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Squid characterization

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Title:
Squid characterization
Creator:
Trujillo, Javier J
Place of Publication:
Denver, CO
Publisher:
University of Colorado Denver
Publication Date:
Language:
English
Physical Description:
vii, 58 leaves : illustrations ; 28 cm

Subjects

Subjects / Keywords:
Superconducting quantum interference devices -- Design and construction ( lcsh )
Superconducting quantum interference devices -- Design and construction ( fast )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references (leaf 58).
Thesis:
Electrical engineering
General Note:
Department of Electrical Engineering
Statement of Responsibility:
by Javier J. Trujillo.

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Source Institution:
|University of Colorado Denver
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|Auraria Library
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
57543183 ( OCLC )
ocm57543183
Classification:
LD1190.E54 2004m T78 ( lcc )

Full Text
SQUID CHARACTERIZATION
by
Javier J. Trujillo
B.S., University of Colorado, 2001
M.S., University of Colorado, 2004
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Electrical Engineering
2004


This thesis for the Master of Science
Degree by
Javier J. Trujillo
has been approved
by
Dr. Hamid Fardi
Dr. Joseph Hibey
/t/fftfL- £ Date


Trujillo, Javier Jesus (M.S., Electrical Engineering)
SQUID Characterization
Thesis directed by Associate Professor Hamid Fardi
ABSTRACT
This thesis addresses the topic of design and characterization of a superconducting
quantum interference device (SQUID) using circuit and device layout simulation
tools. It examines two different methods for characterizing a DC SQUID in an effort
to study the effects of the parasitics of a DC SQUID. The methods used in this report
build on prior research conducted by various sources. This thesis is a starting point
on the many possibilities on modeling a DC SQUID without Josephson junctions.
This abstract accurately represents the content of the candidates thesis. I recommend
its publication
Signed
Hamid Fardi
IV


ACKNOWLEDGMENT
I would like to thank Dr. Martin E. Huber for giving me the opportunity to do
research at the SQUID lab, guidance in planning this thesis, and for helping me
develop my skills as a researcher. I would also like to thank Dr. Hamid Fardi for his
support in this project. In addition, I would like to thank the SQUID lab interns
(2003-2004 group) who gave me assistance ranging from setting up equipment to
discussing SQUID subjects. It has been a pleasure working with each and every one
of you and I wish you luck in your future research projects.


CONTENTS
Figures....................................................................vii
Chapter
1. Josephson Junction Basics.......................................1
1.1 Introduction: What is a SQUID?..................................1
1.2 Josephson Junction Modeling Using the Equivalent RSJ Circuit....3
1.3 Physical Layout of the Josephson Junction at the Micron Level...9
1.4 Two Parallel Josephson Junctions Create the S QUID.............12
1.5 Flux Through the Superconducting Ring..........................14
2. Characterizing the Enpuku SQUID with WRspice...................16
2.1 Introduction...................................................16
2.2 Parasitic Elements in the Enpuku S QUID........................16
2.3 The Feedback Effect Caused by the Input Coil...................17
2.4 Coupled SQUID Using the Enpuku Paper...........................18
2.5 Equivalent Circuit Model.......................................23
2.6 Results Using the Equivalent Circuit Model.....................25
3. Characterizing the CDMS SQUID with WRspice.....................31
3.1 Introduction...................................................31
3.2 Characterizing the CDMS SQUID..................................31
3.3 Discrete Fourier Transform.....................................33
3.4 Obtaining a Full Period/Flux Quantum...........................34
3.5 Matrix Notation for Matlab.................................... 37
3.6 WRspice Results................................................38
3.7 Subcircuit Model...............................................39
4. Conclusion.....................................................42
Appendix.....................................................................44
A. WRspice Code...............................................44
B. Matlab Code............................................... 52
C. Gaussian Source............................................55
D. Piecewise Linear Function..................................56
E. CDMS NIST SQUID...........................................57
References............................................................58
vi


