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DISSIPATIV E QUANTUM TUNNELING OF A SINGLE
DEFECT IN A SUBMICRON Bi WIRE BELOW 1 K

By

Kookjin Chun

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy
and Center for Fundamental Materials Research

1 993

ABSTRACT

DISSIPATIVE QUANTUM TUNNELING OF A SINGLE
DEFECT IN A SUBMICRON Bi WIRE BELOW 1 K

By

Kookj in Chun

The quantum mechanical problem of a particle tunneling in a
double-well potential is of great theoretical and experimental interest.
Interaction of the tunneling system with a dissipative environment can
have a striking effect on the tunneling dynamics. A very interesting
case is that of ohmic dissipation, which occurs when an atom tunnels in
a metal in the presence of conduction electrons.

We have studied the electrical resistance of submicron Bi wires at
low temperature. Due to quantum interference of the conduction
electrons, the resistance is highly sensitive to the motion of even a
single scattering center. We observe discrete switching of the
resistance due to the motion of bistable defects in the sample. We have
measured the tunneling rates of a particular defect over the temperature
range 0.1-2 K and magnetic field range 0-7 T. The energy asymmetry, s,
of this defect varied over the range 40-420 mK depending on the value of

the magnetic field. The temperature dependence of the tunneling rates

is qualitatively different for the cases kBT<<8 and kBT>>£ . We

observe that for kBT <<£ , the fast rate (transition rate from upper

state to lower state) is roughly temperature independent and the slow

rate (transition from lower state to upper state) decreases

exponentially, as expected from a simple picture of spontaneous emission
and stimulated absorption. When k375>>t£, however, both rates increase
as the temperature is lowered, as predicted by dissipative quantum
tunneling theory. We fit our data to the theory and discuss the defect-
electron bath coupling parameter (I, and the renormalized tunneling
matrix element Ar

We have also studied the effect of Joule heating on the dynamics
of the defect in the same sample. The ratio of the fast and slow

transition rates of a defect depends on temperature through the detailed

balance relation, 7f/)g==edhr. We interpret this ratio as a local

thermometer. When E/kb75>l, this ratio is sensitive to small changes
in the temperature T'seen by the defect. since the defect is strongly
coupled to the conduction electrons in the sample below 1 K, we
interpret its sensitivity to drive current as an indication of electron
heating. As the drive current increases, the defect temperature
approaches a power law dependence with the drive current, independent of
the nominal lattice temperature. The data are consistent with a simple
model of heating, and strong thermal coupling between the defect and the
electron bath. We also tried to measure the electron temperature
directly, based on the temperature dependence of the amplitude of the
resistance fluctuations, 5R. Since 5R depends only on the electron
temperature, we expected 5R would serve as a good electron thermometer.

The electron thermometer does not follow the simple heating model.

H

To my dearest wife Young Ae and my children, James and Christine

III

ACKNOWLEDGEMENTS

I would like to thank my advisor Professor Norman 0. Birge for
making my contribution to this research a remarkably cherishable and
enjoyable experience. His invaluable guidance and extraordinary
patience made this work much more fruitful and rewarding. I thank
Professors Michael Dubson, s. D. Hahanti, Daniel Stump and Horace Smith
for serving on my guidance committee. Special thanks go to Professor
Dubson for the use of his lithographic facilities for sample
fabrication.

The friendship and valuable help of fellow students was
indispensible. Paul Mcconville provided a lot of valuable effort on
this work and helped me with many discussions in performing the
experiment. Jeong Sun Moon is also acknowledged for his valuable
opinions about the experiment, and his cheerful personality.

I thank my family for continous aid and support, which I sometimes
take for granted. Special thanks go to my parents who have been
patiently waiting for this moment for a long time. Finally, I thank my
wife Young Ae, who has always been with me, giving invaluable support
both mentally and physically. Without her warm support and patience,
this moment would have never come to pass.

This work was supported by the National Science Foundation under

grant Dun-9023458.

IV

TABLE OF CONTENTS

1. Introduction ........................................................................ 1
References ........................................................................... 6

2. Background and Theory ..................................................... 7
2.1 Classical Transport ........................................................ 7
2.2 Quantum Transport ....................................................... 8
2.2.1 Universal Conductance Fluctuations (static) .................... 9

2.2.2 Universal Conductance Fluctuations (dynamic) ............... 11

2.2.3 Characteristic Magnetic Field Scale .............................. 14

2.3 Defect Dynamics .......................................................... 15

2.3.1 Defect Dynamics in Different Temperature Regimes ......... 16

2.3.2 Theory of Dissipative Quantum Tunneling ..................... 17
References ......................................................................... 23
3. Experimental Methods ..................................................... 30
3.1 Sample preparation ...................................................... 30
3.1.1 Optical Lithography .................................................. 31

3.1.2 Electron Beam Lithography ........................................ 33
3.2 Electronics .................................................................. 38
3.3 Dilution Refrigerator ................................................... 40
3.4 Temperature Calibration of Thermometer ..................... 42
References ......................................................................... 44

4. Tunneling Dynamics of a Single Defect ............................... 55
4.1 Time Trace of a Single Fluctuator ..................................... 55
4.2 Data Analysis .................................................................. 56

4.2.1 Transition Time Histograms and Setting Comparator Levels..57

4.2.2 Debye-Lorentzian Fit to Power Spectrum and

Two Gaussian F it to the Conductance Histogram .............. 59
4.3 Detailed Balance and Energy Asymmetry s ...................... 60
4.3.1 Magnetic Field Dependence of Energy Asymmetry ............. 60
4.3.2 Characteristic Field Considerations ................................. 61
4.4 Temperature Dependence of Tunneling Rates .................... 62
4.4.1 The Defect-Electron Bath Coupling Constant a, and
the Renormalized Tunneling Matrix Element Ar ............... 63
4.4.2 Scaling Function ........................................................ 67
References ............................................................................ 69
5. Electron Heating Experiment .............................................. 86
5.1 Defect Temperature and Electron Temperature ................ 86

5.2 Power Law Dependence of Defect Temperature
on Drive Current ............................................................ 89
5.3 Attempt to measure Electron Temperature based on

Universal Conductance Fluctuations Theory .................... 94
References ........................................................................... 96
6. Summary ............................................................................ 104

Appendices
A. 1/ f noise spectrum .......................................................... 106
B. Thermometer Calibration in a magnetic field .................. 109

VII

LIST OF TABLES

Table 3.1 : The processing steps of optical lithography.

Table 3.2 : The processing steps of electron beam lithography (double layer
technique).

Table 4.1 : The values of a and A, determined from the fits with the different
magnetic fields.

VIII

LIST OF PUBLICATIONS

l. "Dissipative Quantum Tunneling of a Single Defect in Bi"; Kookjin Chun and
Norman 0. Birge, Phys. Rev. B 48, 11500 (1993)

2. "A Single Defect Thermometer as 3 Probe of Electron Heating in Bi"; Kookjin

Chun and Norman 0. Birge, submitted and accepted to Phys. Rev. B.

3. "Dissipative Quantum Turmeling of Defects in a Mesoscopic Metal"; Norman
0. Birge, Kookjin Chun, Glen B. Alers and Brage Golding, to appear in Physica
B, Proceedings of the 20th International Conference of Low Temperature
Physics, Eugene, OR, August 1993.

LIST OF FIGURES

Figure 2.1 (a) shows a schematic diagram of scattering of electrons ofl‘ disorder.
Figure 2.1 (b) shows a simplified graph of resistivity versus temperature.

Figure 2.2 (a) shows a schematic diagram of the phase change of the electron
wavefunction due to a magnetic field.
Figure 2.2 (b) is a schematic diagram of phase shift due to the motion of a scatter.

Figure 2.3 (a) shows the resistance fluctuations as a function of magnetic field.
Figure 2.3 (b) is a schematic diagram of an autocorrelation function.

Figure 2.4 (a) illustrates thermal activation of a defect over the barrier.
Figure 2.4 (b) illustrates tunneling of a single defect between two ground states.

Figure 2.5 is a graphic expression of the probability fimction, P(t).

Figure 3.1 (a) is an SEM picture of sample and leads patterned by electron beam
lithography.

Figure 3.1 (b) is a schematic diagram of a five-terminal sample.

Figure 3.2 (a) is an SEM picture of the primary pads patterned by optical
lithography.

Figure 3.2 (b) is an SEM picture of a sample with secondary pads patterned by
electron beam lithography.

Figure 3.3 shows a schematic diagram of the profile of the wire shadow
technique.

Figure 3.4 shows a schematic diagram of the lithography "lift-oft“ process.

Figure 3.5 is a schematic diagram of the electron interaction volume.

Figure 3.6 (a) is an SEM picture of the profile of the etched area in a
PMMA/MMA bilayer.

Figure 3.6 (b) is a schematic diagram of the interaction volume in the bilayer.

Figure 3.7 shows a schematic diagram of the electronics for our experiment.

Figure 3.8 shows a simplied schematic diagram of the dilution refrigerator.

Figure 4.1 (a) shows a typical time trace of the conductance of the sample.
Figure 4.1 (b) shows two comparator level settings to analyze the time trace.

Figure 4.2 illustrates the fitting of a histogram to a single exponential to determine
the mean transition time.

Figure 4.3 shows a fit of a Debye-Lorentzian function to power spectrum data.
Figure 4.4 illustrates the two gaussian fitting to the distribution of raw of
conductance values, with the two peak values V,, V2 and standard

deviation 0'.

Figure 4.5 shows a plot of ln(rf / 7,) versus 1/ T for the four data sets, along with
linear fits through the origin.

Figure 4.6 shows the transition rates versus temperature at B = 0.14 T.

Figure 4.7 shows the transition rates versus temperature at B = 2.274 T.

Figure 4.8 shows the transition rates versus temperature at B = 2.286 T.

Figure 4.9 shows the transition rates versus temperature for B = 6.997 T.
Figure 4.10 shows the theoretical fits to the data at B = 6.997 T and B = 0.14 T.

Figure 4.11 shows the theoretical fits to the data at B = 2.286 T and B = 2.274 T.

Figure 4.12 is a plot of the values of a and Ar determined from the four fits,
plotted in the log(A,) versus a/ 1- a parameter space.

Figure 4.13 shows the plot y(T/ T0)"2" versus kBT/s, where 7;, = l K and a =
0.195.

Figure 4.14 shows two fits to the data at B = 6.997 T and B = 0.14 T, with the
least squares values of a, and with the global value of a from the

scaling fit.

Figure 4.15 shows two fits to the data at B = 2.274 T and B = 2.286 T, with the
least squares values of a, and with the global value of a from the
scaling fit.

Figure 5.1 shows a schematic diagram of energy flow in the sample.

Figure 5.2 shows In(y, / )9) versus 1/ T at B = 6.997 T,with no significant heating
of the sample.

Figure 5.3 shows the defect temperature versus drive current.
Figure 5.4 shows a fit of the Grabert function to the data at B = 6.997 T.

Figure 5.5 shows the increase of 6 R with decreasing temperature due to universal
conductance fluctuations.

Figure 5.6 shows the electron temperature versus drive current.

Figure A. (a) shows a graph of a flat energy distribution.
Figure A. (b) show the corresponding frequency spectum.

Figure B. shows the fits of the calibration fimction to the carbon thermometer
data at different temperatures.

XII

Chapter 1

Introduction

The conduction electrons in a metal are scattered from lattice
vibrations, impurities and structural defects while traveling through
the sample, giving rise to the electrical resistance of the sample. At
high temperatures, the dominant scattering mechanism is the scattering
of the electrons from the lattice vibrations, which is inelastic
scattering. At low temperatures, on the other hand, the scattering is
from static impurities and lattice defects, which is elastic scattering.
Only recently it has been appreciated that these two types of scattering
mechanisms have fundamentally different effects on the resistance of
metals. Inelastic scattering destroys the quantum-mechanical phase
coherence of the electronic wavefunction, while elastic scattering does

not. Hence quantum interference phenomenon are important at low

temperature, when the phase coherence length, L , is longer than the
elastic mean free path, L. In particular, the exact spatial
configuration of the elastic scattering centers is important because the
interference of multiply scattered electrons depends on the locations of

[1'3]

the scatterers. since the electron wavefunction is phase-coherent

over the distance l., the electrical resistance can not be derived
simply by adding independently the contributions from all the defects
and impurities. Instead, the scattering amplitudes from different

trajectories interfere quantum-mechanically, so that the conductance

(inverse of the resistance) is due to the coherent sum of the

amplitudes.
The consequences of quantum interference are greatly enhanced in

mesoscopic samples, samples with dimensions comparable to [4. Such

samples have properties that differ drastically from those of the
"ensemble average".[4] The effects of quantum interference have been
widely observed in small samples with fixed impurity configuration at
low temperature. By varying a magnetic field, which causes the
interference to change in a complicated way, one observes time-
independent fluctuations of the conductance as a function of the

5
I 1 These conductance fluctuations are aperiodic and

magnetic field.
vary in detail between samples. For mesoscopic samples, the magnitude
of the conductance fluctuations is of order ez/h. The universality of
these fluctuations at low temperature has been confirmed in several
different metallic systems. This phenomenon is called ”universal
conductance fluctuations'.[1-3]
One of the most interesting predictions of the theory of the
universal conductance fluctuations is that the conductance of a
mesoscopic sample can be quite sensitive to the motion of a single

6-7
[ I In a very small sample the motion of a single defect can

atom.
totally scramble the phase of the electron wavefunction and produce as
much conductance change as moving the entire configuration of defects.
For a weakly disordered one-dimensional or two-dimensional mesoscopic

sample, the magnitude of the conductance fluctuations is approximately
ez/h. This dynamic feature of universal conductance fluctuations can
be used to study microscopic phenomena such as the dynamics of defects

8
in metals and their relationship to 1/ f noise. It has been proposed[ 1

that l/ f noise is caused by the linear superposition of a large number
of independent defect motions with a broad distribution of transition
rates.

The dynamics of such defects arises from two different mechanisms:
thermally activated hopping over the potential barrier at high
temperatures and quantum tunneling through the barrier at low
temperatures. At high temperatures, a defect in a double-well potential
hops back and forth between the two wells over the barrier. The
distribution of defect hopping rates is given from the broad
distribution of activation energies. As the temperature is lowered, the
defect dynamics will cross over to tunneling though the barrier.

When the temperature is lower than the vibrational energy in a
single-well, the defect motion is limited to the quantum tunneling
between the lowest levels in each well. In a metal, the defect
interacts strongly with the bath of conduction electrons, leading to
dissipative quantum tunneling.[9] The dissipation has three primary
influences on the tunneling dynamics: (1) the tunneling is an
incoherent rate process rather than a coherent oscillation between the
two wells (due to the strong coupling of the defect with electron bath);
(2) the tunneling matrix element is renormalized due to the interactions
with high-energy electron-hole excitations; and (3) the tunneling rates
can increase with decreasing temperature, in some circumstances. The
tunneling rates of a particular defect can be characterized by three
parameters: the energy asymmetry,8, which is the energy difference
between the ground states in the two wells, the renormalized tunneling

matrix element, A4! and the coupling constant between the defect and

the bath of electrons, an The energy asymmetry,8, for a single defect

[10] [11]

has been observed to vary with magnetic field , as predicted
based on local fluctuations in the electron density that occur in
disordered metals.

