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UNIVERSITY OF CINCINNATI
DATE: May 30, 2003
I, Luis Beltran Gómez ,
hereby submit this as part of the requirements for the degree of:
Doctor of Philosophy
in:
Physics
It is entitled:
High Frequency Electrical Transport Measurements Of
Niobium SNS Josephson Junction Arrays And Niobium
Thin Films With Nanoscale Size Magnetic Dot Array
Approved by:
Prof. David Mast
Prof. Howard Jackson
Prof. Mark Jarrell
Prof. Brian Meadows
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High Frequency Electrical Transport Measurements Of Niobium
SNS Josephson Junction Arrays And Niobium Thin Films With
Nanoscale Size Magnetic Dot Array
A dissertation submitted to the
Division of Research and Advanced Studies
of the University of Cincinnati
in partial fulfillment of the
requirements for the degree of
DOCTORATE OF PHILOSOPHY (Ph.D.)
in the Department of Physics
of the College of Arts and Sciences
2003
by
Luis B. Gómez
Licenciado en Física, Universidad Central de Venezuela, 1992
M.S., University of Cincinnati, 1993
Committee Chair: Professor David B. Mast
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ABSTRACT
High Frequency Electrical Transport Measurements Of Niobium SNS Josephson JunctionArrays And Niobium Thin Films With Nanoscale Size Magnetic Dot Array
By
Luis B. Gómez
Doctor of Philosophy in Physics
University of Cincinnati, 2003
Professor David. B. Mast. Chair
In this dissertation, measurements on two different niobium (Nb) based systems,
namely superconductor-normal-metal-superconductor (SNS) Josephson junction arrays
(JJA), and thin Nb films with nanoscale size magnetic dot arrays, are presented.
Without high frequency (rf) signals at nearly zero magnetic field, these
measurements show similarities for the two systems. These similarities are explained by
proposing that very close to the critical temperature (Tc) of the Nb film sample, the stray
magnetic field of the dots reduces the superconductivity in the film to make it a
superconductor-weaker-superconductor-superconductor (SS’S) JJA.
With rf signals, their dc-voltage-current characteristics (VI’s) show the
appearance of constant voltage steps known as Shapiro steps. For SNS-JJA, the
dependence of the width of the steps (∆ In) on the amplitude of the rf current (Irf ) is
Bessel-function-like. The maximum width of the first step ([∆ I1]MAX) was found to
increase as a function of rf frequency (ν rf ) reaching a maximum for ν rf ~1.8 times the
array’s Josephson frequency (ν c). This maximum was maintained for ν rf >4.9ν c. Also,
in JJA with missing junctions, [∆ In]MAX was smaller, disappearing for ν rf ~ν c.
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For Nb films near Tc, no Bessel-function-like oscillations were observed for ∆ In.
Also, ∆ I0 (or critical current, Ic) showed a maximum for Irf ≠ 0, contrary to what was
found in SNS-JJA, falling to zero for higher Irf . This enhancement of Ic with Irf was
attributed to non-equilibrium superconductivity mechanisms near Tc. Observation of
higher order steps (n=1,2,3,etc) became evident only from the dynamic resistance
(dV/dI)vs.V plots (where a step corresponds to a minima for this curve). The first step
was most evident for Irf values that made ∆ I0=0. At these values, V1 (the voltage that
corresponded to the first minimum in the dV/dIvs.V curve) satisfied the Josephson
relation V1=N(h/2e)ν rf , where N is the number of “junctions” in the direction of the
current. At fixed ν rf , V1 was found to increase with Irf . This result was interpreted as
having more “junctions” participating in the step’s formation, which was attributed to
magnetic interactions among the dot’s magnetic moments and with the I rf to minimize the
total energy of the system.
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© Copyright by
Luis Beltran Gómez2003
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DEDICATION
To my wife and kids.
“Nothing is more powerful than an idea whose time has come”
Victor Hugo
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ACKNOWLEDGMENTS
Many people have helped me throughout my graduate school life. It is impossible
for me to mention them all, but I would like to acknowledge a few of them here.
First of all, the completion of this work has been possible with the incredible help
and love of my wife, Kate and my kids; Florence, Evan and Vera, without which I would
not be writing these lines.
Second, I am grateful to my advisor, Dr. David Mast, for his guidance and support
through my many years of graduate school. Also, I am grateful to the members of my
committee; Dr. Howard Jackson, Dr. Mark Jarrell, and Dr. Brian Meadows, for their help
and guidance. I would like to give special thanks to Dr. Mike Sokollof for his many
hours of mentoring in data interpretation and his personal help in seeing this work
completed.
I would like to thank Dr. Ivan Schuller, Dr. Yvan Jaccard, and Axel Hoffmann for
their help providing the magnetic dots samples used in this dissertation and for their
useful discussions and letters of encouragement. Also, I would like to thank Maribel
Montero, Charles Reichhart, and Lieve Van Look for the useful discussions about their
work. Also, I would like to thank Dr. George Crabtree for his opinions regarding my
work.
I would like to thank Dr. Miguel Octavio and the late Dr. Juan Aponte from IVIC,
who helped me to graduate school and for initiating me into low temperature physics and
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superconductivity. Also, I would like to give special thanks to Dr. Lisseta D’Onofrio and
all my professors from UCV, for their teachings, and support.
I thank the rest of the faculty and staff at UC’s Physics department who helped
and encouraged my continuation in graduate school.
Finally, I would like to thank my family and friends, for their love and for
stopping asking me “when are you getting done?” For all of you a BIG thanks!
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3
TABLE OF FIGURES
Figure 2.1: H-T phase diagram for a bulk superconducting slab in a uniform magnetic field (using Equation
2.1 and parameters found for bulk Al [25])...........................................................................................14 Figure 2.2: Graphical representation of Equation 2.4 for the superconducting half-space (using values found
for bulk Nb [25] at zero temperature)....................................................................................................16 Figure 2.3: Graphical representation of Equation 2.6 (using values for bulk Nb [25]) .................................17 Figure 2.4: H-T phase diagram for a type II superconducting slab in a uniform magnetic field...................23 Figure 2.5: RCSJ model representation of a real JJ. The total current i into the junction is divided into three
channels: the supercurrent (ic) through the ideal junction, the current through the junction capacitor(C), and the quasiparticle current through the resistor (R 0). ..................................................................26
Figure 3.1: SNS Josephson junction array (not to scale) of 5×3 junctions. Current is injected via the Nb bus bars to the sides of the array and the voltage is measured across the whole array..........................33
Figure 3.2: Measuring bridge prototype for the film samples. The contact lead names are assigned asshown. In this photograph, the brown square in the center is where the magnetic dots array is located.The clear pattern is the Nb bridge for transport measurements. The gray area is the Si substrate. ......35
Figure 3.3: Picture of the lower part of the probe without vacuum can, showing the main parts.................37 Figure 3.4: Probe’s lower part, showing vacuum can and superconducting magnet. ...................................38 Figure 3.5: Sample holder chamber showing the electrical connection for current, voltage and high
frequency signals (showing Nb film sample ready to be measured). ....................................................39 Figure 3.6 Picture showing in detail the temperature controlling stage and sample holder...........................41 Figure 3.7: Block diagram for the experimental set up showing dc and rf electrical connections. ...............45 Figure 4.1: Resistive transition for a typical SNS JJA sample. Measurements were taken with a bias current
of 10µA in ambient magnetic field. The resistance error bars are shown in red and the line connectingthe dots is a guide to the eye..................................................................................................................48
Figure 4.2: Resistance vs. current passed through the solenoid. Measurements were taken at a temperatureT < Tc0 of 3.000K. The sample was biased with I < Ic of 200µA. The error bars for the resistance areshown in red, and the line connecting the dots is a guide to the eye. ....................................................50
Figure 4.3: VI curve at ambient magnetic field. The curve was taken for T < Tc0 of 3.000K. The error bars
for V are shown in red, and the line connecting the dots is a guide to the eye. .....................................54 Figure 4.4: Same data plotted in Figure 4.3, but on a log-log scale. The error bars for V are shown in red,and the line connecting the dots is a guide to the eye. ...........................................................................54
Figure 4.5: Dynamic resistance curve calculated from Figure 4.3. The error bars for dV/dI are shown inred, and the line connecting the dots is a guide to the eye.....................................................................56
Figure 4.6: Current vs. normalized-voltage graph for a 100% 300×200 JJA at T=1.8K and zero magneticfield. The black curve is for no rf signal. The red curve shows Shapiro steps when rf signal is applied.The voltage separation between steps is proportional to the rf frequency and the step width is
proportional to the rf power...................................................................................................................59 Figure 4.7: dV/dI vs. V curve shows a series of minima located at the steps’ voltage. The plot shows
dV/dI≠0 at the step location meaning that the step has non zero spread. ..............................................62 Figure 4.8: Detail from Figure 4.7 illustrating how to find the spread of the first step by using the dynamic
resistance of the VI, to indicate the center of the step, and the derivative of dV/dI as a functionvoltage, to find its value. .......................................................................................................................63
Figure 4.9: Widths for steps n=0 and n=1 as a function of applied rf amplitude. Graph shows Besselfunction like oscillations for the step widths. Notice how the oscillations are approximately 180° outof phase. The n=0 and n=1 step’s first maxima ([∆ I0]MAX and [∆ I1]MAX) for small Irf , are later usedto further characterize the JJA irradiated by rf signal............................................................................64
Figure 4.10: Maximum step width for steps n=1 as a function of reduced frequency for a 100% array(Figure 1 in reference [12]. Data points are obtained from similar curves as shown in Figure 4.9. Inhere T=1.91 K and f ≈ 0.........................................................................................................................65
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4
Figure 4.11: Normalized voltage vs. current for a 90% sample (300×300). The first Shapiro step presentsconsiderable reduction in its width and increase in its spread when compared to 100% arrays............67
Figure 4.12: Normalized maximum Shapiro step width for the first step vs. normalized frequencies, for twosamples with site disorder (90% and 80% samples) (from reference [12], inset in Figure 1). The linesare a guide the eye. ................................................................................................................................68
Figure 4.13: Voltage spread as a function of normalized frequencies for site disorder samples (fromreference [12], Figure 3). Results obtained for a 100% sample are shown for comparison. The spreadincreases linearly with frequency as can be seen from the linear fit to the data for the 90% and 80%samples. The line linking the 100% data points is a guide to the eye. ..................................................70
Figure 5.1: Resistive transition for the Nb/Ni square sample. Measurements were taken with a bias currentof 10µA in ambient magnetic field. The resistance error bars are shown in red and the line connectingthe dots is a guide to the eye..................................................................................................................75
Figure 5.2: Low temperature detail from Figure 5.1, showing the local resistance minimum at ST=7.860Kfollowed by an increase in resistance until it reaches a maximum to drop to zero resistance atTc0=7.766K. The resistance error bars are shown in red and the line connecting the dots is a guide tothe eye....................................................................................................................................................76
Figure 5.3: Drawing for the Nb film sample grown on top a square array of magnetic dots. The magneticdots are the gray circles shown in the drawing. The Nb film covers the whole array of dots (the Nbfilm right on top of the dots is not shown for clarity). This drawing is to scale, where the diameter ofthe circles is 200nm. ..............................................................................................................................77
Figure 5.4: Resistance vs. magnetic field perpendicular to the sample. The graph shows data for twoslightly different temperatures below TRMAX. The current bias to the sample (627µA) is the same in
both cases. .............................................................................................................................................79 Figure 5.5: VI curve at ambient magnetic field. The curve was taken T ~7.8K > Tc0. The error bars for V
