ABSTRACT 
 
 
 
 
Title of Document: NANOGAP JUNCTIONS AND CARBON 
NANOTUBE NETWORKS FOR CHEMICAL 
SENSING AND MOLECULAR 
ELECTRONICS   
  
 Gokhan Esen, Doctor of Philosophy, 2006 
Directed By: Associate Professor, Michael S. Fuhrer, 
Department of Physics 
 
 
This thesis work may be divided into two parts. The first part (chapters 2-7) 
focuses on the fabrication of a particular test structure, the electromigration (EM) 
formed metal nanogap junction, for studying the conduction through single molecules 
and for hydrogen sensing.  The second part (chapters 8 and 9) focuses on carbon 
nanotube networks as electronic devices for chemical sensing. 
Chapters 2-4 discuss the formation of nanogap junctions in thin gold lines 
fabricated via feedback controlled electromigration. Using a feedback algorithm and 
experimenting on thin gold lines of different cross sections, I show that the feedback 
controls nanogap formation via controlling the temperature of the junction. Chapters 
5 and 6 discuss the background and my experimental efforts towards fabricating 
superconducting electrodes for single molecule electronics research. 
Chapter 7 discusses the application of the techniques of chapters 2-4 to form 
palladium nanogaps via electromigration. I show that such devices can be used as 
  
hydrogen sensors, but suffer from slow response times (on the order of minutes). The 
results are discussed in the context of the in-plane stress buildup between the 
palladium metallization and the SiO
2
 substrate.  
The use of nanotube networks as chemical sensors is discussed in the second 
part of the thesis (chapters 8 and 9). I show measurements of the resistance and 
frequency-dependent (50 Hz - 20 KHz) gate capacitance of carbon nanotube thin film 
transistors (CNT-TFTs) as a function of DC gate bias in ultra-high vacuum as well as 
low-pressure gaseous environments of water, acetone, and argon. The results are 
analyzed by modeling the CNT-TFT as an RC transmission line. I show that changes 
in the measured capacitance as a function of gate bias and analyte pressure are 
consistent with changes in the capacitive part of the transmission line impedance due 
to changes in the CNT film resistivity alone, and that the electrostatic gate 
capacitance of the CNT film does not depend on gate voltage or chemical analyte 
adsorption to within the resolution of my measurements.  However, the resistance of 
the CNT-TFT is enormously sensitive to small partial pressure (< 10
-6
 Torr) of 
analytes, and the gate voltage dependence of the resistance changes upon analyte 
adsorption show analyte-dependent signatures. 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
NANOGAP JUNCTIONS AND CARBON NANOTUBE NETWORKS FOR 
CHEMICAL SENSING AND MOLECULAR ELECTRONICS      
 
 
 
By 
 
 
Gokhan Esen 
 
 
 
 
 
Dissertation submitted to the Faculty of the Graduate School of the  
University of Maryland, College Park, in partial fulfillment 
of the requirements for the degree of 
Doctor of Philosophy 
2006 
 
 
 
 
 
 
 
 
Advisory Committee: 
Associate Professor Michael S. Fuhrer, Chair 
Professor Ellen D. Williams 
Assistant Professor Min Ouyang 
Assistant Professor John Cumings 
Professor Neil Goldsman 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
? Copyright by 
Gokhan Esen 
2006 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ii 
 
Acknowledgements 
I would like to thank all the friends for helping me to complete this important 
milestone in my life. I especially would like to thank my PhD advisor, Professor 
Michael S. Fuhrer, for helping me to finish my study his lab both with his ideas and 
his friendship. If I was starting my PhD today, I would still join his group. In 
numerous times during my study when we talked about science and physics, I felt the 
curiosity and excitement he had for it. I will always remember him and his wife, 
Cynthia, as being nice and friendly people. 
I would also like to thank the other members of my dissertation committee, 
namely Professor Ellen D. Williams, Min Ouyang, John Cumings and Neil Goldsman 
for honoring me by serving in my committee.  
Through the time I spent in Michael's lab, I had the chance to meet and work 
with several students and postdoctoral associates: Todd H. Brintlinger, Tobias 
D?rkop, Yung-Fu Chen, Enrique Cobas, Adrian Southard, David Tobias, Daniel 
Lenski, Tareq Ghanem, SungJae Cho, Chaun Jang, Shudong Xiao, Alexander 
Manasson, Anthony Ayari, Byong M. Kim and Stephanie A. Getty. I want to thank 
them since they were able to put up with me in the lab. I would especially like to 
thank Stephanie for being a kind mentor. I learned a lot of laboratory skills from 
Todd, Tobias, Yung-Fu and Anthony as well. I would like to thank them for helping 
me in numerous issues. I wish success for the rest of the Fuhrer Group that are 
working on their dissertation, I am sure they will graduate with lots of experience and 
skills. I also would like to thank Masahiro Ishigami and Elba Gomar-Nadal. They 
 
 iii 
 
were good friends and talented scientists.  I also would like to thank all the staff 
working in the department. 
I made so many good friends in Maryland. I would like to thank all of them 
for helping me to deal with the problems and sharing happy moment with me. Among 
them, I especially want to thank Nazif Cihan Ta? and Murat ?m?r. During the time I 
spend in United States, they were like two brothers to me and I hope they will be in 
the rest of my life. Lastly, I would like to thank my family. Their constant support in 
my life helped me to deal with difficulties I faced and the ones that I faced during 
PhD was not an exception. 
 
 iv 
 
Table of Contents 
 
 
Acknowledgements....................................................................................................... ii 
Table of Contents......................................................................................................... iv 
List of Tables .............................................................................................................. vii 
List of Figures............................................................................................................viii 
Chapter 1: Introduction................................................................................................. 1 
Chapter 2: Electromigration.......................................................................................... 4 
2.1: Introduction............................................................................................. 4 
2.2: Microscopic Theories and Empirical Relations...................................... 6 
2.2.1: Microscopic Theories ....................................................................... 7 
2.2.1.1: Ballistic Model......................................................................... 7 
2.2.1.2: Polarization Charge Models..................................................... 9 
2.2.1.3: Local Field ............................................................................. 11 
2.2.2: Empirical Relations......................................................................... 12 
2.2.2.1: Black Equation....................................................................... 14 
2.2.2.2: Effect of Stress Buildup in Electromigration......................... 16 
2.2.2.3: Film Microstructure ............................................................... 18 
2.3: Electromigration in Au ......................................................................... 21 
Chapter 3: Point Contacts and Nanogap Junctions..................................................... 23 
3.1: Introduction........................................................................................... 23 
3.2: Conductance of point contacts of normal metals.................................. 24 
3.3: Methods of Fabricating Point Contacts ................................................ 29 
 
 v 
 
3.4: Fabrication of Nanogaps for Single Molecule Electronics................... 31 
Chapter 4: Formation of Gold Nanogaps via Electromigration.................................. 33 
4.1: Introduction........................................................................................... 33 
4.2: Device Fabrication and Measurement Setup ........................................ 34 
4.3: Feedback Controlled Electromigration................................................. 39 
4.4: Conclusion ............................................................................................ 56 
Chapter 5: Mesoscopic SNS Josephson Junctions...................................................... 57 
5.1: Introduction........................................................................................... 57 
5.2: Basic Properties of Superconductors .................................................... 59 
5.3: Josephson Effect and SNS Josephson Junctions................................... 63 
5.4: Mesoscopic SNS Junctions................................................................... 67 
5.5: Transport Properties of Mesoscopic SNS junctions ............................. 69 
Chapter 6: Preparing Mesoscopic SNS Junctions as Electrodes ................................ 77 
6.1: Introduction........................................................................................... 77 
6.2: Device Fabrication and Difficulties Faced ........................................... 78 
6.2.1:  Thin Au lines and Nb contacts....................................................... 78 
6.2.2:  Shadow evaporation of gold and aluminum contacts .................... 82 
6.2.3: Electron beam deposited thin Au lines ? Aluminum contacts........ 86 
6.3: Conclusion ............................................................................................ 89 
Chapter 7: Palladium Nanojunctions as Hydrogen Sensors ....................................... 90 
7.1: Introduction........................................................................................... 90 
7.2: Device fabrication and Measurement setup.......................................... 91 
7.2: Results and Discussion ......................................................................... 95 
 
 vi 
 
7.4: Conclusion .......................................................................................... 103 
Chapter 8: CNT Thin Films as Chemical Sensors-Fabrication and Measurement 
Technique............................................................................................. 104 
8.1: Introduction......................................................................................... 104 
8.2: Device fabrication and Measurement setup........................................ 107 
8.2.1: CNT Synthesis .............................................................................. 107 
8.2.2: Device Fabrication and Measurement Setup ................................ 110 
Chapter 9: CNT Thin Films as Chemical Sensors-Results....................................... 114 
9.1: Introduction......................................................................................... 114 
9.2: CNT networks modeled as an RC transmission line. ......................... 117 
9.3: Effect of low pressure gas adsorption on the conductance and the 
capacitance of CNT films ................................................................. 126 
9.4: Conclusion .......................................................................................... 135 
Bibliography ............................................................................................................. 136 
 
 
 
 
 
 
 
 
 
 
 
 
 vii 
 
List of Tables 
 
Table 8-1. CNT growth recipe.................................................................................. 109 
 
 
 viii 
 
List of Figures 
Figure 2-1. Void and hillock formation in a polycrystalline metal film....................... 5 
Figure 2-2. Change of the structure of polycrystalline interconnects......................... 20 
Figure 3-1. Conduction through a mesoscopic sample............................................... 26 
Figure 4-1. Schematic of the electron beam lithography technique. .......................... 36 
Figure 4-2. SEM micrograph of a typical device used in controllable electromigration 
experiments............................................................................................... 37 
Figure 4-3. Measurement setup used in controllable electromigration....................... 38 
Figure 4-4. I-V
bias
 curve of  feedback controlled electromigration process................ 40 
Figure 4-5. I-V
bias
 during the feedback-controlled electromigration of a gold wire at 
T=1.3 K..................................................................................................... 42 
Figure 4-6. Power dissipated in the junction vs. the voltage drop at the junction during 
electromigration ........................................................................................ 44 
Figure 4-7. Evolution of junction resistance as a function of junction voltage. ......... 45 
Figure 4-8. Power dissipated in the junction during electromigration vs. nanowire 
cross-sectional area. .................................................................................. 46 
Figure 4-9. Stable and unstable regimes in controllable electromigration. ................ 48 
Figure 4-10. Effect of feedback in electromigration................................................... 50 
Figure 4-11. Electromigration of a device in which the stable branch does not extend 
beyond the unstable branch....................................................................... 52 
Figure 4-12. Conductance change in discrete steps of conductance quantum............ 53 
Figure 4-13. I-V
bias
 curve of a junction where the electromigration is stopped at 1Go.
................................................................................................................ 54 
 
 ix 
 
Figure 4-14. I-V
bias
 curves after junction formation. .................................................. 55 
Figure 5-1. Superconducting energy gap.................................................................... 62 
Figure 5-2. An SNS junction. Two superconductors (SC
1
  and SC
2
) are weakly linked 
via the normal metal.................................................................................. 63 
Figure 5-3. DC Josephson Effect................................................................................ 65 
Figure 5-4. AC Josephson Effect................................................................................ 65 
Figure 5-5. Mesoscopic SNS junction. ....................................................................... 67 
Figure 5-6. Andreev reflection at the superconductor-normal metal interface........... 72 
Figure 5-7. Multiple Andreev Reflection of quasiparticles. ....................................... 75 
Figure 6-1. Metallization for thin gold lines............................................................... 79 
Figure 6-2. Melting of gold metallization due to niobium deposition........................ 80 
Figure 6-3. SEM micrograph of a sample with gold lines and niobium contacts....... 81 
Figure 6-4. Shadow evaporation of gold and aluminum contacts. ............................. 83 
Figure 6-5. A gold-aluminum device fabricated via shadow deposition.................... 84 
Figure 6-6. Kondo effect in gold deposition............................................................... 85 
Figure 6-7. SEM micrograph of a gold-aluminum device.......................................... 86 
Figure 6-8. Post-Mortem SEM micrographs of two devices after pumping the vacuum 
space.......................................................................................................... 88 
Figure 7-1. SEM micrograph of a palladium device before electromigration............ 92 
Figure 7-2. Palladium device where the electromigrated thin line is palladium and the 
rest of the metallization in the device is chromium-gold.......................... 93 
Figure 7-3. Measurement setup used in hydrogen sensor experiments. ..................... 94 
Figure 7-4. Nanogap formed via electromigration in a palladium device. ................. 95 
 
 x 
 
Figure 7-5. Nanogaps formed via electromigration in palladium devices with gold 
contacts. .................................................................................................... 96 
Figure 7-6. Feedback controlled electromigration of a palladium device with gold 
contacts. .................................................................................................... 97 
Figure 7-7. I-V
bias
 curve of a palladium line where the feedback is turned off during 
the electromigration. ................................................................................. 98 
Figure 7-8. Formation of a neck in the junction after electromigration of a gold line..
................................................................................................................... 99 
Figure 7-9. Typical turn-on behavior of sensor devices fabricated with only palladium 
metallization............................................................................................ 100 
Figure 7-10. Typical turn off behaviors of devices fabricated with only palladium 
metallization............................................................................................ 101 
Figure 8-1. Schematic of CNT growth apparatus. .................................................... 108 
Figure 8-2. SEM micrograph of the thin film of CNT network material after CVD 
growth on Si/SiO
2
 substrate.................................................................... 110 
Figure 8-3. SEM micrograph of the CNT network device. ...................................... 111 
Figure 8-4. UHV system used in the experiment...................................................... 113 
Figure 9-1. Capacitance of the CNT network as a function of gate voltage............. 115 
Figure 9-2. Conductance of CNT network as a function of gate voltage. ................ 116 
Figure 9-3. Capacitance as a function of signal frequency....................................... 117 
Figure 9-4. Schematic  of an RC transmission line .................................................. 118 
Figure 9-5. Device capacitance as a function of drive frequency at different gate 
voltages at ambient atmosphere.............................................................. 122 
 
 xi 
 
Figure 9-6. Schematics of source, drain and the gate capacitors.............................. 123 
Figure 9-7. Capacitance and inverse square root of resistance as a function of gate 
voltage..................................................................................................... 125 
Figure 9-8. Capacitance of the CNT network in UHV and 9.5 10 
-7
 Torr acetone 
pressure. .................................................................................................. 127 
Figure 9-9. Resistance of the CNT network in UHV and 9.5 x 10 
-7
 Torr acetone 
pressure. .................................................................................................. 128 
Figure 9-10. Change in the capacitance and resistance due to 9.5 x 10
-7
 Torr acetone 
pressure as a function of gate voltage..................................................... 129 
Figure 9-11.  Capacitance of CNT network in UHV and 2.8 x 10
-8
 Torr water 
pressure. .................................................................................................. 130 
Figure 9-12. Resistance of CNT network in UHV and 2.8 x 10
-8
 Torr water pressure.
................................................................................................................. 131 
Figure 9-13. Change in the capacitance and resistance due to 2.8 x 10
-8
 Torr water 
pressure as a function of gate voltage..................................................... 132 
Figure 9-14. Capacitance as a function of gate voltage in UHV and 3 x 10
-6
 Torr 
argon pressure. ........................................................................................ 133 
Figure 9-15. Resistance as a function of gate voltage in UHV and 3 x 10
-6
 Torr argon 
pressure. .................................................................................................. 134 
 
 
 1 
 
Chapter 1: Introduction 
Since the invention of the transistor [1], the miniaturization trend in 
semiconductor industry has provided us with faster and cheaper computers. But the 
present trend of miniaturization which is based on the scaling of the complementary 
metal-oxide-semiconductor (CMOS) process in silicon is close to its limits due to 
both the technical difficulties such as ultra-thin gate oxides, doping fluctuations, short 
channel effects and the cost of production [2]. Therefore the next step in 
miniaturization will require conceptually new device structures.  
The idea that a single molecule can be used as an active electronic component 
was first introduced by Aviram and Ratner [3] and spurred research activity over the 
last couple decades.  However, the fabrication of complete circuits out of molecules 
still remains challenging (if possible at all) and it will come only after a solid 
understanding of the conduction properties of individual molecules that can perform 
the basic functions such as rectification, amplification and storage. 
Chapters 2-7 of this thesis will focus on the fabrication of a particular test 
structure, the electromigration (EM) formed metal nanogap junction, for studying the 
conduction through single molecules and for hydrogen sensing.  Chapter 2 will give a 
general review of electromigration and points out several metallurgical factors that 
affects the type and rate of electromigration.   
The EM-formed metal nanogap junction is one of a large number of test 
structures for single-molecule electronics currently under study, and Chapter 3 will 
discuss some of the various competing techniques and the advantages and 
disadvantages of the EM-formed nanogap junction for molecular electronics study.  
 
 2 
 
Briefly, the major advantages of the EM-formed nanogap junction are the relative 
ease of fabrication and its thermal stability, allowing low-temperature measurements 
of metal-molecule-metal junctions.  Chapter 3 will also review electron transport 
through a mesoscopic sample and previous efforts in this research area that are related 
to our experimental results.  
Chapter 4 will discuss the experiments I did to understand the formation of 
nanogaps in thin gold lines via electromigration.  The control of the final junction 
resistance through a feedback scheme is demonstrated.  By studying feedback-
controlled EM in wires of various dimensions, it is found that the feedback scheme 
operates via temperature control of the electromigrating junction.    
Chapters 5 and 6 describe my experimental efforts and the difficulties I faced 
in attempting to fabricate superconducting contacts for measurement of 
superconductor-molecule-superconductor junctions. 
In chapter 7, I describe our efforts for making a novel hydrogen sensor using 
EM-formed nanogaps formed in palladium nanowires. Palladium, having high 
hydrogen solubility, is used as transducer element for various hydrogen sensors and 
filters. I attempted, with mixed success, to fabricate nanogap junctions in palladium 
nanowires using electromigration and to use these nanogaps as hydrogen sensors by 
utilizing the hydrogen induced lattice expansion mechanism of palladium.   
The last part of the thesis, chapters 8-9, describes experiments to determine 
the nature of chemical sensitivity in carbon nanotube thin-film transistors (TFTs).  
Chapter 8 will give a brief introduction to carbon nanotubes and nanotube thin-film 
transistors.   
 
 3 
 
Chapter 9 describes the experiments I did to understand the working 
mechanism of carbon nanotube TFTs as chemicapacitative sensors. I carried out 
several experiments to understand the resistive and capacitive response of carbon 
nanotube TFTs in ultra high vacuum, ambient atmosphere and in the presence of 
several analytes. I discuss our results by modeling the nanotube network as a simple 
resistor-capacitor (RC) transmission line, and find that, within the errors of our 
experiment, the measured changes in capacitance may be explained by changes in 
RC-transmission-line impedance induced by changes in the sheet resistance of the 
nanotube film. 
 