This thesis describes measurements of the dynamics of a single
defect in a submicron Bi wire over the temperature range 0.1 - 2.0 K.
The temperature dependence of the defect tunneling rates were measured
for several values of magnetic field up to 7 Tesla. The temperature
dependence of the tunneling rates shows excellent agreement with the
predictions of dissipative quantum tunneling theory for each value of
the magnetic field . Fits of the experimental tunneling rates to the

theory for each magnetic field yield values of 8, a and Ar as a

function of the magnetic field. The value of 8 varies randomly with

[10-11]

field, consistent with previous work. The question of whether (I

and Ar vary with magnetic field is more difficult to address
quantitatively, because the theoretical fits to the data depend only
weakly on a over the experimental temperature range. A scaling plot

[12]

shows that our data is consistent with the hypothesis that a and Ar

are independent of field for this defect, which was assumed in a
previous study.[12]
A study of the effect of Joule heating on the dynamics of the
defect is also included in the thesis. We measured the transition rates
of a single defect and the amplitude of the resistance fluctuations as a

function of the drive current for different lattice temperatures. The

ratio of the two transition rates of a defect depends on temperature

through the detailed balance relation, 7f/)Q==e“hr. We interpret this

ratio as a local thermometer at the defect site. since the defect is

strongly coupled to the conduction electrons in the sample below 1 K, we

interpret its sensitivity to drive current as an indication of electron
heating. The Joule heating data are consistent with a model of electron

13
I 1 However, we could not determine the

heating studied previously.
absolute electron-phonon scattering rate from this experiment, due to
the sample geometry.

The organization of the thesis is as follows: Chapter 2 presents
the theoretical background necessary to understand the experiments;
Chapter 3 contains the details about the experimental techniques and
apparatus used in this work. Chapter 4 is the central part of the
thesis and illustrates the data-acquisition, analysis, and
interpretation in terms of the theory of dissipative quantum tunneling.
chapter 5 describes the effect of Joule heating on the defect dynamics

and its use as a local thermometer to study electron heating. The

thesis ends with a summary in Chapter 6.

References

l. P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622 (1985).

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

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

4. A. D. Stone, Phys. Rev. Lett. 54, 2692 (1985).

5. C. P. Umbach, S. Washburn, R B. Laibowitz and R. A Webb, Phys. Rev. B 30, 4048
(1984)

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

7. S. Feng, P. A. Lee and A. D. Stone, Phys. Rev. Lett. 56, 1960, 2772(E) (1986).

8. P. Dutta and P. M. Horn, Rev. Mod. Phys. 53, 497 (1981).

9. See "Quantum Tunneling in Condensed Matter", ed. Yu. Kagan and A J. Leggett,
Elsevier, New York 1993; also A. J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987) and
references therein.

10. N. M. Zimmerman, B. Golding and W. H. Haemmerle, Phys. Rev. Lett. 67, 1322
(1991)

11. B. L. Al'tshuler and B. Z. Spivak, Pis'ma Zh. Eksp. Teor. Fiz. 49, 671 (1989) [JEPT
Lett. 49, 772 (1989)].

12. B. Golding, N. M. Zimmerman and S. N. Coppersmith, Phys. Rev. Lett. 68, 998
(1992).

13. M. L. Roukes et al., Phys. Rev. Lett. 55, 422 (1985).

Chapter 2

Background and Theory

2.1 Classical Transport

The classical picture of electrical conduction in metals is based
on Drude theoryIl]. It is assumed that when atoms of a metallic element
are brought together to form a metal, the nuclei and the core electrons
bound to them form the immobile metallic ions, while the valence
electrons are allowed to wander freely through the metal. These
electrons, called conduction electrons, are responsible for the
conduction mechanism in metals. The disorder in the lattice gives rise
to electrical resistance. This disorder comes from vibrations of

lattice atoms, impurities and lattice defects. All of these cause

scattering of the conduction electrons, with a total scattering rate

l/‘t't =1/‘t,+l/z'¢. Here, I" is the elastic scattering time, i.e. the time

between collisions with the impurities and defects, while I} is the

inelastic scattering time, i.e. the time between inelastic scattering
events, such as electron-phonon and electron-electron collisions.
Figure 2.1 (a) illustrates the scattering of conduction electrons off
the lattice vibrations, impurities and defects.

In the presence of an applied electric field, the relationship

between the electric field.l§ at a point in the metal and the current

density j is j: 01‘], where o is the conductivity. From kinetic theory,

the current density is given as j==-nev, where n is the electron

carrier density. Assuming that an electron emerges from each scattering

event in a random direction, the average electronic velocity is

eEh'
=-———‘-, where e is the electron charge and m is its mass and

m

we

nezq

 

accordingly, j==[ )E. Equating the two expressions of j, one has

nezq

0': . The conductance is G: O'A/L, where A is the cross-sectional
m

 

area of the sample and L is the length. At high temperatures, inelastic

phonon scattering is the dominant scattering mechanism. As the

temperature decreases, 1} increases due to the decreasing rate of

electron-phonon collisions, but 1; remains finite, leading to a residual

nezr

C

conductance, G=O’0A/L, where 0'0: . Figure 2.1 (b) shows the
m

 

graph of resistivity versus temperature for a typical metal.

2.2 Quantum Transport

Drude theory has a questionable feature when applied to metals at
low temperature, since it treats elastic and inelastic scattering as
having equivalent effects on the conductivity. In a metal at

sufficiently low temperatures, one has to distinguish between the two

different scattering processes because the inelastic scattering time I}
exceeds the elastic scattering time 1; by several orders of magnitude.
In this regime, I; is called the "phase-breaking scattering time"

because an electron loses its phase coherence through inelastic

[2]

scattering events. Quantum interference , whose effect is neglected

in the classical picture, becomes important in a metallic sample

whenever the phase coherence length, I“, is longer than the elastic

mean free path, L. 14 is defined as the distance an electron can

diffuse elastically without undergoing an inelastic collision. It is

related to the phase-breaking time, I}, by the relation lv=sdl)r ,

1
where D=§v,,1e is the diffusion constant and VF is the Fermi velocity.

Since the electronic wavefunction is phase-coherent over the distance
I“, the electrical resistance cannot be derived simply by adding
independently the scattering rates from all the defects and the
impurities. Instead, the amplitudes of multiply scattered waves from
different trajectories interfere quantum-mechanically, so that the
conductance is due to the coherent sum of the amplitudes. The effect of
quantum interference gives rise to quantum corrections to the electrical
conductance.[2]

The consequences of quantum interference are greatly enhanced in

mesoscopic samples where the sample size is comparable to [3. One
consequence of quantum interference on the length scale 14 is the

[3‘5]

phenomenon of "universal conductance fluctuations" to be described
below. These fluctuations can be observed by applying a magnetic field

to a metal sample or by the motion of scattering centers.

2.2.] Universal Conductance Fluctuations (static)

The application of a magnetic field is one way to change the
relative phases of the interfering electron waves, resulting in the
phenomenon of universal conductance fluctuationsl3]. Electron wave
functions enclosing a magnetic flux exhibit a phase shift introduced by

7
the magnetic vector potential A[ 1. Figure 2.2 (a) shows a schematic

diagram of the phase change due to the magnetic field. Consider the

10

complete trajectory A.—+ B -+ C —+ D -+.A, encircling the area through
which a magnetic flux penetrates. Electrons traveling along one
trajectory A.-+ B -+ C will acquire a phase change and electrons in the
other trajectory A.-+ D —+ C will experience a different phase change.

Changing the magnetic flux enclosed by these trajectories will tune the

e
phase change along one trajectory by an amount §1=— jA-dl and
A—+B—>C

e
5 =- IA-dl along the other trajectory. The phase tuning appears as

2 IAaDaC
a cycle of constructive and destructive interference of the wave
functions, the period of the cycle being ¢5==h/e. In a one-dimensional
ring in the mesoscopic size range with a flux ¢ithrough the hole, there
is coherent interference between the waves propagating through each arm

of the ring. The magnetic field changes the relative phase of the

contributions from each arm of the ring by an amount 27t¢/<D0, giving

rise to magnetoresistance fluctuations periodic in d5.

For a mesoscopic wire in a magnetic field, the electrons traveling
along the different paths acquire different phase changes from the field
because trajectories enclose different amounts of flux. Thus, variation
of the field causes the phase dependent contribution to the conductance

[8] of the

to vary randomly, and gives rise to an aperiodic fluctuation
resistance. These conductance fluctuation phenomena are, therefore,
sample specific and aperiodic. The other unique feature of the
fluctuations is that they are time-independent and reproducible within a
given sample. These fluctuations have a universal magnitude 5Gze2/h
for samples with dimensions smaller than [1. The universal conductance

fluctuations induced by the field are static, and are called the

”magnetofingerprint".

11

2.2.2 Universal Congrctance Fluctuations (dynamic)

Another way to observe universal conductance fluctuations is to
change the locations of all of the scattering centers, i.e. the
impurities and defectslg-lo]. Figure 2.2 (b) is a schematic diagram of
the phase shift due to the motion of a single scatterer in the sample.
If the scatterer moves, it changes the phases of all the Feynman-like

paths that scatter from this scatterer, resulting in a conductance

change. For a disordered metal in the diffusive regime (i.e. when

Lg>e>L), multiple elastic scattering is very much like a random walk
process with a step size A. In quasi one-dimensional systems, each
Feynman path actually passes through a given site many times; therefore
the motion of a single scatterer will affect all the scattered paths and
accumulate sufficient phase shift to change the conductance by
5Gze2/h, as much as if all the scatterers are moved around. In two
dimensions the number of sites visited by each Feynman path that
traverses a sample of size L is of order (L/le)2. This means that a
finite fraction of all the Feynman paths pass through a site and the
motion of a single defect will alter the phase of all the paths passing
through that site. Thus, the motion of a scatterer induces a change in
the conductance that is typically a finite fraction of that induced by

the motions of all scatterers. In d22 dimensions, a fraction (.L/le)(2'd)/2

of the Feynman paths pass through a given site and the conductance

change is of order dGzezlh(L/Ic)(2'd)’2. The quantitative descriptions

10-11
are, respectively,[ 1

(1) 1-D (wire) case : 1 << szLy << L: < L, ;

12

 

._ 32.2 1 is.
(501) —Q[h) (lg-[)2 a(kr=5r) L L

c x y

(2) 2-D (film) case : 1¢< Lx<< Ly: L,< L, ;

 

(5(3)2=C 5: 2 1 a(k 6r) 1°-
! 2 h (kF 1c)2 F Lx

(3) 3-D (cubiC) case : I<<L : L = = L<L,;

(soy—C faiz——1—az(k6r)-"=—
"3 h (15.1,)2 F L

where C 's are constants of order unity which depends on dimensionality.
The function a(x)= l-sit‘12(JC/2)/(x/2)2 accounts for the phase shift due

to moving a scatterer a distance 6r. LJr , Ly , Lz , are thickness,

width and length, respectively.

If more than a single scatterer is moved within a single coherence

volume (i.e. a box of dimensions L,), the effect on the conductance

change is additive as long as 50b0, << eZ/h. Thus one has

(501m)2 = (5002 ”3(7) Vbox

where n,(T) is the density of moving scatterers within a single
coherence volume Vim. The magnitude of the conductance fluctuations is

reduced when kBT> 717:], due to "energy averaging”. At finite

temperature the incident electron distribution is smeared out over a

range kBT around the Fermi energy E . Therefore, the conductance is

13

the sum of kBT ”123' uncorrelated interference patterns, where htj' is

the energy over which the conductance fluctuations are correlated. The

conductance fluctuations are reduced in magnitude by (ital/ICED” due to

averaging over these kBT/ ht',‘ uncorrelated patterns. This process is

called energy averaging. In this case the expression of (50W)2 is

multiplied by the factor hzj'/kBT=L}/L2,, where L,=‘/71D/kBT is called

the thermal length. L, is the distance over which electrons differing

in energy by kBT become out of phase with each other.

If only one single defect is moving in a one dimensional sample of

size L,>L,, the conductance change is
(22)-}. 66..
G L, Gbox

One can extend this consideration to a two dimensional sample with

Ly, L, >L,. In this case, the conductance change is given as
[29): Li [50,.)
G LyL, Gbox

The total conductance G is related to Gbox by the sample geometry and

 

the standard rules for combining resistors in series and parallel. The

two results shown above can both be expressed as

L2
_ ¢

14

For a large sample where all coherence volumes fluctuate, the
conductance fluctuations in the different coherence volumes are
statistically independent and one combines them classically to obtain
the conductance variance. Therefore

(19):}. _.s... 2
G N 0,,

X

where AV is the number of coherence boxes in the entire sample.

For a one dimensional or two dimensional sample with dimensions
comparable to l“, the motion of a single scatterer is sufficient to
yield a conductance fluctuation comparable to the universal magnitude,
ezlh. The universal conductance fluctuations due to the motion of
scatterers are dynamic, in contrast to the static universal conductance

fluctuations observed as a function of magnetic field.

_._2.3 Characteristic Magnetic Field Scale

Figure 2.3 (a) shows a magnetofingerprint of a Bi sample with
dimensions 3 tan long, 0.15 ran wide, and 10 nm thick at 0.8 K. The
magnetofingerprint can be characterized by two quantities, the amplitude
of the conductance fluctuations and the characteristic magnetic field
scale. This can be seen most easily by calculating the autocorrelation

function:

C(AB) = (66(8) 6G(B + AB))

15

where the bracket indicates an average over the magnetic field 13. A

schematic diagram of C(AB) is shown in Figure 2.3 (b). When AB=O,
the correlation function C(AB) is equal to the variance,<5G2 >. As
AB increases , the fluctuations in 56 will tend to vary randomly in

sign, and on average, the product will tend to zero at large AM}. The
magnetic field where C(AB) drops to half its maximum value is called
the "characteristic magnetic field scale", ta. la is the typical

spacing of the peaks and valleys in 50(3).

An estimate of It can be obtained by the following argument: the

largest areas between Feynman paths that interfere quantum-mechanically

are of order La. Putting half a flux quantum through that area will

totally alter the interference between those paths. Hence in a two

dimensional sample, B, as (Do / L2 .

2.3 Defect dynamics

If defects in the metal are mobile, they can cause conductance
fluctuations in time. The most commonly believed general mechanism for
1/ f noise in metals is that of defect motion. Such motions can be due
to thermal activation of defects at high temperatures, or to quantum
tunneling of defects through potential barriers at low temperatures.

[12]

Dutta, Dimon, and Horn (DDH) showed that low frequency 1/ f noise in

metals over the temperature range 100 - 500 K could be explained as
arising from a distribution of thermally activated motions of defects,
with typical activation energy scales for hopping in metals (about 1

eV). Feng, Lee, and Stone (FLS) proposed that the sensitivity of the

conductance to atomic motions should give rise to l/ f noise through the

16

10]

UCF mechanism at low temperatures.[ In disordered metals at low

temperatures, l/f noise arises from tunneling defects with a broad

distribution of potential barrier heights and tunneling distances. A
more detailed discussion of l/ f noise is given in Appendix A.

In the mesoscopic regime, a connection between quantum tunneling
of a single defect and conductance fluctuations offers the chance to
study microscopic phenomena which have previously been studied only in

the ensemble average.

2.3.1 Defect Dynamics in Different Temperature Regimes

As described above, the dominant dynamics of the defects at high

temperatures are thermally activated hopping. A single defect hops with

a characteristic time scale 2', where 1’: 2'0 exp(E/kB T), where E is the

13
I 1 A single double-well

activation energy for making a transition.
potential can be characterized by the energy barrier E3, and the energy
difference between the ground states in each well, called the energy
asymmetry, s.[14] Figure 2.4 (a) shows a schematic diagram for the
activation process of a defect over the barrier. At high temperatures,
this thermal activation behavior of the defects results from the
interaction between the defect and the thermal reservoir of electrons
and phonons.