are shown in black. The red curve is the temperature of the sample at each voltage point. The error bars for ST are shown in red. The line connecting the dots is a guide to the eye.................................81
Figure 5.6: Same data plotted in Figure 5.5, but on a log-log scale. The blue line here represents thearbitrary value of voltage chosen to define Ic using the threshold method. The line connecting the dotsis a guide to the eye. ..............................................................................................................................82
Figure 5.7: Dynamic resistance curve calculated from Figure 5.5. The error bars for dV/dI are shown in red(calculated in the same way as in Figure 4.5)........................................................................................83
Figure 5.8: VI curve for the Nb/Ni square sample at T~7.8K and ambient magnetic field. The black curve
is the VI with rf signal. The red curve is the temperature of the sample. No Shapiro steps are apparentfor νrf =2GHz and Irf =123µA..................................................................................................................85
Figure 5.9: dV/dI vs. V curve from VI with rf curve in Figure 5.8. The dynamic resistance is shown in blueand the IV, in the black curve, is plotted for reference..........................................................................86
Figure 5.10: Plot of the voltage location for the minima found in the dynamic resistance curve in Figure 5.9. These voltages are plotted vs. the number of the minima staring with the first minimum at n=1, alinear fit of this data in shown. For the calculations of the linear fit, the (0,0) point is included. ........87
Figure 5.11: Plot of the critical current (defined by the threshold method) vs. Irf . The critical current is the black curve. A plot (in red) of the sample temperature is shown to indicate the increase in temperature produced by the applied Irf .....................................................................................................................88
Figure 5.12: Plot of the dynamic resistance vs. voltage for five different Irf . The green curve is the one plotted in Figure 5.8. The curves have been shifted a fixed amount in the y-axis for clarity. The lineconnecting the dots is a guide for the eyes. ...........................................................................................90
Figure 5.13: Similar plot to Figure 5.12. The range [40-70] dB in the attenuator box was found to deliverthe same amount of Irf as in Figure 5.11. A maximum in the critical current as a function of Irf is moreevident. ..................................................................................................................................................92
Figure 5.14: Plot of the dynamic resistance vs. voltage for four different Irf . The curves have been shifted afixed amount along the y-axis for clarity. The line connecting the dots is a guide for the eyes. ..........93
Figure 5.15: Dynamic resistance vs. voltage plot. Here the dV/dI was calculated from five different VI’sall taken with an attenuation of 51dB. The first curve on top (51D) is the same curve plotted in Figure 5.13 for 51dB attenuation. The curve at the bottom (51H) was measured by allowing the current to
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6
LIST OF EQUATIONS
Equation 2.1 ( ) ( )
−≈
2
10c
ccT T H T H ........................................................................................13
Equation 2.2 E m
en
t
J s sv
v
*
2**
=∂∂
..............................................................................................................15
Equation 2.3 s L J Bvvv
×∇−= 2
0λ µ .........................................................................................................15
Equation 2.4
−=
L
a
x B x B
λ exp)( ...................................................................................................15
Equation 2.5
4
1
−=
cn
s
T
T
n
n..............................................................................................................16
Equation 2.6 ( )2
14
0 1
−
−≈
cT
T T λ λ ............................................................................................17
Equation 2.7el α ξ ξ
111
0
+= ................................................................................................................18
Equation 2.8e
Ll
0ξ λ λ = ....................................................................................................................18
Equation 2.9 ( ) ( ) ( )r ier r vvv φ Ψ=Ψ ........................................................................................................18
Equation 2.10 ( ) ( )[ ] 2/1* r nr svv =Ψ ......................................................................................................19
Equation 2.11 ( ) )()()()(2
1 22
*
*r T r r r Ae
im
rrrrhΨ−=ΨΨ+Ψ
−∇ α β ..............................19
Equation 2.12
Ψ
−∇Ψ= Aeim
e J s
rrhr**
*
*
Re .........................................................................19
Equation 2.13 )(2 *
*
r m
ne J s
s
rrhrφ ∇−= .................................................................................................19
Equation 2.14 ( )
( )2/1
*2 T m
T
α
ξ h
= ................................................................................................19
Equation 2.15 ( ) el 00 ξ ξ = ..............................................................................................................20
Equation 2.16 ( )
cT
T T
−
=
1
)0(855.0 ξ
ξ ..............................................................................................20
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9
Chapter 1 : INTRODUCTION
The main focus of this dissertation is to investigate the dynamic response to an
external radio frequency (rf) signal (in nearly zero external magnetic field) of two
different niobium (Nb) based systems. The systems investigated were: superconductor-
normal-metal-superconductor (SNS) [1] Josephson junction arrays (JJA) [2, 3], and thin Nb
films deposited on top of a nanoscale size magnetic dot arrays [4].
This study is motivated by the similarities observed in the resistive transition
curves (RT’s) of these two systems that allow us to propose a model in which the Nb film
samples are viewed as a superconductor-weaker-superconductor-superconductor (SS’S)
[1] JJA. This model is based on the assumption that the superconductivity in the film is
locally destroyed by the stray magnetic field of the dots [5, 6] and weak links or “junctions”
are created in the film, in the direction of the current.
In JJA, the coherent response to an rf signal is observed as a series of steps in
their dc voltage-current characteristics (VI’s) known as Shapiro steps [7]. The study of
the conditions that give rise to the disruption of this coherence (i.e. the disappearance of
Shapiro steps from the VI’s) is an important issue that can be used to sharpen our
understanding of the dynamic processes occurring in these systems. Also, in the case of
“classical” JJA, these studies will help us in the utilization of these systems in
applications [8, 9, 10].
When “ideal” classical JJA (i.e. JJA in which superconducting islands are
described by a single order parameter, with a pure sinusoidal current-phase-relationship
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(CPR) among their junctions and minimal disorder), in a “zero” external magnetic field,
are irradiated with rf signals, well defined Shapiro steps are observed in their VI’s for
large ranges of external rf signal amplitudes and frequencies[12]
. In these circumstances,
the width of the steps (the range of currents in the VI’s over which the voltage is a
constant, or ∆ I) can be used as a measure of the “strength” of the coherent response of
the JJA to the rf signal.
The conditions that disrupt the coherent response of JJA to an external rf signal
include: unavoidable sample fabrication defects, known as bond disorder [11],
imperfections in the array (the removal or disappearance of junctions from the array)
known as site disorder [12], and having a non-sinusoidal CPR [13, 14] between the junctions.
Non-sinusoidal CRP’s in junctions occur when the coupling between the superconducting
“islands” is not-weak [15, 16, 17] and/or the system is operated in conditions of
nonequilibrium superconductivity [18, 19, 20].
The consequence of disorder in JJA is to create a random distribution of critical
currents (Ic) and shunted resistances (R N) among the junctions in the array that will
diminish the “strength” of the Shapiro steps [21] (i.e. how well they are formed in the
VI’s). More fundamentally, the nonequilibrium processes that can occur in these systems
affects the superconducting state in the islands themselves (described by a order
parameter
φ i
s en=Ψ ), changing both the number of superconducting electrons (ns) as
well as their phases (φ ) [22, 23]. When this occurs, the macroscopic variables Ic and R N,
which are derived from ns and φ , will also change, consequently impairing the ability of
theses systems to respond coherently to the external rf signal.
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11
This dissertation is organized as follows: Chapter 2 covers the theoretical
background of superconductivity, Josephson Effect and Shapiro steps in junctions and
arrays. Chapter 3 covers the experimental aspects of this work. Chapter 4 presents the
experimental results found for SNS JJA. Chapter 5 presents the experimental results
found in the superconducting film samples. Conclusions and suggestions for future
research are given in Chapter 6.
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In pure bulk samples the change from resistive to resistive-less takes place over a
temperature range of only a few hundredths of a degreei [25].
In 1914 Onnes [26] also found that if a superconductor (at T<Tc) is placed in a
strong magnetic field, the zero dc-resistance “state” is destroyed, to be later regained
when the magnetic field is removed. The minimum magnetic field required to destroy
superconductivity is called the critical (magnetic) field “Hc” and experimentally its value
depends on the shape of the sample and its orientation to the external magnetic field ii.
The temperature dependence of H c is represented in bulk superconductors by the
parabolic relationship:
Equation 2.1 ( ) ( )
−≈
2
10c
ccT
T H T H
In 1933 Meissner and Ochsenfield [27] discovered that a superconducting material
was more than merely a perfect conductor. By systematically applying a weak magnetic
field to a lead sample, they discovered that the effective relative permeability of the
sample was zero when the sample was in the superconducting state, concluding that the
magnetic field inside the superconductor must be zeroiii. This perfect diamagnetism, or
Meissner effect , observed in superconductors cannot be derived from its perfect
conductivity and therefore is a unique characteristic of the superconducting state.
i For bulk niobium (Nb), Tc =9.25 K. See reference [25]ii For bulk Nb, µ0Hc(0)=2060 G. See reference [25]iii Later it was found that this is only true in bulk samples, except for a thin layer of material near thesurface of the sample of ~50nm
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15
Equation 2.2 E m
en
t
J s sv
v
*
2**
=∂∂
Equation 2.3 s L J Bvvv
×∇−= 20λ µ
Where ns* is the number of super-electrons per unit volume, m* and e* are
respectively the effective mass and charge of these super-electronsi. The factor
02**
*
µ λ
en
m
s
L = is known as the London penetration depth ii. A one-dimensional solution
of the London’s second equation for a superconducting half-space in a static magnetic
induction is given by:
Equation 2.4
−=
L
a
x B x B
λ exp)(
i It was later found in the BCS theory that the “super electrons” are really a pair of electrons called aCooper pair, so that e* = 2e and m* = 2me.ii For bulk Nb, 0λ = 39 nm.
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Figure 2.2: Graphical representation of Equation 2.4 for the superconducting half-
space (using values found for bulk Nb [25] at zero temperature)
A two-fluid model for superconductors was developed in 1934 by Gorter and
Casimir [29] originally to try to explain the observed thermodynamic properties of
superconductors. The model postulates that below Tc the conduction electrons are
divided into two distinct groups: a superconducting aggregate of super electrons ns
(=2ns*) and a fraction of normal state electrons nn. The best agreement between
experiments and the theory is found if:
Equation 2.5
4
1
−=
cn
s
T
T
n
n
This model, combined with a formulation of the ac electrodynamics of
superconductors based on the London equations, gives accurate predictions for the
London penetration depth Lλ (T) i.e.,
0 39 50 100 150
0
500
1000
1500
2000
2500
Vacuum Superconductor
B(0)
λ 0
B ( X ) ( G )
X (nm)
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18
Equation 2.7el α ξ ξ
111
0
+=
where le is the electron mean free path andα
is a constant on the order of unity.
There are two extreme cases in which Pippard’s non-local relationship reduces to
a simpler local form. The case for λ >> ξ , which is known as the “dirty limit”, is
applicable for very impure specimens, alloys, or thin films. For this limit, le is much
smaller than ξ 0 and ξ → le and the penetration depth is given by:
Equation 2.8e
Ll
0ξ λ λ =
The other extreme case for λ << ξ 0 or the “clean limit”, is applicable for most
pure bulk superconductors at temperatures not too close to Tc.
The Ginzburg-Landau (G-L) theory [32] is a generalization of the London theory to
deal with situations in which ns varies in space, and also to deal with the nonlinear
response to fields that are strong enough to change ns. In this theory, London’s ns is
given by:2
)( xn s Ψ= , where Ψ is a complex pseudo wave-function taken as an order
parameter within Landau’s general theory of second-order phase transitions.