 
 
 
 
 
 
 
 
 
 4 
 
Chapter 2: Electromigration 
2.1: Introduction  
When a thin metal film  is subject to high current densities over a period of 
time, impurity ions and/or atoms of the metal may move diffusively along or in 
opposite direction to the electron flow. This current induced diffusion is called 
electromigration.  
In industry, electromigration is a menace and can be defined as the principle 
metallization wear out failure mode in integrated circuit interconnects [4]. Because of 
the small device sizes in integrated circuits, even a few milliamps of current would be 
enough to produce current densities that will cause considerable electromigration 
damage. As long as the interconnects are wide enough and/or electromigration 
activity is lessened by using capping layers such as TiN electromigration is not a vital 
problem, that is, electromigration does not significantly degrade the reliability of a 
device in a short period of time. But since scaling down of integrated circuits requires 
scaling down of interconnects as well, electromigration stands as a serious future 
problem for industry. 
In scientific research, failure of a narrow metal wire due to electromigration 
has also been utilized extensively to prepare stable electrical contact pairs with 
nanometer separation after the technique was shown to be a practical test bed [5-7] 
for single molecule electrical experiments. These experiments rely on producing 
nanogaps that are a few nanometers across by using electromigration as an 
electromechanical etching process. Later these nanogaps can be used as electrode 
 
 5 
 
pairs for single molecules which are either adsorbed on the surface before the 
electromigration is performed or after the electromigration process. 
Electromigration takes place when the high current density produces a 
pressure on the conductor atoms forcing them to move. Metal thin films contain large 
number of facile grain boundary and interfacial diffusion pathways that allow 
significant mass transport at relatively low temperatures. When there is a net 
divergence or convergence of mass flux, two failure modes of electromigration are 
observed. The first is failure due to formation of a void which causes an open circuit 
failure; and the other is failure due to formation of a hillock, production of excess 
material, which may form a bridge to an adjacent track in integrated circuits and 
cause a short circuit. In less severe cases where there is no failure but considerable 
diffusion activity, an increase in the interconnect resistance can be observed. 
 
 
Figure 2-1. Void and hillock formation in a polycrystalline metal film. Adapted from reference 
[33] 
 
In polycrystalline metallic interconnects; diffusion of atoms in the bulk, grain 
boundary diffusion, and diffusion of atoms at the surface are believed to be the 
 
 6 
 
dominant physical mechanisms of electromigration. But how they cause 
electromigration in different metals and how important they are in different stages of 
electromigration is not well understood. Thus a clear understanding of the processes 
that play role at different stages of electromigration and their relative importance on 
one another is technologically very important. 
Electromigration by its nature is a destructive phenomenon. Although it is 
possible to examine structures after the damage happens with scanning electron 
microscopy and/or transmission electron microscopy, with these techniques it is very 
hard to obtain information about the causes of the failure mechanisms. A good 
understanding of primary causes requires reasonable amount of data to be collected in 
the early stages of the process and/or while the failure due to electromigration occurs. 
 
2.2: Microscopic Theories and Empirical Relations  
Although electromigration can be stated simply as the migration of atoms in a 
preferred direction when current flows through a conductor, physical mechanisms 
that cause electromigration contain subtle issues such as the transport of electrons in 
the vicinity of a diffusing atom and how this transport produces a driving force that 
causes electromigration. 
It is accepted that this force is in part due to the effect of bulk electric field 
inside the material on the bare valence of the migration ion (direct force) and in part 
due to momentum transfer of scattering of conduction electrons (wind force), but 
there is not a general agreement about their relative contribution to the net force. The 
difficulty arises because there is no clear understanding of the basic issues: how the 
 
 7 
 
electric field is screened in the vicinity of migrating ion, what is the nature of 
inhomogeneities of current flow and electric field in the vicinity of migrating ion, and 
how they affect the migration process. 
There are many theoretical models as well as empirical relations developed to 
explain observed electromigration failure. Below I explain some of the mainstream 
theoretical work in order to give the reader a general view of the subject and the solid 
state physics behind the electromigration phenomena so far. 
 
2.2.1: Microscopic Theories  
As stated above, the force causing the electromigration process can be divided 
into two components as shown in Equation 2-1. 
 
EeZFEZeF
wind
nullnullnullnull
?
=+=
                                           Equation 2-1 
 
Where Z is the bare valence of the migrating ion with respect to host lattice, -e 
is the charge of the electron, E
null
 is the macroscopic electric field, and 
?
Z  is the 
effective valence of the migrating ion and contains both the effect of bulk electric 
field on the ion and the dynamic coupling of the electrons. The primary goal in 
electromigration theory is to calculate the wind force and
?
Z . 
 
2.2.1.1: Ballistic Model 
The first model trying to explain the wind force and deviation of 
?
Z  from the 
bare valance, Z, was published by Fiks [8] and Huntington and Grone [9] 
 
 8 
 
independently. In this model the origin of the wind force contribution to the total 
force was taken as the momentum transfer of scattering electrons from the migrating 
impurity ion. This force can be evaluated in the free electron model by using a shifted 
Fermi-Dirac distribution for describing how incoming and outgoing electronic states 
are distributed in k
null
 space and by using a generalized golden rule to evaluate 
transition probability per unit time. As a result of the calculation one can arrive at the 
final result that describes the wind force as  
 
()Ee
n
n
F
i
i
wind
nullnull
?=
?
?
                                                                        Equation 2-2 
 
where n and 
i
n  are electron and impurity densities respectively, ?  is the total 
resistivity and 
i
?  is the contribution of migrating impurities to the total resistivity. 
From the above formula, the wind force is proportional to, and in the opposite 
direction of, the electric field and, due to inverse proportionality with the total 
resistivity, it is expected to increase with decreasing temperature. 
In this model 
?
Z  can be written as    ?nlZZ ?=
?
    where l is the mean free 
path for electrons and includes all scattering effects, Z is the bare valence and ?  is 
the cross section for scattering of electrons from the impurity. 
Within different materials the number of electrons contained in the cylindrical 
volume ?l  might be larger or smaller than Z, and thus plays a dominant role in 
determining the relative importance of the direct force and wind force contributions. 
In simple metals where the free electron approximation is reasonable to use, the latter 
 
 9 
 
contribution is dominant and the wind force is expected to be the more important 
factor. 
Although the model is satisfactory in simple metals, a number of difficulties 
are involved in its generalization. In the Bloch theory of metals it is not an easy task 
to write a closed expression that describes momentum transfer per unit time. In 
addition it is not clear how momentum transfer is partitioned in more complicated 
scattering mechanisms that involve not only one ion but clusters of ions, and it is not 
understood what role the lattice plays in absorbing the momentum. There is also no 
discussion of the effect of screening on direct force. 
 
2.2.1.2: Polarization Charge Models 
An alternative model for calculating wind force was developed by Bosvieux 
and Friedel [10]. In their approach the effect of the scattering was to produce an 
inhomogeneity in the current flow. This inhomogeneity can be considered as a local 
charge polarization which gives rise to an electric field and it is this local field that 
produces the wind force. Since in a stationary situation the force exerted by the lattice 
defect on the electrons for maintaining this inhomogeneity is equal to the force 
exerted on the defect by the polarized charge distribution, calculation of the former 
force circumvents the conceptual difficulties encountered in the generalization of 
ballistic model. Within this approach the expression for wind force can be written as 
 
() rd
R
V
rnF
wind
null
null
null
null
30
?
?
?=
?
                                                                  Equation 2-3 
 
 
 10 
 
Here the electron density, ()rn
null
, depends on both applied electric field and electric 
field produced by current induced polarization , 
0
V  is the interaction potential energy 
between the bare ion and electrons, R
null
 is the position of the impurity ion and the 
integration is over all space. Although this description of the wind force is exact, its 
calculation requires knowledge of the electron density, within the independent 
particle approximation we may write 
 
() ()
2
rfrn
k
k
k
nullnull
null
null
null
?=
?
                                                                  Equation 2-4 
 
Where 
k
f
null
 is the shifted Fermi-Dirac distribution,
k
null
? ?s are scattering state 
wavefunctions and the sum is over all electronic states. ( )r
k
null
null
?  depends on both 
0
V  
and the electrostatic potential energy due to ( )rn
null
, and is determined by self consistent 
calculations. Such self consistent calculations are used to calculate the wind force in 
metals. 
Pseudopotential formalism has been used to calculate the wind force for 
impurities and vacancies [11] and to study the anisotropy in 
*
Z  [12, 13]. Gupta [14, 
15] calculated the wind force within the muffin tin potential formalism in noble and 
transition metals. Lodder [16]  the KKR distorted-lattice approach to make 
calculations of wind force; later Van Ek [17] used this approach to calculate hydrogen 
electromigration in transition metals. In their 1962 paper [10] apart from the 
expression of wind force, Bosvieux and Friedel stated that the direct force on an 
impurity vanishes due to screening by conduction electrons. This began a debate, later 
 
 11 
 
authors like Das and Peierls [18], Schaich [19] and Sorbello [20] gave arguments 
about the existence of the direct force.  
 
2.2.1.3: Local Field 
The form of the local electric field due to scattering from an ionic impurity 
was first studied by Landauer [21]. He found that charge distribution around the 
impurity gives a dipolar potential. This dipole field, called the residual resistivity 
dipole (RRD), is non- oscillatory and second order in potential. The RRD makes a 
significant contribution to the wind force expression of Bosvieux and Friedel when 
strong scattering is considered. Bosvieux and Friedel?s calculation of the wind force 
uses the Born approximation and  ( )rn
null
 is calculated to first order in potential, but as 
mentioned by Landauer [22], when scattering is strong corrections that are second 
order in potential should be taken into account in wind force calculations. 
Landauer mentioned another effect of impurity scattering that contributes the 
direct force [22, 23]. In the presence of an attractive potential electron density in the 
immediate vicinity of impurity is increased (or reduced in the case of a repulsive 
potential). He mentioned that this effect might be seen as a local electric field in what 
he called the carrier density modulation (CDM) field. This field, for an attractive 
potential, is in the opposite direction from the applied bulk field so it gives an 
effective screening mechanism for the direct force. CDM effect is used by Das and 
Peierls [24] to write a formula for total force, EenF
nullnull
))((
0
???= ?? , that Landauer 
called the Das-Peierls electromigration theorem [25]. Here ??  shows the change in 
resistivity caused by a defect density of one per unit volume and 
0
n  is the electron 
 
 12 
 
density of the pure metal. But their theorem does not give a correct explanation of 
experimental data, total force in their expression vanish as 0?? ??  where 
experimentally it does not. 
 
2.2.2: Empirical Relations 
Although, as we have briefly discussed above, the theoretical discussion of the 
electromigration phenomena found considerable interest in solid state physics, 
experimental results are generally discussed by semi-empirical relations both due to 
the fact that it is very hard to quantify and calculate the force acting on the samples in 
real experimental situations and the experimental results seem to follow a semi-
empirical formula first pointed out by Black [4]. Before starting to discuss the Black 
Equation and its relation to mechanical stress (especially in IC interconnects) it is 
fruitful to look at the diffusion mechanism in solids.  
Diffusion of atoms in solids is generally associated with the defects or 
deviations from perfect crystalline structure. The diffusivity, D , follows  more or less 
an Arrhenius type relation  
kT
E
a
eDD
?
=
0
 . The diffusing atoms in the lattice must 
be in the immediate vicinity of a vacancy, grain boundary or surface which provides a 
pathway for atomic transport. Each type of crystal defect has an activation energy 
(
a
E in above equation) that determines the ease by which the diffusion takes place. In 
general lattice vacancies require high activation energy for their formation. 
Comparatively, grain boundaries and interfaces have smaller activation energy and 
surfaces have the smallest and motion along them is the easiest. This order of 
 
 13 
 
activation energies as well as their relative importance on one another in 
electromigration process is, of course, material dependent.  
At constant temperature there is no preferential direction for atomic diffusion 
(this is an implication of Fick?s First Law, see i.e.[26]), but lattice defects such as  
surfaces or grain boundaries break this symmetry by reducing the activation energy 
for motion along the defect. Still, along the defect, diffusion of an atom is random, 
say between forward and backward direction. However in the process of making an 
atomic jump while the atom is in a saddle point between energetically equally 
favorable locations, the momentum imparted by scattered electrons may promote 
diffusion in the direction of electron flow. Because of this property electromigration 
is many times defined as a current-biased diffusion process. 
The electromigration process is often dominated with drift of atoms along a 
certain defect type in an applied field. This can be easily seen in couple of steps by 
using the Equation 2-1 for the total force acting on metal atoms and by using the 
Einstein relation to relate the mobility of atoms to diffusivity. With a simple 
derivation we can show that the relation for
d
V , the drift velocity of atoms is 
 
kT
E
d
a
eD
kT
eZ
jV
?
=
0
*
?
nullnull
                                                                Equation 2-5 
 
The above equation shows that the drift velocity of the metal atoms is 
exponentially related to the activation energy and as long as the activation energies 
for different defect types are not very close to each other, the defect with smaller 
 
 14 
 
activation energy will dominate the electromigration process and will be the 
determining factor in electromigration dynamics.   
 
2.2.2.1: Black Equation  
The discrepancy between the microscopic models of electromigration and the 
experimental results can easily be seen by taking a closer look at the form of the 
driving force in electromigration. If we use 
?
E
j
null
null
=  then one can write the driving 
force as  
 
jeZF
nullnull
?
?
=
.                                                                           Equation 2-6 
 
This equation would seem to predict that the failure should be dependent on applied 
current density, j
null
, and the lifetime should depend on the current density via 
j
null
1 . But the experimentally observed lifetimes in IC interconnects seem to obey a 
different semi-empirical relationship known as the Black Equation [4]. 
 
kT
E
n
a
eAjMTTF
?
=
                                                                  Equation 2-7 
 
In this formula MTTF (or 
50
t  ) is called the median time to failure and it describes the 
time for failure of 50% of a group of samples that prepared under identical 
conditions. "A" that appears in the formula is a sample-dependent constant, j is the 
 
 15 
 
current density, n is a constant, 
a
E  is the activation energy for the electromigration 
process, k is the Boltzman constant and T is the absolute temperature. The current 
density exponent, n, equals to 2 in Black?s original formula, is empirically found 
between 1 and 7 [27].  
In general, experiments for MTTF measurements are done under accelerated 
test conditions, namely under high current densities and temperatures. From the 
measured values of MTTF values the activation energy and the current density 
exponent may be determined. This is done by either holding temperature constant and 
changing the current density for measurement of n, or changing temperature while 
holding current density constant for measurement of activation energy. 
One may qualitatively explain the physical origin of the parameters in this 
equation as follows. A depends on the conditions by which the interconnects are 
prepared because VLSI interconnects have polycrystalline structure and both 
orientation and the size distribution of the crystal boundaries affect the 
electromigration activity, structure of interconnects also affects the wide range of 
measured values of n. Interconnects show more electromigration activity at higher 
current densities and temperatures because motion along the defects become easier 
for electromigrating atoms.  
As discussed in the following section, the disagreement between the 
theoretical understanding of the lifetimes and the experimentally observed behavior 
can be understood considering the stress buildup in electromigrating metal lines. 
 
 
 
 16 
 
2.2.2.2: Effect of Stress Buildup in Electromigration 
The inverse square dependence of the lifetime on current density can be 
resolved when one considers the gradient in the chemical potential, ??
null
, as the real 
driving force for the diffusion of the metal atoms. Under these conditions one can 
write for the mass transport  
 
??=
nullnull
kT
DC
J                                                                                     Equation 2-8 
 
where J
null
 is the mass transport, D is the diffusion coefficient and C is the 
concentration of diffusing atoms and kT is the thermal energy. 
The most important extra contributions to the chemical potential gradient are 
the stress gradient and the concentration gradient of the vacancies. But at thermal 
equilibrium these two are not independent quantities, i.e. in passivated metal lines 
either via their natural oxide or via a refractory capping layer, an increased vacancy 
concentration will produce tensile stress. So it is enough to consider only one of these 
gradients. By introducing a concentration gradient term we write Equation 2-8 as  
    
?
?
?
?
?
?
?
?
?
?
?=
x
C
C
kT
jeZ
DJ
null
null
?
*
                                                                   Equation 2-9 
 
In configurations where there is a blocking boundary, i.e. with a vanishing mass flux 
where there is a diffusion barrier for the mass flux such as a contact to another 
 
 17 
 
material,   ()0,0 =tJ , and via  the continuity equation , 
dt
dC
x
J
=
?
?
,   a solution is 
found [28] where the time to achieve a specific concentration is proportional to the 
inverse of square of the current density as in Black equation but with different 
temperature dependence in the pre-exponential factor i.e. 
2
50
a
E
kT
T
tA e
j
??
=
??
??
. 
In passivated thin films, a large vacancy concentration cannot be supported. 
As electromigration progress, in one end of the conductor excess atoms accumulate 
and produce a compressive stress but in the other end of the conductor the vacancies 
do not exceed the thermal equilibrium value and simply annihilate in convenient 
sinks. But the vacancy annihilation in turn produces a reduction in the volume which 
can not be allowed in a passivated conductor and this volume change is transformed 
into a tensile stress buildup. So as current flows this stress buildup will increase and 
either will failure occur or the stress buildup will produce an equal and opposite force 
that will stop further electromigration activity. Given that the stress gradient 
represents a part in chemical potential one can rewrite Equation 2-9 as 
 
?
?
?
?
?
?
?
?
?
?
??=
xkT
jeZ
DCJ
??
null
null
null
*
                                                              Equation 2-10 
 
where ? is the activation volume and ?  is the hydrostatic component of the 
mechanical stress. If the failure occurs when a critical stress is reached then time to 
reach that stress level can shown to be inversely proportional to the square of the 
current density [29]. So it is this opposite driving force that is responsible for the 
 
 18 
 
failure described by the Black Equation. We can also see from Equation 2-10 that, at 
low enough current density, it is possible that stress gradient may exist in steady state 
without failure and the driving forces due to electromigration and the stress gradient 
may become equal when 
 
?
=
?
? jeZ
x
??
*
                                                                                 Equation 2-11 
 
In this condition the electromigration will cease. This condition was experimentally 
observed initially by Ilan Blech [30-32]. By integrating Equation 2-11 over the stress 
profile one can easily derive the so called Blech length, length at which for a constant 
current density the electromigration activity will stop, and the Blech product 
blechblech
Cjl ?  where 
blech
C
is a material dependent constant. If the product 
of current and length in the conductor passes the value of 
blech
C
 then 
electromigration proceeds, if not than the electromigration does not take place.  
 