At low temperatures, on the other hand, the defect kinetics are
tunneling dominated. In a highly disordered material, one expects the
number of active tunneling defects to depend approximately linearly on

[15]

the temperature. In a mesoscopic sample, the conductance

fluctuations due to a single defect can be resolved.

17

When kBT << 1100, where (00 is the smallest undamped single-well
frequency, the defect tunneling is confined to the lowest states in each
well. This quantum system is called a two level system (TLS). In a
metal, the temporal behavior of a TLS illustrates the main features of

16
dissipative quantum tunneling,[ ]

which will be discussed in the next
section. Figure 2.4 (b) shows the tunneling behavior between the lowest
states in the double-well potential.

For temperatures k,7§>han, the tunneling mechanism of a single
defect is not necessarily described by a simple model. One of the
plausible features is a dominant coupling with the lattice phonons. In
this regime, phonon energies are much smaller than the potential barrier
height, but larger than a%. The defect in a well can therefore make
transitions to the excited state and tunnel through the barrier to
complete the transition to the other well. This process is called
”phonon-assisted tunneling". The tunneling rates rapidly decrease with

decreasing temperature due to the reduction of transitions to the

excited states.

2.3.; Theory of Dissipative Quantum Tunneling

Consider a single defect that moves in an asymmetric double-well
potential with a small bias energy 6 between the two ground states
located at izqo/Z in the defect configuration space. For the case of a
nearly-isolated defect (i.e. only weakly interacting with the

environment), the defect motion is described by the Hamiltonian

18

 

where s is the energy asymmetry, A0 is the bare tunneling matrix element

 

. , , 2mV0
given by the WKB approximation as A0 = 0),, exp - ”2 5r , and the 0', are

Pauli spin matrices. The tunneling dynamics are, in this case,
coherent, and the energy eigenstates are linear combinations of the
states in each well.

For a metal at temperature khTW¢<ha%, the defect tunnels

dissipatively, interacting strongly with the bath of conduction
electrons. The main features of dissipative quantum tunnelingIlG] are:
(1) incoherent tunneling of the defect, arising from the continuous
dephasing of the defect wavefunction by the strong interaction with the
electron bath: (2) the renormalization of bare tunneling matrix elements
(tunneling matrix element without the interaction with the bath) due to
the interaction of the defect with the high-energy (adiabatic) electron-
hole excitations, and (3) the low-energy (non-adiabatic) electron-hole
excitations directly affect the tunneling dynamics, leading to a

striking temperature dependence of the tunneling rates; for k375>8, the

rates increase with decreasing temperature, following a power law,

a- 17 , , , ,
7f, 7,~T2 'I l, where a is a coupling parameter which decribes the

coupling strength of the defect to the bath excitations. From a
theoretical calculation, one has O<:aw<1/2 for metalslla].

The theoretical approach to the dissipative quantum tunneling
problem proceeds as follows: if the barrier height between the two wells

is much larger than the small undamped oscillation frequency, an, in

either well, the bare tunneling rate Ao satifies Ao << me. If, in

l9

addition, kaVa<ha%, then the double-well problem can be reduced to an
effective, biased two-level system.
The bath of electron-hole excitations can be separated into two

classes, which play two quite different roles. The electron-hole

excitations of frequency greater than the cut-off frequency, a5, are

[19]

able to follow the tunneling defect adiabatically. This leads to a

renormalization of the tunneling matrix element A through the relation

A=Ao(a),/a)o)“. (2.1)

The remaining electron-hole excitations with frequency smaller than an

can not follow the defect adiabatically: they are directed toward the
center of the double well. This non-adiabatic character of the bath
affect the tunneling dynamics of TLS. The assumption of the system
being reduced to the effective TLS and the argument of high energy

electron-hole excitations give rise to a truncated Hamiltonian, given by

H=£a —£Aa +0 ZG.(b.+b1’)+thw.bIb.
2 z 2 x z, I J J - J J J
J J
[20]

Caldiera and Leggett showed that the influence of the low energy
electron-hole excitations on the tunneling defect can be modelled by a

bath of harmonic oscillators. The third term describes the interaction

between the defect and the bath, where bf and by are creation and

annihilation operators of the harmonic oscillators, and the (a are

coupling coefficients between the defect and the harmonic oscillators.

20

The fourth term is the Hamiltonian of the harmonic oscillator bath
itself.
The effect of the bath on the defect dynamics depends only on the

21
spectral density[ ]

J(w)=-,2—,XG,26(w—w,)

I

For a metal, the spectral density in the continuum limit takes the form
J(a))= awlzz]. The coupling of the defect to the bath is characterized
in terms of the single parameter (I.

In obtaining the tunneling of the defect in the presence of ohmic
dissipation due to non-adiabatic electrons, it is also necessary to
consider the exclusion of the participation of excitations of arbitrary

[19]

low energies. Since the defect is not stationary in either well,

but spends a finite time 1; in each well, electrons having energies

smaller than w<1/ 1', do not influence the tunneling of the defect.

Therefore, the energy of the non-adiabatic electron-hole excitations has

not only the upper bound an but also the lower bound A». From this

consideration of the lower bound on the energies of the electron-hole

excitations, the tunneling matrix element is renormalized again and the

renormalized tunneling matrix element A, is obtained as:
A, = A (A/m,)“"““’ (2 .2)

If 72A, >> 6‘, akBT, the effects of defect coupling to the electron bath

are relatively weak and can be treated as a small perturbation on the

21

dynamics of the defect. In this regime, the motion of the defect is

[23]

expected to be a coherent oscillation. If, on the other hand,

hA, << 8, akBT, then the coupling of the defect is so strong that the
rapid fluctuations of the electrons act to break the phase of the
tunneling defect so that the tunneling becomes incoherent. The dynamics

of the tunneling transitions are characterized by the occupation

probability of the defect, P(t)=( O',(t) ), given that for (<0, the

18
particle is localized in the +1 state. According to Leggett et.al[ 1,

the calculation of IT!) is based on the consideration of the path
integral over four states (0', =il, 0", =:l:l) and transitions between the
initial and final diagonal states (O,==O;, referred as sojourns) and
the off-diagonal states (O,==-Cfi referred to as blips). Figure 2.5
shows a simplified graphic expression of sojourns and blips. Following

their calculation of IT!) for the double-well potential with energy

symmetry 8, one gets:

P(t)=-tanh[ 8 ]+[1+tanh[ a He‘”
2kBT 2kBT

 

 

 

and a total transition rate of [16'24’25]
2 1cosh —6;— 2
A 27rkBT 0" 21:32" a
7:4 —— F a+i—— (2.3)
2 ”A, ”20) 27rkBT

where ITxHFDO is the complex gamma function. The above equation

describes incoherent relaxation with a total transition rate 7. For a

22

single TLS, the energy asymmetry s is related to the ratio of the

average time spent in each state, averaged over many transitions. This

must obey the principle of detailed balance: 7f/}g==edhr. Here, 7,,

7;, are the fast and slow transition rates, or the reciprocals of the

lifetimes of the excited or ground states, respectively. The total rate

equals the sum of the fast and slow rates, 7==7f-t7,. From detailed

balance, we finally have

 

 

 

a— 2
_ fl ZflkBT 2 ' 98mm. 1.. . 6‘
4 12A, r(2a) 2am“
7.. : 7fe-s/kfl' (2 ,5)

[13]

For 0< a<1/2 , there are two regimes depending on the

relative magnitude of T and a. When kBT>£, the tunneling rates

1[17]

increase with decreasing temperature, 7,,7g~]flw‘ , which is a

characteristic feature of dissipative quantum tunneling. When kBTW<£,
the tunneling rates follow a simple picture of stimulated absorption and
spontaneous emission. The faster rate 7, is determined by the sum of
the spontaneous and stimulated emission rates, while the slower rate is
determined by the stimulated absorption rate. As temperature is
lowered, the faster rate is roughly temperature independent and the
slower rate decreases rapidly as efluhr. Both of these temperature

regimes are observed in the experimental data and are discussed in

Chapter 4.

23

References

99°89

10.
11.
12.
13.
14.
15.

l6.
17.
18.
19.

”Solid State Physics", N. W. Ashcroft and N. D. Mermin, Holt, Rinehart and Winston
1976.

For review, see G. Bergmann, Phys. Rep. 101, l (1984).

P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622 (1985).

B. L. Al'tshuler, Pis'ma Zh. Eksp. Teor. Fiz. 41, 530 (1985) [JEPT Lett. 41, 648

(1985)].

B. L. Al'tshuler and D. E. Khmel'nitskii, Pis'ma Zh. Eksp. Teor. 42, 291 (1985).

[JEPT Lett. 42, 359 (1985)].

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

Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).

A. D. Stone, Phys. Rev. Lett. 54, 2692 (1985).

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

S. Feng, P. A. Lee and A. D. Stone, Phys. Rev. Lett. 56, 1960, 2772(E) (1986).

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

P. Dutta, P. Dimon, and P. M. Horn, Phys. Rev. Lett. 43, 646 (1979).

P. Dutta and P. M. Horn, Rev. Mod. Phys. 53, 497 (1981).

M. B. Weissman, Rev. Mod. Phys. 60, 537 (1988).

See "Amorphous Solids, Low Temperature Properties", ed. W. A. Phillips, Springer-
Verlag, New York, 1981.

A. J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987) and refs. therein.

J. Kondo, Physica 124 B, 25; 125 B, 279, and 126 B, 377 (1984).

K. Yamada, A. Sakurai and M. Takeshige, Prog. Theor. Phys. 70, 73 (1983).

Y. Kagan and N. V. Prokofev, Zh. Eksp. T eor. Fiz. 90, 530 (1985) [JEPT 63,1276
(1986)].

24

20. A. O. Caldeira and A. J. Leggett, Ann. Phys. (NY) 149, 374 (1983).

21. H. Grabert, Phys. Rev. B, 46, 12753, (1992)

22. S. Chakaravarty and A. J. Leggett, Phys. Rev. Lett. 52, 5 (1984).

23. P. Hanggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys, 62, 251 (1990).
24. H. Grabert and U. Weiss, Phys. Rev. Lett. 54, 1605 (1985).

25. M. P. A Fisher and A. T. Dorsey, Phys. Rev. Lett. 54, 1609 (1985).

25

 

(a)

 

 

 

(b)

Figure 2.1 (a) Schematic diagram of the scattering of the conduction electrons off defects,
impurities, lattice vibrations and other electrons while they traverse the sample.

(b) Simplified graph of resistivity versus temperature. Due to the decrease of inelastic
scattering with decreasing temperature, the resistivity reaches the residual resistivity.

impurity

detect

26

 

 

 

 

 

 

 

 

 

 

 

Figure 2.2 (a). The phases of electrons are completely altered by applying a magnetic
field, leading to the static conductance fluctuations, called the "magnetofingerprint". (b)
By the motion of a single defect, the phases are also changed. The motion of a single
defect can induce a conductance fluctuation of order e2 / h.

27

0.03 q-

0.02 -4

0.01 ..

 

0.00 ‘i

ill/R

.o.01 '-

-0.02 -

U

L l j

03:::.L:::-:‘rt:uit.
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.
B-fieldCI'esla)

(a)

 

I

C( As)

 

 

 

(b)

Figure 2.3 (a) Relative resistance fluctuations, 6 R / R = -5 G / G, as a firnction of
magnetic field (magnetofingerprint) for a Bi wire with dimensions of 3 pm long, 0.15 pm
wide, and 10 nm thick. The temperature is 0.8 K. (b) Schematic diagram of the
autocorrelation firnction C ( AB) versus magnetic field. The magnetic field where C ( AB)
drops to half its maximum value is the characteristic field scale, B,

28

Potential
Energy

 

 

 

 

1 2 Defect configuration

Potential 1
Energy

 

 

 

 

 

 

1 2 Defect configuration

(b)

Figure 2.4 (:1) Thomas] activation hopping of the defect over the potential barrier at high
temperatures. (b) At low temperatures (Ir,T < 1101,), the dynamics of the defect are
dominated by tunneling through the barrier between ground states in each well.

29

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

» _J

 

? _. H i i _.
t=0 t1 t2 t3 t4 \\ i // tzj-r t2' t2j-r-1

"Sojourns" jtll sojourn

Figure 2.5 Graphic expression for the probability function P(t). The "sojourn"
represents no change in the state a', , between the initial and final position of the defect
after it performs tunneling. The "blip” represents a change in the state during the
tunneling process.

30

Chapter 3

Experimental Methods

3.1 Sample preparation

The samples for this experiment are polycrystalline Bi films 20 nm
thick, prepared by thermal evaporation at room temperature onto oxidized
Si substrates at a pressure of about 1x10.-6 Torr. The deposition rates
were about 1 nm per second. The films were patterned into five-terminal

[1'3]

devices using electron beam lithography. Each arm of the sample
has nominally identical sample regions with areas of 5x10.2 um
(linewidth of 0.1 um and length of 0.5 pm). Figure 3.1 shows a Scanning
Electron Microscope (SEM) picture of a sample and contact pads patterned
by electron beam lithography. Each arm of the sample has a resistance
of about 1 kCL

Electron beam lithography was used to pattern samples with
linewidths of 0.1 um. Operation of the ISI SC-40A Scanning Electron
Microscope with a magnification of 15,000x yielded the most reliable
and reproducible results. Since the field of view of the SEM is limited
to 70 pm at this magnification, the largest attainable width of a pad
and the spacing between each pad for five-terminal configuration from
electron beam lithography are about 10 um and 20 pm, respectively.
Therefore, it is impossible to make electrical contact between wires of
diameter 25 um and a sample through contact pads patterned only by

electron beam lithography. For this reason, we needed both optical and

electron beam lithography for patterning the sample and contact pads.

31

We made primary and secondary contact pads to connect the sample
to wires, for measurement of the electrical resistance of the sample.
Primary contact pads, patterned by optical lithography, were thermally
evaporated gold of thickness 100 nm. The secondary pads, patterned
together with the sample by electron beam lithography, were thermally
evaporated bismuth 20 nm thick. Figure 3.2 shows SEM pictures of the
primary contact pads patterned by optical lithography, and the sample
with the secondary pads patterned by electron beam lithography. To make
the connection between the primary and the secondary pads, we used a
wire shadow technique to make gentle slopes on the edges of the primary
pads so that the secondary pads could be connected smoothly. Figure 3.3
shows a simplified schematic diagram of profile of the wire shadow
technique. The details about patterning using lithography are given

below.

3.1.1 Optical Lithography

Optical lithography, also called "photolithography", is the most
common commercial method defining the geometry of electronic devices.
Figure 3.4 shows a schematic diagram of the lithography process. We
used positive photoresist (Shipley 1800), which is a photosensitive
polymer that undergoes a chemical change called ”depolymerization" when
it is exposed to ultraviolet irradiation. After exposure, the
depolymerized area can be dissolved away with the appropriate developer,
leading to the generation of the desired pattern.

