = tal where t is a N0 x Nc matrix. Denoting the transmission amplitude
between the channels i and j by tij in a transmission matrix t, the conductance G is

given by

2

Ne
G = 2(e2/h)Tr (tt*) = 2(e2/h) 2|“, (2-1)
131'

where the factor of 2 is due to electron spin degeneracy.

2C. Universal Conductance Fluctuations

The quantum interference among diffusive paths gives rise to extreme sensitivity
of the transmission probability to the detailed configuration of the microscopic
scattering centers, leading to sample-specific transmission values. The resulting
fluctuations of the conductance is a phenomenon called "universal conductance
fluctuations" (U CF), where the conductance fluctuates as a function of a control

parameter by a universal amplitude of order e2/h (z 4E-5 mho), independent of

disorder and sample size (L) as long as L < L¢
50 z eZ/h (22)
So, UCF is most evident in a small sample or mesoscopic system with size

comparable to the phase breaking length and this remarkable universality stands out

as the key result in the mesoscopic physics.

12

The universality of conductance fluctuations can be understood following several
approaches. The original approach[2], by Al’tshuler and by Lee and Stone, is based
on the microscopic theory using impurity-averaged Green’s function technique.
Alternative approaches are built on the statistical properties of the eigenvalues of
either the random Hamiltonian[3] or random scattering matrix[4]. The second
approaches provide the more intuitive understanding of the universal conductance

fluctuations.
2C.l Random matrix approach

. Random matrix theory by Wigner and Dyson[5,6]

Wigner and Dyson introduced the universal random matrix ensembles to describe
the statistics of nuclear energy levels. The Gaussian orthogonal ensemble describes
systems with time-reversal symmetry and spin symmetry, the Gaussian unitary
ensemble is appropriate when time-reversal symmetry is broken, while the Gaussian
sympletic ensemble describes the case of broken spin-symmetry. Wigner and Dyson
found that the level statistics depend on the symmetries of the ensemble, independent

of the microscopic details of the system.

0 Random matrix theory in UCF
Consider the conductance of a metal which is connected to two ideal leads of size
W in dimension d. The sample length is L. The dimensionless conductance per spin

channel, g=G/(e2/h), is given by the transmission matrix of the conductor as

mentioned before, Eq. (2-1),

2

”C
G = (eZ/h)Tr (w) = (eZ/h) thij
i, j

13

where, i and j are channel indices.
It is interesting to see how Landauer’s approach can be connected to random
matrix theory[4]. Consider a scattering medium with incident flux {1], 1,} and

corresponding transmitted flux {0], Or}.

 

The reflection and transmission matrices r and t are NO x NC matrices, where NC is the
number of propagation channels at the Fermi energy. The scattering property of a
sample or scattering medium is described by either the scattering matrix S or the

transfer matrix T as following

2:212] , 12.2121

Following simple algebra with r and t, the conductance is given by

2

g=Tr(tt*)=Tr( . . _1 )
TT +(T'I‘) +21

 

(2-4)

This trace of the transfer matrix can be rewritten in terms of the eigenvalues of a

random matrix in the following way:

N l

g=i

i 1+)»;

 

(2-5)

14

where M are the eigenvalues ofthe matrix, X = [(TT. + (TT' )"'I — 21] / 4.

Eq. (2-5) implies that the conductance is a "linear statistic" of eigenvalues M (The
word “linear statistic” means that the conductance does not contain products of the
eigenvalues, but the fimctional dependence can be non-linear). The distribution of the

eigenvalues is given by[9],

Pm.» zexpi-BHtimn,
Hm.» = — 2 1:42. -kj|+ZV(ki) <2-6)

i<j

where, the number [3 is different for each of the three ensembles of random matrix ; [3
is equal to 1,2 and 4 for the Gaussian orthogonal, Gaussian unitary and Gaussian
sympletic ensembles respectively. The probability distribution has the form of a
Gibbs distribution, with the symmetry parameter B playing the role of the
temperature, and the H containing the logarithmic repulsive interaction between
eigenvalues in addition to a “potential” V. Since the conductance is a linear statistic

of the eigenvalues, its variance is given by[lO]

C!) co

. 1 1 .

Var(g)= [JAM—fleas) (2-7)
0 0 1+7¥1+7t

Here, the two-point correlation function K2().,}.') is defined by

K2 (2.2) = <p<x>p(>»')>- <p<>~>><p<x )> <2-8)

where p(}.)=28(}. —}.,,) is the eigenvalue density. As long as the correlation

n
function K20, N) is known , the variance of the linear statistic can be calculated in a

15

straightforward manner. For the distribution of the eigenvalues in Eq. (2-6), the mean

density of transmission eigenvalues <p().)> is

[(9% ”liar/90:11"°7~N)CXP(-BH)

 

 

(9(1)) = (2-9)
[6141"]de eXIX-13H)
Differentiation of Eq. (2-9) with respect to V0.) yields
502(k)) _ _ . . _
8V(2.') - 13030090» ))+B(P(7»)><p(7L )) (2 10)

Eq. (2-8) and (2-10) give K20., N) = Eli-w. Then, Eq. (2-7) yields Var(g)
oc 1/B, if there is a linear relationship between p0.) and the "potential" V. This
relation has been addressed by Dyson[l 1] and found to be linear as a consequence of
the distribution function Eq. (2-6). Therefore, the variance is independent of the
microscopic details of system and has a universal dependence on the symmetry
constraint [3. The full expression, i.e. Eq. (1-3), of conductance fluctuations probably
can be derived by taking the degeneracy of eigenvalue into account, although no

calculations with the degeneracy have been reported following the random scattering

matrix approach.

2C.2 Random Hamiltonian approach.

Even though the conductance should be viewed in terms of quantum diffusion, it
is heuristically interesting to consider the approach following the Thouless
argument[12]. In a hypercube of size L, the electronic eigenstates are determined by
the microscopic Hamiltonian. From the uncertainty principle, such eigenstates are

associated with a width, E0 = n / t = hD/ L2 where ‘t is the time to diffuse through

16

the sample. Assuming there is an average energy level spacing AE, the conductance

G is given by

G=—-—=—h—-N(EC) (2-11)

N(Ec) is the number of energy levels within an energy range EC. If the energy levels

are randomly distributed (Poisson distribution), one would expect fluctuations 8N z
2
s/N which results in 8G z%~,/N(Ec) , which is the wrong answer. The

amplitude of conductance fluctuations from this point of view are much stronger than
the real fluctuations in Eq. (2-2). The fact that the level number fluctuations for a
given energy hand must be small compared to J7),— indicates a repulsion between the

energy levels. Dyson[6] showed that in a closed system the level repulsion gives

k-s2

 

<[5N(E)]2>~i- 1n<N<E>> (242)

where, B is equal to 1,2 and 4 for the Gaussian orthogonal, Gaussian unitary and
Gaussian sympletic ensembles respectively. The quantity k is equal to the number of
the non-interacting series of levels; indeed, levels with different precisely defined
quantum numbers do not interact with one-another. The quantity s is the degeneracy
factor. Thus, the amplitude of fluctuations depends on BS, and k. Later, Al’tshuler

and Shklovskii[3] showed that in a system connected to the outside with ideal probes,

k-s2

B

 

<16N<E>12>~ (2-13)

filv-i

which is consistent with Bq. (2-2) and directly gives Eq. (1-3).

l7

2.C.3 Diagrammatic approach

The microscopic theory of UCF has been formulated on the basis of impurity-
averaged Green function techniques with kple as a perturbation parameter. This
complicated calculation provides a quantitative description of UCF including the
UCF crossover fimctions between the different random matrix ensembles. Also, the
effect of finite temperature can be handled quantitatively.

Within the diagrammatic calculation, the amplitude of UCF stems fiom two
channels which contribute equally at zero magnetic field: the cooper (particle-
particle) and diffuson (particle-hole) channel. Quantitative understanding of the UCF
crossover function requires the proper handling of such channels with respect to
magnetic field and spin-orbit scattering. There are several calculations reported. Lee,
Stone and Fukuyama[8] first pointed out some of the UCF reduction factors, and
Feng[l3] calculated the UCF reductions in the presence of spin-orbit scattering and
the Zeeman effect. Stone[l4] presented the calculation on the UCF noise crossover.
Chandrasekhar et.al.[15] discussed the effect of spin-dependent scattering on UCF.

A simple way to look at various UCF reduction factors is to describe the
amplitude of UCF in terms of the spin variables of the channels. The spin variables
are total spin J and its projection M2, of electron and hole for the diffuson channel,
and of electron and electron for the cooper channel. We can re-write each channel in
terms of spin-singlet (J=O, M=O) and spin-triplet (J=l, M=il,0) terms. The

conductance variance is given by:

(66)2 = gemswf +§(SG.(B.L..))2],.,. fleece? +%<66.(guBB.L..»21ph

(2-14)

18

where G, and G, stand for the conductance of singlet and triplet part respectively.
Also, the pp and ph mean the particle-particle and the particle-hole channel,
respectively. Strong spin-orbit scattering (LSD—)0) suppresses the triplet contribution
in both channels, leading to a factor of 4 reduction in UCF. The effect of magnetic
field comes from an orbital effect in the cooperon channel and spin effect in the
diffuson channel. An application of weak magnetic field suppresses the B-dependent
cooper channel (both singlet and triplet), and reduces UCF by factor of 2, whether the
spin-orbit scattering is strong or not. The effect of Zeeman splitting can be observed
due to the suppression of the Mz=ztl triplet parts in the diffuson channel only if the

spin-orbit scattering is weak.

2.C.4 The Amplitude of UCF.

The theoretical approaches have shown that the amplitude of UCF depends on the
statistical properties of the random matrix ensembles which can be varied
experimentally by application of magnetic field and spin-orbit scattering. The

predicted reduction factors are summarized in Eq. (1-3) as:

lit-32
4 [3

 

(22>2 e

and are summarized in Table 1-1. The B represents the random matrix ensemble
characterized by time-reversal and spin symmetry. The degeneracy s stems from the
electron spin symmetry in the weak spin-orbit scattering regime, and from the
Kramers degeneracy in the strong spin-orbit scattering regime. The k is the number
of statistically independent channels. The characteristic field scales involved in the

crossover between different regimes are denoted as BC] and Bez-

19

Here we repeat the Table 1-1 for convenience.

Table 1-1: g is defined as G/(e2/h).

 

 

 

 

B<<Bc1 Bcl<<B<<ng B>>B22
weak spin-orbit (6g)2 5 1 (6g)2 2 1/2 (5g)2 5 1/4
limit B=1, k=1, s=2 [3=2, k=l, s=2 [3:2, k=2, s=1
strong spin-orbit (6g)2 5 1/4 (8g)2 5 1/8 (6g)2 5 1/8
limit B=4, k=l, s=2 13:2, k=1, s=1 [3:2, k--1, s=l

 

 

 

 

 

20

2D. Quantitative Analysis of UCF
First, the mathematical definition of important quantities related to UCF (for
example, field correlation, variance, and noise) are introduced. Since our
experiment focused on the relative amplitude of UCF as a function of the external
magnetic field, the theoretical derivation and evaluation of the UCF crossover

functions is presented.

2D.] Field correlation, Variance and UCF noise

0 Field correlation and Variance[8]

UCF theory is based on the ergodic hypothesis -- ensemble averaging of the
conductance is equivalent to averaging the conductance of a single sample over
magnetic field, or over the Fermi energy in a semiconductor. Changing the Fermi
energy or magnetic field randomizes the phases of electrons, leading to chaotic
variation of the conductance. If one changes the field or Fermi energy enough so that
the phases of electrons are totally uncorrelated with those at the original value of the
field or energy, then the sample acts as a new sample in the measurement of
conductance. The correlation function of conductance with respect to the variation of

the field or the energy, F (AE,AB), can be defined by,

F(AE.AB.B)=<5g(EFiB)5g(EF +AE.B+AB)> (2-15)

where 6g: g(Ep,B)-(g(E,.-,B)). The angular bracket () means an ensemble

average. The variance of conductance is given by,

Var<g> = «62)2 > = «g— (M) (2-16)

21

The correlation function with AB = AB = 0 gives the variance.

At finite temperature, conductance fluctuations are reduced by "classical
averaging" once the temperature is high enough that the phase breaking length is
shorter than the sample length. Also, thermal smearing of the Fermi surface reduces
the amplitude of conductance fluctuations due to "thermal averaging" in which the

length scale is the thermal diffusion length LT = {2%. The formula for the field

B

correlation F T at finite T is given by

Fr(Au.AB,T) = 161151 ldEzf'(Et .u)f'(E2 .11 + Art)

(2-17)
x (58(ElaB)58(EZaB+AB))

= [dAE K(AE,Au)Fo(|AE

 

’AB),

df

where f '= 31::- , f is the Fermi distribution function, F 0 is the T =0 correlation

function defined by Eq. (2-15) and K(AE,Au) is the convolution integral
KMEtAH):idErf'(Eritl)f'(E1- AEttl + All)

The variance of g is

Var[g(B.7)] = —K(——)F0(AE.B) (2-18)

£1 dAE AE
“2 21:37 ZkBT

where K (x) = (x cothx — l) / sinh2 (x) and s is the spin degeneracy.

o UCF Noise

22

Since UCF arises from the interference of the electron waves scattered from
the defects or impurities, a change of the random scattering potential yields
fluctuations in conductance (8g')2 or noise. (A brief discussion of the noise spectrum
is given in the appendix.) Denoting the random impurity potentials of many
impurities by V(r)and V(r'), the noise at magnetic field B and temperature T is given
by:

(sew. 7312 = <1g(B.T.V) - g(B. T.V')12>

= 2{Var[g(B. 7)] - (58(3. T.V)58(B. TiV' )>} (2-19)

The sensitivity of UCF to a change of the random scattering potentials was
calculated by Feng, Lee, and Stone[16] and by Al'tshuler and Spivak[17]. The
conductance change, 5g], of a coherent volume due to a single impurity movement by
a distance Sr is:

1

(581 )2 ~ w(%)d'2a(krfir) (2-20)
F e

where a(k,:6r) is the phase shift that the electron experiences due to a single impurity
motion, and a(kF6r) approaches unity for 8r >> kp". Surprisingly, a single impurity
can cause saturation of the conductance fluctuations in the case of kp'e =1, so the

UCF noise shows extreme sensitivity to the motion of defects.
2D.2 Crossover function in particle-particle channel.