This order parameter has a well-defined amplitude and phase angle and represents
a macroscopic many-body wave function of the superconducting electrons inside the
superconductor. It is written as:
Equation 2.9 ( ) ( ) ( )r ier r vvv φ Ψ=Ψ
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19
Where:
Equation 2.10 ( ) ( )[ ] 2/1* r nr s
vv=Ψ
Assuming a series expansion of the free energy in powers of Ψ and Ψ∇ ,
Ginzburg and Landau derived the following equation:
Equation 2.11 ( ) )()()()(2
1 22
*
*r T r r r Ae
im
rrrrhΨ−=ΨΨ+Ψ
−∇ α β
Here, α and β are expansion coefficients. Equation 2.11 is analogous to the
Schrödinger equation for a particle of charge e*and mass m*, which includes a non linear
term. Using Equation 2.10, the supercurrent is calculated as:
Equation 2.12
Ψ
−∇Ψ= Aeim
e J s
rrhr**
*
*
Re
Using Equation 2.9, the current inside a superconductor in the absence of applied
field, is given by:
Equation 2.13 )(2 *
*
r m
ne J s
s
rrhrφ ∇−=
This is found to vary linearly with the gradient of the phase in the superconductor.
From the G-L theory a new characteristic length is introduced as:
Equation 2.14 ( )( )
2/1*2 T mT
α ξ
h=
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2.3 The Josephson junction effect
In 1962 B. D. Josephson [36] suggested that it was possible for a super-current to
flow between two superconducting electrodes separated by a thin insulating barrier at
zero voltage. Soon afterwards, in 1963, experiments conducted by Anderson and
Rowell[37] suggested the observation of the so-called Josephson Junction Effect (JJE).
The JJE is summarized by the following two equations:
Equation 2.20 ( )21sin γ c s ii =
and
Equation 2.21dt
d
eV
)(
221γ h
=
where,
Equation 2.22 ∫ ⋅−−=
2
1
1221
2r d A
e vv
hφ φ γ
is the gauge-invariant phase difference of the Ginzburg-Landau wave-function in the two
superconductors at each side of the separation, Ar
is the vector potential of an external
magnetic field threading the junction, ic is the zero voltage critical current, and V is the
dc voltage drop at each side of the separation.
If a dc voltage is applied to a JJ, integrating Equation 2.21 will show that the
phase difference will increase in time at a rate given by the magnitude of the voltage.
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Then the supercurrent in the junction will oscillate at a frequency (known as the
Josephson frequency) given by:
Equation 2.23 V h
ec
2=ν
The energy stored in the supercurrent in a Josephson junction is given by the time
integral of the product of Equation 2.20 and Equation 2.21. This gives:
Equation 2.24 )cos()cos(2 2121 γ γ J c s E i
e E −=−=
h
where EJ is the Josephson coupling energy.
The JJE can be observed in a large number of superconducting systems that are
weakly coupled, for example: superconductor-insulator-superconductor (SIS) junctions,
superconductor-normal-metal-superconductor (SNS) junctions, superconductor-
constriction-superconductor (SCS) junctions (i.e., a micro-bridges in a superconducting
film), and superconductor-weaker superconductor-superconductor (SS’S) junctions [19].
A weak link is usually modeled using the resistively-capacitively-shunted
junction (RCSJ) model [38, 39]; in this model, a real weak link is represented by an ideal
one shunted by a resistance R and a capacitance C, as is shown in Figure 2.5.
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which is known as the plasma frequency of the junction, the dimensionless time variable
is defined as: t pτ = , and the “quality factor” Q is defined as: RC Q p= . The quality
factor is identical to c β , which is a damping parameter introduced by W. Stewart [38]
and D. McCumber [39] (known as the McCumber constant).
Depending upon the value of the capacitance, a JJ can be classified as
“overdamped” (when C is small so that Q<< 1 as in the case of a SNS JJ and micro-
bridges) or “underdamped” (when C is large so that Q>>1 as in the case of SIS JJ). The
systems studied in this dissertation belong to the overdamped category and it is
customary in simulations of these types of weak links to drop the term with the
capacitance in Equation 2.26 (in this case the RCSJ model is referred to as the RSJ
model).
Equation 2.26, is identical to the equation of a driven-pendulum where the applied
torque [40] is analogous to the current source, the maximum gravitational torque is
analogous to the critical current, the damping constant is analogous to the inverse of the
resistance, the momentum of inertia is analogous to the capacitance, the angular velocity
is analogous to the voltage, and the angle is analogous to the phase difference in the JJ.
2.4 Shapiro steps
The voltage-current characteristics (VI’s) of a JJ, when biased with both dc and rf
signal, show the appearance of step-like structures called Shapiro steps [7]. These steps
occur at voltage multiples of: rf ν 0Φ , where ν rf is the frequency of the rf signal.
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The occurrence of these steps can be derived analytically from the Josephson
junction equations when the junction is biased with both dc and rf voltage sources. For
simplicity it is assumed that the voltage sources are ideal (i.e., they have no internal
impedance). The following expression gives the total voltage bias for the JJ:
Equation 2.28 )2cos()( t V V t V rf rf dc πν +=
Integrating Equation 2.21 to obtain )(21 t γ and substituting this value into Equation 2.20,
an expression for the supercurrent is obtained as a function of time. Then, by using the
standard mathematical expansion of the sine of a sine in terms of Bessel functions, we
have:
Equation 2.29 ])(2sin[)()1( 0
0
γ ν ν π ν
+−Φ
−= ∑ t nV
J ii rf c
rf
rf
n
n
c s
where 0γ is a constant of integration and n J is the nth order Bessel function of the first
kind. From Equation 2.29 it can be seen that for average dc voltages such that rf c nν ν = ,
there will be current steps in the VI’s that have a maximum width given by:
Equation 2.30 )(][0 rf
rf
nc MAX n
V J ii
ν Φ=∆
This result, which is derived analytically for the case of a voltage biased JJ, has been
found [41] to occur in practice (when the JJ are usually current biased due to their small
impedance compared to that of the source) but the dependence of the steps’ widths are
different from the Bessel function dependences found above.
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In physical terms, we can interpret the occurrence of Shapiro steps in junctions by
recalling that the supercurrent through the junction oscillates at a frequency given by
0
2Φ
V π for any applied dc voltage V. If an rf signal is coupled to the junction, this signal
will interfere with the oscillating supercurrent and the time average values of the total
supercurrent in the JJ will vanish unless rf c nν ν = .
2.5 2D Josephson junction arrays
Consider now a set of superconducting islands at the corners of a square lattice
with lattice constant a, and that each island forms a JJ with their nearest neighbors (i. e.
there is a JJ on each vertical and horizontal bond of the square lattice). Such a structure
is known as a Josephson junction array (JJA). The characteristics of a JJA depend upon
the number of superconducting islands in the array. A square lattice with N+1 columns
and M+1 rows will have N JJ in the horizontal direction and M JJ in the vertical
direction, forming an N×M JJA.
In experiments, it is customary to bias these samples with a current flowing along
the rows of the lattice and the voltage is measured between the first and last columns of
the array. If an external rf signal is superimposed on the JJA, each individual JJ will
couple (phase lock) to the external rf signal (when each individual JJ’s Josephson current
oscillates in phase with the external rf signal) and a voltage plateau will form in the time-
averaged VI’s at a voltage which is N times greater than that of an isolated JJ. This is
given by the following equation:
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Equation 2.31 rf ne
hnN V ν
=2
In the literature the voltage steps that occurred in JJA are known as “giant”
Shapiro steps, to distinguishing them from the ones occurring in JJ. In this dissertation,
such distinction won’t be emphasized since the samples studied here are all array types.
A JJA is not the only superconducting system on which Shapiro step structures
will occur if irradiated by an rf signal. In fact, any system with an equation of motion
that can be written in the form of the driven damped pendulum equation will present such
features in their characteristics.
For example, Martinoli et al. [42] found pronounced equidistant steps in the VI’s of
a superconducting film with laterally modulated thickness for several matching magnetic
fieldsi and rf frequencies. This observation is interpreted to be caused by the coherent
motion of the Abrikosov vortex lattice in the one-dimensional periodic potential created
by the thickness modulation. Recently L. Van Look et al. [43] measured Shapiro steps in
the VI characteristics of a superconducting film with a square lattice of perforating
microholes (antidots) of lattice spacing d. They found that in order for the Shapiro steps
to occur, a magnetic field corresponding to202
d
Φ (the second matching field 20 H ),
should be applied to the sample. Under this specific condition of applied magnetic
induction and for particular temperatures [44], it is possible to have a single vortex
occupying every antidot and a vortex lattice located at the center of the array’s cells.
This weakly interstitial vortex lattice can then be easily moved in a coherent fashion by
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Figure 3.1: SNS Josephson junction array (not to scale) of 5×3 junctions. Current
is injected via the Nb bus bars to the sides of the array and the voltage is measured
across the whole array.
This multilayer sample was then taken to the Cornell Nanofabrication Facility (CNF) at
Cornell University for patterning using e-beam lithography to subsequently remove the
exposed Nb by reactive ion etching (RIE) in a CF4 atmosphere. The crosses fabricated
using this method have arms of width 1.2 m and are arranged in a square lattice 10 m
center to center. Samples were made with average gap distances of 0.3 - 0.5 m
between the arms of the crosses (variation in the gap distances, determined from
measurements of SEM pictures, are approximately 10% on average, due to the limitations
of the e-beam lithography + etch process).
There are Nb contact pads in the sample for transport measurements. Leads are
attached directly to the contact pads using indium as solder. The 4-point room
temperature resistance of these samples is on the order of 100’s of milli Ohms. The
contact resistance obtained by pressing indium is usually on the order of a few Ohms at
room temperature.
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Figure 3.2: Measuring bridge prototype for the film samples. The contact lead
names are assigned as shown. In this photograph, the brown square in the center is
where the magnetic dots array is located. The clear pattern is the Nb bridge for
transport measurements. The gray area is the Si substrate.
After the fabrication of the dot array, a Nb thin film (65nm thick) was deposited
on top of it. To perform transport measurements on the sample, a bridge 40 m wide and
50 m long was optically imprinted and subsequently etched on the Nb film. The
measuring bridge consists of a strip of Nb 40 m wide that runs over the magnetic dot
array and is used to supply the current to the sample. Also, six Nb strips (5 m wide)
extend perpendicularly out from the main Nb strip for voltage measurements. One pair
of contacts is used for longitudinal voltage measurements along the array, another pair
could be used for transverse voltage measurements, and the final pair is a backup pair incase any of the other contacts should fail. One of them, the top right contact, and another
contact down to the right, not shown, were used for measuring voltage drop in the Nb
film).
I+ I-
V L+ V L-
V T-
V T+
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The room temperature resistance of these samples is on the order of 10’s of
Ohms. Current self-heating effects played a very important role in the measurement of
these samples, limiting the current bias used to no more than 1mA[52]
. Due to this, the
experiments where performed very close to Tc (no more than 250mK below Tc).
3.2 Experimental setup
The experiments in this dissertation were performed on a standard liquid Helium
(LHe) cryostat. Some experiments on the JJA were performed in a continuous Helium
flow cryostat (Janis ST) manufactured by Janisi
. The cryostat used for most of the
experiments consists of a straight bore Dewar (Cryofabii model CSM) and a probe that
was immersed directly into the LHe bath.
The probe was designed, fabricated and assembled at the University of Cincinnati
Physics department. The main parts of the probe are: vacuum can, He pot, needle valve,
positioning oxygen-free-high-conductivity (OFHC) copper rods, cold finger stage, weak
link, temperature controlling stage, and sample holder. Figure 3.3 is a picture of the
lower part of the probe
i Janis Research Co. Inc.; 2 Jewel Dr.; P.O. Box 696; Wilmington, MA 01887-0696.ii Cryofab; 540 Michigan Ave.; P.O. Box 485; Kenilworth, New Jersey 07033.
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Figure 3.3: Picture of the lower part of the probe without vacuum can, showing the
main parts.