The photolithography used to make patterns in our experiment

consists of two processes. The first process is called ”contact

32

lithography” because the mask for patterning is in firm contact with the
photoresist when it is exposed to the ultraviolet light for
depolymerization. The process steps are outlined in Table 3.1. The
thickness of the spun-on resist layer is about 1.0 - 1.4 pm. The photo-
reduced masks are made from high resolution negative films, and are
pressed firmly against the resist upon exposure. The exposed substrate
is immersed in chlorobenzene to produce a good undercut profile which
results from a slower developing rate of the chlorobenzene-smeared upper
part of the resist. Additional baking for a short period of time to
remove solvent from the resist after immersing is helpful in developing
to obtain the desired pattern. Making a good undercut profile is very
important in lithography because it avoids the trouble of the metalized
pattern being taken off the substrate in the lift-off process. The
resolution of the contact pattern is about 10 um.

If the dimensions of the pattern are near the resolution of the
contact lithography, it is not possible to obtain reliable and
reproducible patterns. This problem can be overcome by optical
projection of the mask onto the resist using an optical microscope.
Projection lithography has the same steps as contact lithography and can
produce patterns whose dimensions are much smaller and edges are sharper
than those of contact lithography. The resolution of projection
lithography is better than 1 pm. In our experiment, we used a
combination of these two processes through the partial development of
the contact pattern, followed by projection exposure and full
development. The final pattern has contact leads close enough to
overlap with the sample and the secondary pads patterned by electron

beam lithography, to be described below.

33

3. l .2 Electron Beam Lithography

Electron beam lithography is an ideal research tool to define
patterns with submicron resolution because of the high quality of
electron optics, the availability of high brightness sources to form
beams of ultra small diameter and the ability to scan the beam under
computer control to define desired patterns. In electron beam
lithography a finely focussed beam is scanned in the desired pattern
over an electron beam resist which depolymerizes upon electron
irradiation. Even though electron optics and electron sources are of a
very high quality, and focused beams less than 1 nm in diameter can be
generated, it is not readily possible to define a structure on the order
of the electron beam size. The resolution of the process is limited by
the range of scattered electrons and the intrinsic graininess of the
electron beam resist.

When the highly focused, energetic electrons are incident on a
solid they undergo scattering within the specimen. Electron scattering
is divided into two categories, elastic and inelastic scattering.[4]
When elastic scattering occurs, less than 1 ev of energy is transferred
from the beam electron to the specimen; this is negligible compared to
the incident energy, which is typically 10 kev or more. The electrons
are, however, deviated from their incident path as a result of
collision. Elastic scattering results from collisions of the energetic
electrons with the nuclei of the atoms, partially screened by the bound
electrons. A significant fraction of the electron beam emerges from the
specimen after having gone through a sequence of elastic scattering
events in which the not change in direction is sufficient to be

backscattered. (Since the thickness of the Silicon substrate is about

34

400 um, it is unlikely for the electrons to pass through the substrate.)
These emergent beam electrons are collectively known as backscattered
electrons. Due to large angle scattering of these backscattered
electrons, the scattering volume of the backscattered electrons is
widely spread. Elastic scattering is more probable in high atomic
number materials and at low beam energy.

During an inelastic scattering event, energy is transferred to the
target atoms and electrons, and the kinetic energy of the beam electron
decreases. There are a number of possible inelastic scattering
processes, but one of the principal processes under consideration here
is the excitation of conduction electrons leading to secondary electron
(low energy) emission. The interaction of the beam electrons with the
solid can lead to the ejection of loosely bound electrons from the
conduction band and these ejected electrons are referred to as secondary
electrons. The secondary electrons have a short range scattering effect
and are largely responsible for the determination of the resolution in
low atomic number materials like polymeric resist.

Polymethl Methacrylate (PMMA) is one of the prime organic resists
for electron beam lithography. This positive resist undergoes a
chemical change during electron bombardment which renders the material
sensitive to developing in a suitable solvent. The developing rate is
controlled by the electron dose, e‘lcnf, and hence developing for
increasing time periods forms contours of electron energy deposition.

4
I l, which can be understood

This contour defines the interaction volume
in terms of the characteristics of elastic and inelastic scattering,
described above. Figure 3.5 is a schematic of the electron interaction

volume. For a low-atomic-number matrix like PMMA, inelastic scattering

35

is more probable, and hence the electron tends to penetrate deeply
through the resist into the substrate with relatively little lateral
scattering initially to form the ”neck" region of the interaction
volume. The penetrating electrons lose energy and elastic scattering
becomes more probable at lower energy. As a result of elastic
scattering, electrons deviate from their initial direction of travel and
the lateral scattering contributes to the formation of the "bulbous"
region of the interaction volume. This naturally allows us to have a
nice undercut profile. However, the generation of the low energy
secondary electrons results in a cylindrical exposed volume around the
incidence of the beam leading to the broadening of the line width.
These are fundamental problems in determining the ultimate resolution,
but such effects of electron scattering can be reduced by using high
energy electrons. The effect of high electron energy is to narrow the
forward scattering distribution and to penetrate through the resist into
the substrate deeply enough to smear out the backscattering so that the
exposure of a given feature is less influenced by the exposure of its
neighbours via the proximity effect.[5]
In our experiment, we fabricated Bi wires with widths as narrow as
100 nm using a bilayer resist process for electron beam lithography.
The bilayer resist process uses a bilayer consisting of a layer of PMMA
and a layer of copolymer, MMA (methylmethacrylate / methacrylic acid).
For high resolution, PMMA is used as the imaging layer and placed at the
top, and the copolymer which is more sensitive to electron beam is used

6
as the bottom layer.[

The processing steps are outlined in Table 3.2.
P(MMA/MAA) (KTI I-9 %) is spun on the substrate to produce the bottom

layer about 500 nm thick. PMMA(KTI 496-2%) is spun on top of the

36

copolymer, leaving a layer 200 nm thick. High resolution patterning can
be achieved through the use of thin PMMA, but good planarization over
the contact pads and better undercut profile are best achieved through
the use of a thicker copolymer. Figure 3.6 shows an SEM picture of the
profile of the electron interaction in PMMA/MMA bilayer, and a schematic
diagram. Electron beam patterning is done with a modified ISI SC-40A
SEM, which operates at 30 kev, with the beam position and a beam blanker
controlled by a personal computer through a Digital/Analog converter
interface. A computer program written in QUICK BASIC was used to

control the speed of electron beam writing. The writing speeds were

12pm/sec for the sample and 18 ,um/sec for the contact pads. The
electron beam current was fixed at I==L81u4.

In the SEM, electromagnetic lenses are used to focus the electron
beam by the interaction of the electromagnetic field of the lenses on
the moving electrons. Commonly used electromagnetic lenses are axially
symmetric condenser lenses which consist of windings that induce a
magnetic field in the iron core. The condensor lens in combination with
the aperture in the probe-forming lens (or objective lens) determines
the current in the final probe spot. Although perfect magnetic lenses
are assumed to have perfect axial symmetry, machining errors and
inhomogeneous magnetic fields within the iron core or asymmetry in the
windings cause loss of symmetry, leading to astigmatism. The effect of
astigmatism is to produce a stretch of the image in two perpendicular
directions. One corrects for the astigmatism in the final objective
lens by adjusting the stigmator. The stigmator supplies a weak
correcting field to produce the desired symmetrical magnetic field. The

stigmator has two major controls, one to correct for the magnitude of

37

the asymmetry and one to correct for the direction of the asymmetry of
the main field. This correction is very important for the patterning of
ultra-small samples and should be done prior to the final focusing onto
the resist. We used a thin Au film, 50 nm thick, deposited onto a glass
substrate as a specimen for correction of astigmatism. After
astigmatism was corrected, we used silver paint dashed slightly on the
surface of the resist close to the sample site as an object to focus on.
Since a small silver cluster at the edge of the silver paint doesn't
have a noticeable difference in height from the resist, it provides a
nearly perfect focus for patterning.

After exposure, the resists are developed at one time with the
same developer [1:3 methyl isobutyl ketone:isopropanol (MIBK :IPA)] at
23 °C for a minute. An argon ion milling cleaning is used prior to
metallization. In ion milling the Ar ions bombard the surface of the
resist including that of the patterned area, and remove any resist
residue after chemical reaction with developer. After the
metallization, the resist surrounding the sample is lifted off by
immersing the metallized pattern into a solution of 4:1 methylene
chloride and acetone. The resolution of the SEM is about 6 nm in its
best operating conditions. With the bilayer process and the
considerations made above, we've been successful in fabricating single
lines as narrow as 0.06 pm. However, since the real sample is patterned
with secondary pads consecutively, we could not totally exclude the

proximity effect. Line widths of our samples came out to be about 0.1

pm.

38

3.2 Electronics

Figure 3.7 shows a simplified schematic diagram of the electronics
for our experiment. The five-terminal sample configuration allows the
sample to be used as two arms of an ac Wheatstone bridge.[7] With this
technique relative changes in the resistance of the two arms can be
measured with high sensitivity.

The differential voltage across the bridge due to the change of
the resistance of either arm is amplified through a preamplifier
(Stanford SR 550), phase-sensitively detected with a lock-in amplifier
referred to the drive frequency, and the output connected to digital
instruments (analog/digital converter, low pass filter and personal
computer) located outside the shielding room. The analog/digital
converter is for converting the analog voltage output from the lock—in
amplifier to a digitized one. The low pass filter is used to remove any

unwanted high frequency noise. When a continuous function is sampled

with a sampling interval.A, its Fourier transform is defined only

between the frequencies —1/ 2A < f <+l/ 2A. This boundary frequency
f,=l/ 2A is called the "Nyquist frequency". If an original function
that is not bandwidth limited is sampled, then, all of the power
spectral density which lies outside of the frequency range -j£<:f“<}@
is moved back into that range. This effect is called ”aliasing". Any
frequency component outside this frequency range is aliased (falsely
transformed) into that range by the act of discrete sampling, leading to
incorrect Fourier transform of the function. Aliasing can be eliminated
only by low pass filtering the original function before sampling. The
low pass filter used in the experiment is a Krohn-Hite eight-pole

butterworth filter which produces the flattest passband (the region of

39

frequencies that are relatively unattenuated by the filter) response and
has steepness in the transition region from passband to the stopband
(the region of significant attenuation). The order of the filter is
eight-pole. Increasing the number of poles flattens the passband
response and steepens the stopband falloff. It is also used for
rejecting line frequency harmonics and higher frequency resistance noise
sources in the sample.

The dilution refrigerator is located inside a Lindgren double-
electrically-isolated shielded room. The shielded room isolates the
refrigerator and electronics from rf (radio frequency) interference,
which could otherwise propagate down the sample leads into the
refrigerator and heat the electrons in the sample. This problem becomes
most severe as the sample size and temperature decrease. The high level
analog signals are filtered as they enter or leave the room. Only low-
level analog instruments (preamplifier, lock-in amplifier) are kept
inside the shielded room. Just inside and outside the shielded room,
there are ground breaking boxes to minimize ground loop interference.

The dilution refrigerator has a top loading probe on which the
sample and thermometer are mounted. Eighteen low-frequency electrical
leads are accessible to the sample region through the top loading probe.
The probe is loaded from the top of the dilution refrigerator and slides
down to the low temperature region. There are also electrical wires
dedicated to measure thermometers near the sample area. One Fisher
connector with eighteen electrical leads was used for thermometers of
the dilution unit (film burner, cold plate, still, mixing chamber, and 1
E pot) and for heaters of the mixing chamber, film burner and still.

All the electrical wires are twisted in pairs or triplets for rejecting

40

induced magnetic fields when current flows for measurement. For the
measurement of temperatures at each unit, home-built conductance bridges
which read the conductance of the thermometers are used.

Due to the small size of the sample extreme care is required in
handling the sample. Even a static charge from a human body is strong
enough to destroy the sample. Therefore, we added sample protection
guards which tie the five pads together in the optical lithography
process. The guards protect the sample from possible burnout during
sample mounting and when leads are attached to the substrate. The
guards are disconnected after the sample is silver painted to a pin
connector which connects the sample with the electrical leads on the top
loading probe. For more precaution, we always wore a grounded wrist

strap whenever working with the sample.

3.3 Dilution Refrigerator

Our low-temperature measurements were performed in an Oxford
Instruments top-loading helium-3/helium-4 dilution refrigerator in our
laboratory. Figure 3.8 shows a simplied schematic diagram of the

[8] of the dilution

dilution refrigerator. The principle of operation
refrigerator is based on the fact that a mixture of the two isotopes of
helium separates into two phases -- a concentrate phase which is rich in
helium-3 and a dilute phase which is rich in helium-4 --when it is
cooled down below a tricritical temperature of 0.86 K. This phase
separation occurs in the mixing chamber and cooling is produced there by

evaporating helium-3 from the concentrate phase across the boundary to

the dilute phase. Helium-3 is pumped away from the still by a room

41

temperature vacuum pump. The still is maintained at a temperature of
0.6 to 0.7 K by the still heater. The helium-3 is compressed to few
hundred milibar and returned to the condenser attached to the 1 K pot
where it is precooled and liquified through a high flow impedance. It
then passes through the heat exchangers and finally arrives at the
mixing chamber. The helium-3 leaving the mixing chamber due to the
osmotic pressure between the mixing chamber and the still is used to
cool the returning flow of concentrated helium-3 in the heat exchangers.
By continuous circulation of helium-3, temperatures as low as 20 mK can
be achieved. Since the cooling power of a dilution refrigerator is
directly proportional to the flow rate of helium-3, a large cooling
power requires very powerful pumps and wide pumping lines. The samples
are loaded into the dilute helium phase in the mixing chamber,
simplifying thermal grounding. The refrigerator is equipped with a 9-
Tesla superconducting magnet for measuring magnetoresistance and
magnetofingerprints. To keep the field low near the mixing chamber (for
thermometry or other experiments) the magnet has compensation coils.

The top loading probe slides into the refrigerator on a special
'0' ring assembly mounted on top of the vacuum lock. Before inserting
the probe, the vacuum lock assembly is pumped using the diffusion pump.
When loading a sample into the refrigerator, circulation is stopped, and
the mixture is left in a state where it is being cooled by the
refrigeration power in the 1 K pot. To introduce the sample, the gate
valve is opened and the assembly is lowered slowly into the dilution
unit. The lower section of the probe is cooled by the mixture as it
passes through the insert and the dilution unit. When the top loading

probe is screwed into position, the rotary pump and booster pump are

42

used to begin circulation of the helium-3 and the dilution unit starts

to cool.

3.4 Temperature Calibration of Thermometer

To achieve adequate temperature stability, it was crucial to
control the temperature in performing our experiments. A temperature
control program was written in Quick Basic and used to control the power
supply that supplies current to a heater wrapped around the glass tail
of the mixing chamber. With this setup, we controlled the temperature
to within 1 % all through the temperature range 0.05 - 1 K. Above 1 K,
we controlled the temperature via the pumping speed on the 1 K pot.

Due to the magnetoresistance of the carbon thermometer which
measures the temperature of the sample, it was required to calibrate the
carbon thermometer as a function of magnetic field. The
magnetoresistance of the carbon thermometer was measured at seven
different temperatures. At each temperature, the values of
magnetoresistance were measured twice at ten different values of the
magnetic field up to 7 Tesla, once while increasing the field and
immediately after, while decreasing the field. The two results were
averaged. The ratio of magnetoconductance to the value of conductance
at zero field shows a sharp increase up to 0.5 Tesla and a slow decrease

thereafter. We fit our data to the function:

.A’L

M =1.00+gfl°(b(7)— B+d(T)BZ) eam/B _1

G(T,0) B

43

where a(T), b(T), C(T) and d(T) are polynomials in the zero-field
conductance, C(T ,0).

A computer program was used to obtain the corresponding C(T ,0)
value from the given value of G(T,B) measured in the applied magnetic
field. The corrected temperature was obtained from G(T ,0) and the
known temperature calibration of the carbon thermometer. The details

about the fitting procedure are described in Appendix B.