2.2.2.3: Film Microstructure 
Microstructure of polycrystalline thin films such as the grain size distribution 
has an important effect on electromigration activity and on main modes of 
electromigration. When Al and/or Al alloys such as Al/Cu or Al/Cu/Si are used in the 
IC interconnects, the main electromigration activity takes place in grain boundaries 
and/or in the bulk of the material. The reason is that, the oxide of the Al,
32
OAl , forms 
a self limiting refractory shell which adheres extremely well to Al and makes an 
 
 19 
 
excellent diffusion barrier. Therefore, the Al surface, when the oxide is present, is 
effectively shut off as a diffusion pathway. So even thought geometrically there is 
plenty of surface available, electromigration activity takes place mainly at grain 
boundaries and in some cases in the bulk of Al.   
It is well known that IC interconnects become less reliable with smaller grain 
sizes [33]. If the grain size is small then there are many diffusive paths for 
electromigration activity and there might be continuous paths for diffusing atoms 
along the film. In interconnects where the width is very large compared to average 
grain size [a) in figure 2], mass flux is fairly uniform and there is not enough 
convergence or divergence of mass flux to produce electromigration failure. If the 
width of interconnect is decreased until it becomes comparable to the average grain 
size, electromigration activity increases. This increase continues until line width is 
approximately the same as the average grain size, it is this region [Figure 2-2 (b)] that 
shows a very high electromigration activity. 
 
 
 20 
 
 
Figure 2-2. Change of the structure of polycrystalline interconnects as grain size/width ratio 
decreases (top view). a) Shows a polycrystalline film where width is much larger than average 
grain dimension. b) Shows a film where average grain dimension  is approximately same size as 
width. c) Shows a metal film in bamboo structure. Adapted from reference [33].  
In this regime both polygranular clusters that contain high diffusive pathways 
for mass flux and grains that span the width of the film coexist. As mass flows 
through the polygranular clusters it stops at these spanning grains which act as a 
diffusive barrier. 
Then stress due to mass accumulation increases, grain boundaries where metal 
atoms accumulate show a high compressive stress and grain boundaries where metal 
atoms deplete shows a high tensile stress. These two points are generally the two ends 
of a polygranular cluster that is terminated by spanning grains. If the produced steady 
state stress is higher than some critical stress for deformation, electromigration failure 
occurs. This phenomenon can also be interpreted as a polygranular cluster that is 
 
 21 
 
longer than Blech length, so mass transport takes place and results in electromigration 
failure. 
With decreasing line widths more spanning grains that act as diffusion 
barriers, but the distance between them on average becomes shorter (or at least 
comparable to) the critical distance of the Blech length. So the stress that develops is 
smaller than the critical stress but big enough to induce a back flux that will stop 
electromigration activity, resulting in an increase in MTTF. If the width is small 
enough, then ?bamboo? (or near bamboo) structures, where the width of the 
interconnect line is smaller than the average grain size, may develop [c) in figure 2]. 
In this case, grain boundary electromigration is no longer allowed and the only 
possible path for migrating atom is through the bulk of the grain. Thus it is very less 
likely that interconnect will fail due to electromigration. This produces a drastic 
increase in the MTTF. 
 
2.3: Electromigration in Au 
As we can understand from the discussion so far, electromigration studies in 
the literature mainly focused on the metals that are used in IC interconnects that is Al 
and/or Al alloys. The seemingly correct interpretation of experimental results depends 
on the consideration of stress buildup in samples which in turn depends on the fact 
that samples used in electromigration research are mainly (as a consequence of 
industrial drive) are made of Al and/or Al alloys which has a tough oxide on the 
surface and effectively shuts of the surface as a diffusion pathway. Even with 
 
 22 
 
changing line widths of samples one can observe that the MTTF changes drastically 
indicating that main diffusion mechanism affects the electromigration dynamics. 
Au, on the other hand, is the metal of choice in the fabrication of nanogaps for 
single molecule electronics because gold-thiol bonds are employed to attach the 
molecules under interest to the Au surface and Au being a noble metal does not form 
an oxide layer similar to aluminum when exposed to ambient conditions. Therefore, 
one might expect the electromigration dynamics to be different from the aluminum-
based metallization. The above expectation is also supported by the fact that there is 
no reported Blech length for Au in the literature. So we conducted experiments (as 
explained in Chapter 4) to understand the electromigration dynamics in thin Au lines 
in the device geometry used in the single molecule electronics research to prepare 
nanogaps. 
 
 23 
 
Chapter 3: Point Contacts and Nanogap Junctions 
3.1: Introduction 
Ohm?s Law,
A
G
L
?= , describes the electrical resistance of everyday 
conductors such as a piece of copper wire. It tells us that the conductance,G  (inverse 
of the resistance) is proportional to the cross sectional area of the conductor, A , and 
inversely proportional to its length, L . The constant of proportionality,? , the 
conductivity, is a material property and is different for different conductors. However, 
as the size of objects approach the atomic scale, the material properties as we 
understand them in the macroscopic world start to break down; Ohm?s law becomes 
invalid and even the concept of electrical conductance changes. The energy scales 
involved are high enough such that we can see visible effects of quantum nature of 
materials even at room temperature. In this very small scale, the definition and 
understanding of the electrical conductance is based on the idea that the conductance 
is equivalent to the transmission probability for incoming waves of electrons [21]. 
Although this idea originated initially from mesoscopic physics, it can be applied to 
almost all nanoscale objects such as nanotubes, single molecules, quantum point 
contacts of two dimensional electron gas (2DEG) and point contacts of metals. 
In this chapter I will start with a review of the electrical transport through 
metallic point contacts and the experimental techniques to produce point contacts. 
Later I will discuss the idea behind the single molecule electronics and how the 
techniques used originally to fabricate point contacts of metals are used to prepare 
 
 24 
 
nanogaps for testing electrical conductance of single molecules. This chapter provides 
a theoretical basis to discuss the results of the experiments we did to understand the 
electromigration properties of thin gold lines. It also provides a standpoint where we 
can relate our results with previous experimental results.  
 
3.2: Conductance of point contacts of normal metals 
In mesoscopic conductors we can divide the transport into two regimes 
depending on the relative sizes of the length of the conduction channel, L , and the 
elastic mean free path, 
e
null . The elastic mean free path can be defined as the distance 
that an electron travels between two scattering events with impurities. We talk about 
the diffusive transport regime or the diffusive conduction when the electron makes 
many scattering events with the static impurities as it crosses the conduction channel 
(that is when
e
Lnullnull ). Diffusive conduction, in analogy with statistical mechanics, 
can be described as a random walk problem with step size
e
null . On contrary, if the 
electron can travel the channel without any scattering (that is when
e
Lnullnull ) than we 
talk about ballistic conduction or ballistic transport regime. 
Another important length scale in mesoscopic systems to name is the phase 
coherence length, 
?
null . Over the phase coherence length an electron preserves its 
quantum coherence that is it "remembers" its past in the conductor. Phase coherence 
cannot be destroyed via scattering through static impurities but it may be destroyed 
via electron-electron scattering, electron-phonon scattering, or scattering through 
magnetic impurities with internal degrees of freedom. Generally in clean conductors 
 
 25 
 
at low temperatures, the phase coherence length is an order of magnitude greater than 
the elastic mean free path. 
Transport in everyday conductors is diffusive, but as the size of the conductor 
approaches the elastic mean free path or the phase coherence length, we can expect to 
describe the conductance different than as it is described via Ohm?s law. We may also 
expect to see exotic effects due to quantum coherence. 
The problem of such conductors was first considered by Sharvin [34]. He 
considered when the size of a metal contact is much smaller compared to the mean 
free path of the electrons such as a fine needle gently pressed into a conducting 
surface. In this case an approximate classical calculation can be made by treating the 
electrons arriving from each side as being in thermal equilibrium. The conductance 
will then be calculated in terms of the net number of electrons and the area of the 
contact. This so called Sharvin conductance,
S
G , will then be given by 
 
2
2
2
12
42
F
SFF
ke
GvNeA A
h
??
==
??
??
                                                    Equation 3-1 
 
where h is the planks constant, 
F
k  is the Fermi wave vector and A is the cross 
sectional area. Note that 
S
G  does not depend on ?  and 
e
null but depends only on 
F
N  
and A . Power  ()PIV=  is not dissipated in the contact but it is carried as the kinetic 
energy of the electrons passing through the contact and is dissipated in the leads. 
The full quantum-mechanical description of conductance in mesoscopic 
devices where the phase coherence is maintained was first considered by Landauer 
[21]. To illustrate this one can assume an ideal one dimensional conductor where only 
 
 26 
 
one channel (such a as a conduction subband) is occupied and contributing to the 
conductance (See figure 3-1). 
 
 
Figure 3-1. Conduction through a mesoscopic sample. (a) Shows the schematics of a mesoscopic 
sample in a channel of length L.  Incident wave (I) is partially reflected through the sample (R) 
and partially transmitted (T). Leads 1 and 2 are connected to electron reservoirs where there is 
no phase coherence between particles. (b) Shows the energetics inside the channel. Due to the 
redistribution of charge, chemical potentials inside the channel are different from the chemical 
potential of the leads. Adapted from reference [35]. 
The right and the left electron reservoirs have their Fermi energies at 
1
? and 
2
? respectively. The chemical potentials in the leads are 
L
?  and
R
?  due to the 
reduction of electron density in the left lead because of scattering and pile-up of the 
charge density at right due to transmitted electrons. The reservoirs randomize the 
phase of outgoing electrons so that there is no phase relation between particles in the 
contacts and the leads open up to the reservoirs adiabatically such that the outgoing 
electrons do not reflect back. In this case we can write the current as [35] 
 
 27 
 
()() ()() ()
12 2
1
00
2edk dk e
I dE v k E dE v k E dE E
dE dE h
?? ?
?
???
?
??
?
?? ??
?=?=
??
?? ??
?? ??
??
?? ?
  
                                                                                                                     Equation 3-2 
 
where the integration is over the left and right going states ( and kk?respectively), 
()vkis the velocity of an electron in state k  and ( )E? is the energy dependent 
transmission probability.  In the linear response regime (for small applied voltage), 
we may assume that ()E? is energy independent ( ( )E? ?= ) and we can write the 
current as 
 
()
12
2e
I
h
? ?=?                                                                              Equation 3-3 
 
The voltage drop across the sample is ( )
LR
eV ? ?=? and is less than ()
12
? ?? due 
to the redistribution of charge. One can show that [35] in the four terminal 
configuration where the voltage drop is measured at a different set of leads 
()
()
()
12
1
LR
?
? ???
?
?= ?
?
 and we can write conductance as 
 
()
2
2
1
e
G
h
?
?
??
=
??
?
??
                                                                           Equation 3-4 
 
 
 28 
 
In the two terminal measurements ( ) ( )
12 LR
? ????=? since the voltage and current 
are measured through the same set of contacts and we can write the conductance as  
 
2
2e
G
h
?=                                                                                         Equation 3-5 
 
The factor 
2
2e
h
is called as the quantum unit of conductance or simply quantum 
conductance,
o
G , and can be defined as the maximum conductance the channel will 
have in the case of perfect transmission (i.e. 1? = ). 
One can generalize the above result into two probe multichannel configuration 
as 
 
2
2
n
n
e
G
h
?=
?
                                                                                  Equation 3-6 
 
where the sum is over all the channels contributing to conductance. The stepwise 
change of conductance in mesoscopic devices first demonstrated in 2DEG devices 
[36, 37]. Along with the mesoscopic 2DEG devices one can apply Landauer formula 
(Equation 3-6) to atomic sized conductors as well. In 2DEG devices the splitting 
between the quantum channels is ~1meV. This means that, to observe quantum effects 
we need to cool down the devices to liquid helium temperatures. On contrary in few-
atom metallic contacts we may estimate the mode splitting from the Fermi 
 
 29 
 
wavelength,
0
5
F
A? null , as 
22
2
~1
2
F
eV
m
?
?
null
. This implies that we may observe the 
quantum effects even at room temperature. But we should keep in mind that as 
oppose to 2DEG devices where we can change the potential profile and the number of 
conduction channels inside the channel smoothly by adjusting the gate field, in 
metallic contacts where the limiting property is the inherent atomic granularity in 
reducing the contact size, we should not expect the conductance to reduce by the 
simple multiples of quantum conductance,
o
G  due to lack of perfect transmission.   
 
3.3: Methods of Fabricating Point Contacts 
Early experimental studies of ballistic contacts of metals are done with so 
called Spear-Anvil technique [38, 39]. The technique consisted of a type of 
differential screw mechanism to press a needle gently on a metallic surface. Using the 
technique it is possible to get contact sizes down to 10 nm (deduced from the 
resistance of contacts using Sharvin formula). But to study contacts in quantum 
regime where one needs to have contact sizes comparable to the Fermi wavelength 
one needs much more stable and precise control over the contact. Two techniques, 
namely the scanning tunneling microscope (STM) and the mechanically controllable 
break junction (MCBJ) gives such precise control, and therefore they are extensively 
used in the study of point contacts. 
 After its invention [40], STM was used extensively  in the study of metallic 
point contacts (see i.e. [41] and the references therein) under variety of experimental 
conditions such as in ambient conditions, in vacuum, under cryogenic temperatures 
 
 30 
 
etc. The inherent stability of the STM environment and ability to measure 
displacement of the STM tip from the surface let measuring of conductance vs. 
displacement measurements in variety of metals. In the research for metallic point 
contacts, even the direct observation of atomic chains of gold are made by fabricating 
an STM with the tip apex at the focal point of an High Resolution Transmission 
Electron Microscope (HRTEM) [42].     
Along with the STM, probably the most extensively used method in the study 
of point contacts is the MCBJ technique [43]. The technique consists of fixing a metal 
wire under investigation on elastic substrate using some form of epoxy. Later by 
using a three point bending configuration where the substrate is pushed in the middle 
from below towards the counter supports which are at the two ends of the substrate 
and then by gradually increasing the bending of the substrate, it is possible to reach 
few-atom configuration. Mostly a notch in the wire is made before bending to assist 
the process of forming contacts. This method produces clean contacts under variety of 
experimental configurations such as UHV, ambient, cryogenics etc. 
Both in STM and MCBJ experiments possible quantization effects are 
customarily investigated via constructing conductance histograms from large sets of 
data (i.e. on the order of  ~ 10000 individual curves see i.e.[44]). The histograms are 
constructed under the assumption that during the contact breaking any effective 
contact size is equally probable. The investigation of gold nanocontacts with the 
histogram method shows a pronounced peak at 1
o
G  and smaller peaks at 2
o
G and 
3
o
G . The peak at 1
o
G survives under most experimental conditions and is attributed 
to single chain of gold atoms [41]. 
 
 31 
 
3.4: Fabrication of Nanogaps for Single Molecule Electronics 
Research activity on single molecule electronics was driven by both the 
scientific curiosity to understand how the conduction through single molecules take 
place and the high expected technological payoff by creating the smallest electronic 
devices in the history of electronics. 
Investigation of the conduction properties of single molecules generally 
requires a metal-molecule-metal configuration where the choice of metal is often gold 
due to the facts that gold surface does not have a natural oxide layer and the 
molecules under investigation are readily bonded to gold via thiol bonds. 
Electrodes for single molecules naturally have to be on order the size of the 
molecule where this requirement singles out conventional lithography methods for 
electrode fabrication such as electron beam and photolithography. Many groups have 
used the techniques borrowed from the point contact experiments to measure the 
conductance of single molecules [45-51]. Both STM and MCBJ give fine control over 
the electrode separation but they have their drawbacks. In STM it is not possible to 
fabricate a gate electrode that might change the electrostatic doping of the molecules 
and in MCBJ it is only recently shown [52] that it is possible to place a gate electrode, 
but this requires complicated fabrication steps and MCBJ show large drifts in 
changing thermal conditions. On the other hand in nanogaps fabricated via 
electromigration a gate electrode is easily incorporated under the gold line and the 
junctions are thermally stable. Therefore nanogaps fabricated by electromigration 
(although the electromigration dynamics are not well understood) have been used by 
many research groups for molecular-scale electrode fabrication. 
 
 32 
 
In the next chapter we will explain our experimental studies of nanogap 
fabrication via electromigration in order to understand electromigration dynamics of 
thin gold lines connected the bulk electrodes and we will discuss our experimental 
results. 
 
 33 
 
Chapter 4: Formation of Gold Nanogaps via 
Electromigration 
4.1: Introduction 
Gaps formed via electromigration have been used in single molecule 
electronics extensively by many groups to measure the conductance properties of 
single molecules. It has also been suggested [5, 6, 53] that the dominant failure 
mechanism in such electrically stressed nanowires is thermally assisted 
electromigration. As we have explained in section 2.2.2.1, the Black formula suggests 
that both current and local temperature (due to Joule heating) affects the 
electromigration process and they both change rapidly during the electromigration of 
short wire. Understanding the role of current and temperature is then would be critical 
to design circuits that will produce desired nanogap junctions controllably. 
It has been recently shown by two groups [54, 55] that by employing a 
feedback loop it is possible to slow down the rate of electromigration and to produce 
nanogap junctions of a desired resistance (and presumably gap size) in a controllable 
manner. It was also proposed that the controllable electromigration occurs at constant 
applied power and it is triggered at a constant temperature. To understand the 
electromigration dynamics of Au lines and possibly to verify the above propositions, 
we implemented this technique and experimented on wires of varying cross sectional 
areas and varying lengths. We also tried to understand the role of the feedback loop in 
controlling nanogap formation that is via a suitable design of devices, whether one 
can produce small nanogaps by using a simple voltage ramp or not.  
 