The vacuum can was designed so its lower tail fits inside the borehole of a
superconducting magnet (Cryomagneticsi
6T magnet with 0.1% homogeneity over a
10mm diameter spherical volume (DSV)). Figure 3.4 shows a picture of the lower part of
the probe, vacuum can, and superconducting magnet.
This configuration provides a homogeneous magnetic field, normal to the
sample’s surface. The magnitude of the magnetic induction was determined by
measuring the current sent to the magnet and calculating the magnetic induction using the
calibration factor provided by the manufacturer (738.5 Gauss/Amp). In addition, the
applied magnetic field was measured independently using a calibrated low temperature
Hall probe (Lakeshoreii HGCT-3020) situated inside the 10mm DSV of the magnet
(4mm above the sample, located outside the sample holder).
i Cryomagnetics; 1006 Alvin Weinberg Drive; Oak Ridge, Tennessee 37830ii Lake Shore Cryotronics Inc.; 575 McCorkle Blvd.; Westerville, OH 43082.
Sample
Holder
Cold
FingerHe Pot
Positioning
Rods
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Figure 3.4: Probe’s lower part, showing vacuum can and superconducting magnet.
The sample holder (made of OFHC copper) was specially designed by Dr. Mast
and his former graduate student Dr. Hyun Cheol Lee [53] to guarantee proper impedance
matching with the rf signals. The sample holder consists of two pieces: a chamber and a
cover. The chamber has four female bulkhead microdot connectors by Malcoi for dc
current connections and dc voltage measurements. Also, it has two male SMA
connectors to allow the application of rf signals (these connectors can not be seen in
Figure 3.5, they are in the other side of the chamber).
The dimensions of the sample holder (width and depth of chamber) were carefully
chosen to keep the electromagnetic field configuration in the sample approximately
equivalent to that for a microstrip line [54].
i Malco Technologies LLC.; 94 County Line Rd.; Colmar, PA 18915.
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For dc measurements, gauge # 34 insulated copper wire was attached directly to
the samples by pressing indium into Nb pads in the samples. The copper wire was
wounded into a small inductor to prevent the rf signals from leaking into the dc
equipment. The small inductors were also thermally anchored to the chamber with GE
varnish to provide an additional thermal link between the chamber and the sample.
Figure 3.5: Sample holder chamber showing the electrical connection for current,
voltage and high frequency signals (showing Nb film sample ready to be measured).
To inject the rf signal to the samples, straight pieces of bare 0.5mm diameter Au
wire were soldered to the SMA’s center conductor and indium was pressed onto the
sample’s current contact pads to provide a connection in parallel to the current contact
pads.
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The samples were electrically isolated from the chamber and thermally anchored
to it by placing a piece of sapphire wafer with a thin layer of Apiezon i N-grease between
the sample, the sapphire wafer, and the sample holder. In order to prevent the sample
from “springing out” of the camber due to the pull of the wires, and to ensure a good
thermal contact between the sample, sapphire, and the camber, a flat piece of teflon with
a copper spring on one side was placed between the samples and the sample holder’s
cover. Finally, the cover was fastened to the chamber by using 2-56 brass screws and the
sample holder cell was mounted in the probe.
The temperature of the sample holder was controlled upwards from the
temperature of the cold finger stage. This was accomplished by sending current to the
heater (a non-inductively wound resistive wire attached to the heater block with GE
varnish) using a temperature controller (Conductusii model LT-21), and monitoring the
temperature with a calibrated Cernox thermometer called the “control thermometer”
(Lakeshore model CX-1070-SD-4L) in a PID control loop.
Using this method, the temperature on the sample holder was regulated to better
than 0.5mK when using our homemade cryostat. But only 10mK (or 2mK by using a
specially designed data acquisition program) when using the continuous flow cryostat.
Figure 3.6 is a detailed photograph of the temperature controlling stage and sample
holder ready for measurement.
i Apiezon Products M&I Materials Ltd; PO Box 136; Manchester M60 1AN, UK.ii Conductus, Inc.; 969 West Maude Avenue; Sunnyvale, CA 94085.
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The basic algorithms for data collection consisted in sending current to the sample
and measuring the longitudinal voltage while the temperature, magnetic field, and rf
amplitude and frequency were kept fixed.
Some of the data presented in this dissertation (specially the measurements
performed in the SNS JJA) were taken using a technique known as the “delta”
measurement. In this technique, the bias current to the sample is reversed in direction
and a pair of voltages is recorded for each direction of the current. Then the average of
the two measurements is taken as follow:
Equation 3.1 ( )2
,,,,,,,,, rf rf
rf
I BT I V I BT I V I BT I V
−−+=
This method was used to cancel slow voltage fluctuations and voltage offsets present at
the contacts, and to mathematically enhance the sensitivity of the measurements by
averaging many of these measurements.
When the delta-method was not employed, voltage measurements at a fixed
current direction were buffered into the Keithley 182’s internal memory and the average
of these voltages was calculated. This method was employed to avoid fast switching of
the current direction in the samples; especially at high currents (I~Ic(T,B)).
Avoiding the fast switching of the current direction may be particularly important
when taking transport measurements in the Nb film samples, since there is the possibility
that the bias current changes the magnetic orientation of individual dots in the array [55].
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Figure 3.7: Block diagram for the experimental set up showing dc and rf electrical
connections.
For the measurements in JJA, the rf signal was provided using the HP8656B
(100kHz-990 MHz) Synthesized Signal Generator. The amplitude of Irf coupled to the
arrays was determined by converting the detected power through the array to current
using a 50Ω load.
For the measurements in the Nb film, the rf signal was provided using the
HP8671B Synthesized CW Generator (2-18 GHz). The large critical current of these
samples required the amplification of the applied rf signal’s power. This was
accomplished by using the HP8349B microwave amplifier (2-20 GHz). This amplified
signal was then passed through the attenuator box, HP11713A in order to change the
amplified rf signal by one dB increments. The power of the rf signal passed through the
sample was finally measured using the HP437B power meter with the HP8485A Power
Sensor and this power was converted to current using a 50Ω load. Applying this high
power rf signal to the sample was found to increased the temperature of the sample stage
by about 4mK (when compared to the zero rf signal case). For this reason, and in order
Indium
contacts
V
I
dc blocks
rf In rf Out
Inductors
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to least affect the state (magnetic or superconducting) of the films, the measurements of
VI’s with rf signals were taken from high power (less attenuation) to less power (more
attenuation) signals, as will be shown in section 5.1.4
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Chapter 4 : MEASUREMENTS IN SNS JJA
In this chapter, a summary of the important results from the research in 2D SNS
JJA is presented. In section 4.1, the resistive transition of SNS JJA, their magneto
resistance curves (RB’s), their VI’s with and without rf signals, and the effects of
disorder on the VI’s characteristics is described. In section 4.2, a brief discussion of the
difficulties encountered in this research is given and in section 4.3, a summary for this
chapter will be given.
4.1 Measurements
As mentioned in section 3.2, most of the measurements on these samples were
performed in a Janis ST continuous flow cryostat. Specifically, all of the VI’s with rf
data presented in this section were taken using this cryostat. The temperature control and
stability achieved with this configuration was on the order of 10mk.
The rest of the data present in this chapter was taken in a similar probe and
cryostat to the one described in section 3.2. The temperature control and stability
achieved with that system was better than 2mK. This early probe had to be replaced
because it developed a cold leak that was impossible to repair.
4.1.1 R vs. T
Figure 4.1 shows a typical resistive transition curve for a SNS JJA:
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Figure 4.1: Resistive transition for a typical SNS JJA sample. Measurements were
taken with a bias current of 10 A in ambient magnetic field. The resistance error
bars are shown in red and the line connecting the dots is a guide to the eye.
Here, the sample resistance is calculated as:
Equation 4.1bias I
V R =
where V is the average of N voltage points taken at a constant temperature and fixed
current bias, as calculated with Equation 3.1.
In Figure 4.1, three distinct regions are evident first, the initial resistance drop at
9.2K, signals the temperature at which the Nb crosses become superconducting (TcNb),
the size of the drop is proportional to the amount of underlayer Au locally shorted by the
superconducting crosses (throughout this dissertation, TcNb is defined as the temperature
that corresponds to a 10% reduction in resistance, from the high temperature resistance
plateau value).
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0
0
100m
200m
300m
LAT Theory
Tc0
∼ 3.2K
TcNb
∼ 9.2K
Proximity-EffectModel
10/15/97Earth's B fieldTemperature controlled better than 2 mK Current bias= 10 µA
Sample: 300 X 50 W6
R ( Ω )
Temperature (K)
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Figure 4.3: VI curve at ambient magnetic field. The curve was taken for T < Tc0 of
3.000K. The error bars for V are shown in red, and the line connecting the dots is a
guide to the eye.
This voltage value is most easily determined by plotting the VI in a log-log scale, as in
Figure 4.4.
Figure 4.4: Same data plotted in Figure 4.3, but on a log-log scale. The error bars
for V are shown in red, and the line connecting the dots is a guide to the eye.
0 200µ 400µ 600µ 800µ 1m
0
50µ
100µ
I bias
= 200 µA
R N
= (0.1581 +/- 0.0006) Ω10/16/97T=(3.0001+/-0.0002)K Ambient magnetic fieldData: IV300KB.dat
No rf signal
Sample 300 X 50 W6
V o l t a g e ( V )
Current (A)
10µ 100µ 1m1n
10n
100n
1µ
10µ
100µ
Noise floor cutoff
I bias
= 200 µA
R N
= (0.1581 +/- 0.0006) Ω
10/16/97T=(3.0001+/-0.0002)K Ambient magnetic fieldData: IV300KB.dat
No rf signal
Sample 300 X 50 W6
V o l t a g e ( V )
Current (A)
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In Figure 4.4, the blue line at 400nV is the cutoff between the data below and
above the noise floor. This cutoff is chosen after examining the data and determining
where it can no longer be represented by a power law V∝ Ia
, where “a” is the slope of the
VI curve (in the log-log graph)i, at that location. The value chosen to bias the sample
(200µA), gives a voltage drop, which is above the noise floor. This voltage is determined
from the data to be: V V µ )01.095.0( ±= ii.
Other important array parameters, such as the total critical current (Ic) and its
normal resistance (R N
), can also be obtained from Figure 4.3. In JJA, a good estimate for
the non-fluctuating critical current Ic, is given by the “peak” or maximumiii, [69] found
from a dV/dI vs. I curve calculated from Figure 4.3. This is shown in Figure 4.5. From
Figure 4.5, Ic = 540µA. Another important value that is obtained from Figure 4.5 is the
vortex depinning critical current (Ic,v)[70]. Ic,v is defined as the current for which the
dynamic resistance stops being zero.
i By studying curves such as Figure 4.4, researchers are able to study phase transitions (like KTB) in JJA ii the error here is the standard deviation of N averages of “individual ” voltage points as calculated usingEquation 3.1iii This inference follows from studies of the effects of thermal fluctuations in single JJ
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Figure 4.5: Dynamic resistance curve calculated from Figure 4.3. The error bars for
dV/dI are shown in redi, and the line connecting the dots is a guide to the eye.
From Figure 4.5, Ic,v=140µA. This value was chosen because it was the first current
value above the noise floor.
It has been estimated [70] that for square arrays, and in the case of very small
external magnetic induction, the depinning critical current is:
Equation 4.5 cvc I I 1.0, =
The value for Ic,v found in here, is not the value given by Equation 4.5. This discrepancy
might be associated with the lack of sensitivity of the voltmeter i.