44

References

l.
2.

G. J. Dolan and T. A. Fulton, IEEE Electron Device Lett. 4, 178 (1983).

D. J. Bishop, J. C. Licini, and G. J. Dolan, Appl. Phys. Lett. 46, 1000 (1985).

M. J. Rooks, S. Wind, P. Mchen, and D. E. Prober, J. Vac. Sci. Technol. B 5, 318
(1987)

"Scanning Electron Microscopy and X-Ray Microanalysis", ed., J. I. Goldstein et al.,
Plenum, New York 1981.

K. H. Nicholas, R. A. Ford, H. E. Brockman, and I. J. Stemp, J. Vac. Sci. Technol.
B l, 1020 (1983).

R. E. Howard, E. L. Hu, and L. D. Jackel, and P. M. Mankiewich, IEEE Trans.
Electron Devices ED-28, 1378 (1981).

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

See "Experimental Principles and Methods Below 1 K", O. V. Lounasmaa, Academic
Press, London and New York 1974.

45

Table 3. 1 The processing steps of optical lithography.

(1) Clean an oxidized silicon substrate with acetone, methyl alcohol and DI (deionized)
water.

(2) Spin photoresist at 5000 rpm for 30 seconds.
(thickness is 1.0 pm)

(3) Bake for 30 minutes in air at 95 °C.

(4) Expose with UV (ultraviolet) light using contact mask.

(5) Immerse in chlorobenzene for 10 minutes.

(6) Bake for 10 minutes in air at 75 °C.

(7) Develop for 45 seconds in developer at 23 °C for partial developing.

(8) Clean the substrate with DI water.

(9) Expose in the light source in the optical microscope using projection mask.

(10) Develop for l nrinute in developer at 23 °C.

(11) Evaporate metal.

(12) Liftoff by immersing into acetone.

46

Table 3. 2 The processing steps of electron beam lithography (bilayer technique).

(1) Clean an oxidized silicon substrate with acetone, methyl alcohol and DI (deionized)
water.
(2) Spin KTI I-9 % copolymer at 4000 rpm for 1 minute.
(thickness as 450 nm)
(3) Bake for 1 hour in air at 180 °C.
(4) Spin 496 K PMMA, 2%, at 5000 rpm for 1 minute.
(thickness z 200 nm)
(5) Bake for 1 hour in air at 180 °C.
(6) Expose with electron beam.
(7) Develop for 1 minute in 1:3 MIBKzlPA at 23 °C .
(8) Clean the substrate with EA and DI water
(9) Clean the substrate with ion milling to remove the resist residue.
(10) Evaporate metal.

(1 1) Lift-off by immersing in a solution of 4:1 methylene chloridezacetone.

47

one:

3
U
0
er
'1

 

 

 

1.0 pm
4———f>
v, v,
0 1 run
L L
f’ 1 \K
I 21 I

(b)

Figure 3.1 (a) SEM picture of a five-temrinal Bi sample patterned by electron beam
lithography. The dimensions of the sample are 1 pm long, 0.1 pm wide, and 20 nm thick.
(b) Schematic diagram of the sample with five leads.

48

585! v

 

Figure 3.2 (a) SEM picture of the primary contact pads patterned by optical lithography.
(b) SEM picture of the sample with the secondary contact pads patterned by electron
beam lithography. The primary contact pads and secondary pads are connected smoothly
by the wire shadow technique.

49

Source of evaporation

 

 

A
M
I i
I l
I I
ll 11
11 ll
11 H
ll ll
11 ll
11 H
H i!
ll H
l I
l i
l l

 

 

7M!” raw/rim

 

Figure 3.3 Schematic diagram of the wire shadow technique. The shadow of the wire
gives rise to gentle slopes on the edges of the primary contact pads. This ensures that the
thin secondary pads can make good contact with the primary pads.

50

SPIN RESIST
(PHOTO OR E-BEAM)

 

EXPOSURE
(UV OR E-BEAM)

DEVELOPMENT

 

METALIZATION

LIFT-OFF A\\\\\\\\\\\\\V

 

 

 

 

Figure 3.4 Schematic diagram of the lift-ofl' process. The process applies to both optical
and electron beam lithography with different types of the resist and different source of
exposure. The resolution of optical lithography is about 1 pm, and 10 nm for electron
beam lithography.

51

incident beam

 

 

 

Electron beam resist

 

0' .‘e
0' ‘s
I s
I

Backseattered ,0 /
Electrons ;' ‘r

K ) substrate

I I
0 O
‘ o'
|

I
\‘ '

Figure 3.5 Schematic diagram of the electron interaction volume. The pear-shaped
interaction volume is formed by the backscattered electrons.

fl

3Bkv leakx

 

PMMA
COPOLYMER

SUBSTRATE

 

(b)

Figure 3.6 (a) SEM picture of profile of the etched area as a result of electron interaction
with PMMA/WA (electron beam resist/copolymer) bilayer. The difference of sensitivity
to the electron beam between the two polymer layers makes a good undercut profile as
above. (b) Schematic diagram of the interaction volume in the bilayer.

53

..—----- ..-.._..---—c.—..-..-..-.._..—.._..—..-.._..-..-..—--—..-.._..-......_,._____,__‘_.._.H_' __

 

 

 

  
 

 

 

 

 

 

 

 

 

 

 

R
A RB
" " i Current leads i Inside dilution
i A 2 i refrigerator
: i Voltage leads E
L ------------- 1 ------ 4) ________________ J:

Low Noise Preamplifier
reference ,, R. F. filter
signal I Lock-in

| Amplifier

 

Shielded room ’ * Low-pass
AID converter filter

 

 

 

 

[Computer]

 

Figure 3.7 Schematic diagram of the experimental circuit. Each lead going into the
dilution refiigerator passes through a filter for rejection of interference. Sample size and
contact pads are not to scale.

 

54

Room Temperature

6— 6—

He4 pump ____| pump j.__

TI

 

 

 

 

T He3 Gas

 

  

 

 

 

 

 

 

 

 

 

 

 

l K Pot
1.2 K
Flow impedance ' i i I Still 0.6 K
, T
/
¢ Heat Exchangers
_é__ _
Concentrated — f — — ,
Stream _Z __ Drlute Stream
_ a _. _
é
/

 

 

 

 

Mixing Chamber
0.004 K

Figure 3.8 Simplified schematic diagram of dilution refiigerator

55
Chapter 4

Tunneling dynamics of a single defect

4.1 Time Trace of a Single Fluctuator

In our experiments, we measured the electrical conductance of
several Bi wires in a five-terminal configuration. In this thesis we
present data from a single defect in a single submicron wire. The
resistance jumps 5R across the bridge due to the motion of a single
defect in either arm of the sample, were digitized as a function of
time. Figure 4.1 (a) shows a typical time trace of the resistance of
the sample. In this wire, the magnitude of 513 is of order 0.282/h at
1 K and 1.0e2/h at 0.1 K, where ez/h is about 4x10"5 mho. Furthermore,
5(3 increased with decreasing temperature and varied in a random manner
with applied magnetic field, as expected from the theory of universal

1-3
I 1 In the experiment, the magnetic field

conductance fluctuations.
was varied very slowly from 0 to 7 Tesla to find the magnetic field
values at which the maximum conductance signal due to the motion of a
single defect was achieved. By the maximum signal, we mean the signal
in which the conductance has only two discrete values, with the total
noise from all other sources being much smaller in size than the signal
from the defect being pursued. Magnetic fields which contain mixed
signals from several defects were not used for analysis. We found four
different magnetic fields where the signal for a single fluctuator was

clean: B - 0.140 'r, 2.274 'r, 2.286 'r and 6.997 T. At each of these

magnetic fields, we measured the transition rates of the defect as a

56

function of temperature. For good statistics, several hundred
transitions of the defect were recorded and analyzed at each
temperature.

The experimental temperature range was determined by two factors.
Above about 2 K several defects were active, so it was difficult to
determine unambiguously the tunneling rates of a particular defect.
Below 0.1 K the signal to noise ratio was very small. Although 61}
increases as the temperature is lowered due to the increasing lg, the
drive current must be decreased even faster to avoid sample heating (See
chapter 5). We used a drive current of 0.7 nA at 0.1 K. The
conductance (measured as a voltage) was sampled by the computer at a
sampling frequency of either 20 or 50 Hz, depending on the fast

tunneling rate 7f of the defect. We recorded 8 to 12 files, each

consisting of 16,384 voltage measurements, for each temperature.

4.2 Data analysis

Raw data of conductance as a function of time were analyzed using
two independent methods. The first method consists of setting two
comparator levels to measure the transitions between the states, and
hence the dwell times in each state. The histogrammed dwell times are
fit to a single exponential decay function to find the mean dwell times,
t in each state and the transition rates, 7==T4. The second method is
a combination of two different fitting processes; fitting the power
spectrum of the time trace to a Debye-Lorenzian lineshape to find the

sum of the two transition rates, 7, and fitting the histogram of the

raw conductance values to the sum of two Gaussians to obtain the ratio

57

of the rates. These two analysis methods yield consistent values of the

transition rates in each state.

4.2.1 Transition Time Histograms and Setting Comparator Levels

The occurrence of the individual conductance jumps follows a
Poisson distribution, which is appropriate for a process described by a
constant rate per unit time interval. Figure 4.1 (b) shows two
comparator level settings in a time trace. We set the two comparator
levels between the two values of conductance corresponding to each
state. The dwell times in each state are found by the following scheme;
if the digitized input signal is below the lower comparator level, and
then goes above the upper comparator level, it is counted as a jump from
the lower state to the upper state. The dwell time in the upper state
for that particular jump is the time elapsed before the signal moves
down below the lower level. This algorithm is identical to a "Schmitt

r41,

trigger“ electronic circuit A computer program written in Quick
Basic was used for the counting and recording processes. Although most
of the time traces of the defect were very clean and bistable, there
were some cases when the effect of a second defect was noticeable.
However, since the second defect was much less active, and had smaller
51; than the dominant defect, it was possible to analyze the data by
breaking the time trace into several segments and setting different
comparator levels in each segment. In determining the two comparator
levels, we used the two peak values Ifi,I; in the distribution of raw

conductance values and standard deviation 0; given from fitting two

gaussians to a histogram of the conductance values (see the next

58

section). We set the comparator levels as V,+O‘, Vz-O' for Vl < V . This
method of setting the comparator levels worked in most cases, but manual
adjustment of the comparator levels was inevitable when the signal to
noise ratio became very small, at temperatures below 0.2 K and above 1.5
K.

Since the dwell times in each state are exponentially distributed,

they were histogrammed and fit to a single exponential decay function,

P(t)oce""', to find the mean dwell time in each state, If, 1,, and the
transition rates 7,,73, where 7i==fi4 (i=f or s). In the data-taking

process, we set the ratio between the sample frequency and the low pass

filter cut-off frequency to be / - 4 to reduce the contribution
sl'nple filter

of aliased signals (See section 3.2). We kept data with frequencies

f S f, where j; s 0.75 fmm. The filter frequency was either 5 or 12.5
Hz, depending on the sample frequency. Due to the filter, transitions
from one state to the other state faster than ~0.1 second were not
always detected. As a result, the first bins in the histograms in each
state were often less than they should have been. This could lead to
errors in determining the mean transition times in each state. To avoid
thess errors, we used a two parameter fitting program in which the total
number of transitions and the mean transition are parameters to be
determined. Figure 4.2 illustrates the fitting of histograms to a

single exponential to determine the mean transition time.

59

4.2.2 Debye-Lorentzian Fit to Power Spectrum and Two Gaussian Fit
to the Conductance Histogram

The power spectral density SbLf) for the single defect was
obtained using a fast Fourier transform (or FFT) of the raw data of
conductance. The fast Fourier transform is a fast computing algorithm
which converts the time series of the experimental measurements to the
frequency domain. The power spectrum is determined as follows: the FFT
converts each time series to a series of complex Fourier coefficients in
the frequency domain. The Fourier coefficients are squared and averaged
over many time series to obtain an estimate of the power spectrum Sbflfi)
in a narrow frequency range centered at_fi. Each data file has 16,384
voltage measurements and was divided into two groups for FFT. The
frequency range was 10.3 st f 510 Hz depending on the measurement time
and low-pass filter setting. At each temperature, the power spectrum
after background substraction was fitted to a Debye-Lorentzian function
to yield the sum of the two transition rates, y==7,-+7,. Figure 4.3
shows the fits of data to fitting function.

Since the histograms of the dwell times in each state show that
the transitions are totally independent random events characterized by a
single rate for each state, the distribution of raw conductance values
in each state is totally independent. Assuming that the distribution of
conductance values in each state follows a Gaussian distribution, the
overall distribution of raw conductance values is the sum of these two
independent Gaussians centered at peak values, I1,I’. Fitting a
histogram of the raw conductance values to the sum of two Gaussian gives

the area and peak values of each Gaussian and standard deviation (7.

The ratio of the two areas is the ratio of the transition rates, 7f/73.

60

The two gaussian fit to the distribution of raw of conductance values

with the two peak values VL I; and standard deviation.(7 are illustrated

in Figure 4.4. It shows good agreement between the data and the fitting
function of the sum of two Gaussians. The value of 7 obtained from the

fit to the power spectrum and the value of 7f/7g from the two Gaussian

fit, together determine both 7,,and 7;. The results are in good

agreement with those obtained from the first analysis method.

4.3 Detailed Balance and Energy Asymmetry a
As was described in section 2.3.3, the two transition rates are

related by the energy asymmetry a of the defect through the detailed

balance, 7f/)g==edhr. To demonstrate this, we examined the temperature

dependence of the rates at fixed magentic field. Figure 4.5 shows a

plot of ln(7f/ 7,) versus 1/ T for the four data sets, along with linear

fits through the origin. The good fits confirm the above relation. The

slopes of these lines give values for E/kB of 402, 213, 92 and 40 mK,

respectively, with a standard deviation of 110 mK.

4.3.1 Magnetic Field Dependence of Energy Asymmetry
It is obvious from Figure 4.5 that the energy asymmetry s is
strongly modified by the application of a magnetic field. The magnetic

field dependence of s was proposed theoretically by Al'tshuler and

5
I l and observed experimentally by Zimmerman, Golding and

6
Haemmerlel 1. The mechanism was explained as follows: the local

Spivak

electron density "(3) is very sensitive to a magnetic field due to

61

long-range interference from the scattering centers (see section 2.2.1).
Then, a charge coupling of the local electron density to the defect
would yield an interaction energy E(B)ocn(B). Since the fluctuation in
the local electron density due to the magnetic field is random and can
have equal probability of lowering or raising the energy asymmetry, the
interaction energy between the defect and the electons should be a
random function of the magnetic field. Therefore, the variation of the
energy asymmetry, 8, should also be random with the magnetic field.
since it will be shown later that the four data sets at different
magnetic fields come from the same defect, Figure 4.5 illustrates this
random variation of E with the magnetic field. In a previous work,

[5]

Zimmerman et al. found that 8 increased at low magnetic fields for

three defects studied. In our experiment, we found that the value E/kB

= 213 mK near zero field is greater than the values of 40 mK and 92 mK

at B - 2.274 and 2.286 Tesla, and that the value a/kb = 402 mK at B a

6.997 Tesla is the largest. This shows that the variation of a with the
magnetic field is random. Attention should be also made to the rapid
varation of 8 with the magnetic field, illustrated by its change from

40 to 92 mK when the field changed from 2.274 to 2.286 Tesla.