 34 
 
4.2: Device Fabrication and Measurement Setup 
We have fabricated our devices by using electron beam lithography and metal 
deposition process as we describe below. 
Electron beam lithography (EBL or e-beam lithography) can be defined as a 
specialized technique for creating extremely fine patterns that could not be possible to 
create otherwise by using optical lithography tools. EBL in brief consists of scanning 
a beam of electrons across a surface covered with a film of resist film that is sensitive 
to the exposure of electrons. Therefore by exposing the surface with a beam of 
electrons it is possible to deposit energy in the desired pattern in the resist film. 
Electron beam resists can be viewed as the reading and transfer media for e-beam 
lithography. The usual resists are polymers that are dissolved in a liquid solvent. The 
resist is dropped onto a substrate that is then spun at high rpm values (~ 4000-6000) 
to form a layer of coating. The rpm value of the spin determines the thickness of the 
coating. After baking out the casting solvent, exposure to electrons modifies the resist 
leaving it either more soluble (positive) or less soluble (negative) in a developer. The 
created pattern is then transferred to substrate either through an etch process (such as 
plasma or wet chemical) or through material deposition and liftoff. In material 
deposition and liftoff process, the material is evaporated (such as thermal evaporation 
or e-beam evaporation) or sputtered (such as SiO
2
 or refractory metals) from a 
material source in a vacuum chamber onto the substrate and resist layer. Later the 
resist is lifted off (or washed away) in a solvent such as acetone. Mostly two layers of 
polymers (bilayer resist) used the bottom layer being more sensitive to electron beams 
 
 35 
 
compared to top layer to generate an undercut resist profile that will aid the liftoff 
process by providing a clean separation of materials.    
Commonly, to produce a slight undercut two layers of PMMA (Poly-methyl 
methacrylate) with different molecular weights (bottom layer being low molecular 
weight compared to top layer) is used. To produce a large undercut the bottom layer 
is spun from copolymer PMMA and top layer from PMMA. In the later case the resist 
after e-beam exposure developed in methyl isobutyl ketone: isopropanol (MIBK: 
IPA) solution where MIBK develops PMMA and IPA develops the copolymer 
PMMA. Therefore by using high ratio MIBK: IPA (such as 1:3 or more) or 
developing in IPA only after an initial MIBK: IPA development structures such as 
resist bridges of PMMA can be made. Figure 4.1 shows the above procedure 
schematically. 
The recipe I used for electron beam lithography in device fabrication is as 
follows. I prepared my samples on SiO
2
 (500nm)/Si substrates. I used bilayer resist 
where the bottom layer is spun from MMA at 4500 rpm for 45 seconds producing a 
300 nm thick coating. After spinning the MMA I baked the chip on a hot plate for 15 
minutes at 150 
0
C to evaporate the solvent (Chlorobenzine in this case). The top layer 
is spun from PMMA at 6000 rpm for 45 seconds producing 100 nm coating. I again 
bake the chip for 15 minutes at hot plate at 150
0
C . I then used the as-prepared chips 
in e-beam lithography. After metal deposition, lift-off of the resist and excess metal is 
performed by immersing the chip in acetone for 45 minutes to 2 hours depending on 
the resist thickness and the type of metallization.  
 
 36 
 
The patterns to be exposed in EBL are designed with CAD software, and the 
Nanoscale Pattern Generation System (NPGS) is used to control the motion of 
electron beam in a Scanning Electron Microscope (SEM, FEI model XL-30). 
 
 
 
Figure 4-1. Schematic of the electron beam lithography technique.  (a) Clean starting substrate. 
(b) Substrate is covered with bilayer resist. Bottom layer is spun from MMA-MAA and the top 
later from PMMA to produce a large undercut. (c) Exposure of resist to electron beam. (d) The 
exposed resist is developed in MIBK-IPA solution (e) Metal deposition on the created pattern. 
Metal is deposited to both the surface of the resist and the substrate through the window opened 
up in the resist after development. (f) After the liftoff, resist is washed off the surface and only 
the metal deposited on the substrate is left behind.   
 
 37 
 
I prepared my devices in two steps of electron beam lithography. An SEM 
microscopy picture of a typical device is shown in figure 4-2. I first fabricated the 
thin gold lines with no adhesion layer. In the second electron beam lithography step 
the bulk of the electrodes and the contact pads are fabricated by depositing typically 5 
nm chromium adhesion layer and 70 nm gold. 
   
 
 
Figure 4-2. SEM Micrograph of a typical device used in controllable electromigration 
experiments. 
After fabrication I cleaned the devices with oxygen plasma typically at 300 militorr 
oxygen pressure and 300 Watts for 15-30 seconds to remove the possible resist 
residues from the surface and to be sure that there are no organic contamination on 
the surface. I later fixed my chips on the chip carriers with General Electric (GE) 
varnish, and connected the electrical features on the chip to the leads on the carrier 
 
 38 
 
using an ultrasonic wire bonder. I used a 
4
He gas-flow cryostat (Desert Cryogenics) 
to take the measurements at temperature ranges from 1.2-325 K. The schematic of the 
measurement setup I used in my experiments is shown in Figure 4-3. I used a 
Keithley 2400 multimeter as voltage source and I measured the current by using an 
Ithaco 1211 current amplifier and an Agilent multimeter. The setup and the feedback 
(as explained in next section) are controlled via measurement programs written in 
LabVIEW language. 
 
 
 
Figure 4-3. Measurement setup used in controllable electromigration. A Keithley 2400 
Multimeter is used as a voltage source and devices are measured in a 4He gas flow cryostat. 
Current is measured using an Ithaco 1211 transimpedance amplifier and an Agilent Multimeter. 
The setup is computer controlled by measurement programs written in LabVIEW language. 
 
 39 
 
4.3: Feedback Controlled Electromigration 
To control the electromigration process, I used a feedback scheme similar to 
Ref. [54] consisting of the following steps: I first measure a reference conductance 
value at a voltage of 100 mV. I then increase the voltage until the conductance drops 
by a set fraction (typically 2-5 %) of the reference conductance value. The value of 
the fraction is chosen such that the change in conductance is due to electromigration 
as opposed to decrease of conductance due to bare joule heating. At this point, the 
voltage is ramped down by 50 to 100 mV (at a rate of 50 mV/s) and a new reference 
conductance value is measured. We repeat this process until the desired conductance 
value is reached. A typical I vs. 
bias
V  curve of such feedback controlled 
electromigration is shown in figure 4-4. Note that the current as a function of bias 
voltage is a multivalued function.  
 
 
 
 
 
 
 
 
 
 
 
 
 40 
 
 
 
Figure 4-4. I vs. V
bias
 curve of a typical feedback controlled electromigration process. The inset 
shows the progression of the electromigration in fine details. 
As we have explained in Chapter 2, considerable electromigration activity 
starts when atoms become mobile enough. If the local temperature increase of the 
electromigrating wire supplies this energy, one may expect the electromigration 
process to be highly temperature dependent and not to start before the wire becomes 
gets hot enough due to joule heating. For a uniform wire one expects the heating to 
start from the midsection, but for a non uniform wire it will start from the point where 
the scattering is concentrated.  Figure 4-5 demonstrates the effect of temperature 
increase on electromigration of our samples. The data labeled A shows a smooth I vs. 
bias
V curve indicating that the electromigration has not begun in the gold wire. 
 
 41 
 
Although we observe a resistance increase with increasing bias, we found that if we 
stop the voltage in bias in region A, this resistance increase is reversible. Such a 
reversible resistance increase shows that the gold wire heats up before 
electromigration begins. The data labeled B shows that after this initial heating, the 
gold wire begins to change resistance irreversibly due to electromigration. The data 
labeled C shows that one can stop and restart electromigration before the gold wire 
totally fails. 
bias
I V? curves of two bias processes perfectly match each other 
indicating that in the second biasing process the gold wire first heats up to the 
temperature where significant electromigration takes place, and the electromigration 
restarts. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 42 
 
 
 
Figure 4-5. Current vs. bias voltage during the feedback-controlled electromigration of an Au 
wire at T=1.3 K. Part A is a smooth curve indicating than the EM has not begun whereas in part 
B the resistance of the line increases irreversibly due to EM. Both parts A and B are recorded in 
a single voltage biasing process, producing a final resistance of ~120 ? .  At this point the voltage 
was reduced to zero for some time.  When the bias process was restarted in C, the wire resistance 
is the same, demonstrating that the EM process may be frozen by turning off the voltage. 
In the analysis of my data, I will assume that the total resistance of the circuit 
R is the sum of two resistors
L
R , the lead resistance (equals to the resistance 
measured at low voltage bias) and
J
R , the resistance of the "junction", the weak spot 
formed in the wire by electromigration; i.e., 0
J
R = initially.  I then can calculate the 
power dissipated in the junction as
2
J J
PIR= .  
 
 43 
 
Note that 
J
R includes a contribution from the resistance change of the leads 
upon heating, and the heating due to the resistance of the nanowire itself is ignored. 
Therefore, 
J
P is a rough estimate of the power that is heating the junction, but should 
be valid when 
J
R is significantly non-zero. 
Figure 4-6 shows 
J
P versus
J J
VIR= , where the inset graph is the 
corresponding 
bias
I V?  curve of the data. After the junction resistance begins to 
increase due to electromigration [labeled as (a) in both the graph and the inset], the 
power dissipated in the junction reaches a relatively constant value. Constant power 
dissipation in the junction is observed until the resistance of the junction becomes on 
the order of several kiloohms. At the resistance values of several kiloohms, one 
should treat the junction as being in the Sharvin or mesoscopic limit and one should 
not expect all the energy of the electrons to be dissipated in the junction. Therefore 
our classical description of power dissipation becomes invalid.   
 
 
 
 
 
 44 
 
 
 
Figure 4-6. Power dissipated in the junction P
J
 vs. the voltage drop at the junction V
J
 (quantities 
defined in text). The irreversible change in resistance due to electromigration starts at the point 
labeled (a). Inset shows the corresponding current vs. bias voltage data.  The starting nanowire 
has dimensions 830 nm long x 60 nm wide x 25 nm thick; the length and width of the nanowire is 
determined using SEM, and thickness by quartz crystal monitor during gold film deposition. 
As we can see, temperature of the wire plays an important role in the 
electromigration process. By looking at the change of 
J
R as a function of the junction 
voltage we can get further insight on whether the electromigration process is 
primarily temperature or bias controlled. A primarily temperature-controlled 
electromigration should produce 
2
J J
R V? but a primarily bias-controlled junction 
should produce
J J
R V? . In figure 4-7 we plot the change of the junction resistance as 
a function of junction voltage during electromigration. The solid lines are guide to 
 
 45 
 
eye and showing 
2
~R V  and the dashed lines are showing ~R V  for comparison. 
Figure 4-7 indicates a primarily temperature controlled electromigration process. 
 
 
 
Figure 4-7. Evolution of junction resistance as a function of junction voltage. In both figure (a) 
and (b) the solid lines are guide to eye showing R ~ V
2
 and the dashed lines are showing R~V. 
If the junction temperature is the main factor in controlling the 
electromigration, one expects for better thermally coupled junctions more power to be 
dissipated at the junction. Since the thermal conductance of the wire is proportional to 
the cross-sectional area, the power required to maintain a given wire temperature 
 
 46 
 
should also be proportional to the area. To test this point we fabricated wires with 
different cross sectional areas. Figure 4-8 shows that the power dissipated in the 
junction during electromigration is proportional to the nanowire area. 
 
 
 
Figure 4-8. Power dissipated in the junction during electromigration vs. nanowire cross-sectional 
area.  The power is the average power in the region of near-constant power seen in  Figure 4-6. 
Therefore, we conclude from figure 4-8 and the relatively constant power observed in 
figure 4-6 that the electromigration rate is dominated by temperature and the feedback 
scheme operates to control the wire temperature. 
 
 47 
 
We estimate the temperature of the junction 
J
T during electromigration as 
follows. We neglect the heat conduction to SiO
2
 substrate by considering the relative 
magnitude of the thermal conductivity of gold and SiO
2
 and due to the thermal 
resistance between the gold line and SiO
2
  substrate [56]. We also consider the 
contacts as infinite heat sinks at T = 1.5 K. The temperature at the midpoint of a wire 
with uniform power dissipation over its volume is given by 
2
8
J
PL
T
V?
= , where P, V, L 
and ? are, respectively , the total power generated in the nanowire, the volume and 
the length of the nanowire, and the thermal conductivity of gold [57]. For the 
nanowire in figure 4-6, the maximum power estimated as ~0.67 mW(including 
J
P at 
point [Fig. 4-6(a)] and the additional power generated due to the estimated resistance 
11 ? of the nanowire at 1.5 KT = . Using the thermal conductivity of gold as 
320
W
mK
? (the room temperature value for bulk gold) we estimate 145 K
J
T ? . If 
we estimate that all the power is generated at the center of the nanowire and carried 
out to the leads by the nanowire, then 
2
290 K
4
J
PL
T
V?
=? ; which is still low enough 
to allow the study of many molecular adsorbates without desorption or dissociation of 
the molecules. Note that if we include the power loss to the substrate as well, then this 
will lead us to a lower temperature value which will not change our main conclusion. 
We will now discuss why the feedback process is feasible. The thermal time 
constant, which can be thought as the characteristic decay time of the temperature 
variations in a one dimensional heat conduction, [58] 
2
p
2
th
LC?
?
? ?
= , where 
p
C is the 
specific heat and ? is the density, is less than 1 nanosecond in our nanowires; much 
 
 48 
 
faster than our feedback circuit. This suggests that the electromigration process must 
occur very slowly. If this is the case, is feedback needed at all?  
 
 
Figure 4-9. Stable and unstable regimes in controllable electromigration of a typical device. 
Following the I-V
bias
 curve of a typical electromigration process, such as 
Figure 4-9, we can see that I , and hence R , is a multivalued function at a given 
V
bias
. Assuming that the junction temperature is proportional to the power dissipated 
in the junction, that is
2
J J
TIR? , we can write  
 
 
 49 
 
()
2
j
j
jL
R
T
RR
?
+
                                                                              Equation 4-1 
 
The change of junction temperature with junction resistance is then given by 
 
 
()()
23
2
1
jj
j
jL jL
dT R
dR
RR RR
??
??
??
++
??
                                                  Equation 4-2 
 
From above formula we can see that once the electromigration starts the thermal 
runaway can be prevented if 0
j
j
dT
dR
< . We call this regime as the stable branch of the 
I-V
bias 
curve (see i.e. Figure 4-9) and it corresponds to the regime where the junction 
resistance is greater than the lead resistance, 
J L
R R> . If the junction resistance is less 
than the lead resistance, that is
J L
R R< , than 0
j
j
dT
dR
> . We call this regime as the 
unstable branch in the I-V
bias
 curve. So when the junction is in the stable regime 
electromigration progress in a self limiting fashion and when the junction is in 
unstable regime electromigration increases the junction resistance and this will 
produce more heat and more increase in the junction resistance. So we may expect a 
transition from stable to unstable regime when we turn off the feedback as 
electromigration progress. Figure 4-10 illustrates this instability. 
 
 
 50 
 
 
 
Figure 4-10. Current vs. bias voltage during the electromigration of three similar gold wires 
(600-700 nm long x 40 nm wide x 15 nm thick) at T=1.3 K.  For the blue and red curves, the 
external feedback is turned off at the points marked by the blue and red arrows respectively.  
For the black curve, no external feedback was used. 
In figure 4-10, we turned off the feedback at various points during the 
electromigration process and solely ramped the voltage upwards at a fixed rate. The 
red curve shows the feedback turned off while on the stable I-V
bias
 branch 
(positive
bias
dV
dR
); the current decreases smoothly with increasing voltage from this 
point.  However, when the feedback is turned off on the unstable I-V
bias
 branch 
 
 51 
 
(negative
bias
dV
dR
; blue and black curves), the current drops rapidly to the stable branch 
at the same V
bias
.  Thus the feedback scheme is only necessary to produce final 
resistances
J L
R R< ; with suitable circuit design (minimization of
L
R ; i.e. short 
nanowires with highly conducting leads) small final
J
R ?s may be produced using a 
simple voltage ramp.  
Note that in some circuits [i.e. see figure 4-11 (a)] the stable I-V
bias
 branch 
does not extend beyond the unstable branch; in such cases a simple voltage ramp 
causes abrupt failure of the wire by melting (as observed via post-mortem SEM 
micrographs [i.e. see figure 4-11 (b)], resulting in large ( 10 nm)> gaps. 
 
 
 
 
 
 
 
 52 
 
 
 
Figure 4-11. Electromigration of a device where stable branch does not extend beyond unstable 
branch. (a) Shows the I-Vbias curve a device where the stable branch does not extend the 
unstable branch. (b) Shows the post-mortem SEM micrograph of a device where simple voltage 
ramp results in a large gap via melting. The scale bar in (b) is 500 nm. Note that the controllable 
electromigration data in (a) and the SEM micrograph are taken from different devices, 
In the last stages of the electromigration process the electromigrating junction 
starts to approach the mesoscopic limit. In this region we don?t observe a smooth 
change of the conductance, but instead we observe change of conductance in discrete 
steps on the order of conductance quantum
o
G . Figure 4-12 shows conductance 
change in units of conductance quantum vs. time in the last stages of controlled 
electromigration process.  
 
 
 53 
 
 
 
Figure 4-12. Conductance change in discrete steps of conductance quantum. The steps are on the 
order of, but not at exact multiples of Go indicating a mesoscopic junction. 
In this regime, we often observed conductance plateaus on the order of but not at 
exact multiples of
o
G . We ascribe the observed changes of conductance to changes in 
the atomic configuration of the contact. Although we don?t observe strict quantization 
of conductance, the conductance is still determined by a limited number of modes 
where not all the modes are showing perfect transparency [41, 59]. The 1
o
G level is 
an exception to this where formation of an atomic chain of Au atoms and single atom 
contacts are observed [42, 60]. Figure 4-13 shows 
bias
VI ?   curve of a junction after 
 
 54 
 
we stop electromigration at ~1
o
G and voltage bias up to failure. We ascribe the 
conductance plateau at ~1
o
G  to a single atom contact due to both linear 
bias
VI ?  
curve and the high voltage value it sustains before failure. 
 
 
 
Figure 4-13. I-V
bias
 curve of a junction where the electromigration is stopped at 1Go. Both the 
linear I-V
bias
 curve and the high voltage value at failure indicates a single atom contact. Numbers 
on the figure shows the direction of voltage bias. 
In junctions where we stopped electromigration when the final conductance is 
less than
o
G , we often observed two types of highly conductive non-linear  
bias
VI ?  
 
 55 
 
curves as shown in figures 4-14 (a) and 4-14 (b). Figure 4-14 (a) shows a junction 
where we observed highly conductive nonlinear 
bias
VI ?  curves indicating quantum 
mechanical tunneling through a thin potential barrier [61] and Figure 4-14 (b) 
shows
bias
VI ?  curves indicating transport through a Coulomb Blockaded junction as 
observed in similar experiments [62]  
 
 
 
Figure 4-14. I-V
bias
 curves after junction formation. (a) Shows highly conductive nonlinear I-V
bias
 
curve with positive curvature. (b) Shows an I-V
bias
 curve indicating Coulomb blockade. 
 
 
 
 
 56 
 
4.4: Conclusion 
In conclusion, we performed controllable electromigration on nanowires with 
different cross sections. We found that the average power dissipated in the junction 
during electromigration increases linearly with the area of the junction indicating 
temperature control of the process and confirming that the mechanism is thermally-
assisted electromigration. Using the maximum power dissipated in a typical device, 
we estimate the junction temperature during electromigration performed at KT 5.1=  
to be only a few hundred Kelvins.  We also note that the role of the feedback process 
in controlling electromigration is to prevent thermal runaway in the region of 
positive
j
j
dT
dR
.  This region can be reduced by reducing the series resistance in the 
circuit, allowing controlled electromigration with a simple voltage ramp. 
 