In this part, the derivation and evaluation of the UCF noise crossover

functions in a quasi-1D sample will be discussed since we measured the l/f noise in

23

quasi-1D samples. (See Chapter 4 & 5.) Stone[l4] calculated the l/f noise crossover
function in 2D. To get the 1D crossover function, one approach is to modify Stone’s
2D calculation. Instead, we used an alternative route in which the 1/f noise crossover
function can be obtained starting with an analytical expression for the field

autocorrelation of conductance, F (AB) = (g( B) g(B + AB)) , given by Beenackker

and von Houton[18].
2D.2.1 Method with field correlation function

0 overview

The amplitude of UCF stems from two channels which contribute equally:
particle-particle and particle-hole channel. Lee etal [8] showed that the magnetic
field enters the UCF calculation by means of the semi-classical approximation for the

Green function, G(r,r'):

. r
G(r,r', B) = exp[31‘ijA-d1]a(r,r')
h/er

Since the product of G(r,r')G(r',r) enters in the particle-hole channel, only AA appears
in the diffusion equation and the correlation function F ’" (AB) is determined by the

eigenvalues and eigenvectors of the diffusion equation[8]:

‘t (-iAE + D(—iV — eAA)’ +t ," )Q," (r) = Ma, (r) (221)

Whereas for the particle-particle channel, (AA+2A) appears in the diffusion equation
which determines F P” (B, AB). The diffusion equation for the particle-particle

channel is:

24
1:(-iAE + D(-iV — e(2A + AA»2 +1: ,4 )Q, (r) = Mpg, (r) (222)

The magnetic field correlation firnction is given by the sum of the two contributions,

which are equal at B=O:
F(B, AB) = FP”(AB) + F””(B, AB) (2-23)

The B-dependence of F comes from the particle-particle channel. (Here, we don't
consider the Zeeman effect.) When there is magnetic flux larger than h/e (one flux
quantum) in a coherent area of the sample, then EDP —-) O and F(B,AB) z FP”(AB), i.e.
the variance drops by exactly a factor of two, i.e. F(B >> Bel, O) z E’h(0) = 1/2
F (0,0).

Stone[l4] pointed out that the crossover function for the variance of g is
identical to the large field conductance correlation function F ”" (AB) if one makes
the substitution eAB —) 2eB. This can be checked from the diffusion equations Eq.
(2-21) and (2-22). The cooperon contribution to the UCF variance at fixed field can
be obtained from the diffusion Eq. (2-22) with AA = 0. Compared to Eq.(2-21), the

only difference in the diffusion equation is the substitution of AA into 2A. Then
Var” (g: B) = F”"(AB —) 2B) (2-24)

Al'tshuler and Spivak[l6] showed that for small variations in the impurity potential V,

and V', we get the noise fimction in the following way:

. 2 ~_ . d -
[5g (B,T)] ~ 2y d(l/r¢)Var[g(B,T)] (2 25)

25

where, 1': [1—((VV')/(V)2)]/te,. This is the case ofunsaturated noise, i.e. 8g' <<

1. The noise crossover function in the cooper channel is simply,
vPPtB,T)=15g'(B.T)12 /16g'<B=o.T)12. <2-26)

0 ID field correlation function
An analytical formula for the field correlation function for conductance in the

diffusive regime was calculated by Beenakkar and von Houten[17], in the case of a
quasi-1D geometry (W << L¢ << L) where L and W are the length and width. The

field correlation function is given by

e2

_ 2i
h

—1
2-27
2“ L12) ( )

L 3
F(AB) e 6( )2 33—0 +
L

Here, the effect of magnetic field is corporated into L,» = JD”, with D = vfle / 3
in the following way:
1/1¢(AB) = l/T¢(O)+l/TAB

1 nWAB
1/ =—D
TAB 3 x( h/e

 

)2 (2-28)

L;- is the thermal length defined as LT = HAD? . From Eq. (2-24),
B

26
VarPP(g:B) = FP"(AB —+ 23)

 

 

2 L 3(8) 9 L 2(2) (2-29)
z 6637? —¢——3——(1 + —Ji——2—)‘l
L 2n LT
where
L¢‘2(B)= 1 l "W3 2 (2-30)

2 +
L,» (o) 3 h/2e

(Note the factor of 2 compared to Eq. (2-28).)

0 1D llf noise crossover function
The l/f noise crossover function, V” (B), in the cooper channel is calculated

from Eq. (2-25) and (2-29):

sg'zw T)~ L. (B) (1 9 L" —(—,))'
2n 1.

2 T 2 (2-31)
11-3“ (B>(1+_9_L. (B)

n L72 2n LT2

 

)“1

where the effect of magnetic field is incorporated into L¢(B) using Eq. (2-30). The

total noise is given by Cooperon and diffuson contributions that are equal at zero

field, so the crossover fimction for the relative noise power at low magnetic field is:

6g'2 (B)

v(B)=-—(1+2
58' (0)

—) (2-32)

Figure 2.1 shows the evaluation of the ID field correlation, variance and l/f noise

crossover function. Here, we use 1 pm for L4,, 0.17pm for L7- and 0.45 um for the

27

sample width W. (These values are appropriate for the sample in Chapter 4 at T = 1.6
K.)

2D.2.2 Method by Stone[l4]
Stone showed that the T =0 energy correlation function F 0(AE,B) is given by

1 2 +1Re(-i2-) (2-33)

Fo(AE.B) = Z 2 pp
la

‘1 Iii”!

 

where, xppa are the eigenvalues of the differential equation,

(—iAE / 21+ D(—iV —2eA)2 ”(1)9, (r) = 21:59, (r) (234)
The variance of conductance is given by Eq. (2-18) and the 1/f noise can be obtained
from Eq. (2-25). The noise (8g')2 of the cooper channel can be written as following,

4:2 dAE AE d
5 '2 B, =—— —K F AE,B 2-35

 

Substitution of F 0 yields numerical formula for the noise crossover. The difference
between the analytical and numerical expressions for the field correlation is less than
10% [18]. We expect that the difference in the noise crossover function is similarly

insignificant.

2.D.2.3 Characteristic field scale

From a measurement of the static variation of G versus magnetic field (the
"magnetofingerprint"), we get the magnetic field correlation range Bc where F (AB)
drops to half of its fully correlated value(i.e. F(AB=O)). Bc represents the typical

28

spacing of the peaks and valleys in 80(3) of the magnetofingerprint and it is
shown[8,l8] that the field correlation range of 80(3) is determined by the phase
breaking length L4, in a quasi 1D sample (thickness t, width W << L¢). The value of

Bc is given by

 

Bc = const x

Beenakker et. a1 [18] calculated the numerical constant and showed that is equal to
0.95 for L¢>>LT and 0.42 for L¢<<LT.

In the 1/f noise measurement the field scale Ed is defined as the magnetic
field where the 1/f noise power drops to 3/4 of that of zero field. BC] is the field scale

required to suppress the particle-particle channel contribution to the 1/f noise, and is

also determined by the phase breaking length L4,. In quasi-1D system[8,l3,14] Bel is

given by

BC] = const fle—
W

where the numerical constant is evaluated as 0.21 for L¢>>LT and 0.16 for L¢<<LT

from Eq. (2-31) and (2—32).
2.D.3 UCF crossover function at strong magnetic field

0 Overview
At strong magnetic field where the cooper channel is completely suppressed,
the magnetic field couples to the diffuson channel due to the Zeeman splitting of the

electron spin state in the weak spin-orbit scattering limit. Because the Mz=i1 triplet

29

states are sensitive to the Zeeman splitting, there will be a second reduction of UCF
from suppression of the contribution of the diffusons with Mz=:t:1. The Zeeman
effect can be incorporated into UCF theory following the work of Stone[l4] or
Feng's[1 3] calculations. According to Stone, the Zeeman effect can be incorporated
by replacing AE by AE+guBB in Eq. (2-33). Stone claimed that for the 1/f noise in all
dimensions and for Var(g) in quasi-1D samples, the characteristic field scale(Bc2) is
determined by the energy correlation length ("Thouless energy") of a phase coherent
area, hD/ L2 , not by the temperature. For Var(g) in 2D- or 3D- samples, Bc2 given
by Bc2=Emax/gu3, where Emax is the larger of {k B [11}. Independently, Feng
calculated the Zeeman effect on UCF, and showed that the values of 3,2 are
determined by the temperature in all situations when LT < L4, < L. We have checked
that F eng's formula in 1D is the same as the 1D formula based on Stone's approach.
We also evaluated the noise crossover function in 2D, following Stone's calculation
(see figure 2.5). We found that Bcz is determined by the larger of {k BR;- } in all
sample dimensions, and our current understanding of the Bcz is consistent with Feng's
predictions. Thus Stone's statement about the Zeeman crossover of the noise is
incorrect, although his calculations are correct. Here, we describe the calculations on
the Zeeman effect on the UCF variance, and present our numerical evaluation of the

Zeeman effect on the 1/f noise.

a 1D variance calculation
F eng has treated the Zeeman effect on UCF diagrammatically by taking the

Zeeman energy into account in the spin-dependent electronic energy

Sky = fizk2 /2m +v§u3H— E].- where v = i1 is the spin index. The complex

calculation of the spin-dependent diffuson diagram in the particle-hole channel is

carried out using a Dyson equation. Here, we skip the details of the algebra but

30

explain the outline to get the variance of conductance in the particle-hole channel.

From the algebraic manipulations, the particle-hole diffusion propagators Aapys are

obtained explicitly. The T =0 energy correlation function can be given, following Lee
et.al.[8] and Al'tshuler and Shklovskii[3], by

2
Mot/115,3). ):[2 2 lAemstqpi,1/rt.guhB)| + 2 Remaeysf] (2-35)
qph (11375 (113%

where, q ph = (rtm/ L)2 , m = 1,2,- - -oo. By substituting the various expressions into

this equation, the T=0 energy correlation function is

 

 

2 4 2 2
FP"0(AE, B) = 2(1)2 L‘4 2{ + +
h .4 £122 .4 AErgflsBz .4 AE+8flsBz
qphq +(D) q +( D ) q +( D )
AB 32 AE B
2(q.4_(_A_E_)2) ta" -(———-—g"” )) (q'4 —(—————+g“” )2
+ 4 Al132 4 M 811332 + 4 AE+2H332} (2-36)
q 42(3) q +(-—-l—)—) q +(———5——)

where q'2 = q: h + ‘12-- The conductance fluctuations at T=0 are given by
11>

F pho (AE = O, B). One can recover the formula of the T=O conductance fluctuations
given by Feng's published result[13] and by the Lee and Stone result[2] for the case of

B=0. From Eq. (2-18), the conductance fluctuations at finite temperature are

Ph __ ph
Var 1g<B n1~ [ST—ABETquBTW (AB B)

A typical term contributing to the conductance fluctuations with the Zeeman effect is,

31

 

w l
Var[g(B. 7)] cc 2 [mm ' <2-37)

.2. 0 11m2 +7)2 +(ni9)2]

2
wherev=<i)2. «FAB/2.. win-22, €=8HBB/(kBT), t="BT.
nL¢ L 8

C

 

Figure 2.3a shows the evaluation of the finite temperature variance in the particle-

hole channel.

0 1D l/f noise crossover function
From Eq. (2-25), the l/f noise amplitude in the particle-hole channel can be

calculated in a straightforward manner. A typical contribution is

 

”’ d 1
5ng (BJ) oc dn Km) {-— 1
.21 0111th +2)2 +(nis)2]

(2-3 8)
Figure 2.3b shows the evaluation of the noise crossover function in the particle-hole

channel.

0 Characteristic field scale

The field dependence of the conductance fluctuations due to the Zeeman effect
in the particle-hole channel can be characterized by the field scale Bc2 where the
variance or the l/f noise drops to the half of the value at zero field. From figure 2.33
for variance and 2.3b for the noise, we conclude that for the case k 3T >> h_D2 in
quasi-1D at finite temperature, Bc2 is given by L6)

kaT
ng

aggzj

32

in all dimensions. Bc2 is the same for measurement of the variance or the 1/f noise.

When kBT << Z—D , B62 is determined by the Thouless energy. Figure 2.4 shows the
4’

evaluation of the UCF noise at different temperatures with a fixed Thouless energy,

and implies that Bc2 is determined by the larger of {k B T, g}.

42

0 Effect of spin-orbit scattering

The calculation of the Zeeman effect shown so far has been limited to the
weak spin-orbit scattering limit. Strong spin-orbit scattering breaks the spin-rotation
symmetry and the Zeeman effect can not be observed. This interplay between the
Zeeman effect and the spin-orbit scattering is demonstrated in figure 2.5. Figure 2.5
shows the relative amplitude of the variance of the particle-hole channel for several

values of the spin-orbit scattering rate. It is shown that for L4, z 3Lso, we are already

in the strong spin-orbit scattering limit.

2E. Weak localization and low field magnetoresistance

The usefulness of weak localization(WL) is that it allows one to extract the
electron transport length scales from the low-field magnetoresistance. One can get
then information about the phase breaking mechanism from the temperature
dependence of the phase breaking rate. Here, we describe briefly the quasi-1D WL
application to the magnetoresistance. A comprehensive overview of WL is given by
the review of Bergrnann (1984)[19].

The 1D WL correction to the resistance can be written in the terms of the spin-

triplet and spin-singlet[20],

33

R(0)-R(B)_ e_2_ R {2[ -2 -2 2 —2 —2-
—— +-— :L +— L +L

_ _ _ —1/2
—--2—[L¢ 2 +2Ls 2 +3LB 2lsinglet}

where the effect of a magnetic field is expressed in terms of a one-dimensional

f},

"magnetic length", L (---—— — ———). L is the spin-orbit scattering length and L
3" 2e BW so sf

is the spin-flip scattering length assuming the spin is static. This formula is valid in

the region of L¢(T) > W and W < L B- In one-dimensional WL, the resistance

correction is sensitive to W, giving an estimation of W of the 1D wire. In the case of

L << L , the tri let term is su ressed, resultin in a sitive ma etoresistance.
so '1 P PP g P0 gn

Strong spin-flip scattering suppresses the WL contribution completely.

34

References

[1] R. Landauer, Phil. Mag. 21,863 (1970).

[2] B. L. Al’tshuler, Pis’ma Zh. Eksp. Teor. Fiz. 41, 530 (1985) [JEPT Lett. 41, 648
(1985); P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55,1622 (1985).