4.3.2 Characteristic F ield Considerations

The amplitude of the conductance fluctuation, 513(3), is a random

function of magnetic field, with a field scale indicating that it arises

from the same mechanism that causes the aperiodic magnetofingerprint

(3(3). The energy asymmetry, 8, on the other hand, can be strongly

affected by magnetic field, with a large increase in higher field, and

62

[5]

smaller reproducible random features . Therefore, the autocorrelation
function (6(3) 8(0)) does not have translational invariance over the
magnetic field, unlike the autocorrelation function for the conductance
fluctuations. Accordingly, it is more difficult to define a

characteristic field scale in this case.

4.4 Temperature Dependence of Tunneling Rates

In this experiment, the defect tunneling rates have been measured
as a function of temperature from 0.1 to 2 K. Figure 4.6 shows how the
transition rates for the defect vary with temperature at B a 0.14 Tesla.
With E/kb a 213 mK, it most clearly illustrates three distinct
temperature regimes. When kaV<£, 7} is roughly temperature

independent, while 73 decreases rapidly with decreasing temperature.

This behavior is in accord with the picture of spontaneous emission and
stimulated absorption described in section 2.2.3. As the temperature is
raised so that kb75>8, the rates cross over to a qualitatively
different regime. The ratio of the two rates still obeys detailed
balance, but both rates decrease with increasing temperature. This
behavior is an essential feature of dissipative quantum tunneling in
metals. Above 1.2 K, the rates increase rapidly, due first to phonon-
assisted tunneling, and eventually to thermal activation over the
barrier. The phenomenon of dissipative quantum tunneling is shown more
vividly in Figure 4.7. In B = 2.274 Tesla, we have the smallest energy

asymmetry E/kB of 40 mK, which is far below the lowest temperature

taken in this experiment. Therefore, at this magnetic field, the data

are always in the regime k375>8, and the rates decrease with increasing

63

temperature as a power law, 77“*. At B= 2.286 Tesla, we have E/kb = 92

mK, which is very close to the minimum experimental temperature. Figure

4.8 also illustrates three temperature regimes, but most of the rates

belong to the power law regime. The largest value of a/kb of 402 mK

for this particular defect is obtained at B = 6.997 Tesla, shown in

Figure 4.9. Due to the size of ElkB, more than half of the data are in

the regime kb75<8 at this magnetic field.

4.4.1 The Defect - Electron Bath Coupling Constanta.
and the Renormalized Tunneling Matrix Element A,

In section 2.2.3, the temperature dependence of the tunneling

rates 7,,79 were calculated in terms of the doubly renormalized
tunneling matrix element, A,, the energy asymmetry, s, and the coupling

constant, a, between the defect and the electron bath. For the case of

incoherent tunneling (hA,<3<kaW, the results for the tunneling rates

are from equations (2.4) and (2.5):

 

 

 

A, 27rkBT 2"" e‘m"T . s 2

7,=— — F a+t—— (4,1)
4 11A, F (2a) 27rkBT

and

7‘ : rfe-s/kfl' (4 .2)

For a defect tunneling in a metal, the coupling constant, a, is
constrained to lie between 0 and 1/2. Since the theory of dissipative

quantum tunneling describes the tunneling of the particle in the

presence of the conduction electrons, we limit quantative analysis to
the temperature range below 1.2 R, i.e . below the onset of phonon-
assisted tunneling. With the energy asymmetry, 8, given by the slope of

the detailed balance plot, the data were fit to the equations for 7,,7,

with two parameter least-squares fits to obtain the values for the
coupling constant, a and the renormalized tunneling matrix element, A,.
The agreement between theory and experiment is excellent for all
temperatures below 1.2 R. The theoretical fits to the data at each
magnetic field are illustrated in Figure 4.10 and 4.11. The values of
a and A” determined from the fits are given in Table 4.1. For fitting
the equations to the data, a program written in FORTRAN with the
subroutine called MINUIT in the VAX 4000 was used, incorporated with the
CERW library to calculate the complex gamma function. The parameters
are completely consistent with the dissipative tunneling model for

7-9
incoherent tunneling, i.e. , hA, << 6‘, akBT and 0 < a <1/2 .[ 1

Given excellent quantitative agreement between theory and
experiment at each value of the magnetic field, the coupling constant a
appears to vary with the magnetic field. Since the random variation of
s with the magnetic field is believed to be due to random fluctuations
in the local charge density near the defect, and since a also depends
on this charge density, it is plausible that a also varies with the

10
magnetic field. Golding et a1.[ ]

argued, however, that the change in
a due to the magnetic field is small because a depends on electron-hole
excitations with energies up to the Fermi energy whereas the magnetic

field affects low-energy processes only. Our attempt to make this

argument clear experimentally leads to some difficulty. The statistical

uncertainties in or. and log A, are only about 0.02 and 0.08,

65

]

11
respectively, estimated from contour plots of 231 . Since a varies

over the range 0.14 - 0.22 for the four data sets presented here, a
statistical analysis suggests that a does vary with magnetic field. We
present an alternative analysis below, based on a scaling fit of the

data, that suggests that.¢1 does not vary with magnetic field.

The parameters a and A5 are highly correlated through the
relation between A, and bare (unrenormalized) tunneling matrix element

A0. From equation (2.1), we have
A = Ao(w,/wo)“

where 0% and cg are the characteristic vibration frequency of the
defect in one of its two wells and the upper cut-off frequency of the
electron-hole excitations, respectively. .A is the tunneling matrix
element renormalized by the adiabatic approximation for electron-hole

excitations. From equation (2.2), we have

A, = A (A / 6980'“)

for the tunneling matrix element renormalized again due to the lower

bound for non-adiabatic electrons. Inserting equation (2.1) into

equation (2.2) yields the relation between Ar and A0:
A, = A0 (Ac/mo)“’("“) (4.3)

where the cut-off frequency, a), has dropped out. The values of d and

C

Ar determined from the four fits are plotted as log(A,) versus a/ l—a

in parameter space (Figure 4.12). It is found that they fall on a
straight line . This correlation is highly suggestive. Taking the

logarithm of equation (4.3), we have
log(A,) = log(A0) + log(AO / w0)-(a/ l — a) (4 . 4)

Assuming that a. varies with magnetic field, but A0 and mo do not,
equation (4.4) implies a linear dependence of log(A,) on a/l—a , with
an intercept and slope equal to log(A,) and log(A0/a%). With this
interpretation and from the plot of our values a and A,, we obtain the
results hAo/k, = 1.8x10-6 R and hag/kg = 0.23 R. While the first of
these values is reasonable, the second is not. Not only is 0.23 K
extremely low for a vibrational frequency (even for a defect that
tunnels at low temperature), but also such a low value violates the

condition k3750<hah upon which equation (4.1) is based.

A more likely reason for the observed correlation between a and

[11]

A, can be explained in terms of I? considerations. In the

parameter space discussed above, the least-squares fits for each value
of the magnetic field determine a line of minimum 1?. Even the
slightest change of a and A, off this line in the parameter space leads
to huge increase in 2?. However, as long as the change of a and A, is
along the line, the fits vary only slightly within the experimental
temperature range. This line of slowly varying 2? turns out to lie
along the same line connecting the four points in Figure 4.12. This

”coincidence” suggests that perhaps all four data sets can be fit with a

unique set of parameters a and A,.

 

67

4.4.2 Scaling Function

To test whether the data are consistent with single values of a

and A, , we plot all the data in the form 7TH“. From equation (2.4),

we have

 

2a—l COST! 8 2
2k,,T

r(2a)

 

 

F a+i g
[ anBT]

_fi 27rkBT
7 2 11A,

 

 

and

2

(4.5)

2a—1
7,“, : $[2nk3) cosh(l/2x)

2 1m, r(2a)

 

 

F(a+i;)
27rx

where x=kBT/£ is defined as a scaling variable. Since 7T Ha depends
on temperature only through the ratio kBT/ 6‘, we can write 7T H" = f (x),
where f(x) is a scaling function. In Figure 4.13, we plot 7(T/T,‘,)Hal

versus kBT/E, where T) = 1 K for or = 0.195. The remarkable consistency
between all these data sets suggests that or and A, are independent of
magnetic field for this defect. The solid line in Figure 4.13 is a
simultaneous least-squares fit to all four data sets, yielding the
values a=0.195 and A, =I.34xl04 S". Figures 4.14 and 4.15 show the
theoretical fits to the data of tunneling rates versus temperature, both
with the values of a and A, obtained at each magnetic field and with
the single values of a and A, given from the scaling plot. The fits

for the case of single values of a and A, to the data are nearly as

good as the other case.

68

The results show that the theoretical fits depend only weakly on
a over the experimental temperature range. To determine the dependence
of a with magnetic field more accurately, we would need either a larger
experimental temperature range, or smaller experimental uncertainties.
We have attemped to minimize obvious sources of systematic error. We
were very careful to limit Joule heating of the sample from the
measurement current. (The tunneling rates provide an excellent measure
of the defect temperature, through the detailed-balance relation.) The
effect of external interference was minimized by rf filters on all

[12]

electrical leads to the sample. It is possible that other defects

in the sample may influence the dynamics of the defect we are

13
studying.[ ] Even small changes in the measured tunneling rates can
lead to significant uncertainty in the evaluation of (I, especially when

kBTESs. In the future, we will pursue further measurements in the

temperature regime kBT>8, where we can most accurately determine a.

69

References

1. P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622 (1985).

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

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

4. "The Art of Electronics", P. Horowitz and W. Hill, Cambridge University Press,
New York, 1980.

5. B. L. Al'tshuler and B. Z. Spivak, Pis'ma Zh. Eksp. Teor. Fiz. 49, 671 (1989).
[JEPT Lett. 49, 772 (1989)]

6. N. M. Zimmerman, B. Golding and W. H. Haemmerle, Phys. Rev. Lett. 67, 1322
(1991).

7. H. Grabert and U. Weiss, Phys. Rev. Lett. 54, 1605 (1985).

8. M. P. A. Fisher and A. T. Dorsey, Phys. Rev. Lett. 54, 1609 (1985).

9. A. J. Leggett et al., Rev. Mod. Phys. 59, l (1987) and refs. therein.

10. B. Golding, N. M. Zimmerman and S. N. Coppersrnith, Phys. Rev. Lett. 68, 998
(1992)

11. See "Numerical Recipes in Fortran", W. H. Press, B. P. Flannery, S. A.
Teukolsky, and W. T. Vetterling, Cambridge University Express, New York, 1988.

12. The success of the rf filtering was confirmed by the reproducible behavior of one
defect over a two-month period.

13. There is a strong evidence for defect-defect interactions at much higher temperatures;

see K. Rails and R. A. Buhrman, Phys. Rev. Lett. 60, 2434 (1988).

70

Table 4.1 Physical parameters of four data sets.

 

 

 

 

 

B(T) s (mK) or A r (s'1 )I log A r
0.140 213 0.16 2.5x104 4.40
2.274 40 0.21 1.0x104 4.01
2.286 92 0.14 3.4x104 4.53
6.997 402 0.22 8.3x103 3.92

 

 

 

 

 

 

 

71

 

 

 

 

1'1
A 30
E
9 .1
E 15 --
_. - r2 . . , . -
O0.01: ' 0.261: T 0.511: T 0.771: 1.021:
Timc(sec)
(a)
E 30 L. Liii‘il .i
9 ‘ III-“IlI—Illllllllll
z
«e 15 1""17 7"”
00.01: i 0.2151: c 0.311: ' 0I771: ' 1.021:
Time(sec)
(b)-

Figure 4.1 (a) Resistance change 6 R as a function of time, at B = 6. 997 T and

T = 0. 22 K. The resistance switches between two values, corresponding to the tunneling
of a single defect between two states. The resistance of each arm of the sample is about

1 K0 and 6 R in this case is about 20 (2. (b) Comparator levels are set between the two
values of the resistance to measure both the number of the transitions and the dwell times
in each state.

72

 

1.0,, Fit to e-t/T to find. 1'

.5

an -

,3 100.0 - , g

a . -

g ° ’ a

3 ° .
. ‘5 10.0 § °

3 " 2 "~
“5

3

'2 1.0

5

Z

I j . I— ] I I I j
5.0 10.0 15.0 20.0
Time (see)

Figure 4.2 The dwell times in each state are histogrammed and fit to a single exponential

decay firnction to find the mean transition time, t, and the transition rate to the other
state, 7 = r".

73

'r ITIrrnl'V'

Sr (1) (V2 ”12)

'l r UUIIIII'V

 

.21 rrrrrrlllar-lr ririLrl-.-ri r rrrrrriL

 

 

3 ALIL . 1.
10-2 2 34, 10-1 2 34 100 2 345 101

f (Hz)

Figure 4.3 The power spectrum of the time trace data The spectrum is fit with a Debye-
Lorentzian function which represents the spectrum of a single two-level system with a
characteristic time t. The amplitude of the resistance jump and the sum of the transition
rates are obtained fiom the fit of the data to the Debye-Lorentzian.

 

74

 

“ v
00
a a. '0
.. 304.0- '2 1 .j
a: u
i ' -'
. l
g “ 1 ..°
11 .0 I .O.
O l
"' 1
3 , ,
3 a“, 800 ' 1 ,
' "5 ° .15 . 1 ,,
b I e“‘ |
3 1 . I
1 ~ °. 1
z ‘ ' I
l 1
l 1
1
1 l
' r
_A. I a
0.0 , , , r
-:.o V,-o.o 0.0 V. 0.0
Voltage bin

.‘ L-Aar'r'rfv'. {‘1

 

I
.1317. JILL.‘ 5."... - x .

Figure 4.4 Histogram of raw values of resistance. The solid line is a fit to the sum of two
Gaussians. The ratio of the areas of the two Gaussian distributions gives the ratio of the
two transition rates. Also fiom the peak values V, , V2 and the standard deviation 0, the
positions of the two comparator levels were determined.

 

 

 

 

 

 

75
_TT 11 r1 1 1 I 111 1 1111 r] I 1 II 111T.
3 _ ——J
- B = 6.997 T ~
’ B = 0.140 T , I
.z 2— m
\ _ ..
5 L . -
_ B = 2.286 T
1 — 5“
,.. 1
- B = 2.274 T -
.. 1" .
I}:
0 gag i r I r r 11 I r 1 r r I I'LL tit 11
0 2 4 6 8 10 12
1/T(K")

Figure 4.5 Ln (7 f / 7,) versus 1/ T, for four values of the applied magnetic field. The

slopes of the lines are 402, 213, 92 and 40 mK, respectively, with an uncertainty of 1:10
mK.

76

 

 

 

 

2.0
5 a
I
10 I {Iii
: j. §§
A i “r if;
H '— ! ii!
is 0.4- l _
I; h s
i‘ 0.2- i
0.1111 I l ILIIIII l
0.1 1.0

T (K)

Figure 4.6 Fast and slow transition rates versus temperature at B = 0.140 T. The value of
s/ 1:, (from Figure 4.5) is 213 mK. The data clearly show three distinct temperature

regimes, T<s/k,, T> slit, and T>1.2 K.

77

 

2.0 - {O

 

 

 

A O

7 Il.:t

33. I

u 1.0: ‘

r~ — I! f

,i‘ _

0.5- g Q
[11 l l I 111141 I
0.1 . 1.0
T(K)

Figure 4.7 Fast and slow rates versus temperature at B = 2.274 T. The value of s/ k, is
40 mK. The data show the regime T > s/ [1, only.