The normal resistance of the array (R N) is determined by doing a linear fit to the
VI for current values higher than Ic. A good way of choosing what points to use in the
i The dynamic resistance error in here was calculated as:
∆
V
V
dI
dV *
0.0 200.0µ 400.0µ 600.0µ 800.0µ 1.0m
0
50m
100m
150m
200m
250m
I c , v
= 140 µA
Noise floor cutoff
Ic = 540 µA
Sample 300 X 50 W610/16/97T=(3.0001+/-0.0002)K Ambient magnetic fieldData: IV300KB.dat
f = 1/300
Current (A)
d V / d I ( Ω )
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linear fit is by selecting, from Figure 4.5, data points to the right of Ic, at values for which
the dynamic resistance is fairly flatii. From Figure 4.3, a linear fit gives:
Ω±= m R N )6.01.158( .iii
The normal resistance of the array and its critical current are essential parameters
that will determine the ability of a JJA to perform as a microwave oscillator. From these
two values the Josephson Frequency νc (given by Equation 2.23) can be calculated.
The single junction parameters R 0 and ic, required to calculate νc, can be obtained
for a N×M JJA, by assuming the JJA to be a lattice of resistors with resistance R 0 and that
the critical current flows uniformly throughout the whole arrayi. This means:
Equation 4.6 M
I i cc =
Equation 4.7 N R N
M R =0
Thus, the Josephson frequency, in terms of the array’s measurable parameters is
given by:
Equation 4.8 N
R I N cc
0
1
Φ=ν
Using the values found for this array, its Josephson frequency is: νc~138MHz.
i In chapter five, a 9G ambient magnetic induction was found to be produced by the temperature controllerheater at the location of the sample. This magnetic field will produce a frustration f ≈ 43, which is enoughto be in the non dilute limit and to make Equation 4.5 not applicable in this case.ii In the data shown in Figure 4.5, the flat range of dV/dI has not yet been reached.iii In here the error in the normal resistance is the error obtained from the linear fit
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nearly flat steps located at voltages that are N times the voltage of an individual JJ. The
width of these steps is proportional to the amplitude of the rf signal.
Figure 4.6: Current vs. normalized-voltage graph for a 100% 300×200 JJA at
T=1.8K and zero magnetic field. The black curve is for no rf signal. The red curve
shows Shapiro steps when rf signal is applied. The voltage separation between steps
is proportional to the rf frequency and the step width is proportional to the rf
power.
For fixed external parameters such as the temperature and frustration, these
Shapiro steps can be characterized as a function of the applied rf signal’s amplitude and
frequency. This characterization consists in measuring the “width” of the step (∆I) as
well as its “spread” (∆V)i for different samples. There are many factors that will
determine the shape and location of Shapiro steps. For example, for “perfect” JJA
(known in here as 100% arrays i.e. samples where none of their crosses have been
removed), the voltage location of the step and its width depends on the size of the
samples (N×M).
0.0 0.5 1.0 1.5 2.0 2.5 3.00
1
2
3
4
n = 3
n = 2
n = 1
I rf =1.4mA
V o l t a g e / V
1
Current (mA)
∆ I
V n= n (N Φ
0ν
rf )= n (31µ V)
T=1.8K ν
rf = 50 MHz
f = 0
Sample: 300 X 200 W6 Zero rf
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Due to the arrays’ fabrication process, it is impossible to create identical JJ in any
100% array. All arrays are subject to a certain degree of difference among their R 0 and ic.
These types of defects, known here as “bond disorder”, diminish the ability of arrays to
keep a coherent response to the external rf signal. The disruption of the phase coherence
can be seen in the VI’s as steps with rounded corners and considerable spread as the rf
signal’s amplitude and frequency increases (as can be seen for the n=3 step in Figure 4.3,
where the spread is larger than for the case of n=1 or 2).
Other types of defects, known here as “site disorder”, occur when junctions are
missing, or have been intentionally removed, from the array. This type of defect will
cause a greater disruption to the phase coherence than bond disorder defects. In section
4.1.7, results showing the effects of site disorder will be presented for samples where
10% and 20% of their crosses were intentionally removed from the array (known as 90%
and 80% samples respectively).
There are other factors that affect the ability of an array to respond coherently to
an external rf signal, these include:
• Performing measurements in arrays near the island’s Tc.
• Performing measurements in arrays for magnetic fields that are strong
enough to drive the material forming the superconducting electrodes or
“banks” into their mixed state.
i The spread of the step refers to how “horizontal” the step is (∆V = 0), or if it has a slope (∆V ≠ 0). Ideallywe want zero spread at a step location so, at the voltage of the step the dynamic resistance is zero.
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• Having an array with a non-sinusoidal current-phase relationship (CPR),
which occur in other weak link structures, such as Dayem bridges, or in
SNS JJA with gaps smaller than the normal-metal coherence length ξ n
where a large critical current can exist.
The factors listed here, affect the superconductivity in the banks that make the JJ.
Under these conditions, each bank can no longer be considered as having a single phase
and there will be a considerable amount of quasiparticles present at the banks. Also, the
possibility exists for Meissner vortices to be present in the banks.
When JJA are operated under these circumstances, they are known to be working
in the “nonequilibrium” regime [73].
Care was taken here to work under “zero” external magnetic field conditions for
which each cross could be modeled as having a “strong” order parameter connected by a
weak link i.e., a purely sinusoidal CPR. Under these conditions, only bond and site
disorder are the main causes for disruption to the coherent response of the arrays to
external rf signals.
To characterize the effect of disorder the width and spread of the steps were
measured directly from the VI’s using a graphical method. This rudimentary method
provided a qualitative way to do this characterization. Another more sophisticated
method that can be used for this purpose involves calculating the dynamic resistance
from the VI’s and plotting it vs. the array’s voltage or current. For example, a plot of
dV/dI vs. V will show a series of minima where each minimum will be located at the
voltage for which the Shapiro steps should occur (as given by Equation 2.31).
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Figure 4.7: dV/dI vs. V curve shows a series of minima located at the steps’ voltage.
The plot shows dV/dI≠0 at the step location meaning that the step has non zero
spread.
This can be seen in Figure 4.7, where the VI curve with rf signal from Figure 4.6
is used to illustrate the method. As can be seen from Figure 4.7, the minima are located
at the step position. Also, the minimum values for the dynamic resistance at the step are
not zero, indicating the spread of the step is not zero (having a non zero dV/dI at a step is
another way to visualized that this array has bond disorder)
The spread could then be further estimated by taking another derivative of the
dynamic resistance and finding the inflection points in dV/dI. This is illustrated in Figure
4.8 for step n=1.
-20µ 0 20µ 40µ 60µ 80µ 100µ 120µ
0.0
500.0µ
1.0m
1.5m
2.0m
2.5m
3.0mT=1.8K ν
rf = 50 MHz
I rf =1.4mA
f = 0
ISample: 300 X 200 W6
C u r r e n t ( A )
Voltage (V)
0
10m
20m
30m
40m
50m
60m
70m
dV/dI
d V / d I ( Ω )
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Figure 4.8: Detail from Figure 4.7 illustrating how to find the spread of the first step
by using the dynamic resistance of the VI, to indicate the center of the step, and the
derivative of dV/dI as a function voltage, to find its value.
Similar procedures, as shown in Figure 4.8, can be followed to find the width of
the step. The next sections, will show results for this 100% sample, which was obtained
using the direct graphical method to characterize the samples in the presence of rf signals.
4.1.5 Dependence of VI’s with rf amplitude
As the amplitude of the applied rf signal is changed, for fixed rf frequency, the
width of the steps oscillates as a Bessel function. Figure 4.9 shows these oscillations for
steps n=0 and n=1.
20µ 25µ 30µ 35µ 40µ
300.0µ
600.0µ
900.0µ
1.2m
n=1
Sample: 300 X 200 W6
C u r r e n t ( A )
Voltage (V)
I
0
30m
60m
dV/dI
Inflection points for dV/dI
T=1.8K ν
rf = 50 MHz
I rf =1.4mA
f = 0
Inflection points for dV/dI
d V / d I ( Ω )
d[dV/dI]/dV
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Figure 4.9: Widths for steps n=0 and n=1 as a function of applied rf amplitude.
Graph shows Bessel function like oscillations for the step widths. Notice how the
oscillations are approximately 180° out of phase. The n=0 and n=1 step’s first
maxima ([∆ I0]MAX and [∆ I1]MAX) for small Irf , are later used to further characterize
the JJA irradiated by rf signal.
As mentioned earlier, Figure 4.9 is obtained from a series of VI curves, like the
one shown in Figure 4.6, where the width is measured directly from the plot. From
Figure 4.9, it can be seen that the maximum width for step n=1 ([ ∆ I1]MAX) is almost as
high as the maximum for n=0.
[∆ I0]MAX, for the case of zero applied rf signal to the array, will represent the
critical current of the array. But, in this case [∆ I0]MAX is smaller because it was not
obtained as the maximum of a dV/dI vs. I curve as defined earlier from Figure 4.5. As
the rf amplitude increases the steps widths diminishes to a point where the graphical
method can no longer be used to characterize the steps from the VI’s.
Curves like the one shown in Figure 4.9 are repeated for different rf frequencies
to also characterize the steps as a function of frequency. The characterization continues
0 100 200 300 4000
100
200
300
400
500
[ ∆ I 0 ]
MAX
[ ∆ I 1 ]
MAX
Sample: 300 X 200 W6
T=1.30K + ∆Tν
rf = 50 MHz
f = 0
n=1
n=0
∆ Ι
n ( µ
A )
I rf
(Arb.)
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until the steps are no longer observed in the VI’s (in the low frequency range), or the
spread is two large (in the high frequency range).
In the next section, the characterization of the steps as a function of rf frequency
is presented. The maximum width value for each step is used as an indication of the
ability of the array to maintain a coherent response (i.e., wide, horizontal steps) to the
external rf signal.
4.1.6 Dependence of VI’s with rf frequency
The dependence of the maximum of the step width ([ ∆ I1]MAX) for steps n=1 is
shown in Figure 4.10.
Figure 4.10: Maximum step width for steps n=1 as a function of reduced frequency
for a 100% array (Figure 1 in reference [12]. Data points are obtained from similarcurves as shown in Figure 4.9. In here T=1.91 K and f
≈ 0.
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This Figure shows how [∆ I1]MAX increases as a function of reduced frequency
(c
rf
ν
ν =Ω ). For frequencies higher than Ω ≥ 1.8, [∆ I1]MAX reaches a “plateau” which is
maintained up to Ω =4.9 and possibly for higher Ω . This indicates the ability of the
arrays to oscillate coherently with the external rf signal as the frequency increases. The
saturation value of the steps at high frequency is understood [12] in terms of the single JJ
behavior, where the width of the steps oscillates as a Bessel function. But in contrast to
what is found in single JJ where the saturation is reach for Ω =1, here it is obtained at
almost twice this value. This is consistent with what has been found by others [74] and by
simulations using the RSJ model [75,]. Higher order steps (i. e. n=2, 3, etc) were not
characterized here because they presented considerable spread (R ≠0), as can be seen
slightly in step n=2 and more pronouncedly in step n=3 in Figure 4.7.
Continued characterization of the steps for Ω > 4.9 were limited by the frequency
range and power output capabilities of the rf signal generator used (HP8656B).
In the next section, the coherence response to rf signals for arrays with site
disorder are presented.
4.1.7 Effects of site disorder
The effects of site disorder on the width and spread of Shapiro steps are shown in
Figure 4.11. As mentioned in the previous section, removing crosses from the array
cause a much greater disturbance on the coherence response of arrays to an external rf
signal, compared with the response of arrays with only bond disorder.
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In Figure 4.11, the first Shapiro step is shown for a 300×300 JJA in which 10% of
the crosses have been removed at random from the sample. An array such as this is
known as a 90% sample.
These arrays with site disorder were fabricated by Dr. Donald C. Harris, former
student of Dr. J. C. Garland, from OSU. Details on the fabrication of these arrays can be
found in Dr. Harris’ doctoral dissertation [76]. These site disorder arrays had a slightly
smaller gap (0.45 µm compared to 0.50 µm for the 100% samples shown earlier).