[3] B. L. Al’tshuler and B. I. Shklovskii, Zh. Eksp. Teor. Fiz. 91, 220 (1986) [Sov.
Phys. - JETP 64, 127 (1986)].

[4] Y. Imry, Europhys. Lett. 1, 249 (1986).

[5] For a review of random matrix theory, see T. A. Brody et al., Rev. Mod. Phys. 53,
385 (1981).

[6] F. J. Dyson, J. Math. Phys. 3, 140 (1962); 3, 166 (1962); F. J. Dyson and M. L.
Metha, ibid. 4, 701 (1963).

[7] D. S. Fisher and P. A. Lee, Phys. Rev. 323. 6851 (1981).

[8] P. A. Lee, A. D. Stone, and H. Fukuyama, Phys. Rev. B 35, 1039 (1987).

[9] R. Balian, Nuovo Cimento 57, 183 (1968); K. A. Muttalib, J .-L. Pichard, and A.
D. Stone, Phys. Rev. Lett. 59, 2475 (1987).

[10] C. W. J. Beenakker, Phys. Rev. B47, 15763 (1993).

[11] F. J. Dyson, J. Math. Phys. 13, 90 (1972).

[12] D. J. Thouless, Phys. Rev. Lett. 37, 1167, (1977).

[13] S. Feng, Phys. Rev. B 39, 8722 (1989).

[14]A. D. Stone, Phys. Rev. B 39, 10736 (1989).

[15] V. Chandrasekhar et. al, Phys. Rev. B 42, 6823 ( 1990).

[16] S. Feng, P. A. Lee, A. D. Stone, Phys. Rev. Lett. 56, 1960 (1986)

[17] B. L. Al'tshuler and B. Z. Spivak, Pis’ma Zh. Eksp. Teor. Fiz. 42, 363 (1985)
[JETP Lett. 42, 447 (1985)]

[18] C. W. J. Beenakker and H. van Houten, phys. Rev. B 37, 6544 (1988)

[19] G. Bergman, Phys. Report 107, 1 (1984).

35

[20] B. L. Al’tshuler and A. G. Aronov, Pis’ma Zh. Eksp. Teor. Fiz. 33, 515 (1981)
[JETP Lett. 33, 499 (1981)]

36

 

()9 ~13 i
0.8 ~ 2
0.7 C 5
0.6 L j
()5 E 1
()4 . _
()3 l D J
()2 E 3
04,. u“

0.0 b 1 l 1 l 1 L 1 l
0.01 0.02 0.03 0.04

AB(T)

F‘corr(g)

 

 

 

 

I'I'I TII1711]"'I IITITIII‘FTI ITlllll["‘1 ll

1'0. ‘ 5 A CooperChanneli
0.9? C A

0'8 f O l/tNotse
()7 ~ 0 A

()6 L
()5 l A
0.4 L o _
()3 l A ;
()2 5 i

0.1 - A —
00 .111411111111111 11411Q111Wu4

10‘4 2 10'3 2 10"2 2 to"1 2
I30?)

A Variance

11%;141

1 l

 

 

Figure 2.1 Evaluation of (a) ID field autocorrelation function F (B=0, AB)
calculated by Beenakker and von Houton and (b) variance 5g2(B)and 1/f noise
crossover vPP(B) in particle-particle channel. The parameter values are L4, = 1 pm, LT

=0.17um.

37

 

 

 

 

TIII] I I IIITTIT 1 IIIIIIII I TTTTIITI I IIIITTT] T TTlllll
LO — .OO.........666“ r
0.9 - 0 A ~
0 A
0.8 T o ‘ T
A 0.7 *— O A J
8 o 1
d? (16 - 00A —
> 0.5 — obOMAAAAAAA —~
E Cooper Channel
0 0.4 —
U)
0.2 - A L¢=0.65pm “
OJ ~ 2
00 11111 1 11111111 1 11111111 1 11111111 1 1111111141 1111111
10‘5 10" 10'3 10‘2 10'1 10°
B(T)

Figure 2.2 Evaluation of the 1D 1/f noise crossover function vPP(B) in particle-
particle channel at L4, = 1 pm and 0.65 pm. The LT is fixed as 0.17 pm.

38

 

1.0 T1] "1 "T IIlTll[ 'I'I'I'l II‘IIIIIfi

0.9 Diffuson Channel;
0.8 _ 9
0.7 o Variance _*
0.6 9
0.5 I
0.4
0.3
0.2

0.1

0.01LL11.1.1.11111111.1-n1111 1111111.‘
10'1 2 346 10° 2 345 101

13(1)

ll'T'

l

1
O
O

—4

l

__4

.\€

1

1

Normalized Var(g)

l

—4

 

 

 

 

1.0 IT] ’I'I'TYI 1 11111] "'l"‘l I TIITTI '

0.9 Diffuson Channelj
0.8 9
0.7
0.6

0.5 O o
O
0.4 00

)- O -<
0.3 r 0% 7,
0.2

0.1

0011111.1.11111J_LL11111111.1 1
10"! 2 3 46 10° 2 a 45 101

ND

1

l

O l/f Noise

1 l
l l

1
O
O

l

Semi/30(0)

I
1 1

T T—TT
1

 

 

 

Figure 2.3 Evaluation of the finite temperature (a) 1D variance and (b) l/f noise
in particle-hole channel. The parameter values are L¢ = 1.0 pm, T= 1.9 K, and

D(diffusion constant) = 7.8x10'3 m2/s.

39

 

1.0 llTT"'l ITIflTTT'YVT l IITIIII'1'7 Tfillllllr"l TIIIITTT’T

0.9 — Diffuson Channel
0.8 —

0.7 ~

1 I l 1444

0.6 t —

0.5 - —

’1— I
1

0.4

30(B)/Sc(0)

l

0.3

0.2 T=3.3 mk < Ec T=1.6 K > Ec

Y 17 fi fij‘

¥Lmli

0.1 F

 

 

00 1111...1 Ill—1J1'1+LAL lLlllLlJ_rLL1 11111111.1.: 11111111.1

10'3 2 10‘2 2 10'1 2 10° 2 a 101
B(T)

 

Figure 2.4 Evaluation of the l/f noise crossover in the particle-hole channel for T
= 3.3 mK and 1.6 K. In both cases Ec (Thouless energy) = 59 mK. For the case kBT
> Ec, Bc2 4 T and is determined by the temperature. For the case kBT < Ec, Bc2
~0.04 T and is determined by the Thouless energy

40

 

 

 

 

 

1.0 ”II "'T ITVTWII‘T'I IIfiIHI] '1'] i IIIIHI 'I'T
1 .
0.9 _ Diffuson Channel _
0.8 - _
0.7 — a
E? ' '
‘6 0-6T ~
m 1'
\ 0.5 e —
ES 1
'75 0A-- _
m _
0.3 P 0 1,5101, —
0.2 b A “0:1? ‘4
0-1 j 1: L,°=o.31., j
o 1111 .1.14 1111111 11.1 11111111.1.1 L 1111111.111
10'2 2 10'1 2 10° 2 101 2

B(T)

Figure 2.5 Evaluation of the 2D 1/f noise crossover in the particle-hole channel
with different spin-orbit scattering rate. The parameter values are L1) = 1.0 pm, LT =

0.17 pm and Ec =59 mK. At Lso = 0.3 L¢, the Zeeman effect on the l/f noise is
almost invisible.

41

Chapter 3

Experimental Techniques

In this chapter, I will discuss the experimental requirements and techniques to
carry out quantum transport in a low spin-orbit metal. Since the spin-orbit scattering
rate in metals depends on the atomic number(Z), we studied quantum wires and films
made of the lightest of metals, lithium. For this experiment, we utilize quench-
condensation of films, sub-micron electron beam lithography, optical lithography and
in-situ measurement of the l/f noise. To recover the small 1/f noise, we utilize a two-
phase digital lock-in amplifier which allows simultaneous measurement of total
noise(llf noise + background) and background. A magnetic field up to 9T is applied
with superconducting (SC) magnet outside the vacuum-can of a He-4 pumped

cryostat.
3A. Quench-Condensation

There are several ways to fabricate a film on a substrate; thermal evaporation,
electron-beam evaporation, sputtering, chemical vapor deposition and molecular
beam epitaxy. Different methods give different film quality from single crystal to
highly disordered films. The thermal evaporation used in this work produces
polycrystalline films and the grain size can be controlled by the substrate temperature
at the evaporation stage.

The metallic lithium is very reactive in air, so, it must be handled in an inert
gas (He, or Ar) atmosphere or in a high vacuum. Metallic lithium is thermally
evapbrated onto a cold substrate kept at liquid helium temperature and the deposited

metal atoms are "quench-condensed" due to low diffusion mobility. All the electrical

42

measurements are performed in-situ, with electrical wires connected already on the
substrate before the Li evaporation. The important factors in the quench-
condensation are the following; fine alignment between the Li source and the
lithographed mask on the substrate, a scheme for holding the Li source inside the
filament, a pre-evaporation to remove the contaminated surface of Li source, and
minimization of thermal load on the substrate by providing a good heat-sinking
scheme and a fast shutter control during the evaporation.

Figure 3.1a shows a schematic set-up for the quench condensation with a
cryo-evaporator inside the vacuum-can. The diameter of the vacuum-can tail is set as
2.5 inch due to a geometrical restriction given by the 9T supercondcting NbTi
magnet. The substrate, shown in figure 3. lb, has a sub-micron mask fabricated with
electron-beam lithography and is heat-sunk to the 1K pot. The RF -filtered electrical
wires are for in-situ measurement.

Lithium has a high vapor pressure and is easy to evaporate. The melting point
is 180°C. Lithium reacts with some metals, (for instance, W or Au) and forms alloys,
so, materials such as Ta, Ni-Cr and Fe based alloys are used for the heating elements.
Be aware that Cr or Fe impurities are strongly magnetic in the Li matrix[l]. To
evaporate both directions, a filament geometry is used with a scheme to keep a Li
blob from falling out of the filament. A blob of lithium is inserted into the filament in
an Ar gas-filled glove bag. The Li blob is somewhat contaminated at its surface
during loading into the cryostat. Several stages of pre-evaporation of the Li blob are
carried out during the cooling of the cryostat. A spring-loaded shutter and radiation
baffles are located between the filament and the substrate. The fast shutter control
helps to minimize thermal shock to the mask on the substrate.

A film thickness monitor, located at the bottom of the vacuum can, is heat-
sunk at 4.2 K to minimize drifi of the temperature, and hence the resonance frequency
(~6 MHz) of the Quartz sensor crystal. A small area on the back of the sensor crystal

43

is greased down to the housing without deteriorating the gold-plated electrical
contact. All the electrical connections are made with a solid Copper miniature coaxial
cable (capacitance z 30 pF/ft at 5 kHz) except the connection just below the vacuum
can lid where 304 stainless steel mini-coax (capacitance z 50 pF/fi) is used.
Hermetically sealed microdot miniature connectors are used. The noise level in the
film thickness and rate readings is not problematic in the case of evaporation of heavy
material, for example, Ag or Au. In the case of Li evaporation (lithium has a very
low mass density only about half that of water), the signal to noise ratio is worse and
sometime problematic in terms of reading the evaporation rate. Checking all the
electrical contacts including the spring contact at the back of the sensor crystal and

proper grounding of the electrical circuits are helpful.

3B. Electron-beam Lithography

The demand of large scale integrated circuit devices has driven the
development of nricrofabrication technique. Nowadays, state-of-art lithography
produces nano-meter scale structures which can be smaller than the electron phase
coherence length.

In low temperature l/f noise experiments, signal to noise ratio (SNR) is
somewhat problematic. According to UCF theory[2], the quasi one—dimensional (1D)
geometry (width < L¢) has better SNR than 2D film. A typical value of the phase
coherence length is of the order of one micron or less at low-temperature, so the
transverse dimension of the sample must be in the sub-micron or nano-meter scale.
Fabrication of sub-micron sized samples requires lithographic technique beyond
optical. Several methods are known to fabricate sub-micron structures; X-ray
lithography[3], deep UV lithography[4], step-edge technique[5] and electron-beam
lithography[6,7]. For this work, we utilize electron-beam lithography (BBL). EBL is

44

done in a scanning electron microscope (SEM) system built with control for electron-
bearn writing. The focused electron beam of diameter a few nanometers or less, is
scanned in the desired pattern over an e-beam sensitive resist. Then, the exposed
resist is developed and the resulting image of the resist pattern is used as a mask.

The SEM is a JEOL JSM84OA microscope with a tungsten source filament.
Exposure onto the e-beam sensitive resist is done by the focused e-beam with energy
of 30 keV (the maximum for this system is 35 keV). The e-beam current, used for
small structure patteming, is 5 pA, and a beam-blanking shutter controls the exposure
time. The important factors to get optimum performance in EBL are the following;
(1) generation of a well-focused e-beam with fine tuning of astigmatism in the beam
lenses, and sharp focusing of the high energy e-beam on top of the imaging layer, (2)
correct beam dose distribution to minimize the proximity effect, i.e. overexposure of
nearby parts of the pattern, (3) correct aligmnent of lithographic patterns, (4) correct
procedure for developing the exposed resist with control of developing time and
environmental conditions, (5) formation of the “undercut” profile of the developed

area, (6) proper set-up of SEM electronics to minimize the noise from electrical lines.

33.1 Resolution, Undercut profile.

The most popular electron-beam resist used as the imaging layer is PMMA
(Polymethl Methacrylate) resist which has an inherent resolution of about 20nm[7].
Finite thickness of the resist leads to multiple scattering of the electron beam,
resulting in a bulbous interaction volume. If the resist thickness is kept low, then the
reduced scattering volume and shorter developing time lead to higher resolution.
Even though the thin resist scheme can improve the resolution, thick resist is better to
get the proper vertical profile, i.e. "undercut" profile, of the developed resist for the

device applications. The undercut profile of the resist keeps the evaporated metal

45

through the mask from electrically shorting to the outside, so is crucial in the device
fabrication. Bilayer resist systems[8,9] are used to improve the resist profile of the
mask. A bilayer scheme consists of a thin PMMA layer on the top of a thick
copolymer layer, MMA. The bottom layer planarizes substrate topography and
minimizes backscattering at the top imaging layer. Because the copolymer is more
polar than PMMA, the proper choice of a single strong developer such as
methylisobutylketone (MIBK) produces the desirable undercut profile[6,7]. Although
the undercut profile can be obtained in the bilayer resist system, the control of the
resist thickness and developing rate for both layers is important. If the bottom layer is
considerably thicker than the top layer, then developing of both layers at the same
time could result in widening of the pattern.