78

 

 

2.0-‘1’”;
:‘ 1”"
'3; 1.0- 1'5'32
5 Ir! lg,
3: Z
12’

 

 

III I I IIIIIII l

0.1 1.0

 

Figure 4.8 Fast and slow rates versus temperature at B = 2.286 T. The value of s/ k, is

92 mK. The data show three temperature regimes, but most of the data are in the regime
T > s/ k,.

 

79

 

 

 

 

2.0
1.0: f,§.§§ 9* f
A E **§**
'1' — i
3, 0.4- ' i,§;iI
IN .—
,., r
P 0.2- '
I
0.1: I
0.1 1.0
T(K)

Figure 4.9 Fast and slow rates versus temperature at B = 6.997 T. The value of s/ k, is
402 mK About halfofthe dataare inthe regime T< s/kB.

80

 

 

 

.1"

I

\W: 2.0 11 111111 1
i" 0

$3 1.0

I IIIIII
"b

 

 

 

0.4

0.2"

0.111 1 11111111
0.1 1.0

T (K)

Figure 4.10 Fast and slow transition rates versus temperature at B = 6.997 T and 0.140 T.
Solid lines are fits of Equations 4.1 and 4.2 to the data for T < 1.2 K. The values of s/ k,
are 402 and 213 mK, from top to bottom. The values of a obtained fiom the fits are 0.22
and 0.16, respectively, with corresponding values of A, of 8.3 x103 5" and 2.5 x 10‘ s".

Uncertainties are discussed in the text.

81

 

Illl

 

.0
(’1
l

 

 

 

 

 

0.1 1.0
T (K)

Figure 4.11 Fast and slow transition rates versus temperature at B = 2.286 T and 2.274 T.
Solid lines are fits of Equations 4.1 and 4.2 to the data for T < 1.2 K. The values of s/ k,

are 92 and 40 mK, fi'om top to bottom. The values of a obtained fi'om the fits are 0.14
and 0.21, respectively, with corresponding values of A, of 14 x10“ .r'and 1.0x 10‘ s".

Uncertainties are discussed in the text.

82

 

llll]lrlllilll[ITIllllll

 

 

 

 

9°
or

Figure 4.12 Plot of the values of a and A, given from the four fits in the log(A,) versus
a/ l - a parameter space. ’

 

83

 

2-0 r1111111 111111111 11

 

 

 

 

. 0 e A A
A A
Tm 1.0: A :
z: — 4° 4
:3 _. _
,1,- _ _.
g. 0.5— . —
\ __ __
9..
V I
02 11111111 1 11111111 1 1
0.3 1.0 3.0 10.0 30.0
kaT/E

Figure 4.13 Log-log plot of (T / TQM“ 7 versus IrBT/ s for all four data sets, with

7; =1 K and a = 0.195. The solid line is a fit oquuation 4.5 to all the data, with

A, = 1.3 x 10‘ s". The four data sets are B = 6.997 T (I), 0.140 r (0), 2.286 T (o), and
2.274 T(A). .

84

 

0.1

 

 

 

 

 

 

'1?
\‘n/ IIIIII
g
93 1.02
+-
C
0.4-
0.2-
I
I
0.1 I I I 111111]
0.1 1.0

T (K)

Figure 4.14 Fast and slow transition rates versus temperature at B = 6.997 T and 0.140 T.
Dashed lines are fits of Equations 4.1 and 4.2 to the data for T < 1.2 K, with the values
a: 0.22 and 0.16 with corresponding values of A, 0f8.3 x10‘ 5'" and 2.5 x10‘ 5". The

solid lines are fits with the global value of a = 0.195and A, = 1.34 x 10‘ s".

85

 

 

 

 

 

L11 1‘ 1 1 1 11111
0.1 1.0
T (K)

 

Figure 4.15 Fast and slow transition rates versus temperature at B = 2.286 T and 2.274 T.
Dashed lines are fits of Equations 4.1 and 4.2 to the data for T < 1.2 K, with the values
a = 0.14 and 0.21 with corresponding values of A, of 3.4 x10‘ 3" and 1.0 x10‘ 3". The

solid lines are fits with the global value of a = 0.195and A, = 1.34 x 104 s".

86

Chapter 5

Electron Heating Experiment

5.1 Defect Temperature and Electron Temperature

We have studied the effect of Joule heating on the dynamics of a
single bistable defect in a submicron Bi wire below 1 K. At high
temperatures, the dominant inelastic scattering mechanism is electron-
phonon scattering. At low temperatures, the electron-phonon scattering
decreases, and electron-electron scattering becomes the dominant source
of inelastic scattering. Due to the weak electron-phonon coupling in
this temperature range, the energy transfer from the electrons to the
phonons is much slower than the energy absoption of the electrons from
the applied current drive. The electron-electron scattering, unlike
electron-phonon scattering, acts to thermalize the electrons internally,
but does not contribute to the energy transfer to the lattice. The
difference in energy flow between the two transport processes gives rise
to electron heating to a temperature above the phonon temperature. This
temperature difference reflects the rate of energy transport between the
electrons and lattice. By applying the idea of electron heating to our
sample and measuring the temperatures of the electrons, the defect and
the lattice as a function of drive current, we investigated the energy
transport between the hot electrons, the defect and the phonons. Figure
5.1 illustrates the energy transport of the system.

This experiment was motivated by the observation that defects with

small duty cycles (i.e. with large ratio of the fast transition time and

 

87

slow transition time, tf/z,) are quite sensitive to the amplitude of

the measurement current. The ratio of the fast and the slow transition

rates of a defect depends on temperature through the detailed balance

relation 71/ 7, =e‘lk‘T. When E/kBT> 1, this ratio is sensitive to small

changes in the temperature seen by the defect. since the defect is

strongly coupled to the conduction electrons in the sample below 1 K, we

’1

interpret its sensitivity to drive current as an indication of electron
heating and the ratio of the two rates as a local thermometer. The

defect temperature is defined from the detailed balance relation as

_ slit, .1
111(7] Ira) '1'

 

defect

Using the "defect thermometer” to study heating is attractive because it
is based purely on statistical mechanics - a feature it shares with the
Johnson noise thermometer used by Roukes et al.[1] in a previous
electron heating study. An electron heating study based only on the
defect thermometer relies on the assumption that the defect is more
strongly coupled to electrons than to phonons. We will show that this
assumption breaks down for temperatures above about 1 K, based on fits

2-3
of our data to dissipative quantum tunneling theory.[ I

The data
presented here were taken well below 1 K, where we expect the assumption
to hold. We would have liked to measure the electron temperature
directly, and thereby deduce the relative thermal coupling between the
defect and the electrons and phonons as a function of temperature. We

did not achieve this more ambitious goal, for two reasons. We tried to

measure the electron temperature from the amplitude of the resistance

88

fluctuations induced by the moving defect, but this thermometer did not
obey a simple heating model. Also, we could not extend our heating
measurements above 1 K due to interference from other defects in the
sample.

The sample for this study is the same Bi wire discussed in Chapter
4. Among several different values of the applied magnetic field where
the resistance of the sample shows clear transitions between two states F
of a bistable defect, the data were taken with a magnetic field of 7

Tesla, where the energy asymmetry of the defect was largest, hence the

*"afln 9.”.

ratio of transition rates was most sensitive to temperature. At each

 

value of temperature and drive, several hundred transitions were
recorded and analyzed. The same analysis method as described in the
previous chapter was used to obtain the fast and slow transition rates,
7f: 7: °

Figure 5.2 shows In(7f/ 7,) versus inverse temperature, taken with
a low enough drive current that no significant heating of the sample
occurs. The data can be fit by a straight line through the origin, with
a slope equal to the energy asymmetry of the defect, 8/kb = 0.420i0.02
K. We performed the heating experiment by increasing the amplitude of
the AC drive current used to measure the resistance. We checked that
using a large DC current for heating plus a small AC measurement current

produced consistent results.

89

5.2 Power Law Dependence of Defect Temperature on Drive
Current

Figure 5.3 shows the results of heating the sample with large
drive current, for lattice temperatures of 0.132, 0.247, and 0.490 K.
Rather than plot 7f/7g, we plot the defect temperature. For low drive
current, the defect temperature rises slowly above the lattice
temperature. As the drive current increases, the defect temperature
approaches a power law behavior that depends only on drive current,
independent of the initial lattice temperature. Behavior similar to
that shown in Figure 5.3 has been observed by Roukes et al.[1], who
derived an expression for the electron temperature as a function of
drive current based on an earlier model of Anderson, Abrahams and

[4]

Ramakrishnan. (The concept of an "electron temperature" far from

5
equilibrium was justified by Arai.[ 1) The model is based on the

observation that the thermal conductance between electrons and phonons

in a sample of volume V is Kai-p11 = ,,r;',_,,,, where Ccl =7,,TV is the

electronic heat capacity and 7,1,, = aoT" is the electron-phonon
scattering rate. The quantity 0% is the strength of the electron-

phonon coupling. The exponent p varies between 2 and 3 for typical

[5’7]

disordered films. For low drive current, the temperature offset

between the electrons and phonons is proportional to the power
dissipated in the sample, Q=12R. As current increases, the electron
temperature rises, and the electrons become more efficient at cooling

themselves by emitting phonons. Eventually the electron temperature

approaches the asymptotic dependence on drive current 7;, oclzmfi’),

independent of the initial lattice temperature 7;. The rate of the

power given to the electrons from the drive current is

 

d'Q, =21Rd1

and the rate of energy loss to phonons is

40... = ..-..(I)dT= a.r..VT‘*PdT

where T'is the electron temperature. Energy balance requires

de =de, in the steady state. Integration of the relation from

78107;, yields the expression:

 

(2+ p) . 12R
aorcl V

 

 

T2+p _ 762+p :

cl

(5.1)

The solid lines in Figure 5.3 represent a single global fit of equation
(5.1) to all the data shown in the figure, with two free parameters p
and 057d. The values of the parameters obtained from the fit are
p=2.0:t0.2, and a0701=(4.5:i:1.0)x101° ulx‘nfi .

Before interpreting these results, we examine the assumptions made
regarding heat transfer in the sample. First we consider the thermal
coupling of the defect to the electrons and phonons. Figure 5.4 shows a
log-log plot of the fast and slow transition rates of the defect versus
temperature. The behavior of these rates below 1 K is described by

2
I l and was the subject of Chapter

dissipative quantum tunneling theory,
4. In our experiment, the temperature and energy asymmetry are much

smaller than the vibrational level spacing in a single-well, and much

larger than the renormalized tunneling matrix element. In this regime,

 

91

the only property of the heat bath that enters into the theory is the
low-energy spectral density of excitations, J(a0==af, where n = 1 or 3
for electrons or phonons, respectively. We showed in chapter 4 that for
T <1.2 K the data could be fit with a model that considered only
electrons in the heat bath. To fit the data above 1.2 R, one must
include the phonon contribution. The solid line in Figure 5.4 is a fit
of the data to a function calculated by Grabertl3] that includes the
effect of both electrons and phonons. The agreement between the

function and the data is quite good. However, this does not mean that

the fit shown in Figure 5.4 confirms the formula of Grabert. We clearly

 

need more high temperature data points to confirm it. We are interested

in the value of Grabert's parameter T;, the crossover temperature

between electron-dominated and phonon-dominated behavior. For the data

shown in the figure, Tph =0.95:i:0.03 K. A fit to data from the same
defect measured at a different value of the magnetic field (0.14 T)
yields Tph =0.9l:t0.03 R. since the lattice temperatures in our heating

study are well below 38' we are justified in neglecting the thermal

coupling between defect and phonons.[3]

Another crucial assumption in the analysis leading to equation
(5.1) is that the electrons heat while the phonons remain at the
temperature of the cryostat. This assumption, already described in
section 5.1, is valid if the thermal link between the electrons and
phonons is much weaker than that between the phonons and substrate and
between the substrate and the cryostat.[8] The latter condition is
easily fulfilled. The Si substrate has a surface area 108 times larger

than the sample area, and is immersed in liquid helium in close

proximity to the cryostat thermometer. We believe that the former

 

92

condition is also satisfied, by the following argument. The thermal

conductance between film and substrate is about Kubl T3A 21500 W/mZK‘,

9
where 24 is the area of the film in contact with the substrate.[ 1 We

estimate Kh,h in our Bi film as follows: The density of electron and

hole states at the Fermi level is generally larger in thin Bi films than

in bulk Bi. We take the value 7,1=2 J/K2m3, which is twice the value

[10]

for bulk Bi. The electron-phonon scattering rate was measured by

[5]

Dorozhkin et al., who obtained rp,h==ag78 with p = 3 and

ao=3x109 s'1K'3. (Our own measurements suggest a smaller value of p, but
we will use this as a starting point.) Since the thickness of our
sample is 20 run, we have x,1_ph/T‘A ~120 W/Rsm’. The above estimates

give x.ub/x.1,,,h~4 at 1 K and 40 at 0.1 R.

A final assumption usually made in a heating experiment is that
the electron and phonon temperatures are uniform throughout the sample.
Unfortunately, this assumption is not valid in our experiment, due to
the very small sample size. (This small size is necessary to enable us
to detect the motion of a single defect.) In our sample, shown in
Figure 2.1, heat dissipated in the sample can flow within the electronic
system directly out the leads to the large pads, without first
equilibrating with the phonons in the sample. We estimate the thermal
conductance out the leads using the Wiedemann-Franz relation:

K/ 07’ =2.5x10‘3 WQ/Rz, where K, 0’ are thermal and electrical
conductivity, respectively. The longest distance from any part of the
sample to the large pads is about 0.5 um, corresponding to about half
the sample resistance, i.e. 1 kn. Hence the thermal conductance of the

electrons from the hottest part of the sample to the cool part is about

25x104'T WIRE. In comparison, the thermal conductance between the

 

j

 

93

electrons and phonons inside one arm of the sample (A2 =0.05 umz) is
only 6x1042T4 ifl/K5 according to the estimate made above. These numbers
show that the Joule heat generated within the sample diffuses out the
leads at least as quickly as it transfers to the phonon system, giving
rise to a temperature gradient in the sample.

This last consideration precludes us from obtaining an absolute
estimate of the electron-phonon scattering rate from our data. Equation T”“
(5.1) was derived assuming that all the Joule heat dissipated in the

sample transfers to the phonons within the sample volume. If we try to
interpret the value of ab obtained from Figure 5.3 this way, we find

czo=2><10lo s'1R‘2, which is a factor 7 larger (1 K) than the value

 

6
obtained by Dorozhkin et al.[ 1

The discrepancy reflects the fact that
the heat in the electron system diffuses out the leads, and therefore

has a larger volume in which to transfer to the phonon system.

Nevertheless, our estimate of the temperature dependence of the

electron-phonon coupling may still be valid. Our value p=2.0:l:0.2 is

[5]

lower than the value p a 3 obtained by Dorozhkin et al. in a low-
temperature heating experiment, but it is consistent with the values
between 2 and 2.5 obtained from weak-localization studies carried out at
higher temperatures,[7] including those in reference [6].