Measurements on these arrays were performed at temperatures T=3.34K and near zero
magnetic field.
Figure 4.11: Normalized voltage vs. current for a 90% sample (300×300). The first
Shapiro step presents considerable reduction in its width and increase in its spread
when compared to 100% arrays.
In order to make comparisons between measurements on samples with site
disorder and 100% samples, careful considerations need to be taken in estimating the
current for the disordered arrays (for details see reference [12]).
0.0
0.5
1.0
1.5
2.0
0.0 0.2 0.4 0.6 0.8
IV with dc + rf bias, 90% Sample
n=1
∆I1 / I
c
∆V/ V1
I / Ic
V
/ ( N Φ 0
ν r f
)
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Following the same procedure described previously, the width and spread of the
steps were characterized for these site disorder samples. This characterization procedure
continued until enough points were taken to plot a graph similar to Figure 4.10. This is
shown in Figure 4.12.
Figure 4.12: Normalized maximum Shapiro step width for the first step vs.
normalized frequencies, for two samples with site disorder (90% and 80% samples)(from reference [12], inset in Figure 1). The lines are a guide the eye.
Figure 4.12 shows the width of the first Shapiro step maximums as a function of
normalized frequencies for two 300×300 samples for which 10% and 20% of the crosses
were removed at random from the arrays (these samples are known as 90% and 80%
samples respectively). It is clear from the figure that the values for the maximum width
of the steps are smaller than for the 100% sample, indicating that site disorder disrupts to
a larger extent the coherent state in the arrays, and that the greater the amount of site
disorder, the greater the disruption. Also, no saturation was observed for the maximum
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74
Chapter 5 : MEASUREMENTS IN NB FILMSWITH MAGNETIC DOT ARRAY
In this chapter, transport measurements performed in Nb film samples with
magnetic dot arrays will be presented. Specifically, the data presented here are obtained
from a Nb film deposited on top of a square array of nickel (Ni) dots. These results will
be compared to those for the SNS JJA in chapter 4. In section 5.1, measurements of the
resistive transition, magneto-resistance curves, and VI’s with and without rf signals for
this sample are presented. In section 5.2, a discussion of the model proposed in section
5.1 is given with more detailed explanations and interpretation of the data. A summary
of these results is given in section 5.3.
5.1 Measurements
All the data collected on the sample mentioned above (known here as the Nb/Ni
square sample) were taken using the cryostat described in section 3.2. Other data was
also collected on an Nb/Fe square sample and on an Nb/Ni hexagonal sample using the
first probe mentioned in section 4.1
5.1.1 R vs. T
A typical resistive transition curve for the Nb/Ni square sample is shown in
Figure 5.1.
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Figure 5.1: Resistive transition for the Nb/Ni square sample. Measurements were
taken with a bias current of 10µA in ambient magnetic field. The resistance error
bars are shown in red and the line connecting the dots is a guide to the eye.
The resistance of the sample plotted in Figure 5.1 was calculated in the same way
as in Chapter 4, using Equation 4.1. For this curve, the voltage across the sample was
also measured using the delta method, since the current used to bias the sample was
10µA<Ic.
Figure 5.1 shows an initial resistance drop to zero resistance starting at
TcNb=7.926K and finishing at ST=7.860K, only 66mK wide. Under close inspection of
Figure 5.1, it was found that the resistance of the sample increased from a local minimum
(at ST=7.860K), reached a maximum (at a temperature known in here as TRMAX) then
decreased again until it became zero (within the experimental errors) at a lower
temperature Tc0=7.766K, as can be seeing in Figure 5.2.
7.70 7.75 7.80 7.85 7.90 7.95 8.00 8.05 8.10
0
1
2
3
4
5
6RT-10uA-f0-Is0-BoRF-121dB-A.dat
TcNb
= 7.926K
ST=7.860K
Nb/Ni Square SampleTemperature controlled better than 0.5mK [8.10-8.50] K, ∆T =0.01K 9/26/01Ambient B-fieldI bias
= 10 µA
Tc0
= 7.766K
R ( Ω )
Sample Temperature (K)
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At temperatures below where the local minimum occurs, the resistance of the
sample increases. An “alternative” explanation (other than flux flow resistivity) for this
increase in resistance could be attributed to a process similar to the Kondo[80]
effect, in
which a source of electrical resistance is produced by the scattering of normal electrons
with scattering centers that have a magnetic momentii. This process is thought to give a
contribution to the resistivity that increases with decreasing temperature. Similar
behavior in RT’s (to Figure 5.2) has been observed in the dilute iron alloy Fex NbSe2,
where for zero applied magnetic field the alloy can become superconducting [81].
As the temperature continues to decrease the resistance stops increasing when the
portion of the Nb film at the magnetic dot interstitial area becomes superconducting (this
is represented by the darker blue diamond shaped area in Figure 5.3). At this point, a
process similar to the proximity effect in JJA takes place, and the resistance starts to drop
again when Cooper pairs couple across the “normal” regions between the dots.
If this model is correct, these types of samples can be visualized as a JJA for
temperatures to the left of the resistive maximum in Figure 5.2. In the following sections,
other transport measurements performed on these samples will be presented which show
the similarities between the Nb film samples and the JJA studied in chapter 4.
i The regions of the samples with only Nb have a lower temperature transition than expected for pure Nb possibly due to the processing of the samplesii Kondo discovered, from perturbation theory, that the magnetic scattering cross section is divergent,giving an infinite resistivity
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5.1.2 R vs. B
For the curves shown below, the temperature of the samples is controlled at
values which are below TRMAX. At this temperature, a small current is chosen to bias thesample and the voltage across the sample is measured using the delta method. The
resistance across the sample is then calculated using Equation 4.1.
A typical magneto-resistance curve on these samples is presented in Figure 5.4.
A series of minima are observed that correspond to locations of magnetic field induction
that correspond to integer values of the frustration in the array.
Figure 5.4: Resistance vs. magnetic field perpendicular to the sample. The graph
shows data for two slightly different temperatures below TRMAX. The current bias
to the sample (627µA) is the same in both cases.
For this particular array, the lattice constant is: nma )10400( ±= [82]. Thus, from
Equation 4.2, G B )7129(0
±=Φ . However, the experimental magnetic induction value
found from Figure 5.4 is G Ba 116= [83]. Using a calibrated Hall probe the ambient
0 50 100 150 200 250 300
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2
I bias
= 627µA
ST=(7.8065 +/- 0.0005) K ST=(7.8089 +/- 0.0003) K
Nb/Ni Square Sample
f = 2
Ba=116 Gauss
f = 1
f = 1/2
f = 0
RB820KNoRF-627uA-A.txt 12/21/01 RB820KNoRF-627uA-A.txt 12/14/01
R ( Ω )
B (G)
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magnetic induction on the sample for zero current through the superconducting magnet
was ~ 9G. This extra 9G at the sample’s location was found to be in the same direction
as the magnetic induction provided by the superconducting magnet, so the difference
between a B and0Φ B cannot be attributed to the ambient magnetic field found at the
sample’s location. Further characterization of this sample in the presence of a magnetic
field should be performed in order to find the cause for this discrepancy. A possible
explanation is that the array’s lattice constant is a little larger than 400nm [83]. Also, there
exists the possibility that the plane of the sample was not completely perpendicular to the
applied magnetic field, but this possibility should give larger values of a B not smaller.
The source of the ambient magnetic induction was determined to be the temperature
control heater. Using Equation 4.3, the frustration created by 9G is:
07.043
3
129
9≈== f .
This level of ambient magnetic field in the sample was not considered significant
in experiments on Nb film samples, but they are an important unaccounted factor in the
measurements on JJAi.
The magneto-resistance curves taken near 7.8K were found to be very sensitive to
small changes in temperature. In Figure 5.4, the small temperature difference (<3mK)
between the two curves, resulted in a difference in resistance on the order of 200mΩ near
the first minimum. The two curves shown in Figure 5.4 were acquired on two different
days and no special procedures were taken to control the magnetic state of the dots
i See section 4.2 for implications of having 9G of ambient magnetic induction in the measurements of JJA.
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Figure 5.7: Dynamic resistance curve calculated from Figure 5.5. The error bars for
dV/dI are shown in red (calculated in the same way as in Figure 4.5)
The large dynamic resistance fluctuation found around 800µA in Figure 5.7 is due
to a sudden variation in the slope of the VI curve. These sudden variations should be
regarded as an artifact of the measurements.
In Figure 5.7 two current values are shown: the value at the maximum of the
dV/dI curve (IMAX=800µA), and a current (called Imin=413µA), which is the lowest
resistance before the dynamic resistance begins to increase. Using the values found for
IMAX, R N, and knowing that in the sample there are 125×125 magnetic dots (i.e. there
could be a maximum of 124 “junctions” in the sample), the Josephson frequency is
calculated to be: νc~36GHz.
In the next section this value for νc is used as a starting point to look for evidence
of Shapiro steps occurrence in the presence of rf signals.
0.0 200.0µ 400.0µ 600.0µ 800.0µ 1.0m
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
IMAX
= 800 µA
Imin
= 413 µA
Nb/Ni Square Sample
IV-120dBm8200K-IMag0mA-Frq2000MHz-Att121dB-OFF-A.dat12/14/2001 11:15 PMST = (7.8095 +/- 0.0003) K Ambient B-field = (9.3 +/- 0.2) G
d V / d I ( Ω )
Current (A)
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Figure 5.8: VI curve for the Nb/Ni square sample at T~7.8K and ambient magnetic
field. The black curve is the VI with rf signal. The red curve is the temperature ofthe sample. No Shapiro steps are apparent for rf =2GHz and Irf =123µA.
Figure 5.8 shows a VI with a 2GHz, 123µA rf signal applied to the sample. At
first glance, no evidence for Shapiro steps is apparent. However, when the dynamic
resistance of the VI curve is taken, a series of minima appear in the dV/dI vs. V curve.
This is shown in Figure 5.9.
0.0 200.0µ 400.0µ 600.0µ 800.0µ 1.0m
-500.0µ
0.0
500.0µ
1.0m
1.5m
2.0m
2.5m
3.0m
3.5m
4.0m
4.5m
5.0m
5.5mI
AC= (123.0 +/- 0.3) µA
ST = (7.8095 +/- 0.0003) K 4KT = (4.01 +/- 0.00) K B = (9.3 +/- 0.1) GI
c [5µV] = 1.43 µA
V
V o l t a g e ( V )
Current (A)
7.800
7.802
7.804
7.806
7.808
7.810
7.812
7.814
7.816
7.818
7.820
12/14/2001 12:13 PMIV10dBm8200K-IMag0mA-Frq2000MHz-Att30dB-ON-B.txt
ST
S a m pl e T e m p e r a t ur e ( K )
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Figure 5.9: dV/dI vs. V curve from VI with rf curve in Figure 5.8. The dynamic
resistance is shown in blue and the IV, in the black curve, is plotted for reference.
The voltage values for the minima found in Figure 5.9 were plotted versus the
number of the minima, starting with the first minima being n=1, this plot shown in Figure
5.10, shows that the values for the voltage minima follow a linear relationship when
plotted with respect to the number of the minima. From a linear fit to this data assuming
Equation 2.31 for an array is applicable, the number of junctions in the direction of the
current can be calculated. The value found from the linear fit is: 384 ±=∆± N N , which
is different from the value expected for this sample (N=124). The reason for this
difference is discussed below.