We use 9°/o-copolyrner dissolved in 2-ethoxyethanol solvent as a bottom layer
and 2%-PMMA dissolved in chrolobenzene solvent as a top layer. A single
developer, MIBK, for the both layers is used. Figure 3.2 demonstrates the line width
versus line dose obtained from the fabrication of l rim-long line with five-terminals
attached. Figure 3.3 shows the SEM photograph of the bilayer resist pattern just after
it is developed. The line doses vary from 0.6 nC/cm to 1.4 nC/cm. To minimize the
proximity effect on the sample line, critical doses are exposed in the big areas nearby
the sample line. Two different sets of data in figure 3.2 correspond to the results
obtained from two different thickness of the copolymer layer. Total thickness of each
bilayer system is about 320 nm and 400 nm respectively. A line resolution as small
as 30 nm is achieved. Thinning the bottom copolymer improves the line resolution
with better yield of fine line fabrication.

For the pattern to work as a mask, the profile must be "undercut". Figure 3.4a
shows both the undercut profiles and line resolutions, obtained in the ~400 nm thick
bilayer system. Several lines, separated by 1 pm, are patterned with the line doses

from 3 nC/cm at the left to 7 nC/cm at the right. Line resolution as small as 40 nm is

46

obtained. The SEM photo in fig. 3.4b shows that the undercut profile expands ~80nm
deep into each side wall. Figure 3.5 also shows the ~100 nrn deep undercut profile of
the bilayer resist exposed by an area dose of 225 uC/cmz. From this bilayer scheme,
we are able to obtain the resolution and a reasonable amount of undercut in the sub-
micron mask.

This bilayer resist scheme works in most applications but has limitation in
terms of controlling the amount of the undercut without losing the line resolution.
One approach to avoid the competition between resolution and undercut profile, is to
use mutually exclusive developers for each layers[9]. For example, the copolymer is
insoluble in a non-polar solvent such as chlorobenzene, which is a solvent for
PMMA. PMMA is insoluble in a polar solvent such as ethoxyethanol, which is a
solvent for the copolymer, MMA. In this scheme, the amount of the undercut is

limited by the e-beam sensitivity of the bottom layer.

3B.2 sub-micron metal stencil.

Fabrication of a sub-micron mask for low-temperature quench-condensation,
combined with in-situ measurement, requires a major change of the conventional
lithography because the final "lift-off" can not be carried out. The mask should
provide mechanical rigidity and electrical isolation between the deposited material on
the substrate and that on the mask surface. Also, the mask must be thermally resistant
during the evaporation or thermal cycling. Low-temperature application of the
bilayer resist scheme is generally painful unless some extra treatment of the resist is
done - for example, evaporation of all the solvent out from the resist matrix.
Application of a metal stencil is the best, and several techniques to produce sub-
micron metal stencil have been reported, including the "brush-fire" technique[10], and

trilayer lithography[l 1]. The brush-fire technique, developed by G. Dolan, produces

47

the metal stencil without losing the e-beam resist resolution. Instead, we utilize the
trilayer structure, i.e. PMMA/metal/copolymer structure. A 50nm thick layer of
aluminum is used for the metal layer because the aluminum can be etched easily.
Figure 3.6 shows a schematic procedure for trilayer lithography and the details are
given in Table 3.1. Although the trilayer process provides us with the easy control of
the undercut and the relatively easy process in metal stencil fabrication, the resolution
of the final mask is, however, determined by the isotropic chemical etching process
on the metal layer, resulting in a wider pattern than the imaging layer. An alternative
approach to avoid this problem is to do anisotropic etching of the metal layer; an
aluminum layer can be etched using Cl-based reactive ion etcher(RIE).

Figures 3.7a, 3.7b show SEM photographs of the top view and the side view
of a sub-micron metal stencil on top of the copolymer resist. The aluminum metal
layer is etched in OH-based solution and the stencil opening is 0.3 pm wide. The
0.3~0.4 um deep undercut into a side wall is shown and is quite uniform throughout
the patterned edges. A stencil opening as small as 0.2 pm has been achieved.

Figure 3.8 shows a schematic diagram of the sample geometry, which has five
probes attached. For the l/f noise measurement, the two sides of the sample form the
two lower arms of the Wheatstone bridge circuit shown in figure 3.9.

The characteristics of the samples used in this thesis are summarized in Table
3.3. Samples #5 and #6 were fabricated using Ta filament and are discussed in
Chapter 4. Samples #14 were fabricated using Ni-Cr filament and are discussed in

Chapter 5.

48
Table 3.1. Procedure for trilayer e-beam lithography.

1. Copolymer coating
Spin(static dispense) 9%-copolymer (casting solution is 2-Ethoxyethanol) @ 4000
RPM and 1 min., then bake the substrate about 1 hour @ 137~140°C. (The air oven
should be dehydrated.)
Hints:
(1) The baking temperature could be lower than 140°C, but it is better to be above
124°C which is the glass temperature of the copolymer.
(2) The sensitivity of copolymer has small dependence on the baking temperature.
(3) For cryogenic usage, it is better to get the solvent out as much as possible.

The melting point of 2-Ethoxyethanol is ~ -90F

2. AL layer coating: use thermal evaporator with water-cooled stage, and deposited

thickness is ~50 nm with a chamber pressure, ~2E-6 torr.

3. PMMA layer coating.
Spin 2%-PMMA (casting solution; chlorobenzene) @ 4900 RPM, 1min, then bake
the substrate about 1hr @ 125°C in air convetion oven. Don't bake the substrate at
higher temperature - aluminum metal layer will severely crack.

Hint:
(1) The glass temperature of PMMA is ~ 104°C; as long as it is basked above the
glass temperature, the resolution of polymer matrix is fine.
(2) Eventually the PMMA layer is stripped out by 02 RIE to maximize the

cryogenic application.

4. Sample mounting in SEM chamber

49

Put a small drop of Ag paint on the substrate for the E-beam focusing and wait
until it is completely dried. Using a clean dry gas, blow off dirt, if any. Mount the
sample onto a stub. The sample must be flat against the stub. Load the sample
into the chamber and wait until the pressure inside the chamber reaches the base

pressure.

5. Electron- beam exposure.
(1) Set the SEM to the following configuration:
Voltage: 35 kV
Gun Bias: Auto
Coarse Probe Current: 6e-12 (the smallest)
Aperture: 4 (the smallest)
Working Distance: 15 mm
EOS Mode: SEM
Image Select: SE1
(2) Obtain the proper saturation current and align the gun.
(3) Adjust the aperture centering and alignment mark.
(4) Focus the beam at as high as magnification. Correct the astigmatism in the
beam using the standard. Repeat this step until an optimum image is achieved.
(5) First, the sensitivity test related to e-beam exposure needs to be done critically.
(6) Do coarse alignment. Coarse alignment is not a problem either SE mode or SEM
mode for Al 50 nm case because Au contact pads underneath Al layer give enough
contrast.
(7) Do fine alignment between the contact pads and e-beam pattern.
(8) Expose the e-beam.
(9) Develop the resist about 45 see with developer, MIBK:IPA(1 :3) at room

temperature.

50

Hint: the solubility ratio between the exposed and unexposed PMMA layer is high
for MIBKzlPA developer.
(10) Rinse with IPA (30 sec), and blow dry with N2 gas

6. check pattern carefully with OPTICAL microscope (X1000)

7. Al layer Wet-ETCH
Prepare fresh etching solution(for example, diluted KOH) , monitor humidity and
temperature.
Calibrate the etching time everyday and etch the substrate.
See the distinct color changes.
Rinse it very carefully with DI water.
Dry the substrate either the N2 gas blow dry or spinner dry(20 sec @ 4000 rpm).
Run a low power RIE process (19 W, 50 mtorr, 30 sec, 12 dc-bias)
Recheck the etching status of the substrate. -- The substrate should have a very nice
edge and some shadow. Be sure to etch completely through out the pattern.
Develop the copolymer layer; as long as it etches through completely, the

copolymer is developed homogeneously and reasonably fast.

8. Develop the COPOLYMER
Develop the copolymer resist with MIBK : IPA (1:3) developer in 90 sec, and rinse
it with IPA, and rinse it with DI water. Dry the substrate (spinner dry or blow dry).
Take a look and the substrate should be clean and uniformly developed. The nice,
optically-visible undercut can be defined with the continuous developing as long as

3 min. Be aware that MIBK itself will dissolve the butylacetate based Ag paint.

9. RIE pattern transfer.

51

Use high power 02 RIE (99W, pressure ~ 75 mtorr, processing time ~ 90 sec). It
will take the top PMMA out and transfer the etched pattern in the metal layer down
to the bottom copolymer layer. Proper control gives a well-defined undercut and
clean substrate. A 3 min. long RIE process gives about 0.5 pm undercut, with

optical halo.
10. SEM check
Check the metal stencil opening with e-beam current of 5 pA, at magnification around

SOOOX ~15000X.

11. check connections between Al layer and Au pads electrically

52

3C. Optical Lithography

Microfabrication of features bigger than 1pm can be done optically. A photo-
sensitive resist (for example, Shipley 1813) undergoes a chemical change after it is
exposed optically. When developed, it forms a mask for microfabrication.

Defining a vertical profile of the mask is generally important in the
application of photolithography and can be controlled easily using several different
approaches. We use a bilayer photoresist scheme where the sensitivity of each resist
is controlled by the different UV exposure on the layers. This bilayer scheme
produces a 1 ~ 1.5 pm deep undercut profile at the bottom layer with proper exposure
and developing. Evaporation of metal through the mask is done at an angle of 45°
while spinning the substrate. This process allow us to get a well-tapered contact leads

for the next step. Table 3-2 summarizes the procedure of bilayer optical lithography.

53

Table 3.2 Bilayer Optical Lithography

1. Make a contact printing mask and prepare clean substrates.

2. Spin the photoresist (Shipley 1813) @ 4900 rpm during 30 Sec. Watch out for the
edge bead. Bake the substrate @ 90°C during 30 min. inside air convection oven.
(The oven must be pre-baked and dehydrated). The resulting thickness of the
photoresist layer is about 1.1~1.3 pm

3. Warm up the Hg UV lamp (100 watt ,9» = 240~350 nm) more than 10 min.. Do
blank exposure about 10~12 second.

4 Spin and bake the top layer of photoresist (Shipley 1813) at the same condition.
Cool the substrate down.

4. Do contact printing with mask; expose the UV on the substrate during 11 second.
5. Develop the substrate during about 45-50 second.

6. Check the developed pattern and amount of the undercut.

7. Evaporate metal onto the substrate; tilt the substrate at 40°~45° from the
evaporation source and spin during the evaporation.

8. Soak into Acetone and lift off the resist.

Table 3.3: Sample characteristics. The samples discussed in chapter 5 are 1,2,3 and 4.

54

Sample 5 and 6 are discussed in Chapter 4.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 20 0.11 13 5.75 13 0.3
2 20 0.11 13 11.5 4.4 0.18
3 5800 205 7.8 0.26
4 5800 205 1.7 0.57
5 20 0.2 33 0.9 32 1.3
6 20 0.45 54 0.45 40 1.4
Bishop 0.074 25 2.1 20 1.0

 

 

55

3D. l/f Noise Measurement.

The standard procedures for measuring resistance fluctuations (1/f noise)
involve passing a current through the sample, thereby converting resistance noise to
voltage noise by Ohm’s law. (See also figure A in the Appendix.) The chief
difficulty in such measurement is that the noise from the sample may be hidden by
large background noise» either Johnson noise or preamplifier noise. Of course the
signal from the sample can be increased simply by increasing the measurement
current, but this is ofien unacceptable due to Joule heating. Several methods have
been used to reduce the background noise such as an ac bridge technique[12], or use
of a cooled step-up transformer[l3]. Recovering the l/f noise from the background
requires a second measurement without current, then the background is subtracted
from the first total noise measurement. This procedure relies on stable background
noise.

An alternative approach is to measure noise and background simultaneously.
Several such techniques have been discussed. The two amplifier cross-correlation
technique[l4] rejects the preamplifier noise in the background, but does not reject the
sample Johnson noise. The double-frequency ac method[15] requires that the bridge

circuit must be in balance at both frequencies.

3D.] dual-phase technique.

There are two elegant methods that subtract background noise using the ac
bridge method with a single drive frequency. First, one can simultaneously measure
the in-phase (0°) and quadrature (90°) signals fi'om the bridge using a dual-phase
lock-in amplifier. The power spectrum of the former contains both sample resistance
noise and background, while the power spectrum of the latter contains only
background. Subtracting the two spectra yields the sample noise alone.

Unfortunately, commercial lock-in amplifiers tend to have large phase noise, so this

56

technique is limited in practice. Verbruggen et al. showed that the background is
eliminated by correlating the two orthogonal outputs of a dual phase lock-in amplifier
set to phases of plus and minus 45° with respect to the bridge current[16]. The 45°
cross-correlation method also requires extremely good orthogonality and low phase
noise in the lock-in amplifier.

To achieve the extreme phase stability and orthogonality need for these
methods, we use a completely digital measurement system[l 7]. In addition to these
requirements, a digital measurement system demands a large real-time calculation
bandwidth, since it acts as both lock-in amplifier and spectrum analyzer. The recent
development of the digital signal processor (DSP) integrated circuits makes it
possible to implement such a system using only a DSP board with a small amount of

memory, a digital-to-analog interface, and personal computer.

3B.2 Digital lock-in amplifier.

(This part is based on our article published in Review of Sicentific
Instruments in 1992.)

We have developed a noise measurement system based on the Motorola DSP
56001 digital signal processor. Figure 3.8 shows our noise measurement system
using the Motorola DSP boards and personal computer(PC). The digital signal
processor and several kilobytes of memory lie on one board. A second board contains
16 bit D/A and A/D converters which operate from a single clock, assuring their
synchronization. A sine wave generated by the D/A converter excites the bridge. By
choosing the frequency of the excitation signal to be commensurate with the D/A
clock frequency, total harmonic distortion (THD) 140 dB below the carrier is
achieved. A third board is an interfacing board with the PC.