 57 
 
Chapter 5: Mesoscopic SNS Josephson Junctions 
5.1: Introduction 
Systems such as ferromagnets or superconductors that may go through a phase 
transition from a disordered to an ordered phase show many exotic physical behaviors 
in the ordered phase which is absent in the disordered one. Such systems may be 
treated phenomenologically after defining an order parameter, F, which shows how 
the order in the system varies. In the disordered state the order parameter vanishes 
showing complete lack of order and in the ordered state it takes nonzero values. In the 
case of ferromagnets F can be taken as the magnetization and in the case of 
superconductors as the effective Cooper pair density (density of paired of electrons). 
Typically such order parameters are continuous functions in space and their variations 
occur smoothly within the scale of the coherence length of the system. Therefore by 
placing an ordered and a disordered system in contact with one another, one may 
induce the spread of order from the ordered system into the disordered one. In the 
case of superconductivity the spread of superconducting order into normal metals that 
are in electrical contact with superconductors is called the proximity effect as 
discovered by H. Meissner [63].  
At low temperatures for most superconductors the coherence length is a few 
hundred nanometers, and devices with these length scales can routinely be fabricated 
with present day experimental techniques. Therefore one may use this effect to 
fabricate exotic mesoscopic devices where superconductivity is induced in metals, 
such as gold, which lack intrinsic superconductivity. Such superconducting gold 
 
 58 
 
mesoscopic devices have been fabricated and tested [59] where superconductivity is 
induced in a mesoscopic gold point contact, and from the shape of current-voltage 
curves the number of conduction channels and their individual transmission 
probabilities can be determined. 
As we have explained in Chapter 3, gold nanogap junctions (either prepared 
via electromigration or via experimental techniques that are commonly used in 
preparing point contacts) are used extensively as a test bed for single molecule 
electronics. In principle one may produce gold nanogap junctions that are proximity-
effect-induced superconductors and use these junctions as test beds for single 
molecule electronics. Such superconductor-molecule-superconductor (SMS) junctions 
can then be used to study a number of fundamental properties of the conductance of 
single molecules, such as the number of conductance channels the molecule has, the 
individual transmission probabilities of each conduction channel, and whether the 
molecule will carry supercurrent or not. 
  Chapter 5 and 6 summarize our efforts to construct SMS junctions and explore 
these issues. In Chapter 5, I will review the transport properties of superconductor-
normal metal-superconductor (SNS) mesoscopic junctions. I will then explain in 
Chapter 6 the experimental approaches I took in fabricating mesoscopic SNS junction 
devices that would be suitable to use as a test bed for fabricating single molecule 
SMS junctions.  I will explain the experimental difficulties we faced in device 
fabrication, and conclude with a discussion of what might be done to improve our 
experimental efforts. 
 
 
 59 
 
5.2: Basic Properties of Superconductors 
Superconductivity, in its manifestation of perfect conductivity, was discovered 
at Leiden by H. Kamerlingh Onnes in 1911 [64]. He observed that when pure 
mercury is cooled it looses its electrical resistance at a critical temperature, T
c 
.Below 
T
c
 the superconducting state can be destroyed at a high enough magnetic field, H
c,
 
where metal enters to normal state.  
In addition to the perfect conductivity, the second seminal property of 
superconductors, namely the perfect diamagnetism, was discovered by W. Meissner 
and R. Ochsenfeld [65]. They observed that when placed into magnetic field, or when 
the system becomes superconducting in magnetic field, superconductors expel 
magnetic flux. It is understood later that magnetic flux is completely expelled below 
H
c
 in type I superconductors which are mainly the pure elementary superconductors 
such as tin and aluminum. Type II superconductors (many of which are alloys) on the 
other hand allow partial penetration of magnetic flux in flux lines or vortices for H
c1
 < 
H < H
c2
. 
 Superconductors show an energy gap, ?, in their quasiparticle excitation 
spectrum. Establishment of the existence of an energy gap was one of the key steps in 
the understanding of the superconductivity in classical superconductors. First 
experimental indication of this gap came from the peculiar jump of the specific heat 
at the transition temperature [66, 67]. Later much direct evidence came from the 
measurements of electromagnetic absorption using infrared light [68, 69] and electron 
tunneling experiments pioneered by I. Giaever [70]. Furthermore the infrared 
 
 60 
 
measurements gave indications of pairing of quasiparticles and they turned out to be 
one of the early verifications of the microscopic theory of superconductivity.      
In 1950, before the appearance of the microscopic theory of superconductivity 
(The BCS theory, due to Bardeen, Cooper, and Schrieffer [71]), V. L. Ginzburg and 
L. D. Landau gave a phenomenological description of the superconducting state that 
is based on the Landau?s theory of phase transitions. [72] Being phenomenological, 
G-L theory is based upon the generally observable features of superconductivity. The 
theory introduces a complex pseudo-wavefunction ( )r?  as an order parameter that 
describes the superconducting state, and ()
2
r?  is taken as to represent the density of 
superfluid. One may then expand the free energy of the superconductor in terms of 
()
2
r? and ()
2
r?? with expansion coefficients ? and ? and arrive at two 
differential equations for 
()r? and the supercurrent 
s
J  .  
 
2
*
2
*
1
0
2
e
A
mi c
?? ?? ? ?
??
++?? =
??
??
nullnull
null
                                       Equation 5-1 
()
**2
***
2
s
ee
JA
mi mc
? ??? ??=????
nullnullnullnull
null
                                        Equation 5-2 
 
 
Where c is the speed of light, e
*
 is the effective electronic charge, null is Planck?s 
constant, m
*
 the effective electronic mass, A
null
 the vector potential, J
null
the supercurrent 
density. G-L theory introduces an important length scale called the Ginzburg-Landau 
 
 61 
 
coherence length,? , given by 
2
2
*
()
2()
GL
T
mT
?
?
=
null
. G-L coherence length is the 
characteristic distance over which the spatial changes on 
( )r?  may occur and the 
order parameter may vary. Ginzburg-Landau theory is very useful in describing the 
situations where there is spatial inhomogeneity and the order parameter varies, such 
as superconductors in a magnetic field or the penetration of order parameter into other 
metals as it occurs in some proximity effect configurations. But it concentrates only 
on the superconducting electrons rather than the excitations. Therefore to analyze 
mesoscopic Superconductor-Normal Metal-Superconductor (SNS) junctions one 
generally takes a different approach using the non-local form of BCS theory.  
Before the analysis of mesoscopic SNS junctions in section 5.4, I will note 
main features of BCS theory since most of the main concepts like the existence of 
quasiparticle excitations, the energy gap etc are retained in non-local form of BCS 
theory as well. BCS theory was published in 1957 by Bardeen, Cooper and Schrieffer 
[71]. BCS theory is based on the idea that in superconductors, electrons near the 
Fermi level experiences a mutually attractive force and this attractive force gives rise 
to a new form of quantum state at temperatures below T
c
. In this new quantum state, 
some portion of electrons near the Fermi level are bound together in pairs which are 
called Cooper pairs. In classical superconductors such as lead, tin and aluminum, the 
Cooper pairs? size, the coherence length, is several hundred nanometers, i.e. for 
aluminum the coherence length is approximately 1600 nm, for tin 230 nm and for 
lead 83 nm [73], and the Cooper pairs strongly overlap in space. The external center 
of motion of pairs is coupled together and each pair is in the same state. The paired 
 
 62 
 
state, like the Fermi sea, still has excitations which are electron-like for momenta just 
outside the Fermi surface and hole-like for momenta just inside ( In 
superconductivity, the hole state refers to an empty state below the Fermi level ). 
These single particle excitations are still fermions, with a Fermi distribution at 
temperature T, but in some respects such as their dispersion relation, they differ from 
ordinary electron and hole excitations. 
To break up a pair, one need to supply energy that amounts at least 2? and 
that produces two quasiparticle excitations in the superconductor, therefore the 
excitation energy, E
k
, cannot be less than ?. 
 
 
 
 
Figure 5-1. (a) Shows difference in the dispersion relation of normal and superconducting 
metals. (b) Shows the density of states of superconductor. There are no states for quasiparticles 
in the energy gap. Energy of Cooper pairs is measured from the chemical potential. Adapted 
from reference [73]. 
 
 63 
 
We may add electrons to the superconductor in two ways. One may add 
electrons as bound pairs with energy 2? , or one may add a single electron by 
producing a quasielectron excitation. There exist no states for the quasielectron 
excitations for energies below the gap, so to add an electron to the superconducting 
system from outside, one need to supply it with energy at least the value of the gap. 
Although the BCS theory revolutionized the understanding of the 
superconductivity, in its original form it can only handle translationally invariant 
superconductors containing excitations of definite momentum. Therefore to analyze 
the superconducting-normal metal-superconducting junctions in mesoscopic limit, the 
nonlocal form of the BCS theory is employed. 
 
5.3: Josephson Effect and SNS Josephson Junctions 
Josephson Effect was predicted theoretically by B.D. Josephson in 1962 [74]. 
Although his predictions were originally for tunnel junctions it turned out that they 
applied to a wider class of structures including SNS junctions. 
 
 
 
Figure 5-2. An SNS junction. Two superconductors (SC
1
  and SC
2
) are weakly linked via normal 
metal. 
 
 64 
 
The Josephson Effect is seen in two weakly interacting superconductors, that 
is two superconductors close enough that coherent Cooper pairs may travel between 
the two. The weak link that connects the two superconductors might be a tunnel 
barrier (like Nb-AlO
x
-Nb), a superconducting constriction, or a normal metal. Current 
through the weak link is carried by coherent interaction of electrons which is 
tunneling for a tunnel barrier and coherent Andreev reflection at the two interfaces for 
an SNS junction. (We will explain the mechanism of Andreev Reflection on section 
5.5). The resulting structure is called a Josephson Junction. 
The Josephson Effect is one of the macroscopic quantum phenomena resulting 
from superconductivity. One can separate the effect in two parts called as DC and AC 
Josephson effect. The DC Josephson Effect states that two superconducting 
electrodes may contain a component of supercurrent which does not depend on the 
voltage across the electrodes, but rather the phase difference between the two 
superconductors. Experimentally this states that one can push current through the 
weak contact (up to a limit namely
csc
I II? ??) without building up a voltage 
gradient. The critical current,
c
I , depends on the geometry of contact,  temperature, 
the material of the contact and other factors.     
 
()
12
sin
sc
II ? ?=?                                                                          Equation 5-3 
 
 
 65 
 
 
 
Figure 5-3. DC Josephson Effect.  
 
AC Josephson Effect states that, if a constant voltage is applied between the two 
weakly contacted superconductors, then the phase difference evolves with time and 
produces an AC supercurrent. 
 
()
12
2
d
eV
dt
???
=
null
                                                                           Equation 5-4 
 
 
 
 
 
Figure 5-4. AC Josephson Effect 
 
 66 
 
The supercurrent in SNS weak links flows along a conducting material. If we 
have an SNS sandwich junction (or similar constrictions), the supercurrent flows 
through this junction due to the proximity effect. In the normal metal there arises a 
nonzero order parameter,?, which exponentially decreases within the metal over a 
distance of the order of ?normal coherence length? or ?the decay length?
n
? . 
n
?  is 
given by 
0
2
F
nn
B
kT
?
??
?
==
null
in the clean limit that is when the electronic mean free 
path,null, is greater than 
0
n
? . In the dirty limit ,
0
n
?nullnull , where the electronic motion is 
diffusive, than 
n
?  becomes
0
1
3
n
?
??
??
??
null . On the other hand, the value of ? in the 
superconductor becomes less than the equilibrium value in the superconductor over 
distances on the order of coherence length in the superconductor. 
Transport properties of such junctions are well studied for various geometries 
and the temperature dependence of the critical current is calculated. The interested 
reader may see the review by K.K. Likharev [75]. For my purpose (to use such 
constrictions as test beds for single molecule electronics) I tried to build a mesoscopic 
SNS point contact junctions which I explain below. 
 
 
 67 
 
5.4: Mesoscopic SNS Junctions 
 
 
 
Figure 5-5. Mesoscopic SNS junction. Both the width and the length of the normal region are 
much smaller than the coherence length of superconductor. 
Mesoscopic SNS junctions, as distinguished from other junctions such as SNS 
sandwich junctions, refer to the class of junctions where both the length of the 
junction, L, and the width of the junction, W, are much smaller than the coherence 
length of the superconductor i.e. 
0
,WL ?null . A point contact or a microbridge between 
two superconductors belongs in general to this class of junctions. In the theoretical 
analysis of these junctions the suppression of the 
( )r?
null
approaching the junction can 
be neglected and one generally uses a step function model: 
 
()
1
2
2
0
2
0
        if 0
            0             if 0   
       if 
i
i
ex
rxL
exL
?
??
?
?<
?
?
?= <<
?
?
?>?
?
null
                                          Equation 5-5 
 
 
 68 
 
where ?  is the phase difference between two superconducting electrodes (assumed as 
the same superconductor here) and x is the coordinate along the junction. The above 
condition holds when the width of the junction is small compared to the coherence 
length, since the non-uniformities in 
( )
r?
null
 extend only to distances of order W from 
the junction. This is because of the geometrical dilution of the influence of the narrow 
junction in the wide superconductor. Since the non-uniformities on length scales ?null  
don?t affect the quasiparticle dynamics these can be neglected and the step function 
model holds. In the literature above approximation is referred as ?rigid boundary 
conditions? [75]. Experimentally, to satisfy the above requirement, one needs to 
prepare devices where the junction has size , 100 nmWL?  
For supercurrent flow in the junction the electron flow through the junction 
should be phase coherent that is the electrons should move in the junction without any 
phase breaking scattering. Therefore experimentally one needs ?clean? deposition of 
the metals containing no magnetic impurity which is the main phase breaking 
scattering mechanism at low temperatures. 
In addition to above the interface between superconductor and the normal 
metal should be in good electrical contact. Experimentally this requires either to 
deposit both metals without breaking the vacuum, or if the fabrication requires two 
steps of lithography, to clean the surfaces via plasma etching or ion milling and 
without breaking the vacuum to deposit the second metal. I will point out these 
requirements in Chapter 6 again. 
 
 
 
 69 
 
5.5: Transport Properties of Mesoscopic SNS junctions 
I-V characteristics of superconducting point contacts with finite conductance 
in the normal state can be calculated by adopting a non-perturbative approach using 
the Bogolibov-de Gennes (BdeG) equations. BdeG equations describe the 
quasiparticle excitations (which consist of electron and hole like states) in non-
uniform superconductors. BdeG equations could be written as [76] 
 
() () ()( )
() () ()( )
22
22
*
,(,,
2
2
F
F
EurE Ur E urE r rE
m
E rE Ur E rE rurE
m
?
??
??
?
=+? +?
??
??
??
?
=+? +?
??
??
null
nullnullnullnullnullnull
null
nullnullnullnullnullnull
              Equation 5-6 
 
Here ()Ur
null
and ()r?
null
are the effective potentials which are determined self 
consistently, E is the excitation energy and 
( ),urE
null
and ( ),rE?
null
are the amplitudes of 
electron-like and hole-like states. When 
( ) 0r? =
null
 the two equations become 
uncoupled and we get the normal state excitations with no gap at Fermi level, when 
()
r?
null
 and the potential are translationally invariant that is 
( )
0
r? =?
null
and 
()
Ur U=
null
, 
the solution can be made in terms of plane waves 
( )
.ik r
k
ur ue=
null
null
null
null
and ()
.ik r
k
re??=
null
null
null
null
, and 
we recover the usual dispersion relations for the quasiparticle excitations in BCS 
theory i.e.
()
2
2
0F
kk
EE?=?+?
nullnull
where 
( )
2
2
k
k
m
? =
null
. Inverting the dispersion 
relation, one can get k
?
null  in terms of the energy of the excitations (Equation 5-7) 
where excitations with k
+
are the quasielectrons and with k
?
are the quasiholes. 
 
 70 
 
 
22
2
F
kmEE
?
??
=???
??
null                                                           Equation 5-7 
 
As we mentioned in section 5.3, Andreev reflection is the main mechanism for the 
current flow in an SNS junction by converting electron current in the normal metal to 
supercurrent in the superconductor. Andreev reflection can be understood using the 
BdeG equations as follows. 
Assuming (for simplicity) a single channel connecting both superconductors along the 
coordinate x  and also assuming rigid boundary conditions at the interface between 
superconductor and normal metal (as we have explained in section 5.4), the 
wavefunctions on both sides of the interface for an electron with energy E can be 
written as  
 
()
()
()
()
()
()
101
,  for 0
010
,     for 0
iq x iq x iq x
eh ee
ik x ik x
ee eh
xE e r e r e x
uE E
xE t e t e x
Eu
?
?
?
?
+? +
+?
?
?
?? ?? ??
=+ + <
?? ?? ??
?? ?? ??
?? ??
=+ <
?? ??
?? ??
                 Equation 5-8 
 
Here 
()2
F
qmEE
?
=?null are the momenta of electron and hole excitations in the 
normal metal and the two elements in the column vector represents the electron and 
hole components of the quasiparticle excitation in the superconductor. The 
coefficients , , ,
eh ee ee eh
rrtt represent the four processes that may occur to the incident 
electron in the interface, namely reflection of the incident electron as a hole, 
 
 71 
 
reflection as an electron, transmission as an electron and transmission as a hole. They 
are determined from the boundary condition that both the electron wavefunction and 
its derivative should be continuous at the interface. 
When there is no electrostatic potential mismatch between the normal metal 
and the superconductor, that is the case of a perfect interface, calculation leads to 
0
ee eh
rt==and 
()
()
()
eh
E
rE
uE
?
= where 
eh
r is the probability amplitude for an electron 
to reflect as a hole or to make an Andreev reflection. In the case of a BCS 
superconductor, if the energy of the incident superconductor is less than the 
superconducting gap 
eh
r  can be calculated as 
()
exp arccos
eh
E
ri
? ?
=?
?
? ?
 and for 
energy values greater than the gap 
eh
r decays exponentially. So the electron incident 
from the normal side reflects as a hole and in the whole process two electrons are 
added from the normal metal to superconductor as a cooper pair. This is the basic 
mechanism for the conversion of normal current to supercurrent in the normal metal-
superconductor interface as first pointed out by Andreev [77]. 
 
 
 
 
 
 
 
 
 
 72 
 
 
 
 
Figure 5-6. Andreev reflection of an electron as a hole from the superconductor-normal metal 
interface. As a result of the reflection, two electrons are transferred from the normal metal to the 
superconductor as a Cooper pair. Adapted from reference [41]. 
An imperfect interface can be analyzed by representing the electron potential 
in the interface with a delta function potential 
( ) ( )Ux H x?=  [78]. For energy 
values E <?, 
eh
r  is calculated as 
()
2
2
2
222
12
2
eh
F
F
r
kH
EE
E
?
=
??
??
+?? +
??
??
??
??
.  Using 
eh
r then one can calculate the linear conductance of a normal-superconductor junction 
[79]. For a single conduction channel at zero temperature the conductance is given by 
()
2
24
0
NS eh
e
Gr
h
= , i.e. twice the value of the Landauer formula reflecting the fact 
that there are two electrons transmitted through the interface as a result of the 
Andreev reflection. 
 