-500.0µ 0.0 500.0µ1.0m 1.5m 2.0m 2.5m 3.0m 3.5m 4.0m 4.5m 5.0m
3.0
3.5
4.0
4.5
5.0
5.5
6.0
d V / d I ( Ω )
Voltage (V)
V'
-100.0µ
0.0
100.0µ
200.0µ
300.0µ
400.0µ
500.0µ
600.0µ
700.0µ
800.0µ
900.0µ
1.0m
1.1m
n=6
n=5
n=4
n=3
n=2
V6 = 2090.4 µV [47]; N ≅ 84
V5 = 1709.1 µV [40]; N ≅ 83
V4 = 1451.8 µV [35]; N ≅ 88
V3 = 1208.0 µV [30]; N ≅ 97
V2 = 755.8 µV [20]; N ≅ 91
I AC= (123.0 +/- 0.3) µ AST = (7.8095 +/- 0.0003) K
4KT = (4.01 +/- 0.00) K
B = (9.3 +/- 0.1) G
Ic [5µV] = 1.43 µ A
12/14/2001 12:13 PM
IV10dBm8200K-IMag0mA-Frq2000MHz-Att30dB-ON-B.txt
V1 = 346.4 µV [10]
ν = 2 GHz
2e/h = 4.836 x 1014
N = V1/ν * 2e/h ≅ 84
C ur r en t ( A )
I
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Figure 5.10: Plot of the voltage location for the minima found in the dynamic
resistance curve in Figure 5.9. These voltages are plotted vs. the number of the
minima staring with the first minimum at n=1, a linear fit of this data in shown. For
the calculations of the linear fit, the (0,0) point is included.
In the next sections, more measurements in the presence of rf signals are
presented.
5.1.5 Dependence of VI’s with rf amplitude
Other VI’s were taken for different rf amplitudes, but for the same conditions of
temperature, ambient magnetic field, and frequency, used in taking the VI shown in
Figure 5.8. From these VI’s, the critical current (defined by the threshold method) was
0 1 2 3 4 5 6
-200.0µ
0.0
200.0µ
400.0µ
600.0µ
800.0µ
1.0m
1.2m
1.4m
1.6m
1.8m
2.0m
2.2m Vn = [N(h/2e)ν] * n
ν = 2GHz
N +/- ∆ N = 84 +/- 3
Nb/Ni Square SampleFrom: Data of 12/14/2001 12:13 PMIV10dBm8200K-IMag0mA-Frq2000MHz-Att30dB-ON-B.txt
Linear Fit: Vn = A + B * n
Parameter Value Error ----------------------------------A 4.17214E-5 4.88858E-5B 3.46164E-4 1.35585E-5
Vn
Fit
V n
n
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plotted as a function of Irf this is shown in Figure 5.11. In this figure, the critical current
first increases at small Irf , reaches a maximum at Irf ~60µA, and then monotonically falls
to zero at larger Irf . Also in Figure 5.11 is a plot of the sample temperature as a function
of Irf , which indicates an increase in sample temperature as Irf increases. It is quite
apparent that the critical current does not show the Bessel-function like dependence that
is observed in JJA under applied rf currents, see Figure 4.9.
Figure 5.11: Plot of the critical current (defined by the threshold method) vs. I rf .
The critical current is the black curve. A plot (in red) of the sample temperature is
shown to indicate the increase in temperature produced by the applied Irf .
40µ 60µ 80µ 100µ 120µ 140µ 160µ 180µ 200µ
-20µ
0
20µ
40µ
60µ
80µ
100µ
120µ
140µ
160µ
180µ
I c [ w i l l b e t a k e n
a s ∆ I 0 h e r e ] ( A )
Irf (A)
Ic
7.8086
7.8088
7.8090
7.8092
7.8094
7.8096
7.8098
7.8100
7.8102
7.8104
7.8106
39 dB
26 dB
30 dB Attenuation
Nb/Ni Square Sample, 12/14/01V-I's as a function of RF power, Attenuation = [26-39] dBν = 2GHZ, CT = 8.200K, f = 0, Ambient B field ≅ 9 Gauss
Critical current determinedusing the 5µV criteria
S a m pl e T e m
p e r a ur e ( K )
ST
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In Figure 5.11, the Irf dependence of other Shapiro steps are not shown since they
were not observed from the VI’s. Only the zero-steps width was unmistakably linked to
the applied rf signal’s amplitude. The point marked in Figure 5.11 with the label “30 dB
Attenuation”, corresponds to the critical current found from the VI curve shown in Figure
5.8.
Strictly speaking, the critical current determined using the threshold method
plotted in Figure 5.11 is not the same as ∆I0 determined using the graphical method in
Figure 4.9. And also, it is clearly not the same as the maximum in the dV/dI vs. I curve,
The critical current (defined by the threshold method) was used for Figure 5.11 because
many of the VI curves with rf signal where almost straight lines and also because this is
the usual value reported for Ic in the literature (although for the data presented in this
dissertation, the voltage threshold value used is at least one order of magnitude higher
[84]).
To illustrate the difficulties encountered when trying to observe Shapiro steps the
dynamic resistance calculated from VI curves with different Irf , are plotted on the same
graph in Figure 5.12. Here the green curve is the one plotted in Figure 5.8, while the
other curves are for two different Irf above and below to the curve in Figure 5.8.
For all of these VI’s, the critical current is near “zero” as shown in Figure 5.11.
These VI’s were particularly chosen because of what is known to happen in JJA; the
maximum step widths for the first Shapiro step occurs near the zero values of ∆I0.
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Figure 5.12: Plot of the dynamic resistance vs. voltage for five different I rf . The
green curve is the one plotted in Figure 5.8. The curves have been shifted a fixed
amount in the y-axis for clarity. The line connecting the dots is a guide for the eyes.
In Figure 5.12, plots with more curvature represent the VI’s with more of a
superconducting current component in them. As the Irf increases the level of Ic in the
sample diminishes and the curve flattens (ohmic behavior). The minima in the dV/dI vs.
V curves shown in Figure 5.12 do not seem to appear at the same voltage as would be
expected in a JJA for fixed rf frequency. The only minima that seems to be a constant in
all five curves is the “n=1” from Figure 5.8. Data for this step is shown in Table 5.1.
0.0 1.0m 2.0m 3.0m 4.0m 5.0m 6.0m0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
32dB
31dB
30dB
29dB
28dB
Nb/Ni Square SampleData of 12/14/01ν
rf
=2.000GHz
d V / d I ( Ω )
Voltage (V)
Irf =(149.5 +/- 0.1)µA, 28dB Attenuation
Irf =(132.8 +/- 0.2)µA, 29dB Attenuation
Irf =(123.0 +/- 0.3)µA, 30dB Attenuation
Irf =(109.4 +/- 0.5)µA, 31dB Attenuation
Irf =( 97.7 +/- 0.5)µA, 32dB Attenuation
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“glitch” in the data, which often occurred when the data acquisition program was delayed
by the computer.
Figure 5.14: Plot of the dynamic resistance vs. voltage for four different I rf . The
curves have been shifted a fixed amount along the y-axis for clarity. The line
connecting the dots is a guide for the eyes.
Table 5.2 shows the corresponding data set for Figure 5.14 as Table 5.1 for Figure
5.12. From this data, the increase in N as Irf increases is observed again. In JJA, the only
way V1 would increase for fixed rf frequencies is by having more junctions in the current
direction participating in the formation of the Shapiro step.
0.0 1.0m 2.0m 3.0m 4.0m 5.0m 6.0m
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
52dB
51dB
50dB
49dB
Nb/Ni Square SampleData of 12/14/01ν
rf =2.000GHz
d V / d I ( Ω )
Voltage (V)
Irf =(132.7 +/- 0.2)µA, 49dB Attenuation
Irf =(122.8 +/- 0.3)µA, 50dB Attenuation
Irf =(109.2 +/- 0.5)µA, 51dB Attenuation
Irf =( 97.6 +/- 0.6)µA, 52dB Attenuation
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Figure 5.15: Dynamic resistance vs. voltage plot. Here the dV/dI was calculated
from five different VI’s all taken with an attenuation of 51dB. The first curve on
top (51D) is the same curve plotted in Figure 5.13 for 51dB attenuation. The curve
at the bottom (51H) was measured by allowing the current to decrease after
reaching the maximum, then increase back again from zero and so on until
completing 3 round trips. In this plot the x-axis was limited to show data only up to
1.5 mV. The curves were shifted a fixed amount in the y-axis for clarity.
5.1.6 Dependence of VI’s with rf frequency
Using the technique described in the previous section, a series of VI curves were
taken for four different rf frequencies. The results are shown in Figure 5.16. In this
figure a series of minima were observed at different voltage positions as a function of rf
0.0 500.0µ 1.0m 1.5m
2
4
6
8
51H
51G
51F
51E
51D Nb/Ni Square Sample
Data of 12/14/01ν
rf =2.000GHz
Irf =(109.2 +/- 0.5)µA, 51D dB Attenuation
Irf =(109.4 +/- 0.2)µA, 51E dB Attenuation
Irf =(109.5 +/- 0.1)µA, 51F dB Attenuation
Irf =(109.5 +/- 0.1)µA, 51G dB Attenuation
Irf =(109.5 +/- 0.1)µA, 51H dB Attenuation
d V / d I ( Ω )
Voltage (V)
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frequency. For the curve at 3GHz a minimum can be observed if the y-axis scale is
enlarged.
Figure 5.16: dV/dI vs. V curves as a function of rf frequency. The curves were
shifted a fixed amount in the y-axis for clarity.
The values found in Figure 5.16 marked with the label “V1” are plotted as a
function of rf frequency in Figure 5.17.
0.0 200.0µ 400.0µ 600.0µ 800.0µ 1.0m 1.2m 1.4m 1.6m 1.8m 2.0m
3.0
4.0
5.0
6.0
7.0
8.0
V1= 965.4µV [27]
V1= 758.4µV [38]
V1= 562.5µV [25]
V1= 363.4µV [20]
5GHz 16dB [A]
4GHz 51dB [A]
3GHz 46dB [A]
2GHz 51dB [A]
Nb/Ni Square SampleData of 11/02/01
d V / d I ( Ω )
Voltage (V)
5GHz, Irf =(110.3 +/- 0.1)µA, ST = (7.8058 +/- 0.0003)K
4GHz, Irf =( 76.0 +/- 0.2)µA, ST = (7.8064 +/- 0.0003)K
3GHz, Irf =( 86 +/- 1 )µA, ST = (7.8063 +/- 0.0003)K
2GHz, Irf =( 22.6 +/- 0.4)µA, ST = (7.8061 +/- 0.0003)K
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Figure 5.17: Plot of the voltage points label “V1” from Figure 5.16 as a function of rf
frequency. The (0,0) point has been included in the calculation.
The data follows a linear relationship. Using Equation 2.31, the number of
“junctions” found in the direction of the current is: 193 ±=∆± N N . This value is
different from previous values.
It should be mentioned that if slightly different values of rf amplitude would have
been used in taking the VI’s shown in Figure 5.16, it is likely that different values for V1
would have been found and the linear fit would have given a different slope, and hence a
different N than reported here. An explanation for this rf current dependence on N is
given below.
0.0 1.0G 2.0G 3.0G 4.0G 5.0G
-100µ
0
100µ
200µ
300µ
400µ
500µ
600µ
700µ
800µ
900µ
1m
1m
V1 = [N(h/2e)] * ν
N +/- ∆N = 93 +/- 1
Linear Fit: V1 = A + B * ν
Parameter Value Error
-----------------------------------
A -1.02973E-5 1.00958E-5
B 1.92892E-13 3.07206E-15
V1
Fit
From VI's taken on 11/02/01Nb/Ni Square Sample
V 1
( V )
RF Frequency (Hz)
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99
Equation 5.1τ ω
t
et Bt B−
= )sin()0()(
This curve is very smooth and different from the green and blue curve.
Figure 5.18: Transverse voltage vs. current in the Nb/Fe square sample.
The curves shown in Figure 5.18 are non-linear. The non-linearity in these VT vs.