The difference signal from the bridge is amplified by a low-noise preamp,
such as the Stanford SR 560 or the PAR 116. The output of the preamp is low-pass

57

filtered to prevent aliasing, then digitized by the A/D converter. A single-pole RC
filter is sufficient to prevent aliasing with the sigma-delta-type A/D converter which
has the high sample rate. The sigma-delta A/D converter samples the incoming signal
at a rate 128 times faster than the normal data conversion rate, then averages groups
of 128 points to get the output. Since the Motorola A/D converter is supplied with dc
power levels of 0 and 5 V, the A/D inputs are dc biased to 2.5 V with a simple home-
built difi'erential-to-differential level shifter from the standard op-amp circuits. Hence
the signal can be ac coupled.

Figure 3.9 shows the block diagram for the digital signal processing of the
input signal. First, the phase of the signal is shified to compensate for the overall
phase shift of the experiment. The signal is digitally mixed (multiplied) by two
orthogonal sine waves chosen to be either at 0° and 90° or at i45° with respect to the
reference. The mixed signals are digitally filtered[18] and the sampling rate is
decimated to reduce the number of points stored in memory on the DSP board and
later Fourier transformed by the PC. The processing of filtering and decimation is

represented by the following equation:

N
y(M) = 12 {100Mmn - k)

where n is the decimation ratio, h(k) are the filter impulse response coefficients, N is
the number of filter taps, and x and y are the input and output data streams,
respectively. We performed the digital filtering in three nested stages to reduce the
number of filtering taps in any one stage. The advantage of a multistage digital filter
is that the initial stages may have wide transition bands as long as aliased signals do
not appear in the final passband of the complete three-stage filter[19]. Only the final

stage need have a sharp transition band to minimize the number of unusable points in

58

the power spectrum. We designed three Chevychev equiripple FIR filters with
maximally flat passbands, using the Monarch software package. Three filters have
75, 51, and 40 taps, respectively. Transfer functions of the three filters are shown in
figure 3.10. When used with a 10-5-2 decimation scheme, these filters provides over
100 dB attenuation of the signals above the Nyquist frequency of the final
(decimated) sample rate.

The PC starts the DSP system to collect the data, and simultaneously analyzes
the data from the previous run. The PC performs fast Fourier transforms (F FT) of
each data channel to get the l/f noise spectrum by cross correlation in the case of $45
° method, or it calculates two power spectra and subtracts the results in the case of 0°-
90° method. Our program for the real-time signal processing is written in assembly
language. Table 3.3 shows an assembly code used in the implementation of a dual-
channel digital lock-in-amplifier.

The performance of our system is demonstrated in figure 3.11. The noise
measurements are obtained by averaging 128 runs (~ 35 minuites). Our system
suppresses the background noise by a factor of 100, and substantially out-performs
commercial analog lock-in amplifier. The limit is set by statistical errors due to the
finite measurement time.

After we developed our DSP-based noise measurement system, Stanford
Instruments introduced a commercial DSP lock-in amplifier which is convenient to
vary the experimental set-up, for example, drive current and phase shift. We used a
Stanford SR850 DSP lock-in for the actual noise measurements in this thesis. Since
the Stanford DSP lock-in amplifier works as a voltmeter basically, the output signal is
processed with the Iotech filter and digitizer to get a bandwidth-limited signal for the
Fast-Fourier transform (FF T). The personal computer interfaced with Iotech

instruments performs the FFT to extract the noise power.

59

References

[1] N. Papanikolaou et al., Phys. Rev. Lett. 71, 629 (1993).

[2] S. F eng, P. A. Lee, A. D. Stone, Phys. Rev. Lett. 56, 1960 (1986)

[3] D. C. Flanders, Appl. Phys. Lett. 36, 93 (1980).

[4] B. J. Lin, J. Vac. Sci. Technol. 12, 1317 (1975).

[5] D. E. Prober et al., Appl. Phys. Lett. 37, 94 (1980).

[6] A. N. Broers, J. M. E. Harper, and W. W. Molzen, Appl. Phys. Lett. 33,392
(1978)

[7] R. E. Howard et al., J. Vac. Sci. Technol. B1, 1101 (1983).

[8] R. E. Howard, E. L. Hu, and L. D. Jackel, Appl. Phys. Lett. 36, 141 (1980).

[9] M. Hatzakis, J. Vac. Sci. Technol. 16, 1984 (1979).

[10] G. J. Dolan and J. H. Dunsmuir, Physica B 152, 7 (1988).

[11] B. J. Lin, and T. H. P. Chang, J. Vac. Sci. Technol. 16, 1669 (1979). B. D. Hunt,
and R. A. Buhrman, J. Vac. Sci. Technol. 19, 1308 (1981).

[12] J. H. Scofield, Rev. Sci. Instrum. 58, 985 (1987).

[13] D. E. Prober, Rev. Sci. Instrum. 45, 849 (1974).

[14] A. Van der Ziel, Noise (Prentice Hall, Englewood Cliffs, 1970); see also S.
Demolder et al., J. Phys. E 13, 1323 (1980).

[15] M. B. Weissman, Ph. D. Thesis, University of California, San Diege (1976); H.
Stoll, Appl. Phys. 22, 185 (1980).

[16] A. H. Verbruggen, H. Stoll, K. Heeck, and R. H. Koch, Appl. Phys. A 48, 233
(1989).

[17]P. A. Probst and B. Collet, Rev. Sci. Instrum. 56, 466 (1985); N. O. Birge and S.
R. Nagel, Rev. Sci. Instrum. 58, 1464 (1987); P. K. Dixon and L. Wu, Rev. Sci.
Instrum. 60, 3329 (1989).

60

[18] For an introduction to digital signal filtering, see Oppenheirn and Schafer,
Digital Signal Processing (Prentice-Hall, Englewood Cliffs, 1975).
[19] For a discussion of multistage digital filtering, see R. E. Crochiere, Multirate

Digital Signal Processing (Prentice-Hall, Englewood Cliffs, 1983).

61

Set-up for quench-condensation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Vacuum can-‘-
ma et 9 T
substrate gn ( )
radiation baffle
shutter ,
evaporation lfiA/VVV—
filament
film thickness
monitor -
Substrate with metal stencil
evaporation
Baffle with Slit
metal stencilN /sub-micron opening

    
 
  

electrical
Si substrate)

Figure 3.1 A schematic set-up for quench-condensation including (a) a cryo-
evaporator and (b) the substrate with sub-micron metal stencil mask.

62

 

 

 

 

I ’ I ‘ I ' I ‘ I T I f I ' I ' I
110 - C1 4
100 LArea1_dose : 90pC/cm2 a
_ Area2_.dose:225;.1C/crn2
A 90 — ~
8 _
5 80 _ 0 320 nm uuck Bilayer 7
.1: P C]
8 70 — [:1 400 nm thick Bllayer a
3| 60 — o —
a) L .
5 50 — 0 ~
.—1 1 ~
40 - ~
- a .
30 T . . T
20 + 1 1 1 l 1 1 l 1 1 1

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4
Line_dose (nC/cm)

Figure 3.2 Line dose versus line width obtained by the bilayer electron-beam
lithography. Thinning the resist gives the better control of line width well below 100
nm.

63

8828 2st 8188.888 1881?: 11010

 

Figure 3.3 Scanning electron microscope photograph of the patterned bilayer
resist itself. The developed lines are about 40 mm wide.

 

Figure 3.4 (a) The undercut profiles obtained from ~400 nm thick bilayer resist
are shown. (b) Cross-section of the e-beam exposed line is shown with the ~80 nm
deep undercut profile.

65

 

Figure 3.5 SEM photo of the area exposed by the area_dose, 225uC/cm2.

66

Trilayer Electron beam Lithography
E-beam exposure : electron-beam

K
\ yresist
metal —> :\\\\\\\\\\\\\\\\\\\ /

Substrate

 

Developing :

 

Chemical etching :

§\\\\\\\\\\\\\\\V

 

RIE pattern transfer : . .
,/ Sub-mrcron metal stencrl

M

Figure 3.6 Schematic procedure of the trilayer electron beam lithography.

67

8884 zsxu 8'3‘8881 11v» H010

 

Figure 3.7 SEM photograph of a sub-micron metal stencil. (a) The top view
shows the uniform undercut profile developed underneath of the metal stencil. (b)
The side view shows the metal stencil on top of the resist.

Figure 3.8

 

68

 

 

 

 

 

 

/\
\/

Schematic diagram of the sample geometry.

69

 

   

  

 

 

   

 

8 h0‘
DUI. I
n»~n¢~
nxww
mace“,

 

 

$1
_ -
H , _ ox.“ oboe—L

 

 

 

 

 

 

 

 

Ewen" Po morn—sumo Emma—B on :38 Boom—.333. £83. .55 $316 moan
:8 .922 25 ~35 3. :5 43.88838 Gamma. £2? :5 :32 ~35 Ed 2:...an

73.3. Rama?

an”

 

 

 

70

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.10 Block diagram of digital signal processing system. The mixing and
digital filtering and decimation are performed in real time by the Motorola DSP
56001. '

Transfer Function (dB)

71

 

 

 

 

 

 

25 TFI I I I I I I I I I I l I II I I I I I I I
: I I I I 1
01 ‘2
-25 :_ _:
1- .1
1- -1
" "1
-50 _— J
C I
b c1
-75 .3.
1- «It
- -1
-100 [— —'j
I 1
1- -I
-125 [- —‘1

I- ] 7L] 1 [I I l l l 1 LL!

 

 

 

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Normalized Frequency

Figure 3.11 Transfer function ofthe three FIR digital filters preceding the three
decimations. The frequency scale is referred to the original sample rate of 14.4 kHz
before decimation. The sample rate is reduced by factors of 10, 5, and 2 in successive

decimation stages.

72

 

 

 

 

10-15 c I I I IIIIII I I I IIIIII I I I III _I

': 1
~ 4

A o

g 10 13 ..—- o —=

N\ : 0 I

> ~Dnngaouaaaoonoono 1

v 1- c1

.2 - . . ,

m _ o

c: 10 ‘7 :- 0 d.

0 : 0 :

c: : :

—e l- 0 -1

fl 1- ° .1

z»; - ° -

o

o o

o. -18 ._

m 10 i.— 0 o E
C o I

10-19 I l l Llllll 1 l I 1Lllll l l I Illll
0.1 1.0 10.0 100.0

Frequency (Hz)

Figure 3.12 Noise measurements of a pair of 480-0 carbon resistors. The
background (:1) was measured with the 90 method, and the sample noise (0) was
measured with the 45 cross-correlation method.

input_0

predo

input

endo_1
;perform

calcsl

calccl

73

Table 3.4 The part of Assembly code for implementation

of dual-phase lock-in amplifier using DSP 56000.

do #prepts,predo

jclr #7,x:$ffee,input_0

move p:(r7)+n7,x1

move x1,x:$ffef

nop

do #pts,endo_out

do #dec3,endo_3

do #dec2,endo_2

do #dec1,endo_1

jclr #7,x:$£fee,input

move x:$ffef,x0

move #ahift,n7

move p:(r7),x1

move x1,x:$£fef

move (r7I-n7

move #delta,n7

move x:(r7).x1

mpy x1,x0.a

move a,x:(r0) y:(r7)+n7,y1

mpy y1,x0,a

move #shift,n7

move a,y:(r0)-

move (r7)+n7

nop

first filtering for each carriers

move (r0I+

clr a x:(r0)+,x0 y:(r4)+,y0

rep #ntabsl-l

mac x0,y0,a x:(r0)+,x0 y:(r4)+,y0

macr x0,y0,a

move a,x:(r1l

clr a y:(r0)+,y1 x:(r4)+,x1

rep #ntabsl-l

mac x1,y1,a y:(r0)+,y1 x:(r4)+,x1

macr x1,y1,a (r0)-

move a,y:(r1)-

nop

endo_2

;perform second filtering for each carriers

calcsz

calcc2

endo_3

calcs3

move (r1)+

clr a x:(r1)+,x0 y:(r5)+,y0

rep #ntabaZ-l

mac. x0.y0,a x:(r1)+,x0 y:(r5)+,y0
macr x0,y0,a

move a.x:(r2) '

clr .a y:(r1)+,y1 x:(r5l+,x1

rep #ntach-l

mac x1,y1,a y:(r1)+,y1 x:(r5)+,x1
macr x1,y1,a (r1)-

move a.y:(r2)-
nop

;perform third filtering for each carriers
move (r2)+

clr a x:(r2)+,x0 y:(r6)+,y0

rep #ntabs3-1

mac x0,y0,a x:(r2)+,x0 y:(r6)+,y0

74

Chapter 4
Zeeman effect on UCF

4A. Introduction.

In this chapter, we describe our electron transport measurements on a 1D metallic
Li wire in the regime of negligible spin-orbit and spin-flip scattering, which has the
feature of maximum symmetry (GOE)[1] in the absence of a magnetic field.
Application of a magnetic field causes a transition from GOE to GUE, and hence a
reduction of conductance fluctuations (66)2 by exactly a factor of 2. Application of a
much larger field breaks the electron spin degeneracy and creates two independent
eigenvalue sequences, thereby causing a second factor of 2 reduction. Our data show
good agreement with the theory[2,3,4] for the complete crossover function, both for
the GOE to GUE transition and for the splitting of the Zeeman degeneracy. Our
results show that the magnetic field scale for the Zeeman crossover is determined by
the sample temperature, rather than by the Thouless energy as has previously been
suggested[3,5].

4B. Experiment.

We have studied metallic Li films for minimal spin-orbit (SO) scattering with
sample dimensions in the quasi-1D regime (thickness t and width W < L¢) because
this restricted geometry further enhances the noise power via UCF[6]. Samples were
patterned with five leads on silicon substrates using electron-beam lithography and a
trilayer resist/metallresist structure as discussed in Chapter 3. The lateral dimensions
of the sample were 0.44 x 20 pm2 as determined from a scanning electron microscope
(SEM) picture before the sample was loaded into the 4He cryostat. The Li film was
quench-condensed in-situ at 4.2 K through the metal stencil, then annealed at 35 K.