 73 
 
The above result can be generalized for the multi channel case [79] 
()
22
2
4
2
n
NS
n
n
e
G
h
?
?
=
?
?
where 
n
? is the transmission probability of the nth conduction 
channel. An important point about above result is that the normal conduction 
channels are not mixed with Andreev reflection process. That is, the electron incident 
to the interface in the nth conduction channel reflects back as a hole in the same 
channel. This property is very useful in determining the number of conductance 
modes participating in conduction.  
Similar to tunneling junctions, mesoscopic SNS junctions can carry 
supercurrent. In the normal metal channel (of length L) between two superconducting 
electrodes, an electron with E <? will make an infinite series of Andreev reflections, 
reflecting back and forth as electron and hole at the interfaces. This resonance 
condition gives rise to the energy values cos
22
F
F
kL
E
En
E
?
?
? ?
??
=? ? ?
??? ?
??
??
 inside the 
channel, where ?  is the phase difference between the electrodes. The solutions of 
above equation are the bound states inside the channel and they are known as 
Andreev-Kulik states [80]. 
In the short junction limit where
0
L ?null , there are only two states occurring in 
the energy values cos
2
E
?? ?
=??
? ?
??
 [81]. These two states are responsible for carrying 
the Josephson current in the SNS junction. The current for N conductance channels 
can be calculated as [79]  
 
 
 74 
 
() sin tanh
22
B
Ne E
I
kT
?
?
???
??
=
????
??
??
null
                                                   Equation 5-9 
 
The case of the arbitrary normal transmission,? , can also be analyzed using BdeG 
equations. In this case in the short junction limit there are again two bound states at 
energy values 
2
1sin
2
E
?
?
??
=?? ?
??
??
 and the supercurrent is given by [79, 82]  
 
()
()
2
sin
tanh
22
1sin
2
B
eE
I
kT
?
?
?
?
?
???
=
??
?? ??
?
??
??
null
                                      Equation 5-10 
 
In voltage biased junctions the total current through the contact is [83, 84] 
() () ( ) ( ) ( )( )
00
,2Recos2Imsin
o
in t
nomo o
nm
I Vt I V e I I m t I m t
?
? ?
>>
==+ ?
?? ?
. Here 
the dc component of the current,
o
I , is the experimentally more accessible part and 
the part of interest for determining the number of conductance channels. 
In I-V curves of the junction, along with the supercurrent at zero voltage,  for 
voltage biases less than 
2
e
?
one observes nonlinear behavior consisting of sharp 
current steps at voltage values 
2
V
ne
?
= . These current steps are called the 
subharmonic gap structure or subgap structure (SGS) [41, 85, 86]. In SGS, 
1n = corresponds to single quasiparticle transport between superconducting 
electrodes. The phenomenon of higher n values is called Multiple Andreev 
Reflections (MAR). 
 
 75 
 
  
 
 
Figure 5-7. Multiple Andreev Reflection of quasiparticles. At each reflection electron (hole) 
reflects as a hole (electron) and each time the quasiparticle travels the normal region, it gains an 
energy amount eV. Adapted from reference [41].  
For voltage values less than 2? , a quasiparticle can contribute to current via 
making multiple Andreev reflections between the superconducting electrodes to reach 
the states above the gap value by gaining energy amount eV  each time it crosses the 
normal region. For 2n = we observe current increase at the threshold eV =? , 
showing that  the electron (hole) travels the normal region from one SC electrode, 
gains energy by eV , it Andreev reflects back as a hole (electron) travels the normal 
region again and gains another eV  and scatters to a state at 2eV . Similarly we 
observe another current increase at threshold 
2
3
?
via two Andreev Reflections and so 
on for higher orders. The magnitude of the current depends on the number of 
reflections as well since each time the quasiparticle travels the normal channel it is 
 
 76 
 
more probable that it will scatter in the channel and not contribute to current. Current 
increase due to n
th
 current step can be calculated at low transmission [87] 
()
2
21
2
4!
nn
n
o n
enn
I
n
?
?
?
???
??
=
????
??
??
null
. For higher transmission values close to one the SGS 
structure starts to wash out. 
For N independent channels the total current can be written as a sum of 
individual channels () ( )
1
,
N
i
i
IV iV?
=
=
?
 (since the Andreev reflection does not mix the 
channels). The above result is very important because the analysis of the subgap 
structure permits one to obtain information about the conductance modes. Although 
one can be sure that there is more than one channel contributing to the conductance 
for
o
GG? , we cannot be sure about how many modes are contributing for the 
case
o
GG? . It is a possibility that several poorly transmitted channels may add to 
give a total transmission less than one. By fitting to experimental data, the number of 
conductance channels as well as their transmission probabilities can be obtained. The 
above property of mesoscopic superconducting junctions can be used in molecular 
electronics by measuring conductance properties of individual molecules in between 
superconducting leads so that the number of conducting channels i.e. how many 
molecules are participating in conduction as well as their transmission probabilities 
can be measured. 
 
 77 
 
Chapter 6: Preparing Mesoscopic SNS Junctions as 
Electrodes 
6.1: Introduction  
Single Molecule Electronics research concentrates largely on the possibility of 
using single molecules as active electronics components such as a rectifier, diode or a 
transistor. Such drive mainly comes from the expected end of the silicon era in 
electronics. Achieving this goal requires understanding their electronic properties and 
finding ways to stably anchor them to the interconnecting leads. Many research 
groups have made an effort to understand the conductance properties of single 
molecules both due to the above industrial drive and also just due to the intellectual 
curiosity to understand the physics and chemistry of molecular conduction. The 
experiments are done mostly via producing a nanogap either via an STM, via using a 
MCBJ or via electromigration. But in many of these experiments it is not possible to 
be sure how many molecules are conducting and how many channels per molecule 
are participating in the conduction. There are also questions that arise from the 
academic curiosity such as whether a molecule will be able to carry supercurrent or 
not, or if it does whether this current can be understood with known theories.  
To possibly answer these questions, I tried to fabricate a mesoscopic SNS 
junction where the mesoscopic normal constriction is made out of gold (both due to 
easy electromigration and due to the fact that most molecules tested are anchored to 
gold via sulfur bonds). Our aim was, after being sure that we can induce 
superconductivity in the normal layer, to use the junction produced in the normal 
 
 78 
 
constriction (via electromigration) as a test bed for molecules. In this chapter I will 
explain the experimental approaches I took for fabrication and the difficulties I faced, 
and I will conclude with what might be done to improve our experimental efforts. 
 
6.2: Device Fabrication and Difficulties Faced 
6.2.1:  Thin Au lines and Nb contacts 
As I have explained in section 5-4, to build a mesoscopic SNS junction one 
needs to prepare the normal constriction of the device on the order of 100 
nanometers. To fabricate devices in such small dimensions, EBL is commonly used 
(see i.e. section 4-2). I first attempted to fabricate our devices using two steps of EBL 
where in the first step I fabricated the gold lines, a wheel-shaped test pattern (used to 
diagnose problems in EBL), and alignment markers, typically with 20 nm of gold 
metallization  with no sticking layer (See figure 6-1). In the second step I fabricated 
the leads and the contact pads using RF sputtering of Nb typically 70-80 nm of 
metallization.                            
 
 
 
 
 
 
 
 
 79 
 
 
 
Figure 6-1. Metallization for thin gold lines. The alignment markers are prepared for the second 
step of lithography and the wheel pattern is for the check of astigmatism during lithography 
process. 
It is well known that metal deposition using sputtering deposits more energy 
to the substrate compared to both electron beam deposition of metals and deposition 
via thermal evaporation. Although thermal evaporation is the most commonly used 
method of deposition of metals, it is extremely difficult to deposit refractory metals 
such as Nb via this method. Electron beam deposition can be used as well in some 
cases, but the most common approach (and the experimental capability that I had 
available) is to deposit Nb via sputtering. Unfortunately, in my samples sputtering 
resulted in melting of the thin gold lines I prepared in the first step of metallization. 
 
 
 
 80 
 
 
 
Figure 6-2. Melting of Au metallization due to Nb deposition (a) Shows the thin gold line after 
the deposition of Nb contacts (b) Shows a thin gold line from the wheel pattern where Nb 
deposition is made on a resist layer covering the gold line. Scale bar in (a) shows 200 nm length 
scale and in (b) shows 100 nm length scale. 
 Figure 6-2 (a) shows an SEM micrograph of a device after Nb deposition on 
the surface and 6-2 (b) shows an SEM micrograph of a line from the wheel pattern 
after Nb deposition where the unexposed resist shields the gold line from Nb 
metallization. From the SEM micrographs I conclude that there is local heating due to 
Nb deposition and this causes melting of gold lines. One solution to this problem 
might be to cool the sample before and during Nb deposition, such as using a cold 
finger. Unfortunately the deposition chamber I used was not built with such capability 
and it was not possible to make such an addition to the sample stage. Therefore I 
decided to use a wetting layer for making gold less mobile on the silicon dioxide 
surface. I knew that this approach would make preparing the nanogap junction via 
electromigration harder, but I still wanted to see whether it would stop the problem or 
not. The most widely used wetting layer for gold is chromium, but I rejected this due 
to its magnetic properties (which would be incompatible with superconductivity). I 
 
 81 
 
instead used titanium (Ti) as a wetting layer for gold. Although this approach worked 
in some samples, it failed in the majority of cases where I still observed local melting 
of the gold lines. Figure 6-3 shows a sample in which this approach barely worked. I 
still wanted to test samples for which Nb deposition did not melt the gold to see 
whether superconductivity would be induced in them or not. I cooled my samples 
down to 1.2 K using a 
4
He cryostat (monitoring resistance as I lowered the 
temperature) but I did not observe any clear effect of a superconducting transition in 
the gold constriction possibly due to electrically dirty contacts (as explained below) 
or due to magnetic impurities in the gold metallization (as explained in section 6.2.2).  
 
 
 
 
Figure 6-3. SEM micrograph of a sample with thin Au lines and Nb contacts. 
 
As I have explained in Section 5-4, for superconductivity to be induced in normal 
metal the superconducting metal should be in good electrical contact with the normal 
 
 82 
 
metal such that the incoming electrons are Andreev reflected. In metallization which 
requires two different steps of lithography this is typically achieved by cleaning the 
surface of first layer either with oxygen plasma or ion milling before the deposition of 
the second. The sputtering chamber I used did not have the capability for oxygen 
plasma or ion milling, so I was unable to perform any cleaning step before the second 
metallization. Thus I considered that electrically ?dirty? contacts between the normal 
and the superconducting layers were the likely cause of the lack of observed 
supercurrent through the gold bridge.  Therefore I decided to change our sample 
fabrication procedure and decided instead to use aluminum as the superconducting 
metal and to use a thermal deposition chamber which would allow both metallization 
to be performed in one process without breaking vacuum. 
 
6.2.2:  Shadow evaporation of gold and aluminum contacts 
As a second method I tried to fabricate our samples using a ?resist bridge? 
technique followed by evaporation of metals at different angles.  First I ?shadow 
evaporated? the gold (deposited it from two different angles to the bridge) and then 
without breaking the vacuum I deposited the aluminum layer at normal incidence to 
the sample i.e. See figure 6-4.  
 
 
 
 
 
 83 
 
 
 
Figure 6-4. Shadow evaporation of gold and aluminum contacts. (a) Shows the resist bridge (side 
view) after development (b) Shows the shadow evaporation of gold from two different angles to 
the substrate (c) Shows the evaporation of aluminum from direct angle to the substrate. 
As I have explained in section 3-2 one can produce resist bridges out of a 
bilayer electron beam resist via selectively developing P(MMA-MAA) layer with 
Isopropanol (IPA) after an initial MIBK:IPA development. My idea was that by not 
breaking the vacuum between two metal depositions, I would get a ?clean? electrical 
contact between the two metallization layers. I used a thermal deposition chamber 
(deposition chamber in the Center for Superconductivity Research) that has a sample 
stage where one can adjust the tilt angle. I first deposited two gold layers (typically 15 
 
 84 
 
nm each) from two different angles (by tilting the sample stage) to form a continuous 
gold metallization and then I deposited aluminum (typically 70-80 nm) with sample 
being perpendicular to the source.  Figure 6-5 shows an SEM micrograph of a device 
as prepared using this technique. 
 
 
 
 
Figure 6-5. An Au-Al device fabricated via shadow deposition of gold using a resist bridge. 
 
The device in Figure 6-5 would seem to satisfy all the requirements for 
observing supercurrent through the gold bridge.  However, along with the clean 
contacts between two metals another requirement which needs to be satisfied is to be 
able to deposit the metals without having magnetic impurities. Magnetic impurities 
cause phase-breaking scattering and destroy the superconducting correlations in the 
normal layer. When I cooled my samples and measured their temperature-dependent 
 
 85 
 
resistance, I observed the well-known Kondo effect, the signature of magnetic 
impurities in a non-magnetic host metal, in our gold metallization. Figure 6-6 shows 
the resistance vs. temperature curves (Obtained via a standard 4-probe current 
reversal technique) of the raw gold material used in deposition [Figure 6-6 (a)] and 
the metallization done in the deposition chamber [Figure 6-6 (b)].  
 
 
 
Figure 6-6. Kondo effect in Au deposition. (a) Resistance vs. temperature curve of the Au 
material used in metal evaporation (b) Resistance vs. temperature of the deposited material. 
Here the increasing resistivity with decreasing temperature is due to the 
Kondo effect [88] and indicates the presence of magnetic impurities. The possible 
cause of this is the high contamination of the chamber with chromium which is used 
extensively in this chamber as a wetting layer for gold. I tried to clean the current 
leads to the evaporation sources, but this didn?t solve the problem and I still observed 
the Kondo effect in my samples. Since there were no other chambers that I would be 
 
 86 
 
able to use for this technique, I decided to change my sample fabrication approach 
again. 
6.2.3: Electron beam deposited thin Au lines ? Aluminum contacts 
As a third approach, I decided to return to two steps of electron beam 
lithography where I first prepare thin gold lines via e-beam evaporation (which I 
tested and which showed no signature of the Kondo effect) and later prepare 
superconducting contacts out of aluminum in the second step where, before aluminum 
deposition, I clean the gold surface with ion milling. An SEM micrograph of a device 
prepared with such a technique is shown in Figure 6-7.  
 
 
 
Figure 6-7. SEM micrograph of a device where the Au line is deposited via electron beam 
evaporation and superconducting Al contacts are deposited in second step metallization after 
cleaning the surface with ion beam milling 
 
 87 
 
When I prepared the superconducting metallization out of aluminum which 
has relatively low temperature of superconducting transition (~1.2 K), I had to use 
our 
3
He cryostat (Desert Cryogenics) which can achieve temperatures as low as 280 
millikelvins. Preparing the sample for measurement in this instrument requires 
pumping the vacuum space (?can?) down to low (~10
-6
 Torr) pressure to be sure that 
there is no leak into the vacuum space. This is done to be sure that there is no leak in 
the seal of the vacuum can since it is immersed into liquid 
4
He which is superfluid 
and might easily leak into the can if there is a leak. Such a leak both stops reaching 
the base temperature and might be dangerous since, upon removal fro the bath, the 
trapped liquid 
4
He would expand suddenly and cause the ejection of the can which 
could damage both the instrument and possibly the user.  In my lab pumping is 
accomplished using a turbo pump assembly and generally by pumping on the vacuum 
can overnight.  
When I prepared my samples for measurement in 
3
He cryostat I first tested the 
electrical conduction of my devices before pumping the vacuum can. When I was 
sure that my devices were conducting, I electrically grounded the lines and connected 
the pump line [which is electrically isolated from the cryostat assembly during the 
pumping by using an insulating (plastic) vacuum connection]. After pumping 
overnight I tested my devices again, and if they were still conducting I immersed the 
cryostat into the 
4
He bath to lower the temperature and start to take the 
measurements.  
 
 
 
 88 
 
 
 
Figure 6-8. Post-Mortem SEM micrographs of two devices after pumping the vacuum space. 
Both devices were conducting before pumping the vacuum space. Scale bar in (a) shows 500 nm 
length scale and in (b) 1 micrometer length scale. 
 When I prepared my samples by this method for measurements, I observed 
that during the pumping on the can open circuits were produced via breaking the lines 
or by totally destroying them. [See i.e. post-mortem SEM micrographs Figure 6-8 (a) 
and 6-8 (b) taken after pumping].I concluded that this is because the gold lines are not 
adhering well to the SiO
2
 surface and the mechanical vibration from turbo pump is 
causing the breaking of the lines. To overcome this I could have prepared gold lines 
by a nonmagnetic adhesion layer such as Ti, but this in turn would impede the 
electromigration of gold lines for preparing nanogap junctions so I rejected this 
approach. Cleaning the SiO
2
 surface via oxygen plasma before gold deposition would 
be another possible solution, but I didn?t have access to such an instrument. Another 
solution would be to identify the problem caused by the turbo pump (perhaps 
replacing by a diffusion pump or other quiet pumping system).   
 
 89 
 
6.3: Conclusion 
Although I put a considerable amount of effort into this project, I was 
unsuccessful in fabricating the samples that would be useful for my experiments. 
Most of my problems were due to not having access to the right sample fabrication 
instruments.  For example, a thermal deposition chambers free of magnetic 
contamination, an electron beam deposition instrument with a tilting stage, or a 
deposition chamber with an oxygen plasma cleaning facility would have helped 
greatly. One can also perform this experiment using MCBJ technique or STM as well 
but this would require first developing expertise on those experimental techniques. 
Therefore I had to channel my experimental efforts to other questions/experiments in 
my field that might give more fruitful results.    
 
 
 90 
 
Chapter 7: Palladium Nanojunctions as Hydrogen Sensors 
7.1: Introduction 
Hydrogen gas is widespread use in many industries such as chemical, 
metallurgical, and electronics. It is thought to be one of the clean energy sources for 
the future and is being used in hydrogen-powered vehicles. An important 
technological hurdle during production, storage, and transportation of hydrogen is 
that the leaks should be monitored continuously since when mixed with air above 
4.65 % in volume, hydrogen gas becomes explosive [89]. Therefore it is clear that 
developing fast, highly sensitive hydrogen sensors is technologically very important. 
It is well known that palladium has a high hydrogen solubility [90] and thus is 
a material of choice as the active element for hydrogen sensors or as a hydrogen filter. 
In the presence of hydrogen, palladium forms palladium hydride, PdH
x
, 0<x<1, where 
the content of the hydrogen dissolved in palladium depends both on the temperature 
and the partial pressure of hydrogen. Palladium hydride may be considered as an 
interstitial alloy where the octahedral sites of the face centered cubic metal lattice are 
occupied by hydrogen. The palladium-hydrogen system shows structural phase 
transition under increasing hydrogen content from the low concentration hydrogen-
saturated phase called the ? -phase, to the high concentration defective phase called 
the ? -phase. Upon the phase transition to ? -phase, the resistivity shows an increase by 
a factor of up to 1.8 at room temperature [90]. This resistivity increase is utilized in 
macroscopic palladium resistors as hydrogen sensors. 
 