The above analysis presents a consistent, but not unique,

explanation of the data in Figure 5.3. An alternative explanation for

an observed value of p a 2 has been proposed recently by Ranshar,

[11

 

Wybourne and Johnson 1, who point out that the heating model leading
to equation (5.1) assumes good thermal coupling between the phonons in
the films and substrate. By allowing the coupling to vary, these

authors show that data from a heating experiment can obey a power law

94

varying anywhere from p = 2 to p = 3. According to this work, the 100
nm thick oxide on our substrate may reduce the high-frequency component

of the phonon coupling between film and substrate. Another concern is

our estimate of the electron-phonon scattering rate, 1:,h==abTT, based

on reference [6]. The experiments of reference [7] give p = 2, and

ao=2x10‘° s'1R'2. With this larger value of a0, we find x,,,be0.6 x,,1_,,h

independent of temperature, so our assumption that the electrons heat

while the phonons stay cold may be questionable. We also find with this
value of a, that the thermal conductance out the leads is less

important, and our measured value ao=2.2x10'° s'lrt’2 agrees fortuitously

 

with the values from reference [7]. We cannot distinguish between these 1
interpretations of our data, but we can assert that our measured value

of 0, represents an upper bound on the electron-phonon scattering rate

in our sample.

5.3 Attempt to measure Electron Temperature based on
Universal Conductance Fluctuations Theory

In our experiment, we tried to measure the electron temperature
directly, based on the temperature dependence of the amplitude of the
resistance fluctuations, 57?. We can compare the observed behavior of
51?, shown in the Figure 5.5, with that predicted by the theory of

12-13
I 1 Wear 1 R, the phase

[14]

universal conductance fluctuations.

coherence length, L,, in our sample is between 0.1 and 0.2 pm.

Thus the sample is just barely in the 1-D limit, defined by

L,,l,<<[,<:l,, where 1,, I, and I, are again the sample thickness,

min min

width and length, respectively. In this limit, 5R0CL2 -L',/2, where L

95

is the smaller of I, and thermal length [7. At these low temperatures

we expect L,0CT_”2, due to electron-electron scattering, hence

572a27hy4. The observed dependence is about 7“], which is rather good

agreement. As the temperature is lowered, the sample dimensionality

will cross over to 0-D when I, becomes comparable to the sample length

of 0.5 um. In this regime the theory predicts 5R0CL;"2 OCT—m, which is

consistent with the roll-off seen in the Figure 5.5 at the lowest
temperatures.

Since 5R depends only on L, and LT, both of which depend only on
the electron temperature, we hoped that 5R would serve as a good
electron thermometer. Figure 5.6 shows a plot of the electron
temperature, deduced from 5R, versus drive current. These data were
taken simultaneously with those shown in Figure 5.3. Unlike the data in

Figure 5.3, however, the data in Figure 5.6 can not be fit with equation

(5.1), because the electron temperature, 7], does not asymtotically vary

as a power law with drive current. In addition, 7; appears to increase

with drive current initially much faster than the defect temperature,

7' . This latter observation would be plausible if the defect were only
weakly coupled to the electrons, but in the present circumstance we find
it disturbing. Given the choice between our two thermometers, we trust
more the reliability of the defect thermometer because of its connection
to statistical mechanics. We do not know why 5R does not serve as a
reliable electron thermometer, but we mention incidentally that Bergmann

et al.[15]

found the Coulomb-interaction resistance anomally to be a
poor thermometer in a heating study of gold films, possibly due to

violations of Ohm's law.

 

 

Reference

1.
2.

M. L. Roukes et al. Phys. Rev. Lett. 55, 422 (1985).

See "Quantum Tunneling in Condensed Matter", ed. Yu. Kagan and A. J. Leggett,
Elsevier, New York 1993; also A. J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987) and
references therein.

H Grabert, Phys. Rev. B 46, 12753 (1992). Grabert's form for the spectral density of f—
bath excitations can be written as J(w) = Kw(1 + (hw/ k, 1;, )2 /27rK), where ‘
K e 0. 2 is the coupling parameter between the defect and electron-hole excitations.
The ratio of phonon coupling to electron coupling is then roughly O.25( Tm,” / Tph )2 ,

where T...

 

is the actual lattice temperature in the experiment, and Tph = 0.95 K,

from Figure 5.4.

P. W. Anderson, E. Abrahams and T. V. Ramakrishnan, Phys. Rev. Lett. 43, 718
(1979)

M. R. Arai, Appl. Phys. Lett. 42, 906 (1983). Arai justifies the concept of "electron
temperature" without assuming that electron-electron collisions occur more
fi'equently than electron-phonon collisions. This latter condition is nonetheless true in
disordered metals below a few K.

S. I. Dorozhkin, F. Lell and W. Schoepe, Solid St. Comm. 60, 245 (1986).

F. Komori, S. Kobayashi, and W. Sasaki, J. Phy. Soc. Jap. 52, 4306 (1983); also P.
H. Woerlee, G. C. Verkade and A. G. M. Jansen, J. Phys. C: Solid St. 16, 3011
(1983)

We follow the point of view of refs. 1, 4 and 5 here, where the phonon-substrate
coupling is characterized by a thermal conductance, 1cm. At higher temperatures, it
may be necessary to consider a more sophisticated model that treats each phonon
frequency independently. See ref 11 , based on the model of N. Perrin and H. Budd,
J. Phys. (Paris) Colloq. 33, C4-33 (1972).

97

9. We use the same value for the thermal boundary resistance as in ref. 1, taken from W.
A. Little, Can. J. Phys. 37, 334 (1959). Many solid-solid interfaces have similar
values of thermal boundary resistance, of. E. T. Swartz and R. O. Pohl, Rev. Mod.
Phys. 61, 605 (1989).

10. N. E. Phillips, Phys. Rev. 118, 644 (1960). Our guessfor 7cl in our Bi film is based
on the discussion in ref. 14 and refs. therein.

11. M. Kanshar, M. N. Wybourne, and K. Johnson, Phys. Rev. B 47, 13769 (1993). See
also J. C. Nabity and M. N. Wybourne, J. Phys. Cond. Matt. 2, 3125 (1990).

12 B. L. Al'tshuler and B. Z. Spivak, Pis'ma Zh. Eksp. Teor. 42, 363 (1985) [JEPT Lett.
42, 447 (1985)]; S. Feng, P. A. Lee and A. D. Stone, Phys. Rev. Lett. 56, 1960, 2772
(1986)

13. The UCF calculation is strictly valid only for an ensemble of defects.

14. N. O. Birge, B. Golding, and W. H. Haemmerle, Phys. Rev. B 42, 2735 (1990).

15. G. Bergmann, W. Wei, Y. Zou and R. M. Mueller, Phys. Rev. B 41, 7386 (1990).

 

 

98

 

Joule heating
electron bath

 

 

 

 

 

 

 

cI-def
1c em, defect
v K def-ph -
l Phonon bath I
K sub
I substrate I

 

Figure 5.1 Simplied schematic diagram of energy transfer fi'om the electrons to defect and
to the phonon bath. K is a thermal conductance.

 

99

 

. 5 I
A
I .. .-
?~,
\2" —
:‘2 b _
v, -

1— a

 

 

 

1/1' (K“)

Figure 5.2 Ln (7, / 7,) versus 1/ T at B = 6.997 T. The transition rates obey the
detailed balance relation with a slope of 0.42:1:0.02 K.

 

100

 

 

 

 

 

1.0 1llll l IIFIIHI I IIIHI
_- //
0.5L I II II II Q n H. d
A _ V _.
5 ../
g» _ x x / _
s x
E" .. -
0.2“ ’ " ‘
a Q
01 11111 1 llLlLlll 1‘1 111111
1 ‘ 10 100

Idrive (DA)

Figure 5.3 Log-log plot of defect temperature versus drive current, for three difl‘erent
values of the lattice temperature, Tm = 0.132, 0.247 and 0.490 K. The defect
temperature is determined from the ratio of fast and slow transition rates via the detailed
balance relation. The solid lines represent a global least-squares fit of the data to Equation
5.1 with two parameters, p = 2.0 and “07.1 = 4.5 x10'° W/ K‘m’.

 

lOl

 

l llllllll l

 

lllll

f
1.0 E’

7!! 7s (sec-1 )
O
to
l
I

0.1

lllll
lllll

 

 

1 1 1 1 1 1 1 11 1
0.1 ' 0.3 1.0
T (K)

 

Figure 5.4 Log-log plot of fast and slow transition rates versus temperature at B = 6.997
T. The solid line is a least-squares fit to the theoretical fi1nction calculated by Grabert in
ref. 3. The crossover between electron-dominated and phonon-dominated tunneling

occursat Ty. 210.95 K.

 

 

 

102

 

 

 

 

I l r T l llll
3.
_. . . -
O
30" . —
O
’{3‘ 0
i o
O
I 0 . .—
- 0"
3 l l J ll Llll
0.1 0.3 1.0

Tlattice (K)

Figure 5.5 Plot of the amplitude 6 R of the defect resistance jump versus temperature.

The increase of 6' R with decreasing temperature is due to the universal conductance
fluctuation mechanism

 

103

 

 

 

 

1.0 IIIIF T Ilffllli I r 1011111
1— u —1
.. X _.
03
.. 1:1 _
a 9
0.5- a 1: 0° $2 -
A U
a _ X _
X
1:
3 1- X 0 _1-
3
'3 x X x x
F
0.2b -
0
O
-0 0 0
0111111 1 11111111 1 1111111
1 .10 100
Idrlve(n-A)

Figure 5.6 Log-log plot of electron temperature versus drive current, for the same three
lattice temperatures shown in Figure 5.3. The electron temperature is deduced from the
amplitude of the defect resistance jump 6 R .' These heating data do not obey Equation
5.1.

 

 

104
Chapter 6

Summary

We have measured the tunneling rates of a single defect in a metal
over a broad range of temperature and magnetic field. Our work covers
the temperature range from 0.1 K to 2 K, which allows us to make a
thorough comparison between experiment and theory. Above 1 K, the rates
increase rapidly, due presumably to phonon-assisted tunneling. Below 1
K, the temperature dependence of the defect tunneling rates agrees
quantitatively with predictions of dissipative quantum tunneling theory,
spanning the temperature range from kBT<<8 to kBT>>£. We also
observe a random variation of the defect energy asymmetry 8, with
magnetic field, but the minimum value of 8 for this particular defect
occurs for B¢O. Our data are consistent with the hypothesis that the
defect-electron bath coupling constant, (I, is independent of field.
These results illustrate the power of conductance measurements of
mesoscopic samples to obtain microscopic information about individual
defects. Such experiments are especially useful in the study of
disordered materials, where the microscopic parameters characterizing
the defects are broadly distributed.

In an electron heating experiment, we have shown that the
transition rates of a single defect can be used as a local thermometer
under non-equilibrium conditions, via the detailed balance relation. We
have used this defect thermometer to study electron heating in a

submicron Bi sample, and we find the results consistent with a simple

 

 

105

heating model, where the electron-phonon scattering rate is proportional
to 7? with p a 2. An electron thermometer based on the the amplitude
of the dynamic universal conductance fluctuations does not follow the
simple heating model. We can not obtain the absolute electron-phonon
scattering rate from this experiment, due to the nonuniform temperature

profile of the hot conduction electrons in the sub-micrometer sample.

1’11 ° 3‘ .2.m:¢. “4.13.12: ‘ a.

 

I_.
D

 

106

Appendix A

1/ f noise spectrum

The standard way to measure resistance fluctuations of a resistor
is to apply a constant current through the sample, and monitor the
voltage across the sample as a function of time. The noise is usually

characterized by an autocorrelation function

K,(t)=(5V(r)6V(r+z))

 

where 5V(t)=V(t)-( V >, and the brackets refer to a time average over
all times T. This autocorrelation function contains information about
the noise in the time domain. The power spectrum of the voltage noise
is then given by

Sy(w) = it 19(1) e-w d1

'8‘—.3

At high frequencies, the noise spectrum is roughly flat, and its

magnitude is determined by the Nyquist-Johnson formula, S$==4k37"(1?).

At low frequencies, a noise called ”flicker noise" of the spectral form
Sy(a))0Cl/a) (w=2 Itf, Sy(f)=2 IrSV(w)) is observed in most of the

metallic systems. This is the l/ f noise. The l/ f noise is due to

fluctuations in the sample resistance, and obeys ohm's law:

S,,(a)) = 12SR(a)) .

 

107

The resistance noise spectrum is given as a superposition of
contributions from all moving defects, each having some typical time

scale In One has

SR(a))oc [dz 0mm

where LXI) is the distribution of characteristic time constants of the

moving defects. The function 21[1+(a2tf] is the Debye-Lorenzian power

spectrum of a fluctuating two-state defect having a characteristic time
scale 1'.

At high temperature, the motion of defects is thermally activated,
so for a given defect one has 1': To exp(E/kB T), where E is the
activation energy barrier for the defect to hop from one site to another
and 1/15 is the attempt rate. If there is a distribution of such
activation energies, D(E), which has a regime where D(E) is constant
for Emin <E<Em, then one has D(T)0Cl/t for the time interval
Tm < 1' < I'm, where “I'm,“ = To exp(Emin/k3 T) and Tm = To exp(Em/k3 T).
This then leads to a 1/ f spectrum of SV( f ) for 1:“ <<w << 7:,“ . This
behavior is shown in schematic diagrams in Figure A.

At low temperatures, defects move via quantum tunneling. By a
similar argument as the one described above, the conductance
fluctuations have low frequency 1/ f noise due to the distribution of
tunneling rates in a disordered material. The tunneling time scale

depends exponentially on the barrier height and barrier width. Assuming

that the barrier height is broadly distributed, the same argument as

above leads to l/ f noise.

 

 

 

 

 

108
Constant
D (E)
E. b; E
(a)
Constant
In S f ( a)

 

 

 

Ina)

 

(b)

Figure A (a) ”Flat" energy distribution of thermally activated process. (b) Frequency
spectrum resulting from this distribution.

L l‘flifl'finl‘ ‘M:.nl\n'-1

 

109

Appendix B

Thermometer Calibration in a magnetic field

The fitting function for the correction of magnetoconductance of the

carbon thermometer is

G(T, __B_) a__(T) , c__(___7)
G(T 0) =1.+00 (b(T)— B+d(T)B)———-— emu]

The coefficients a(T), b(T) ,c(T) and (1(7) are given as polynomial

functions of magnetoconductance of zero field C(T ,0) such that

a(T) =a0 +a,~G(T,0)-1-a2-G(T,0)2

b(T) = bo +bl -G(T,0)+b2 ~G(T,0)2 +b3 -G(T,0)3 +b4 -G(T,0)‘
C(T)=Co

(1(7) = d0 +dI -G(T,0)+d2 -G(T,0)2 +al3 -G(T,0)3 +d4 -G(T,0)“

where the coefficients are

a0 = 0.2476, a, = —0. 08853, a2 = 0.02137
bO = 10.0561, 6, = —8.5684, b, = 3.5406, b, = —0.65894, 6, = 0.04894
= 0.0114
do = —0. 0270, d, = —0.0857, d, = -0.0418, d, = -0. 0089, d, = —0.0007

Figure B shows the good fits of G(T,B)/G(T,0) to the data for three

different nominal temperatures.

_-_.-._ _ -1

 

 

110

1.04

G(T,B)
G(T .0)

 

0.96
o B (Tesla) 7’03

1.04

G(T,B)
G(T,0)

 

0.96
0 B (T6818) . 7.03

1.04

G(T, B)
G(T,0)

 

0.96
° B (Tesla) 7'”

Figure B. Plot of magnetoresistance of the carbon thermometer versus magnetic field for
three difl‘erent values of the nominal temperature, 565, 400 and 283 mK. The solid line is
the fimction G(T, B) / G(T, 0) calculated with difi‘erent values of the coefficients a(T),

b(T) 6(7) and d(T)

 

"‘1111011111011“