The low-temperature sheet resistance was 0.46 O. The film thickness was obtained at

75

the conclusion of the experiment by a comparison of the linear temperature
dependence of the resistance above 80 K, shown in figure 4.1, with that of bulk
lithium[7]. By this method, we estimated the thickness of the film to be 54 nm. This
implies le = 40 nm , kFIe = 450 with free-electron values for the electronic constants.
The resistance and resistance fluctuations (l/f noise) were measured using a low
frequency ac bridge method[8], with a liquid-nitrogen cooled Triad G-S transformer
to increase the sample-to-preamplifier noise ratio. To compensate for possible
background fluctuations during the l/f noise measurement, the total noise (l/f noise +
background) and the background were measured simultaneously with a two-phase
digital lock-in amplifier. Figure 4.2 demonstrates the sensitive detection of the l/f
noise with a large background. In this measurement, background noise is mostly
from Johnson noise of the cooled transformer.

A magnetic field perpendicular to the film was provided by a 9 T superconducting
magnet in the liquid He bath outside the vacuum can. We measured the 1/f noise as a

fimction of magnetic field at two fixed temperatures, 1.6 K and 4.2 K.

4C. Magnetoresistance measurement.

Figure 4.3 shows the magnetoresistance at different temperatures and the fit based
on one-dimensional weak localization theory[9]. (The fits were done with 130'] = 0.)
Figure 4.4 illustrates the effect of SO scattering rate in the fit. Fits to these data
provide an upper bound on the SO scattering rate of 1;; < 02-131 at 1.6 K; hence we
are always in the low SO scattering limit. Since the 1D weak localization correction

to the magnetoresistance depends on the width of the sample, it gives another
estimate of film width, W as 0.45 pm, close to the SEM observation. L. is 1.4 11m

and 0.75 pm at 1.6 K and 4.2 K, respectively, hence the sample is quasi-1D (L. >
W).

76

Figure 4.5 shows the temperature dependence of the electron phase breaking
length, L. assuming no spin-dependent scattering in the magnetoresistance fits. The

straight line, L¢'2 = (0.096714 0.20)um'2, is a linear least-squares fit to the data. This

linear behavior, although unusual for metal films, has been observed previously for Li
films over a wide temperature range[10] and will be discussed further in Chapter 5.
We believe that the finite intercept at T =0 arises from the presence of a small amount

of spin-dependent scattering (either spin-orbit or spin-flip)[10].

4D. l/f noise measurement.

Figure 4.6 illustrates the 1/f noise power spectra at 1.6 K at three different values
of magnetic field ; B = 0 T, 1.0 T, 8.8 T. At least 64 simultaneous spectra for total
noise and background were averaged to reduce the statistical uncertainty at each data
point. The relatively large error bars at high fiequencies are due to the error
propagation from the background subtraction. The straight lines are least-squares fits
to a power law. The noise slopes are very close to —1.0 d: 0.03 at all values of
magnetic field. The data in fig. 4-6 demonstrate clearly that the amplitude of the l/f

noise power drops as the magnetic field increases.

4D-l Quantitative analysis of GOE-GUE crossover

Figure 4.7 shows the relative l/f noise power at 0.1Hz as a function of the
perpendicular magnetic field at temperatures of 1.6 K and 4.2 K. The relative noise
power at 1.6 K shows a first reduction to 1/2 with characteristic field Bcl z 22 G,
where Bcl is defined as the field where the noise is 3/4 of its zero-field value. Bc1 is

temperature dependent, increasing to 34 G at 4.2 K. Bcl corresponds to the
h / e

O

 

penetration of a flux quantum through a phase coherent region, BC, = A for a

quasi-1D sample, where A is a numerical constant that depends weakly on the ratio of

77
L. = W to LT = W , where D = vpl, I3 is the electronic diffusion
constant. For this fihn, D is 7.8x 10"3m2 /s and the LT is 0.17 pm at 1.6 K.

For a detailed comparison of theory and experiment, we use Eq. (2-31) for the
GOE-GUE crossover function. (See Chapter 2 for the details.) For pure UCF
fluctuations, one would expect Eq.(2-31) to describe the low-field data in figure 4-7,
with L1, as the only free parameter. We observe, however, that the noise does not
quite drop a full factor of 2. A similar behavior is observed in other Li samples,
shown in figure 4.8, and in Ag fihns[11], and apparently arises fi'om a small
contribution of local interference (LI) type noise[12] that is magnetic field
independent. We therefore include a second parameter to account for this small

effect. The function fitted to the data is:

30(3) _ _ -
_SG(0) _ c+(1 c)v(B) (4 1)

where V(B) is given by Eq. (2-31). The solid line in figure 4-7 shows the fit to the
data for B < 0.1 T. The fit yields c=0.1 and 0.2 and L4, =1.0 11m and 0.65 pm at 1.6 K
and 4.2 K, respectively. The latter are remarkably close to the values 1.3 pm and 0.75
pm obtained from magnetoresistance measurements taken at the same drive level as
the noise measurements. (The noise measurements at 1.6 K were taken with a drive
level that caused sample heating to about 1.9 K, as determined from the

magnetoresistance.)

4D-2 Quantitative analysis of Zeeman crossover

As the magnetic field increases above 1 T, the noise power begins a second
decrease. At 1.6 K, the characteristic field scale for this second drop is B¢2 z 4 T,
with saturation at about 7 ~ 8 T. The solid line at high field in fig. 4.7 is our

78

numerical evaluation of the noise crossover function, Eq. (2-3 8), for the Zeeman
effect, valid for B >> BC]. In this computation, we use our experimental values for D,
kBT, and L. , and the free electron value of the g-factor (=2). We emphasize that there
are no fiee parameters in the theory for the Zeeman splitting; the constant c in Eq. (4-
1) was already determined by the fit to the low-field data.

It is of interest to ask whether the Zeeman crossover occurs when Ez exceeds the
Thouless energy, 13c = hD/ L1,2 or the temperature, k,713,4]. At 1.6 K, the value of
Ec/kB in our sample is about 59 mK, about 30 times smaller than the temperature;
hence we can clearly distinguish which of these two energies governs the Zeeman

crossover. In the former case, Bc2 z 2°— = 0.04 T, which is clearly incompatible
8113

with the data. (The uncertainty in our estimation of D due to uncertainty in the fihn

thickness is not enough to account for the discrepancy). In the latter case,

Bc2 z 513-: =15 T , which is not far fi'om the experimental value Bcz z 4 T. When
8113

the correct numerical prefactor is put in, we indeed find the excellent agreement

between theory and experiment shown in fig. 4-7. The result holds only in the regime

k 3T >> Ec ; numerical evaluation of the theoretical crossover function shows that it is

always the larger of the two energies that governs the Zeeman crossover.

4E. Comparison with earlier work.

Mailly et al. [13] measured the variance of conductance jumps induced by applied
voltage pulses in GaAs/AlGaAs heterojunctions at zero magnetic field and 0.2 T. At
50 mK, a decrease in the variance by a factor of 3 was observed at 0.2 T.

Debray et al. [5] studied UCF using a static method in a quasi-1D GaAs/AlGaAs
heterostructure. They measured the conductance variance at sample temperature 1.3
K at several values of magnetic fields from 0 to 0.4 T and observed a first reduction
of (8G)2 by a factor of 2 below B ~ 0.001 T (with one data point!), and then a second

79

factor of 2 reduction near B ~ 0.06 T. They estimated the Thouless energy as Ec z 88
mK and concluded that the field scale for the Zeeman crossover is determined by the
Thouless energy and the measured field scale is consistent with the theoretical
prediction by Stone[3]. We mention that (1) they couldn't fit the magnetoresistance,
which provides the phase breaking length and an estimate of the Thouless energy, (2)
there are no data points showing the crossover regime of the first reduction in the
variance due to the suppression of the cooper channel, and (3) most of all, we found
that the prediction by Stone is wrong! (See Chapter 2 for further discussion on the
field scale of the Zeeman crossover.) If we accept that their measurement is correct,
then they observed a reduction by a factor of 4 in the relative UCF amplitude with the
value of Bc2 ~ 0.02 T, using our definition of Ba as the midway point of the second
reduction. Also, since kBT>> EC, they should get Bc2 = 2.7kBT / (gu B)= 2.6 T,
which is about 100 times larger than the value observed. The value of Ba above is
calculated with the free electron value of g(=2). We note that several values of g-
factor,~ 0.4[14] and l3[15], in GaAs/AlGaAs heterostructure have been reported. It
is, however, unlikely that the g-factor in GaAs is ~100 times the free electron value.
An important difference between those measurements and ours is that the GaAs
experiment measures the saturated UCF amplitude, rather than unsaturated l/f noise.
But our numerical evaluation of the Zeeman crossover function for the saturated case
also shows that B62 is determined by the larger of kBT and EC. Therefore, we find it

difficult to believe that their measurement and analysis were carried out correctly.

80

References

[1] Y. Imry, Europhys. Lett. 1, 249 (1986) ; P. A. Mello, Phys. Rev. Lett. 60,1089
( 1988) ; K. A. Muttalib, J .-L. Pichard, and A. D. Stone, Phys. Rev. Lett. 59, 2475
(1987)

[2] B. L. Al’tshuler and B. I. Shklovskii, Zh. Eksp. Teor. Fiz. 91, 220 (1986) [Sov.
Phys. JETP 64, 127 (1986)], P. A. Lee, A. D. Stone, and H. Fukuyama, Phys.
Rev. B 35, 1039 (1987).

[3] A. D. Stone, Phys. Rev. B 39, 10736 (1989).

[4] S. Feng, Phys. Rev. B 39, 8722 (1989). The calculation of the full crossover
function for ID was sent to us by Feng, private communication.

[5] P. Debray, J.-L. Pichard, J. Vicente, P. N. Tung, Phys. Rev. Lett. 63, 2264 (1989).

[6] S. Feng, P. A. Lee, A. D. Stone, Phys. Rev. Lett. 56, 1960 (1986).

[7] J. S. Dugdale, D. Gugan, and K. Okrnura, Proc. Roy. Soc. London, Ser. A 263,
407 (1961).

[8] J. H. Scofield, Rev. Sci. Instrum. 58, 985 (1987).

[9] B. L. Al’tshuler and A. G. Aronov, Pis’ma Zh. Eksp. Teor. Fiz. 33, 515 (1981)
[JETP Lett. 33, 499 (1981)].

[10] J. C. Licini, G. J. Dolan, and D. J. Bishop, Phys. Rev. Lett. 54, 1585 (1985).

[11] Extensive low-temperature noise measurement on Ag films by our group show
that the local interference contribution to the noise becomes significant (10 %) in
both metals when L¢,/le z 25. See also, P. McConville and N. O. Birge, Phys.
Rev. B 47, 16667 (1993).

[12] J. Pelz and J. Clarke, Phys. Rev. B 36, 4479 ( 1987).

[13] D. Mailly, M. Sanquer, J.-L. Pichard, and P. Pari, Europhys. Lett. 62, 195
(1989).

[14] S. Narita et. a1, Jpn. J. Appl. Phys. Part 2 20, L447 (1981).

81

[15] C. Weisbuch and C. Hermann, Phys. Rev. B 15, 816 (1977); M. Dobers, K. v.
Klitzing, and G. Weimann, Phys. Rev. B 38, 5453 (1988).

82

 

48-
46:
44;
42FL
40L
38—
36:-
34:-
32:-

Resistance ((1)

 

28—

1_ J 1 J 1 I 1 l 1 l 1 I 1 I

50 60 70 80 90 100 110 120 130

 

 

 

Temperature (K)

Figure 4.1 Sample resistance versus temperature obtained from quasi-one-
dimensional metallic Li wire. The change in the linear temperature coefficient of the
resistance is shown around 83 K due to martensitic structural transformation. We
compare the linear temperature dependence of the resistance above 90 K with that of
bulk lithium.

83

 

 
 
  
   

V T 7

.11111111..

TVIIIIII]
.1411

8.,(V2/Hz)

11111

B =3.9 T

l

a: O.99:1:0.04

v—rvIvfiT IIIITII'Y' V]

y...

11.11111111111111'11.11111111.1.1.11

 

 

-21 .
1O 10" 2 3 10" 2 3 10° 2

 

u...

Frequency(Hz)

Figure 4.2 1/f noise power spectrum and background Johnson noise obtained
simultaneously by two-phase digital lock-in amplifier. The spectral slope or is 0.99 i
0.04.

84

 

 

 

    
      

I I I I I
1.6 K
— g?‘ d
' .
O O
‘ ., o a
>- 1 0 j
- ‘
2x10 5 r a
t 2 B K .
- . ‘O.“ .
0:: - ' -° ~ ° 3
. . ‘\
*- . I, ‘ . 4
m e~ ,’ a e.
'0 t ‘ ~
< L- ’93' 2 400 Q ‘33\ -‘
0‘ r9 ,- °\ ‘;$~
r - _- 0' c’ 5"; "o
a.-- u:"'.. -6' .€\ ...an-‘ ~
. "H" e»: ‘c "H'-
4

a. ._ .
o-oa;;:9.a....

 

-
"‘ :“-o.
_____

Figure 4.3 Magnetoresistance data at temperature, T = 1.6 K, 2.8 K, 4.0 K and 5.6
K. The solid line is the fit to the quasi-1D weak-localization theory with zero spin-

orbit scattering.

85

 

_ ._

r L¢=1.4p,rn (Ts-01:0) a
s...
0"
m

'3 5 i
L)

1— . —1

. 4

e e e e e e 0‘

 

 

 

0.10 0.15 0.20 0.25 0.30 0.35 0.40
1/Lso (106m'1)

 

 

 

 

0.8 T I T I ' I F I E I

T" 0.7 — 0 ~
0
E 1
‘00 o
e
S 0.6 F . —
c: o
S 0.5 — 0 ~
0
1 e.o
o4 1 1 1 1 1 l 1 1 1 1

0.10 0.15 0.20 0.25 0.30 0.35 0.40
l/Lso (106m-1)

Figure 4.4 (a) The values of x2 in the fit of the magnetoresistance data at 1.6 K as a

function of the spin-orbit scattering rate assumed in the fit. The upper bound of spin-
orbit scattering is set to be ~ 0.2 14,4. (b) the phase breaking length as a fimction of L30.

86

 

 

 

 

3 r _
1" 2
E
3 .
N
L?

1 — -

o l l l

0 10 20 30
'1‘2 (K2)

Figure 4.5 The electron phase breaking length versus temperature, obtained from

the weak-localization fits of low field magnetoresistance.