 91 
 
At room temperature, the transition from ?  to ?  phase occurs at 8 torr of 
hydrogen partial pressure (~ 1-2 % volumetric concentration of atmospheric pressure) 
[90] and the phase transition leads to a change in the lattice constant from 3.895 
angstroms to (the maximal value for the ?  phase) to 4.025 angstroms ( the minimal 
value for the ?  phase). This leads to an increase of ~ 3.5 % in lattice constant or 11 % 
increase in volume [91, 92].The swelling of the Pd lattice is utilized by researchers to 
prepare hydrogen sensors from electrodeposited palladium arrays [93] or palladium 
thin films near the percolation threshold [94, 95]. 
In my research, I tried to fabricate nanogaps in palladium nanowires using 
electromigration and use them as hydrogen sensors by utilizing the hydrogen induced 
lattice expansion of palladium to close the gap, i.e. my sensors would work as 
hydrogen-activated relays that would consume no power in the absence of hydrogen 
but would conduct when there is hydrogen present in the environment. My sensors 
were indeed sensitive to the presence of Hydrogen but unfortunately they showed 
slow response times due to in plane stress of the metal film on the substrate. 
  
7.2: Device fabrication and Measurement setup 
I prepared my devices using EBL (See i.e. section 4.2). I prepared two types 
of devices.  In the first type, after a single step of EBL I deposited approximately 40 
nm palladium at a rate of 0.4 Angstroms per second using a thermal deposition 
chamber, forming both the palladium nanowires and the rest of the device including 
the contact pads in the same thickness of metallization. In the second type of devices, 
I performed two lithography steps where in the first step I deposited Pd to form 
 
 92 
 
nanowires of thickness 20 to 40 nm and in the second lithography step I deposited 
Cr/Au metallization typically 50-100 nm to form the contacts to the nanowires and 
the bonding pads. SEM micrographs of both types of devices are shown in Figure 7-1 
and 7-2 below.       
 
 
 
 
Figure 7-1. A palladium device (before electromigration) where the thin palladium line and the 
rest of the device have the same thickness of metallization. The scale bar in the figure is one 
micron.  
 
 
 93 
 
 
 
 
Figure 7-2. Palladium devices where the electromigrated thin line is palladium and the rest of the 
metallization in the device is Cr/Au. The device was fabricated in two steps of lithography. The 
palladium metallization in the figure is 20 nm and the Au metallization is 55 nm. The scale bar is 
two microns. 
I measured the hydrogen sensing properties of the devices under different 
volume percentages of hydrogen in nitrogen using a custom-built measurement setup 
as shown in Figure 7-3 (a) and (b) below. I regulated the volume percentage of 
hydrogen in nitrogen gas by regulating their flow using two flow meters purchased 
from Omega Inc. Model FL-3861 ST is used for controlling the H
2
 flow, and Model 
FL-3805 ST is used for controlling the N
2
 flow. While keeping the nitrogen flow 
constant, I started and stopped hydrogen flow using an electrically controllable 
On/Off solenoid valve purchased from Teknocraft Inc. Figure 7-3 (b) shows the 
measurement box.  
 
 94 
 
 
 
 
Figure 7-3. Measurement setup used in hydrogen sensor experiments. (a) shows the gas flow and 
mixing schematically before it flows into the box. (b) shows inside of the box and On/Off solenoid 
valve used to start and stop H2 flow. The dimensions of the box are 4.80 X 4.72 X 3.15 inches. 
 
 95 
 
7.2: Results and Discussion 
 I will first discuss the electromigration characteristics of the devices and the 
nanogap formation via electromigration. I performed electromigration in a nitrogen 
environment at room temperature prior to sensing experiments.  Electromigration was 
accomplished by increasing the voltage bias until the device failed and the current 
reduced to zero. In the devices where I had one layer of Pd metallization, I attempted 
electromigration using feedback, but I couldn?t succeed in controlling the rate of 
electromigration. In this case, nanogaps in Pd lines were formed by abrupt failure of 
the lines. Figure 7-4 shows the nanogap formation after electromigration.   
 
 
 
Figure 7-4. Nanogap formed via electromigration in a Pd device. The gap is several hundred 
nanometers wide and formed via electromigration after increasing the voltage bias until failure 
occurs. The scale bar in the figure shows one micron length scale. 
 
 96 
 
In devices where I had two layers of metallization (The Pd nanowire and Au 
leads) I was able to control the rate of electromigration and perform electromigration 
controllably. Figure 7-5 shows post-mortem SEM images of nanogaps formed via 
controllable electromigration.  
 
 
 
 
Figure 7-5. Nanogaps formed via electromigration in palladium devices with gold contacts. The 
nanogaps are formed via controllable electromigration. In the SEM micrograph the 
electromigration through the grain boundary and the void formation between grains is observed. 
The scale bar in the figure shows 200 nm length scale.  
 
Below Figure 7.6 shows a typical I-V curve taken during the controllable 
electromigration of a Pd line with gold contacts. 
 
 
 97 
 
 
 
Figure 7-6. Controllable electromigration of a Pd device with gold contact. Data is taken at room 
temperature in nitrogen environment at room temperature.  
  The I-V curve of electromigration is similar to the gold lines indicating that 
power dissipated in the junction increases the local temperature until the 
electromigrating atoms acquires enough mobility. However, unlike the 
electromigration in gold, turning off the feedback loop during electromigration 
resulted in abrupt failure of the wire rather than a jump to the stable branch. Figure 7-
7 shows the I-V curve of a sample at which the feedback is turned off during the 
electromigration.  
 
 
 98 
 
 
 
Figure 7-7. I-V curve of a palladium line where the feedback is turned off during the 
electromigration. The arrow shows the point of turning the feedback off. Unlike the 
electromigration in gold lines, electromigration does not evolve smoothly after turning off the 
feedback. 
This is not unexpected considering that palladium has a surface oxide that 
blocks the surface electromigration, leaving the bulk and the grain boundaries as the 
possible pathways for the electromigration. Blocking the surface electromigration 
causes the stress buildup in the electromigrating line and abrupt failure instead of a 
smooth self limiting electromigration as observed in gold [See i.e. the discussion in 
sections 2.2.2.2 and 2.2.2.3 in Chapter 2]. Comparison of post mortem SEM pictures 
 
 99 
 
of devices also shows the difference of electromigration in gold and palladium.  
Figure 7-8 shows an SEM micrograph of a gold sample electromigrated using a 
feedback loop. Comparing with figure 7-5, we may observe the neck formation in the 
gold sample, indicating that electromigration removes atoms from the surface form all 
sides and acts as an electromechanical etching process. But in figure 7-5 instead of a 
neck formation, we observe formation of a void after electromigration.  
 
 
 
 
Figure 7-8. Formation of a neck in the junction after electromigration of a gold line. The scale 
bar in the figure shows 200 nm length scale. 
We will now discuss the sensor responses of both types of devices. We first 
fabricated and measured the sensing response of first type of devices where we had 
only palladium metallization. Figure 7-9 (a) and (b) below shows the typical turn on 
responses we observed upon introduction of Hydrogen into the Nitrogen flow. Figure 
 
 100 
 
7-10 (a) and (b) shows the typical turn off responses we observed upon turning off the 
Hydrogen flow. 
 
 
 
 
Figure 7-9. Typical turn-on behavior of sensor devices fabricated with only palladium 
metallization. (a) shows turn-on behavior for 5.8 % volume percentage of hydrogen in nitrogen 
and (b) shows the behavior for 3.9 % hydrogen. The arrows indicate when hydrogen is 
introduced into nitrogen flow. 
 
 
 
 
 
 
 101 
 
 
 
Figure 7-10. Typical turn off behaviors of devices fabricated with only palladium metallization. 
The arrows in the figures show when the hydrogen flow is turned off. 
In our devices we have observed response times to the presence of hydrogen 
typically ranging from tens of seconds to hundreds of seconds. In general, the turn on 
response of our devices become relatively slower in lower concentrations of hydrogen 
compared to higher concentrations. We also observed higher conductance values and 
more stable contacts for higher concentrations of hydrogen.  Similar slow response to 
hydrogen is observed previously in ultra thin palladium films [95-97]. It is known that 
for a thin film attached to a hard substrate the in-plane stress for expansion is on the 
order of several gigapascals [98, 99]. So when such structure is loaded with hydrogen, 
the lattice expansion is directed out of plane  [100-102], but our devices rely on the in 
plane expansion of the nanogaps produced in palladium nanowires. We see the effect 
of high in-plane stress in turn off responses of our devices as well. Contact making 
and breaking in our devices occurred sometimes instantaneously and sometimes 
stepwise as shown in Figures 7-9 and 7-10 independent of the concentration of the 
 
 102 
 
hydrogen used. We also observed both behaviors for the same concentration of 
hydrogen for different turn-on and turn-off cycles.  
To overcome the above difficulties I tried to fabricate smaller gaps by using 
controllable electromigration and hence I fabricated the devices with palladium 
nanowire and gold contacts and bonding pads in order to reduce the series device 
resistance. In these devices I was able to fabricate nanogaps on the order of tens of 
nanometers. But their hydrogen response was worse than the first type of devices; the 
devices were not sensitive to presence of hydrogen for less than 5 % volume 
percentage and their response occurred more slowly. This indicates that the palladium 
contacts as well as the palladium nanowires in the first type of devices participated in 
overcoming high in-plane stress. We also tried to etch the SiO
2
 from the sides of the 
nanowire for approximately 50 nanometers to reduce the effect of the substrate but 
this did not improve the hydrogen response of the devices.     
 
 
 
 
 
 
 
 
 
 103 
 
7.4: Conclusion 
In conclusion, I have fabricated hydrogen sensors out of palladium nanogaps 
fabricated via electromigration. For both turn-on and turn-off behavior, our sensors 
showed response times on the order of minutes. Such slow response is due to direct 
attachment of thin film to the hard substrate as was observed before for ultra thin 
palladium thin films used for hydrogen sensing. [95]. For substrates such as 
cyanoacrylate [93] response times as fast as 75 milliseconds were measured. It is also 
shown that [94] by using a self assembled monolayer as a  lubrication layer the 
response times for ultra thin palladium films can be improved by several orders of 
magnitudes from minutes to tens of milliseconds. Unfortunately the fabrication 
process we used for device fabrication is not compatible with using such lubricant 
layers. Our results might be improved by fabricating nanowires without using a resist 
layer such as shadow masking, on a hard substrate covered with a lubrication layer 
such as siloxane. 
 
 
 104 
 
Chapter 8: CNT Thin Films as Chemical Sensors-
Fabrication and Measurement Technique 
8.1: Introduction 
Carbon CNTs (CNTs) were discovered in 1991 by S. Iijima while studying 
the fullerene production methods [103]. After their discovery, this new material has 
been studied intensively [104, 105] and has become the material of choice of many 
experimental groups due to its superior electrical [106-113], mechanical [114, 115] 
and chemical [116-118] properties. In many ways CNTs are unique materials. They 
are inherently one dimensional which makes them ideal for studying phenomena 
unique to one dimensional systems such as the Luttinger liquid ground state of 
conduction electrons [107, 119]. Another unique property CNTs possess is that all 
their atoms are on the surface and hence their conduction properties may be affected 
grossly by the chemical environment.  
The structure of a CNT can be thought as a single two-dimensional sheet of 
graphite (termed graphene) seamlessly rolled up to form a tube. CNTs can be multi 
walled (MWNT) or single walled (SWNT) where the MWNT consist of several 
concentric shells of SWNT.  SWNT can be metallic or semiconducting depending on 
their chirality as theoretically predicted [106, 108] and experimentally observed [109, 
120]. 
Although individual CNTs can be either semiconducting or metallic, with the 
growth and separation methods known today, it is not possible to selectively grow 
only one kind (metallic or semiconducting) or successfully separate semiconducting 
 
 105 
 
and metallic CNTs from each other after the growth. Therefore the experiments done 
on individual CNTs are made in general by first growing or depositing sparse density 
of CNTs on the substrate and later finding individual CNTs (mostly with the help of 
an alignment pattern) and connecting them with metallic contacts followed by 
identification as metal or semiconductor using their electrical characteristics such as 
their transconductance.  
Therefore considerable attention has been given to the utilization of thin films 
of CNTs as electronic material in many applications such as thin film transistors on 
both hard and flexible substrates [121-123], diodes [124] or as transparent conductive 
coatings [125]. The CNT thin film consists of a random network of CNTs bound to a 
substrate, and films considerably thinner than a single monolayer are routinely 
studied.  CNT thin films have the advantage that although the electronic properties of 
single CNT devices depend on the chirality and the diameter of each CNT and the 
conduction characteristics of each device is not exactly reproducible, in thin films the 
effects of individual variations are suppressed by the ensemble averaging over a large 
number of CNTs. 
Thin films of SWNTs will be electrically continuous provided that the density 
of the CNTs in the film is high enough, i.e. above the percolation threshold.  The 
percolation threshold is defined as the critical density above which there exists a 
continuous electrical path connecting arbitrarily distant points. In the case of CNT 
thin films it can be estimated as the density at which the average distance between the 
CNTs will be equal to their average length [123]. The electronic properties of the film 
then depend on the electronic properties of the individual CNTs in the film and the 
 
 106 
 
degree of connectivity between the CNTs. This system can be seen as analogous to 
the two-dimensional random resistor networks studied in percolation theory. It is also 
pointed out[126] that above the percolation threshold the behavior of such a network 
can be understood as a consequence of the mathematical properties of the finite sized 
percolation of CNTs with variable intertube coupling.  
Given that, in a SWNT, all the carbon atoms are on the surface, it is 
reasonable to explore the use of SWNTs as active transducer elements for chemical 
sensing applications.  Both devices using isolated CNTs [116, 117] and thin films or 
bundles of CNTs [127-129] have been used for chemical sensing as chemiresistors 
via monitoring their resistance change upon exposure to analyte under test. The 
observed conductance change is attributed to the doping upon gas adsorption and/or 
increasing scattering on the CNT surface. Theoretical studies also predict significant 
changes in the conduction properties of CNTs upon gas adsorption [130, 131]. 
Another possible use of thin films of CNTs is to utilize them as electrodes for 
capacitative gas sensors [132-134]. In these works the CNT film is utilized as one 
plate of a parallel plate capacitor that where the gate layer (generally highly doped 
conductive silicon) is used as the other plate. The presence of the analyte is detected 
by monitoring the change of the capacitance between the gate and the CNT film. The 
change of the capacitance is attributed to the polarization of the adsorbed molecules 
on the CNT surface and the change in the quantum capacitance due to charge transfer 
from analyte to CNT network. 
But little or no attention has been given so far to the fact that the CNTs will 
have a layer of water on their surface in ambient conditions and analysis in terms of 
 
 107 
 
doping and charge transfer should take the water layer into account [135]. In addition, 
the CNT network will consists of interconnected CNTs each with capacitance to the 
gate and resistance that comes from scattering centers such as defects and/or CNT-
CNT junctions. Such a network will have frequency-dependent capacitance (i.e. out-
of-phase component of the impedance) that will be affected by the changing 
resistance of the CNT network.  
To elucidate this point further, I have taken capacitance and conductance 
measurements of CNT networks at frequency range between 50 Hz to 20 KHz. The 
experiments were performed in ambient conditions, ultra-high vacuum (UHV), and 
under several analyte pressures up to 10
-6
 Torr. The results are described in Chapter 9. 
 
8.2: Device fabrication and Measurement setup 
8.2.1: CNT Synthesis 
Although CNTs were discovered as products of arc discharge between two 
graphite rods [103], other methods such as laser ablation [136], the high-pressure 
carbon-monoxide (HIPCO) process [137, 138], and chemical vapor deposition (CVD) 
[139, 140] were soon discovered to synthesize CNT material.  
I used the CVD method to prepare the CNT thin films. The growth of CNTs is 
catalyzed by iron nanoparticles [141] where the size of the catalyst determines the 
diameter of the resulting CNTs [142]. The recipe I followed for the CNT growth is 
adapted from reference [141, 143], and is as follows. I first dissolve ferric nitrate 
(Fe(NO
3
)
3
) in isopropyl alcohol (IPA). To ensure that the (Fe(NO
3
)
3
) salt is well 
 
 108 
 
dissolved, I stir the solution using a magnetic stirrer bar for ~ 4-5 hours or overnight. 
Prior to CVD process, (Fe(NO
3
)
3
) salt is precipitated on the substrate by dipping it 
into the (Fe(NO
3
)
3
)-IPA solution for approximately 10 seconds and into hexanes for 
approximately 2 seconds. 
The substrate is then placed into a quartz boat and then the boat is placed into 
a quartz tube in the center of the growth oven. (i.e. see the figure 8-1).  Following this 
the gas lines are purged with argon and the oven is heated to 850 degrees. After the 
temperature is stabilized, the growth using the feedstock gases are flowed according 
to the recipe in table 8-1.  
 
 
 
 
Figure 8-1. Schematic of CNT growth apparatus. Samples were placed in the middle of the 
quartz tube and feedstock gases are flowed while the oven was at 850 degrees for CVD growth of 
CNTs. 
 
 
 
 109 
 
 
 
 
Table 8-1. CNT growth recipe 
Figure 8-2 shows the SEM image of the CNTs [144] grown on 500 nm thick 
silicon dioxide substrate on degenerately doped silicon according to the recipe in 
table 8-1. To ensure that I get dense and well connected networks of CNTs above the 
percolation threshold, I used fairly high catalyst density of 200 g ml?  ferric nitrate 
solution in IPA.  
 