I curves is yet to be understoodi. Various models have emerged to explain similar non-
linear curves observed in superconductors. One explanation might be given by
i See comments in chapter 6 regarding possible explanation for this curves and future work on this samples
0.0 200.0µ 400.0µ 600.0µ 800.0µ 1.0m
-20.0µ
-15.0µ
-10.0µ
-5.0µ
0.0
5.0µ
10.0µ
15.0µ
IVTP635A.dat, 100 pts. 2 Avg. before taking a R vs. B IVTP635B.dat, 200 pts. 2 Avg. before taking a R vs. B IVTP635C.dat, 100 pts. 2 Avg. after taking a R vs. B
IVTP635D.dat, 100 pts. 2 Avg. taken next day IVTP635E.dat, 100 pts. 2 Avg. taken next day,
after demagnetizing the samplke with B=B0e
-t/τSin(ωt)
08/26/97T=6.350K Controlled@ "Zero" B field
Sample: Square Nb/Fe Dot
T r a n s v e r s e V o l t a g e ( V )
Current (A)
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considering the motion of vortices in the flux flow regime [88]. Another explanation
might be given in the context of the Kondo effect [89, 90]. Other possibility is in the
context of nonequilibrium superconductivity processes in mesoscopic SNS Josephson
junctions [91], where the dynamics of the system are understood in terms of Andreev
reflections [92].
A possible argument that might explain why the curves shown in Figure 5.18 go
from being noisy to smooth can be given in terms of magnetic moment fluctuations in the
magnetic dots. To understand the argument, the following assumptions will be made
regarding the magnetic state of the dots: the magnetic dots are single domain and their
magnetic moments lie in the plane of the dots [93], the magnetic arrangement of the dot’s
magnetic moments in the array (for minimum energy) can be either ferromagnetic or anti-
ferromagnetic [94].
Knowing this, the noise (the fluctuations in resistance) observed in the figure
above could be interpreted to be produced by the flipping of individual magnetic
moments “trying” to achieve either minimal energy configuration. Evidence for these
ferromagnetic or anti ferromagnetic alignments has been observed while taking magneto
resistance curves (similar to the one shown in Figure 5.4) on dot arrays with magnetic
moments “randomly ordered”. Figure 5.19 and Figure 5.20 are examples of these curves.
Here a jump or depression in the temperature of the sample has been observed when the
perpendicular magnetic field is an integer number of the magnetic field produced by a
quantum flux. A jump in temperature (or warming of the sample) could be associated
with the magnetic moments of the dots acquiring a ferromagnetic alignment (the dots
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101
minimized their energy and the excess energy warmed up the sample), as seen in Figure
5.19, and a depression in temperature (or cooling of the sample) could be associated with
an anti-ferromagnetic alignment of the dots (in this case the adjacent magnetic moments
anti-align by acquiring the energy from the surroundings producing a cooling in the
sample), as shown in Figure 5.20.
Figure 5.19: Magneto resistance curve for the Nb/Ni square sample. The curve in
black is the magneto resistance of the sample (the error bars are shown in blue).
The curve in red is the temperature of the sample (the error bars are shown in dark
yellow). The error bars in this curve are the standard deviation of N averages taken
at each data point. The error of the mean value is at least 10 times smaller. An
increase of sample temperature is observed at f =2. This warming is thought to be
associated with a ferromagnetic alignment of the magnetic dots in the sample.
0 50 100 150 200 250 300 350
0.1
1
R
R = V / I ( Ω )
External magnetic field (gauss)
7.848
7.849
7.850
7.851
7.852
7.853
7.854
7.85507/24/01
Data: RB820KE.DAT
Ibias
=650 µ A
Avg. ST=(7.8503 +/- 0.0002) K
Niobium film on top of square array of Ni dots ST
S am pl e t em p er a
t ur e ( K )
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102
Further characterization is needed, with control over the magnetic ordering of
dots. It is important to mention that in Figure 5.19 and Figure 5.20, each individual point
is the average of many voltage readings taken using the delta method, and the number of
averages (N) varied from data point to data point. N depended on the absolute error
V
V e
∆=σ
desired for the run; the average was continued until σe was lower than a
desired error (usually 5%). If this accuracy was not achieved before reaching a fixed
maximum number of averages (usually set < 300) the average was stopped in order to
keep the measurements of individual points from taking more than 5 minutes. Each data
point took in the order of minutes to be acquired and the whole data set was acquired in
several hours (with the new probe, He was kept in the bath for ~ 8 hours).
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103
Figure 5.20: Magneto resistance curve for the Nb/Ni square sample. The curve inblack is the magneto resistance of the sample (the error bars are shown in blue).
The curve in red is the temperature of the sample (the error bars are shown in dark
yellow). The error bars in this curve are defined as ( N /σ , the standard deviation
of N averages divided by the square root of N). A lowering in sample temperature is
observed at f =1. This cooling is thought to be associated with an anti-ferromagnetic
alignment of the magnetic dots in the sample.
Another explanation for the fluctuations observed in Figure 5.18 could be
attributed to an erratic flux flow motion of vortices (where the vertices can be pinned orde-pinned by the magnetic dots or by other imperfections in the film). However, this
possibility would not explain the absorption or liberation of energy observed in the
previous two figures, unless the dot’s magnetic moments undergo the lattice ordering
0 50 100 150 200 250 300 350
0.1
1
R
R = V / I ( Ω )
External magnetic field (gauss)
7.886
7.887
7.888
07/30/01
Data: RB824KC.DAT
Ibias
=500 µ A
Avg. ST=(7.8872 +/- 0.0001) K
Niobium film on top of square array of Ni dots ST
S am pl e t em p er a
t ur e ( K )
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105
This has been discussed by Tilley [96] to be analogous to what happens in an optical
system of radiators.
It can be seen that for small or no rf amplitude, N will be small (there will be JJ
only in the center of the array where the dots are more ordered). As Irf increases, more
and more junctions are formed and N increases as has been found. The discrepancy
found with the number of junctions expected for these samples (N=124) and the
maximum found to contribute from the experiments (N=118) can be attributed to the
location of the longitudinal voltage pads with respect to the dot array. It can be seen from
Figure 3.2 that the longitudinal voltage pads are not centered with respect to the dot array
and this could lead for the lost of 10±1 junctions.
From these arguments, it can be understood the importance of keeping the dots’
magnetic order stable from measurement to measurement. To achieve this, the
temperature of the sample needed to be controlled to a high level of accuracy. Due to the
self heating effects only a limited parameter space in temperatures, rf signal amplitude,
and frequency were accessible for experimentation.
From the study of JJA the impossibility of seeing the steps directly from the VI’s
can also be understood. The experimental requirements of taking measurements close to
TcNb (in reduced temperatures t=T/TcNb~0.98) made the system operate in the non-
equilibrium regime (this is also evidenced by the enhancement of Ic with Irf ). In this
region more complicated CPR than sinusoidal are expected. Also, the lack of control
over the coupling strength of the junctions and the possibility of encountering a different
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106
magnetic dot configuration between measurements accounts for the weakening of the
steps’ widths.
Finally to have encountered regions where the system behaves accordingly to the
Josephson relations is given as indication of the validity of the model.
5.3 Summary
In this chapter, transport measurements for a Nb film sample deposited on top of a
square array of Ni dots were presented. The dc RT’s, RB’s, and VI’s measurements
performed in this sample show that there is a similarity between this sample and a JJA. A
model was developed to explain these similarities. The model is based on an
interpretation for the features observed in the RT’s of the samples. In order to test the
model, measurements with applied rf currents were performed to see the formation of
Shapiro steps in the VI characteristics. According to the model the sample will behave as
a JJA only in a small window of temperatures. Indeed, VI measurements were in
agreement with Josephson’s second relation but only within a very narrow range of
experimental conditions. Verification of these results was complicated by the interaction
of the magnetic dots moments, but once greater control on the system was achieved the
sample were found to follow the Josephson relations (the sample was magnetized at room
temperature with a 0.26T permanent magnet in the direction of the current and parallel to
the surface of the sample).
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According to what was learned in this dissertation, it is in principle possible that
this short but wide bridge will be capable of maintaining a sinusoidal CPR in the junction
better than a Dayem bridge, where all three dimensions (width, length and thickness)
must be comparable to ξ(T) [13, 19], in order to have a sinusoidal CPR. This new structure
will be “cooler” than a Dayem bridge [17] and it is also possible that the presence of the
magnetic dot will improve the self heating problems of continuous film bridges by
allowing the structure to be thicker than ξ(T).
The width of the bridge needs be determined experimentally (in order to obtain
the maximum power output with the least amount of phase coherence disruption). The
length of the bridge can be made shorter or longer depending upon the temperature of
operation intended for the structure and still maintained a sinusoidal CPR. If the
temperature is close to Tc, the bridge could be made longer and in this way increase the
contribution to the power output from the normal resistance, but the proximity to Tc
might affect the superconductivity of the islands (due to the nonequilibrium processes
that take place near Tc) and the power out or coupled could be smaller (due to phase
disruption effects).
The ideal JJA made out of one of these structures will operate well below Tc and
will have a “plaquette” area as small as possible in order for the structure to be the least
affected by ambient or self induced magnetic fields, which have a negative influence on
the phase coherence in JJA [75].
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113
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Bruynseraede, Phys. Rev. B 60, R6998 (1999)
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[48] Details on the fabrication process of these samples can be found in Dr. Hyun-Cheol
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dissertation; Ohio State University (1989)
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Schuller, J. Appl. Physics 84, 411 (1998)
[51] Details on the fabrication process of these samples can be found in Dr. Hoffman’s
Ph. D. dissertation, University of California, San Diego (UMI#: 9917952)
[52] See Dr. Hoffmann’s dissertation (UMI # 9917952) pg. 29
[53] Hyun-Cheol Lee doctoral dissertation, University of Cincinnati (1991)
[54] Samuel Y Liao, “Microwave Devices and Circuits” 2nd ed., Prentice-Hall, 1985
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[56] V. L. Berezinskii, Zh. Eksp. Teor. Fiz. 59, 907 (1970) [Sov. Phys. JETP 32, 493
(1971)]
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117
[90] J. C. Chen, S. C. Law, L. C. Tung, C. C. Chi, and Weiyam Guan, Phys. Rev. B 60,
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Smith, W. Schwarzacher, F. M. Ross, M. Redjdal, and F. B. Humphrey, Phys. Rev. B 65
144417 (2002)
[94] Dr. Axel Hoffmann, private communication (2002)
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120
The logic behind this program is seen in the following diagram: Three main “for-
loops” are used to control the way current is send to the sample. The inner most for-loop
is where all the data is colleted using the subroutine:
AVG-I-V-VH-CT-ST-RF-Buffer-LV6i.vi
All the data is formatted and the rows of data are saved to disk into a designated
data file created at the beginning of the run.
Block Diagram
The following is a list of the main subroutines used in this diagram:
List of SubVIs
Find First Error.vi
Write To Spreadsheet File.vi
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Write Characters To File.vi
InitializationDCIVT8Z1-LV6i.vi
DCIV-Run-Info.vi
Iformat.vi
Errwait.vi
Beep.vi
Std Deviation and Variance.vi
Avg-I-V-VH-CT-ST-RF-Buffer-LV6i.vi
Wait for Temp stabilization.vi
I-Up-Down-LV6i.vi
Log-I-Up-Down-LV6i.vi
Log-I-LV6i.vi
LSHG10316-Cal-LV6i.vi
3D-1D-LV6.i.vi
derivative.vi
Critical-Current.vi
Shapiro step voltage.vi
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The following diagram shows the dependence of these subroutines with respect to
the main program.
Position in Hierarchy
Some lower level subroutines are not mentioned in this appendix, only the first
row of programs is presented.
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