87

 

 

 

 

TV I 1 I 1 11171 1 1 I I 1 r111 T 1 1 I

10" — E
a ’ -
a _ . .
‘2': 10 z 1 j
V : j
c: \ .
v 7 1 . O 3
ea ~ ' ~
10'11 g ‘\ 9 E
; N. :

1111 1 1 1111111 1 mn111111 1 11

10" 10'" 10°
Frequency (Hz)

Figure 4.6 The resistance noise power spectra, SR(f), versus frequency at 1.6 K
with fits to a power law, at three values of magnetic field, B = 0 T (0), 1.0 T (A) and
8.8 T(I). The spectral slopes for the three data sets are -1.03, -1.04, -1.01,

respectively, with an uncertainty of £0.03.

88

 

  

11ml 1 111nm I 1 IIIIIII I Irrlnr I I rnmr I IllIlIl|
1.0 -
A 0.8 - ~
0
II 1
Ca
V 006 — I "'
U in .
m ~ ~ .. .
\ 1 1";
_ ..
m 004 _ J‘ .~ —1
V ‘
U
U)
0.2 - -
O 0 111111 1 11111111 1 11111111 1 11111111 1 11111111 1 1 1111111

 

 

 

10-4 10"3 10-2 10'1 100 101
B(T)

Figure 4.7 Conductance noise power, SC,(0.1Hz), as a function of magnetic field
at 1.6 K(-) and 4.2 K(I). The data are normalized by the noise power at zero field.

The solid line is the theoretical expression for the noise crossover function. The

theory for the low-field crossover is fit to the data with two fi'ee parameters, L1, and c,

as discussed in the text. The theory for the high-field crossover has no free

parameters.

89

 

L0 1

 

0.8 ~

%UCF

(17 c 3

 

 

 

0.5 1 I 1 1 1 1

10 20 30 40 50
L90 / 1el

Figure 4.8 Contribution of %UCF to total noise power versus L¢,/le in Li samples.

The contribution of local interference type noise becomes significant (10%) when L4,

/16 z 25.

90

Chapter 5

SPIN-FLIP SCATTERING on UCF

5A. Introduction

In mesoscopic systems, the electrical conductance depends on the microscopic
configuration of scatterers due to long-range quantum interference[l]. If the
scatterers have free spin states (for example, paramagnetic impurities), then
conduction electrons will experience the exchange interaction with the paramagnetic
impurities, leading to relaxation of the electron spin states by spin-flip scattering[2].
Spin-flip scattering breaks the phase coherence and reduces the amplitude of
conductance fluctuations. This picture is valid in a magnetic field that is not strong
enough to align the impurity spins. When pimpB > k BT , spin-flip scattering is
suppressed and the system behaves similarly to one without magnetic scattering. The
magnetic scattering effect on weak localization[3] and conductance fluctuations [4]
has been observed recently.

In this chapter, we report a quantum transport experiment on quasi-1D Li
wires with a finite spin-flip scattering rate which is comparable to the electron phase
breaking rate. The magnetoresistance measurement shows low-temperature residual
scattering and suppression of the phase coherence. The l/f noise measurement versus
magnetic field shows that the phase coherence is recovered at high magnetic field
causing a dramatic increase of conductance fluctuations. The noise power at low
magnetic field shows a reduction by a full factor of 2. These observations are

consistent with the effect of paramagnetic scattering on UCF.

91

5B. Experiment

Our samples were patterned with five leads on silicon substrates using
electron-beam lithography on a bilayer resist structure, PMMA/Copolymer. The
characteristics of the samples are summarized in Table 3.3. In this chapter, we
discuss samples #1 - 4, which show the residual magnetic scattering. We presume
that magnetic impurities such as Cr from the Ni-Cr filament wire contaminated the
samples since Cr impurities in a Li matrix can be strongly magnetic[5]. The Li films
were quench-condensed as described in Chapter 3 and magnetoresistance and l/f

noise were measured in-situ.

5C. Weak localization[6,7]

The inelastic scattering of conductance electrons comes from several
mechanisms: electron-electron and electron-phonon scattering. The electron-electron
scattering rate in a clean metal is determined by collisions with energy transfer of
order of the temperature and momentum transfer of the order of the inverse screening
length, which leads to a 72 dependence, independent of the sample dimensionality. In
disordered systems, the electron-electron scattering rate depends on the
dimensionality of the sample which is set by the thermal diffusion length. In the 2D
case, the electron-electron scattering rate is dominated by small energy transfer
collisions, leading to a linear temperature dependence. Experimental observation of
the dependence 11;] at T are generally agreed to be due to the 2D electron-electron
interaction. If the film is three dimensional with respect to electron-electron
interaction one expects Tin—l at T3”.

The electron-phonon scattering is governed by the electron-phonon interaction
and leads to a 7‘" dependence in the clean-limit with dimensionality d. In dirty metals

= M [e << 1, where S is the velocity of sound , impurity scattering

92

causes the electron-phonon interaction to change and yields the different temperature
dependence of the electron-phonon scattering rate: 7“ [8], or T2[9]. Thus,
understanding the electron-phonon scattering requires more careful comparison
between experiment and theory. Several experimental factors need to be considered
carefully: the film homogeneity, the film-substrate boundary and phonon spectrum of
the film. Despite experimental disputes, there are several cases reported, including
experiments on relatively clean and thick aluminum films[10], thin and quench-
condensed films[11].

Figure 5.1 shows our magnetoresistance data of quasi-1D and 2D samples and
the fits to weak localization theory[6,12]. (The fits were done with rso“=0 because
our data at the lowest temperature couldn't determine rso'l unambiguously.) From the
fit we extract the electron phase breaking rate and the temperature dependence of L4,
is shown in figure 5.2a for the quasi-1D wires and in figure 5.2b for the 2D films.
The straight lines are linear least-squares fit to the data. For the temperature range T =
1.6 ~ 16 K, 11".] at T2 is observed with a typical experimental value of tin (1.6 K) ~
10'10 s. (This inelastic scattering time is estimated from sample #1 which has the
diffusion constant D = 2.5 x 10'3 mz/s.) Licini et al.[13] also measured weakly
localized behavior in quasi-1D Li films and reported the rin'l at T2 dependence
down to 0.25 K with Tin (1 K) ~ 1 x 10'10 s. The observed single power law of
Tin-1 at T 2 behavior is consistent in form with clean-limit electron-electron
scattering, the pure electron-phonon scattering involving two-dimensional phonons or
the 3D electron-phonon scattering in the dirty metal. The inelastic lifetime due to the
clean-limit electron-electron interaction is order of 10'7 ~10‘8 s[14] and is too big to
explain the experimental values. The electron-phonon scattering modified by the
impurities yields the lifetime of order of 10'11 ~ 10'12 s[6] and the application of the

dirty-limit electron-phonon scattering is questionable because the product 4112 is not

93

less than 1 in the temperature range of interest. Belitz et al. [15] found the asymptotic
expansion used for the electron-phonon scattering to be invalid above helium
temperature and explained experimental results showing I'm-1 at T2 for T = 4 ~20 K,
from Bergmann and collaborators as a superposition of Coulomb and a two-
dimensional electron-phonon contribution. However, the observation of 1: in" at: T2
in Li fihns down to the temperature 0.25 K is quite puzzling at this moment and
further investigation of the phase breaking mechanism in Li films needs to be carried
out to clarify this problem.

From the weak localization phenomena, we were able to identify the relative
strength of the spin-orbit scattering compared to the inelastic scattering. The negative
magnetoresistance shown in figure 5.] confirmed that our Li films were in the low SO

scattering limit. We observed, however, the suppression of the phase coherence

compared with samples discussed in Chapter 4 and saturation of L4, at low—
temperature, which results in the finite intercept at T=0 in the plot of L¢'2 versus 72
shown in figure 5.2. The sample #1 yields L4,“2 = 0.58 T2 + 9.8 (um‘z) and sample #2
yields L52 = 0.79 T2 + 27.6 (um’z). We believe that the finite intercept at T =0 arises

from residual magnetic scattering with very weak spin-orbit scattering[l 3].

5D. 1/f noise and UCF
5D.1 Orbital effect on UCF

Spin-flip scattering breaks the phase coherence of the electron, and leads to
the suppression of conductance fluctuations. Theoretically, it is shown that spin-flip
scattering suppresses the cooper and diffuson channels equally[16], so one should be
able to observe the noise reduction by a factor of exactly 2 with application of

magnetic field.

94

Figure 5.3a shows the relative l/f noise power of sample #1 as a function of
the perpendicular magnetic field at temperatures of 1.6 K and 4.2 K. As the theory
predicted, the noise drop by a full factor of 2 at 1.6 K. The noise at 4.2 K does not
quite drop to a full factor of 2 because there is a small contribution of local-
interference type noise which is magnetic field independent, as discussed in Chapter
4. Figure 5.3b also shows the l/f noise power versus magnetic field at temperatures
of 1.6 K, 4.2 K and 10 K, measured in sample #2. We also observed the full factor of
2 reduction in the noise at 1.6 K and a small contribution of LI type noise at the other
temperatures.

The l/f noise crossover function due to the suppression of the orbital
contribution ( cooper channel) is governed by the phase breaking length, L¢[l6,17].
Therefore, we can extract another reliable estimate of L¢ from the noise measurement

and compare the value of L¢ with that from magnetoresistance. In the fit of the noise

data, we set rso'l = 0 and include a second parameter to account for the L1
contribution[18]. (See Eq. (4.1) in Chapter 4) The solid lines in figure 5.3 are the
theoretical evaluations of the l/f noise crossover and show excellent fit to the data.
The fit parameters for sample #1 are c = 0 and 0.2 and L4, = 0.28 and 0.22 pm at 1.6 K
and 4.2 K, respectively. The fit parameters for sample #2 are c=0, 0.14 and 0.16 and
L4, = 0.19, 0.14 and 0.09 pm at 1.6 K, 4.2 K and 10 K, respectively.

In figure 5.4 the values of L¢ obtained from the noise data are compared with

those from the magnetoresistance . Both measurements yield very close values of L¢

even though we observed finite spin-flip scattering in our samples. Chadrasekhar et

al. [19] show that the spin-flip scattering effect on the conductance fluctuations can
be incorporated via L4, in the singlet(s) and triplet(t) channel:

95

L313”; [£1152 + L114 (8)1“2

_ _ 4 _ _ (5.1)
L” =1L1. 2 +11; 2(B)+3-Lso 2] “2

8,:

where Lin is the inelastic length and Lsf is the spin-flip scattering length. In the
magnetoresistance the spin-flip scattering rate enters into the phase breaking length of

the singlet and triplet channel in a slightly different way[6,10,13]:

LWLS = [Lin-2 +2Ls f—2(B)]-l/2

2 (5.2)

L? =1L1.‘2+— 3§L1f2<B>+ 41.41"”

In the strong SO scattering regime, a finite spin-flip scattering yields a big difference
in the measured value of L4, from the two methods because the triplet channel is

suppressed. In the negligible SO scattering limit, the singlet and triplet channel have
the same diffusion length in the noise measurement but not in the magnetoresistance

measurement. Since magnetoresistance measurements yield the longer diffusion
length in the triplet channel and the shorter diffusion length in the singlet channel

compared to that fi'om the noise measurement, it is not surprising to get a very close

value of L1, from these two measurements on our samples.

Since the fit parameter L¢ in the low-field noise data is

L¢= Lg?— = L“? = [L,-,,'2 + L s", 2'1 1’ 2, the spin-flip diffusion length L,, can be

estimated if we know the inelastic length Lin- From the slope of the L¢'2 versus 72
plot for sample #1 in figure 5.2, the inelastic diffusion length Lin is estimated as 0.78
pm at 1.6 K. Then L sfiS estimated as 0. 30 um, given L1, = 0.28 um from the fit to the

noise data and we get L31 = 0. 2 1.1m,L¢ t = 0.33 pm. The noise measurement

provides Lg? = gar: 0.28 pm. The inferred length scales for singlet and triplet

channel from two measurements are larger than the sample width (0.11 urn), ensuring

96

that quasi-1D localization theory is valid in our sample. For sample #2, we get Lsf =

0.2 pm.

5D.2 Spin effect on UCF

Figure 5.5 shows the magnetic field dependence of the relative l/f noise
power up to 9T obtained from sample #1. (Data below 0.5 T has been discussed in
figure 5.3.) The noise power increases with field dramatically when B > 0.5 T at 1.6
K, and slightly when B > IT at 4.2 K. This behavior is in contrast to the reduction,
we saw in Ch. 4, due to the Zeeman splitting of the conductance electrons. Figure 5.6
shows similar behavior of the relative l/f noise power from sample #2.

A possible explanation of the behavior shown in figures 5.5 and 5.7 is based
on the assumption that there is a finite residual spin-flip scattering in the sample, as
observed in the magnetoresistance. The strong magnetic field aligns the magnetic
impurities and suppresses the spin-flip scattering, leading to the restoration of the
phase coherence and the amplitude of conductance fluctuations. The magnetic field
scale needs to be of order k 3T / 11 mp [2], which corresponds to 1.1 T and 3.1 T at 1.6
K and 4.2 K respectively if we assume Ilimp = 2113. The estimated magnetic field
scales are close to the experimentally observed values.

To analyze the data quantitatively, spin-flip scattering needs to be
incorporated into UCF. Benoit et al. [20] assumed that (1) only the lowest two
magnetic energy levels of the impurity would be important, so that the magnetic
impurity could be treated as spin 1/2 and (2) the magnetic scattering rate would be
proportional to Pul’d where Pu and Pd are the probabilities of the spin 1/2 impurity
being in the up or down state. The calculation of the magnetic field dependence of

the spin-flip scattering rate gives:

97

_ _ _ 1
L4, 2 = L," 2 + L,, 2(13 = 0) 111m B (5.3)
cosh2 (—p—)
kBT

 

At high magnetic field, the electron Zeeman splitting needs to be considered in the
system with electron spin symmetry[16,21]. (See also Chapter 4) Unfortunately, the
situation is more complicated due to the finite spin-flip scattering, Lsf/Lin ~ 0.4 and
0.3 at 1.6 K for sample #1 and #2, respectively. Since strong spin-flip scattering
breaks the electron spin symmetry, we assume that the Zeeman effect on UCF is
negligible in our samples. (See figure 4.7 for the Zeeman effect on UCF.) Then the
theoretical function for the noise with the spin flip scattering is obtained following
Eq. (2.31) and (5.3), where the parameters are Lin, Lsf(B=0) and llimp~ We used the
values of Lin and Lsf(B=0) determined by the low-field fit. The solid lines at high-
field in figure 5.8 show our fits to the theoretical expression with only a single fit
parameter Ilimp = 0.7 113. The simple theoretical function matches the data fairly well
and describes the temperature and field dependence of the noise power at high-field.
Even though not all of the features in the noise data are explained from the simple
model, we clearly observe the effect of the spin-flip scattering on the conductance
fluctuations, and our data at high- magnetic field are consistent with previous
work[4,20]. All the previous work related to the effect of the spin-flip scattering on
the conductance fluctuations have been carried out at high magnetic field and found

that the amplitude of conductance fluctuations increases with field.