 
 
 
 
 
 110 
 
 
 
 
Figure 8-2. SEM micrograph of the thin film of CNT network material after CVD growth on 
Si/SiO
2
 substrate. The CNTs are grown with relatively high catalyst density (200 microgram/ml) 
to ensure a well connected network. 
8.2.2: Device Fabrication and Measurement Setup 
After the CNT material on Si/SiO
2
 substrate is grown, I prepared my devices 
with two steps of photolithography. The recipe I used in device fabrication is as 
follows. In the first step, I spun photoresist (S1813) on the sample for 45 seconds at 
4500 rpm and baked it at hot plate for 60 seconds, producing approximately 1.4 
micrometers thick resist layer. After exposure to ultraviolet light (CL2 for 15 
seconds) and developing the resist layer (in CD-30 for 45 seconds following rinse 
with DI water and N
2
 blow dry), I deposited the metal layer (5 nm Cr / 55 nm Au) 
 
 111 
 
that produces the interdigitated electrode pairs for conductance measurements. In the 
second lithography step I covered with resist the region inside the interdigitated 
electrode pairs to form the device and removed the CNTs from the other areas on the 
surface by using oxygen plasma etch (50 second etch at 100 W power and 300 
militorr  oxygen pressure). Figure 8-3 shows the SEM micrograph of the completed 
device. 
 
 
 
Figure 8-3. SEM micrograph of the CNT network device. The interdigitated electrode pairs form 
the source and drain contacts for conductance measurements. For capacitance measurements, 
both the electrode pairs and the CNT network form one plate of the parallel plate capacitor 
where the other electrode of the pair is formed by the degenerately doped Si underneath the SiO
2
 
dielectric.  
 
 112 
 
After the device fabrication I glued my samples onto a sample holder using 
silver epoxy and placed them into a UHV chamber equipped with a leak detector and 
a mass spectrometer to calibrate the partial pressure of the analyte under 
measurement, i.e. see figure 8-4. I conducted my experiments as follows. I first 
pumped the UHV chamber containing the sample to pressure less than 10
-10
 Torr 
(including baking the chamber using heating tape to remove adsorbates from the inner 
walls of the UHV chamber). Later I attached the gas container or the vial containing 
the analyte to the leak valve that is attached to the UHV chamber. I purified the liquid 
analytes such as acetone using the well known cold freeze-pump-thaw cycle via the 
following procedure. I first solidified the liquid to be purified using a liquid cryogen 
bath such as liquid nitrogen. Once the analyte was frozen, I pumped the residual gases 
in the vial (that are not frozen) using a sorption pump. Following this I closed the 
valve that connects the vial to the sorption pump and thawed the frozen liquid in the 
vial using a water bath at room temperature. I repeated this cycle approximately four 
or five times. To check the purity of the analyte that I introduced into the chamber, 
and the partial pressure of the analyte in the chamber, I measured and calibrated the 
leak using a mass spectrometer (SRS RGA 200). After the calibration step I baked my 
sample in UHV above 200  
o
C  to be sure that the adsorbed molecules and the water 
vapor were desorbed from the surface. After my sample was cooled down to room 
temperature again I turned off the pressure gauge and the filament of the mass 
spectrometer, and introduced the analyte using the pre-calibrated position of the leak 
valve.  I took capacitance measurements using an Andeen Hagerling 2700A 50Hz-20 
KHz Ultra-Precision Capacitance Bridge.   
 
 113 
 
 
 
 
Figure 8-4. UHV system used in the experiment 
 
 
 114 
 
Chapter 9: CNT Thin Films as Chemical Sensors-Results 
9.1: Introduction 
First I will discuss the qualitative behavior of the capacitance and conductance 
as a function of gate voltage and frequency.  Figures 9-1 and 9-2 show the general 
behavior of the capacitance and conductance of the CNT thin film device as a 
function of gate voltage.  The capacitance of the CNT network was measured using 
an AC signal applied between the source and the gate electrodes and using 0.1 V 
RMS amplitude of the drive frequency, and the conductance of the CNT network 
using DC voltage bias. Although capacitance and conductance are measured using 
two different techniques, the similar shape of the curves (including the 
hysteresis[145]) hints that the capacitance and conductance are possibly interrelated.  
 
 
 115 
 
 
 
Figure 9-1. Capacitance of CNT network as a function of gate voltage. Capacitance is measured 
in UHV by applying 14 KHz sinusoidal drive voltage at 0.1 V RMS.  The gate voltage is swept 
from 10 V to +10V and back to -10V; both the up sweep and down sweep are shown. 
 
 116 
 
 
 
 
Figure 9-2. Conductance of CNT network as a function of gate voltage. Conductance is measured 
in UHV by applying 500 mV source-drain bias. 
 
Figure 9-3 shows the measured capacitance as a function of frequency of the 
applied signal at zero gate voltage. In general we observed a decrease in the 
capacitance with increasing frequency. In order to understand and explain the results, 
we model the CNT film over the back gate electrode as an RC transmission line as 
discussed below. 
 
 
 
 
 
 117 
 
 
 
Figure 9-3. Capacitance as a function of frequency. The capacitance is measured at zero volt gate 
voltage in ambient conditions. 
9.2: CNT networks modeled as an RC transmission line. 
The CNT thin film is composed of individual CNTs that are well connected, 
with each CNT having resistance due to scattering centers such as defects and CNT-
CNT connections, and having capacitance to the gate. Such network is very similar to 
an RC transmission line. Therefore before I discuss the experimental results, I first 
will discuss the analysis of such a transmission line. In my analysis I neglect the 
conductance to the gate due to high dielectric constant of the SiO
2
. I also neglect the 
inductance of the CNTs which is reasonable at low frequencies, where the resistance 
dominates the inductive reactance. In the discrete case the analysis of such a 
transmission line can be made by using basic electrodynamics. In figure 9-4 I 
 
 118 
 
schematically describe such a transmission line composed out of capacitors and 
resistors. 
 
 
 
 
Figure 9-4. Schematic  of an RC transmission line. 
Using Kirchov`s current law, an equation for the currents at the branch point can be 
written as 
 
1nnn
R RC
III
?
=+
                                                                        Equation 9-1
  
or 
 
 
11nnnn n
VVVV dV
C
R Rdt
?+
??
=+                                                Equation 9-2 
 
The above equation may be rewritten as  
 
 
 119 
 
11nnnn n
VVVV dV
C
R Rdt
?+
??
?=                                                           Equation 9-3 
 
Dividing by 
2
L and define 
R
r
L
= as the resistance per unit length and 
C
c
L
= as the 
capacitance per unit length, one arrives at a diffusion equation for the voltage: 
 
2
2
1dV d V
dt rc dx
=                                                   Equation 9-4 
 
For a sinusoidal drive voltage ( )0,
it
o
VtVe
?
=  one expects a decaying 
solution
()
,
it x
o
Vxt Vee
? ??
= . Inserting into the diffusion equation one finds the 
characteristic decay constant:  
 
()1
2
i
rc? ?
+
=                                                                        Equation 9-5 
 
This indicates that signals will decay over a length  
 
2
o
l
rc?
=                                                                                   Equation 9-6 
 
 
 
 
 
 120 
 
To derive the continuous case from the discrete case the current may be written 
()
0
1
0,
x
dV
It
rdx
=
=?                                                  Equation 9-7 
Inserting the decaying solution into the equation one finds  
 
()
0,
it
o
It Ve
r
?
?
=
                                                                           Equation 9-8 
 
The complex impedance of the transmission line may then be written as 
 
()1
2
i
rr
Z
c? ?
?
==                                                                         Equation 9-9 
 
Comparing the above result with the impedance of the circuit composed of a series 
capacitor and resistor, that is 
.
.
Eff
Eff
i
ZR
C?
=? , I define the effective resistance and 
capacitance of my device as: 
 
.
1
22
Eff o
r
R rl
c?
==                                                                     Equation 9-10 
 
and  
.
2
Eff o
c
Ccl
r?
==                                                                          Equation 9-11 
 
 
 121 
 
Note that R
Eff
 and C
Eff 
 may be thought of as the resistance and capacitance of a 
section of the transmission line equal to the decay length l
0
. 
Figure 9-5 shows the change of the device capacitance as a function of the 
drive frequency. I measured the capacitance by using AH 2700 Ultra-Precision 
Capacitance Bridge that allows simultaneously application of both an AC voltage 
between gate and the device for capacitance measurement as well as a DC voltage 
between the device and the gate and hence allows changing the carrier concentration 
in the CNT mat. R
Eff
 and C
Eff 
correspond to the measured signals by the capacitance 
bridge, which interprets the circuit as a series capacitor and resistor.  By looking at 
the effective capacitance, 
.
2
Eff
c
C
r?
=  we expect the capacitance to decay with the 
inverse square root of the frequency. The solid line in Figure 9-5 is guide to eye and 
shows the slope of
1
2
? . 
 
 122 
 
 
 
Figure 9-5. Device capacitance as a function of drive frequency at different gate voltages at 
ambient atmosphere. The RMS amplitude of applied voltage during measurement was 0.1 Volts. 
The solid line is a guide to eye showing -1/2 slope. 
The capacitance of my device decays with increasing frequency where the 
roll-off of the decay starts at higher frequency values for higher carrier concentration 
(lower gate voltages). I explain this effect by considering all the capacitors in the 
device and the propagation of the applied AC signal (See figure 9-6). The device 
capacitance to the gate is composed of three capacitances: the capacitance of source 
to the gate which is independent of the frequency, the capacitance of CNT mat to the 
gate which depends on frequency, and the capacitance of drain contact to the gate 
which is independent of the frequency. The device capacitance is measured by 
applying AC voltage between source contact and the gate. For low frequency values 
 
 123 
 
the measured capacitance includes all the three capacitances, but as the frequency 
increases, the contribution of CNT network to the capacitance decreases since the 
signal decays in the CNT network, and the contribution from the drain capacitance 
also decreases, since a smaller signal reaches the drain.    
  
 
 
Figure 9-6. Schematics of source, drain and the gate capacitors. Capacitance measurements are 
taken by using Andeen-Hagerling 2700A Ultra-Precision Capacitance Bridge with 0.1 Volts AC 
voltage applied to source electrode.  
 At low frequencies, the capacitance of the nanotube film should be 
given by the capacitance per length c times the length L, i.e. 
NT
CcL= .  At high 
frequencies, the capacitance is give by equation 9-11 
NT o
Ccl= .  Thus the crossover 
occurs roughly when 
o
lL=  or at a roll off frequency
2
2
o
rcL
? =
.  Thus the roll off 
frequency will change with gate voltage as the resistance per length changes; at high 
negative gate voltages, when the film is less resistive, the roll off frequency is higher. 
The DC device resistance per length r varies monotonically from 7.6x10
7
 to 1x10
9
 
? /m for V
g
 = -10 V and +10 V respectively, which should correspond to a roll-off 
 
 124 
 
frequency (using l
0
 = L) of ? /2?  = 3.5 ? 50 kHz, which is approximately one order of 
magnitude higher than the observed roll-off.  The source of this quantitative 
discrepancy is not clear, but likely reflects that the average resistance and capacitance 
per length of the film do not reflect the resistance per length and capacitance per 
length of the individual conducting paths that dominate the conduction. 
From equation 9-11 we can see that the effective capacitance is proportional 
to the square root of conductance, and therefore the variation of the square root of 
conductance with gate voltage should produce a proportional change in the 
capacitance. Figure 9-7 below shows the capacitance and the square root of 
conductance as a function of gate voltage.   
The change in the capacitance of similar CNT network devices as a function 
of gate voltage has been observed before [134], and attributed to the change of the 
quantum capacitance due to gate field induced doping as follows.   The capacitance of 
the CNT is given by 
111
tot el q
CCC
???
=+ where 
el
C  is the electrostatic capacitance, 
()
2
q
CeDE=  is the quantum capacitance, e is the electronic charge and ()DE is the 
density of states.  Thus, in the off-state at positive gate voltage, the Fermi level 
presumably lies in the semiconducting CNT gap, and 
( )DE is small so the quantum 
capacitance dominates and 
tot
C is small.  However, in Figure 9-7 the proportionality 
of capacitance and the inverse square root of resistance indicate that the change in the 
capacitance is almost entirely explained by the change in the resistivity of the CNT 
network, with no need to resort to the quantum capacitance to explain the change.  I 
do not understand completely the absence of a change in capacitance between on and 
 
 125 
 
off state, but possible origins are a finite density of states in the semiconducting CNT 
gap, or on/off behavior occurring before the semiconducting CNTs are fully depleted.    
 
 
Figure 9-7. Capacitance and inverse square root of resistance of the CNT thin film device as a 
function of gate voltage. Both data are taken in UHV (< -10 Torr pressure). Capacitance is 
measured by applying 14 KHz sinusoidal drive voltage at 0.1 V RMS, and the resistance is 
measured by applying 500 mV source-drain bias. A constant 195 pf (approximately equal to the 
capacitance of the contacts) has been subtracted from the capacitance. 
 
 
 126 
 
9.3: Effect of low pressure gas adsorption on the conductance and the 
capacitance of CNT films 
The effect of gas adsorption on the electronic properties of CNT networks and 
individual CNT devices has been investigated both experimentally and theoretically 
[127-134]. The observed changes of the electronic properties are in general attributed 
to effects of charge transfer and doping in the CNTs. But careful explanations of the 
experimental results and the theoretical conclusions should take into account the 
effects of water vapor and the gases such as oxygen that are present in ambient 
atmosphere. So to investigate this point further we conducted our experiments in 
partial pressures of several analytes such as acetone and water.  
Figures 9-8 and 9-9 show the effect of acetone on the capacitance (measured 
at 20 kHz) and resistance (measured at DC) respectively of the CNT network as a 
function of gate voltage. It is not immediately clear from Figures 9-8 and 9-9 that the 
resistance and capacitance changes are related in any way.  However, we may look at 
the predictions of the RC transmission line model.  From the effective 
capacitance,
.
2
Eff
c
C
r?
= , if we look at the effect of small changes of resistance, we 
can derive the relation between the change in the capacitance and the change in the 
resistance as 
1
2
CR
CR
??
=? . 
  
 
 127 
 
 
 
Figure 9-8. Capacitance of the CNT network in UHV and 9.5 10 
-7
 Torr acetone pressure. 
 
 
 
 
 
 
 
 
 
 
 128 
 
 
 
Figure 9-9. Resistance of the CNT network in UHV and 9.5 x 10 
-7
 Torr acetone pressure. 
 
 In figure 9-10 we plot 
C
C
?
 and 
1
2
R
R
?
?  (where we define the capacitance 
change as 
Analyte UHV
CC C?= ?  and the resistance change as
Analyte UHV
R RR? =?) due 
to acetone introduction in the chamber as a function of gate voltage. The vertical axis 
in the figure shows the changes of both capacitance and resistance.  Note that the 
resistance is measured at DC, and therefore is a true probe of the film resistivity, and 
is not altered by the RC transmission line impedance. The resistance of the CNT 
network upon acetone introduction is presumably changing because of increased 
scattering due to physically adsorbed molecules on the CNT surface, or possibly 
through modification of the CNT-CNT junction resistance. From figure 9-10 we 
 
 129 
 
conclude that the effect of acetone on the capacitance change of the CNT network is 
mainly through changing the resistivity of the film. 
 
 
 
Figure 9-10. Change in the capacitance and resistance due to 9.5 x 10
-7
 Torr acetone pressure. 
Capacitance data is taken at 20 KHz using 0.1 V drive voltage. 
 Figures 9-11 and 9-12 show the effect of 2.8 x 10
-8
 Torr of water on 
capacitance and on resistance. Similar to the case of acetone, in figure 9-13 I plot 
C
C
?
 and 
1
2
R
R
?
?  due to water introduction in the chamber as a function of gate 
voltage. The figure indicates that the capacitance and resistance changes are related to 
 
 130 
 
each other and the capacitance change due to water introduction into the chamber is 
induced by the resistance change of the CNT network. 
 
 
 
 
Figure 9-11.  Capacitance of CNT network in UHV and 2.8 x 10
-8
 Torr water pressure. 
 
 
 
 
 
 
 
 131 
 
 
 
Figure 9-12. Resistance of CNT network in UHV and 2.8 x 10
-8
 Torr water pressure. 
 
 
 
 
 
 
 
 
 
 
 
 132 
 
 
 
Figure 9-13. Change in the capacitance and resistance due to 2.8 x 10
-8
 Torr water pressure as a 
function of gate voltage. 
Figures 9-14 and 9-15 below shows the result of the control experiments I 
performed with argon. Argon, being a nonpolar and noble gas, is not expected to have 
any effect on either the scattering or the polarization, therefore one expects to see no 
change in capacitance or resistance upon argon introduction to the chamber. 
 
 
 
 
 
 
 133 
 
 
 
Figure 9-14. Capacitance of CNT network as a function of gate voltage in UHV and 3 x 10
-6
 Torr 
argon pressure. The capacitance data is taken at 20 KHz and by using 0.1 RMS amplitude of 
drive voltage. 
I indeed observed no change in the capacitance of the CNT network in the 
presence of 3 x 10
-6
 Torr of argon pressure. By comparing figure 9-14 with figure 9-8 
and figure 9-11, I conclude that the effects of acetone and water are not simply due to 
pressure alone, or to other gases that are introduced through the leak valve.  The 
effects of acetone and water are also reproducible upon cycling; i.e. the different 
signatures of ? C and  ? R vs. V
gate
 for acetone and water are reproducible (at least for 
this device) showing that the device is able to discriminate between acetone and water 
at pressures < 10
-6
 torr. Figure 9-15 shows the resistance as a function of gate voltage 
 
 134 
 
in UHV and in 3 x 10
-6
 Torr argon pressure. Similar to the capacitance I observed no 
change in the resistance of CNT network in the presence of argon. 
 
 
 
 
Figure 9-15. Resistance as a function of gate voltage in UHV and 3 x 10
-6
 Torr argon pressure. 
 
 
 
 
 
 135 
 
9.4: Conclusion 
In conclusion I have fabricated CNT network devices and measured the effect 
of several analytes on capacitance and conductance. I modeled my device as an RC 
transmission line and derived an effective capacitance for the device. I showed that 
the measured capacitance is frequency-dependent due to the decay of the applied AC 
signal in the CNT network. I also showed that the capacitance and the square root of 
conductance are proportional as predicted by the RC transmission line model. My 
results show that the resistance changes of the network is capable of producing a 
change in the capacitive part of the impedance at finite frequency, and that the 
measured capacitance change due to chemical analyte is due to this effect. My results 
also indicate that the resistance of the CNT networks is enormously sensitive to small 
partial pressure (< 10
-6
 Torr) of analytes, and the gate voltage dependence of the 
response may be used to discriminate between analytes. 
 My results indicate the importance of the RC transmission line impedance in 
understanding chemicapacitive effects in highly resistive semiconducting films.  A 
previous report[134] of chemicapacitive and chemiresistive effects in CNT films at 
ambient pressure also found that resistance and capacitance changes were 
proportional, though the constant of proportionality sometimes differed from -1/2. 
The authors of reference [134] did not take into account RC transmission line effects, 
though for the values of r and c which may be extracted from the paper, their devices 
are in the regime where l
0
 << L for their measurement frequency of 30 kHz, and 
hence the RC transmission line model is important. 
 
 
 136 
 
 
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