5.E Summary
We have fabricated mesoscopic Li films using quench-condensation and
measured both the weak-localization contribution to the average conductance and the

conductance fluctuations in-situ. From the magnetoresistance, we are able to extract

98

the electron phase breaking length and its temperature dependence. The
magnetoresistance data confirm the low spin-orbit scattering in our Li films but show
the suppression of the phase coherence due to the finite spin-flip scattering.

We observe that the noise power is reduced by a full factor of 2 with an
application of weak magnetic field. This observation is consistent with the theoretical
prediction that the spin-flip scattering suppresses the conductance fluctuations equally
in the cooper and diffuson channel and an application of magnetic field suppresses the
cooper channel contribution to UCF. This is one of our important results which has
not been observed previously. We used the noise reduction versus magnetic field to
measure L¢ and found that the value of L4, is very close to that from the weak-
localization measurement.

The amplitude of conductance fluctuations increases dramatically with high
magnetic field. We attribute this behavior to the alignment of the magnetic impurities
at the high magnetic field and the suppression of the spin-flip scattering. We
developed a simple model following Benoit et al. [20] to fit our data over the magnetic
field and the temperature. This paramagnetic model describes the data fairly well

with a single value of the temperature-independent parameter ”imp'

99

References

[l] Mesoscopic Phenomena in solids, edited by B. L. Al’tshuler, P. A. Lee, and R. A.
Webb (North-Holland, New York, 1991).

[2] B. L. Al'tshuler and B. Z. Spivak, JETP Lett. 42, 447 (1985); P. A. Lee and T. V.
Ramakrishnan, Rev. Mod. Phys. 57, 287 (1987); V. I. Fal'ko, JETP Lett. 53, 340
(1991); A. A. Bobkov, V. I. Fal'ko, and D. E. Khmel'nitskii., Sov. Phys. JEPT 71,
393 (1990)

[3] Wei W, G. Bergmann and R-P. Peters, Phys. Rev. B 38, 11751 (1988).

[4] S. Washbum and R. A. Webb, Adv. Phys. 35, 375 (1986); A. K. Geim et al., JETP
Lett. 52, 247 (1990); W. F. Smith et al., Phys. Rev. B 43, 12267 (1991).

[5] N. Papanikolaou et al., Phys. Rev. Lett. 71, 629 (1993).

[6] For an excellent review, see G. Bergman, Phys. Report 107, 1 (1984).

[7] B. L. Al'tshuler et al., Sov. Sci. Rev. A. Phys. 9, 223 (1987).

[8] A. Schmid, Z. Phys. 259, 421 (1973).

[9] G. Bergmann, Z. Phys. B 48, 5 (1982).

[10] P. Santhanarn, S. Wind, and D. E. Prober, Phys. Rev. B 35, 3188 (1987).

[11] R. P. Peters and G. Bergmann, J. Phys. Soc. Jpn. 54, 3478 (1985).

[12] B. L. Al’tshuler and A. G. Aronov, Pis’ma Zh. Eksp. Teor. F iz. 33, 515 (1981)
[JETP Lett. 33, 499 (1981)]

[13] J. C. Licini, G. J. Dolan, and D. J. Bishop, Phys. Rev. Lett. 54, 1585 (1985)

[14] N. W. Ashcroft and N. D. Merrnin, Solid State Physics (Holt, Rinehart and
Winston, New York, 1976), P. 346 ff.

[15] D. Belitz, S. D. Sarma, Phys. Rev. B 36, 7701 (1987).

[16] A. D. Stone, Phys. Rev. B 39, 10736 (1989); P. A. Lee, A. D. Stone, and H.
Fukuyama, Phys. Rev. B 35, 1039 (1987).

100

[17] C. W. J. Beenakker and H. van Houten, phys. Rev. B 37, 6544 (1988)

[18] J. Pelz and J. Clarke, Phys. Rev. B 36, 4479 (1987)

[19] V. Chandrasekhar et. al, Phys. Rev. B 42, 6823 (1990).

[20] A. Benoit et. a], in Anderson Localization, edited by T. Ando and H. Fukuyarna
(Springer-Verlag, Berlin, 1988), p. 346.

[21] S. Feng, Phys. Rev. B 39, 8722 (1989).

 

AR/R

 

 

 

 

 

AG, /(ez/1rh)

 

 

 

L 1 L 1 A 1 A 1 A; l A l

 

.oo 0.02 0.04 0.06 0.08 0.10 0.12 0.14
B(T)

Figure 5.1 (a) Magnetoresistance of quasi-1D Li sample #1 (W = 0.11 pm, R0 =
5.7 (2) taken at T = 2 K. The solid line is the fit to the quasi-1D weak localization
theory. (b) Magnetoresistance of 2D Li sample #3 (W = 205 um, R0 = 7.8 (2) taken
at T = 3.5 K. The solid line is the fit to the 2D weak localization theory.

 

   
  

 

 

 

 

     
   
   

 

50 -
5‘
IE ‘
Sam lel
3 P
N 25 «
1 a
._‘1
Sample 5
O 1 I 1 l 1 l 1 l 1 l 1 1 1 d
0 10 20 30 40 50 60 70
T2 (K2)
2D films
100 t
A 75 ’-
N
l
g 1
1.. 50 Sample4
'3
25 ~
0 ..

 

 

0 100 200 300

1e (K2)

Figure 5.2 The electron phase breaking length versus temperature, obtained fi'om
the weak-localization fits of low field magnetoresistance (a) for quasi-1D wires and
(b) for 2D films. The data for sample 5 (I) in fig. 5.2a are discussed in chapter 4 and

are shown for contrast since they have no residual spin-flip scattering.

=0)

SG(B)/SG(B

=0)

30(3)/ 3003

Figure 5.3

0.3
0.2
0.1
0.0

 

I

7 Sample 1

I

 

11111 L L

 

L41

 

 

10-3 2 a

111 1 1 1
10-2 2 s

13(1‘)

 

 

 

 

 

0.3 —
0.2 u
1 Sample2
0.1 1 a
21‘ e 1“ . s‘
B(T)

Conductance noise power as a function of the perpendicular magnetic

field, normalized by its zero-field value. (a) The data for sample 1 are taken at the

temperatures 1.6 K(o) and 4.2 K(cr). (b) The data for sample 2 are taken at the

temperatures 1.6 K(o), 4.2 K(EI) and 10 K(v). The solid lines are the least-squares fits

to Eq. (4.1), as discussed in the text.

104

 

 

 

 

I I T I 'T
3 — O a
X
5
2 r —1
E a
3-
.43 W
0 Sample 1. MR
-1 X Sample 1. l/f noise
10 ‘ D ‘1
9 ” 0 Sample 2. MR A
8 " _1
A Sample 2. 1/1 noise
7 _ —1
o 1 1 1 1 1 1 1 1 1
100 101

T(K)

Figure 5.4 Values of the phase breaking length, obtained from the fits to the data
of noise vs magnetic field in figure 5.3, and obtained from the magnetoresistance data

in figure 5.2.

105

 

 

 

 

1,2 117T l I 111111] I I 111111] I I 1111”] 1 rTIInII
1.0 f— . a .J ‘ ‘
a 09:. ‘-
|| 0.8 - o -
m or:
a? 0.6 - :35 00000 a@ D -
> «.0
co 0 ,
7, 0.4 — 9,0 -
CD
0.2 - Sample 1 -
O O 1111 1 1 1 111111 1 1 111111l 1 1 1111111 1 1 1 111111
10-3 10'2 10"1 10° 101
EU”)

Figure 5.5 Noise power as a function of magnetic field from sample #1 at
temperatures 1.6 K(o) and 4.2 K(n). (The data are normalized by the noise power at
zero-field.)

106

 

 

 

 

2,0 111] 1 1 1 r1111] 1 1 Tr11111 1 1 1111111 1 1 1 111111
1'8 _ SampleZ 1:1 '
1.6 - -
€13: 1.4 - D -
CO 1.2 - -
‘6 D
U) 1.0 +3 D 0 -
> ‘0 [:1 $215 ;& [:1 m
m 0.8 " DOD . >§< ‘
V x
”3" 0.6 - ao’ ! xgtxi’ezo‘.’ ‘
DD DD
0.4 - cu: ._
0.2 - -
O O 1111 1 1 1 111111 1 1 1 111111 1 1 1 111111 1 1 1 111111
10'3 10'2 10'1 10° 101
B(T)

Figure 5.6 Noise power as a function of magnetic field from sample #2 at
temperatures 1.6 K(D), 4.2 K(x) and 10 K(o). (The data are normalized by the noise

power at zero-field.)

107

 

1.2 1111 W 1 1111111 1 1 111111] 1 1 111111] 1 111111

=0)

 

SG(B)/SG(B

0-2 " Sample 1 '

 

 

 

0 0 1111 1 1 1 111111 1 1 1111111 1 11111111 L 1 1111111
I

10'3 10"2 10'1 10° 101
B(T)

Figure 5.7 A fit to the noise power as a function of magnetic field from sample #1
at temperatures 1.6 K(o) and 4.2 K(D). The low field reduction fit to the quasi-1D
theory is discussed in figure 5.4a. The noise data at high field is fit to the simple

paramagnetic impurity model as discussed in the text.

108

Chapter 6

CONCLUSIONS

We have studied quantum transport in the low spin-orbit scattering limit by
measuring the universal conductance fluctuations as a fimction of magnetic field and
the weak-localization correction to the average conductance in mesoscopic Li wires
and films. We were able to obtain several significant results which provide us with a
better understanding of UCF and provide a chance to study random matrix theory
pioneered by Wigner and Dyson.

First, we observed a factor of 2 reduction in the amplitude of UCF with
application of weak magnetic field. We attribute the reduction to the suppression of
the orbital effect on UCF which can be viewed as the crossover from Gaussian
orthogonal ensemble to the Gaussian unitary ensemble due to broken time-reversal
invariance. We calculated the UCF crossover function and found very good
agreement with our data. We confirmed that the crossover field scale is determined
by one flux quantum over the phase coherent area.

Second, we observed a second factor of 2 reduction in the amplitude of UCF
with application of strong magnetic field. We attribute this reduction to lifting the
Zeeman degeneracy of the conductance electrons, and found that our data are
consistent with diagrammatic calculations. Our results show that the magnetic field
scale for the Zeeman crossover is determined by the larger of the sample temperature
and Thouless energy. This is very different fiom previous work, where the field scale
was presumed to be determined by the Thouless energy alone.

Both observations are consistent with the predictions of the random matrix

approach, which provides a more fundamental understanding of UCF. The

109

conductance is determined by the transmission matrix, the conductance fluctuations
are universal in the diffusive regime and ultimately determined by the ensemble
symmetry. Since the "universality" is generic for a whole class of transport properties
in mesoscopic conductors and superconductors, future experiments can be carried out
in several directions, including shot-noise, normal-superconductor interfaces and
Josephson junctions.

We also studied the effect of spin-flip scattering on UCF in the low spin-orbit
scattering regime. We confirmed the theoretical prediction that the spin-flip
scattering suppresses the conductance fluctuations equally in the cooper and diffuson
channel, by measuring the noise reduction by a full factor of 2 with application of a
weak magnetic field. At high magnetic field, we observe that the amplitude of
conductance fluctuations increases in a dramatic way and the increase in the noise
power is highly temperature sensitive. We used a simple model based on residual

paramagnetic scattering and found that the model describes our data fairly well.

110

Appendix

A. l/f noise.

In disordered metals, defects or impurities can move either by tunneling at low
temperature or by hopping at high temperature. This spontaneous rearrangement of
scattering centers gives rise to temporal fluctuations in the interference pattern among
all the paths in a coherent volume. This causes dynamical fluctuations of the
conductance. F eng, Lee and Stone showed that the low-temperature conductance is
extremely sensitive to a single impurity movement by a distance 5r 2 kp'l. In the
case of kple=l (close to strong localization), a single impurity can cause saturation of
the conductance fluctuations, i.e. 86w ez/h. In most metals, kfle is bigger than 1, so
the conductance change due to the motion of a single impurity (801) is much smaller
than ez/h. This case is called unsaturated noise.

The standard way to measure resistance noise is to apply a current through the
sample, and monitor the temporal fluctuations of the voltage across the sample.
Figure A shows a typical l/f noise and background Johnson noise in the time domain.

The power spectrum of the voltage fluctuations is given by

SV ((1)) = if(8V(t')5V(t'+t))cosmtdt

where 8V(t) = V(t) — (V), and the brackets refer to an average over t'.
To understand the power spectrum of the noise, consider a random process
with a single characteristic time r from a defect moving between two or more

positions. The power spectrum of that process is a Debye—Lorentzian spectrum:

I
Sd(03) °C 1:03—2:17me

111

For the slow dynamical rearrangement of many impurities, we should consider the

distribution D(t) of the characteristic times of the impurities within a sample. Then

‘1:

1+ (1)212

 

56(1)) at (50V; -D(1:)d‘t

The D(r) depends on the microscopic details of the disordered system. For the cases
of thermal activation or ttmneling, the characteristic time for a defect motion depends
exponentially on parameters which are distributed broadly, so D(ln(1:)) is constant (D(
1:) ~ 1‘1) over a wide range of 1:. The resistance fluctuations show the "1/f' power

spectrum.

112

A simple circuit for noise.
V0

l | 1——

RB

 

\
W
R, S

3“”?
V

V(t)
(sw--open) : Johnson noise.

 

V(t) :

(sw--closed) : Johnson noise + l/f noise.

V(t)

l 1 _J

o 250 500 750 1000
Time (mallisecondsl

Figure A (a) Schematic diagram of noise measurement set-up.
(b) Temporal presentation of background Johnson noise.
(c) Temporal presentation of 1/f noise and background.

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