FIGURES
Figure 1.1: Geometry of a Josephson Junction................................4
Figure 1.2: Josephson Junction and its Equivalent Circuit Model.............5
Figure 1.3: Current-Voltage Characteristics at a Given Value ...............8
Figure 1.4: Physical Layout of the Resistively Shunted Josephson Junction...9
Figure 1.5: Model for the DC SQUID..........................................12
Figure 2.1: Single-Washer SQUID used in the Enpuku Model....................16
Figure 2.2: Feedback Loop for the Magnetic Flux caused by the Parasitic
Capacitance.................................................................18
Figure 2.3: Coupled S QUID Circuit Model....................................19
Figure 2.4: Enpuku Results Showing Two Full Periods of a Degraded Voltage-Flux
Curve with Feedback Effect..................................................21
Figure 2.5: 8th Order Butterworth Filter....................................22
Figure 2.6: Enpuku Results Showing Two Full Periods of a Voltage-Flux Curve
without Feedback Effect.....................................................23
Figure 2.7: Circuit Model for Studying the Behavior of (t) and (?)......24
Figure 2.8: Enpuku Equivalent Circuit Model in WRspice......................26
Figure 2.9(A): Voltage-Flux Relation of Equivalent Circuit Model without the
Feedback Effect.............................................................27
Figure 2.9(B): Voltage-Flux Relation of Equivalent Circuit Model with the
Feedback Effect.............................................................28
Figure 3.0(A): AC Analysis Plot of the Equivalent Circuit Model without
Feedback....................................................................29
Figure 3.0(B): AC Analysis Plot of the Equivalent Circuit Model with Feedback ...30
Figure 3.1: Simulated Voltage-Flux Curves for a Characterized CDMS SQUID ....32
Figure 3.2: Curve Fits in Kaleidograph......................................35
Figure 3.3: Truncated Voltage-Flux Relations................................36
Figure 3.4: Matlab Results for Testing Fourier Coefficients of a CDMS SQUID....38
Figure 3.5: WRspice Results for Ib = 35 pA to 90 pA in 2.73 pA steps........39
Figure 3.6: SQUID Subcircuit................................................40
Figure A. 1: WRspice Results for the NIST SQUID.............................51
Figure B. 1: Fourier Coefficients Tested in Matlab without DC value.........53
Figure B.2: Tested Fourier Coefficients with DC value.......................54
Figure D. 1: Piecewise Linear Function Used in the Enpuku SQUID.............56
Figure E.l: NIST CDMS SQUID Physical Layout.................................57
vii


1. Josephson Junction Basics
1.1 Introduction-What is a SQUID?
A Superconducting Quantum Interference Device (SQUID) is a sensor that uses
Josephson junctions to measure extremely weak magnetic signals. A Josephson
junction is made up of two superconductors, separated by an insulating layer so thin
that superconducting electron pairs (cooper pairs) can pass through. A SQUID
consist of tiny loops of superconductors employing Josephson junctions in parallel to
achieve superposition: each electron pair moves simultaneously along two opposite
paths. SQUIDs have been used for a variety of applications that demand extreme
sensitivity, including engineering, medical and geological equipment. Because they
measure changes in a magnetic field with such sensitivity, they do not have to come
in contact with the system that they are testing. In this report we introduce a SQUID
that is used as a pre-amplifier for a dark matter search, called a Cryogenic Dark
Matter Search (CDMS) SQUID.
The SQUIDs that are studied in this report are made of niobium and are furnished by
the SQUID lab at the University of Colorado at Denver. A direct current (DC)
SQUID, which is very sensitive consists of two Josephson junctions employed in
parallel so that electrons tunneling through the junctions demonstrate quantum
interference, dependent upon the strength of the magnetic field within the loop. DC
1


SQUIDs demonstrate voltage change in response to even tiny variations in a magnetic
field, which is the capacity that enables detection of such minute changes.
In this report we are interested in exploring the parasitic elements of a coupled DC
SQUID in an effort to better represent and explain what causes non-ideal SQUID
behavior. Our goal is to reproduce results provided by Enpuku et al. [1-2], and build
similar models that will take into account variations in bias current and dynamic
resistance without using Josephson junctions. The problem that arises from using
Josephson junctions is that the device is non-linear and to get results with available
SPICE (Simulation Program for Integrated Circuit Engineering) programs can take
anywhere from hours to days. The SPICE program that we use in this report is
Whiteley Research spice (WRspice) which provides Josephson junctions in their parts
library. This device (in WRspice) is only compatible with time transient analysis and
it is incompatible with other analysis such as AC analysis.
Once the equivalent SQUID model is created we will combine this model in an array
consisting of 100 single SQUIDs connected in series to form an array. In this array
connection there are parasitics that we are interested in estimating so that we can
create a better SQUID, in terms of the SQUIDs operation and sensitivity to parasitic
effects such as feedback and other unknown elements. The parasitics are in terms of
capacitance, inductance, and resistances.
2


We have also studied a NIST SQUID (National Institute of Standards and
Technology SQUID) in theory by using a paper by GUTT et al [3], which
characterizes a NIST DC SQUID using the Discrete Fourier Transform (DFT) in
SPICE. This SPICE model is useful in understanding how the SQUID works and
estimating its parasitic effects. From this procedure we have created a circuit that
accesses a subcircuit through WRspices device library.
Chapter 1 introduces basic SQUID concepts that will be necessary to understand the
analysis in later chapters. It starts with a basic description of a Josephson junction
and leads up to a bare SQUID. Chapter 1 should give the reader who is not familiar
with SQUIDs a basic idea of what goes on in a SQUIDs junctions.
1.2 Josephson Junction Modeling Using the Equivalent RSJ circuit
A Josephson junction is composed of two superconductors separated by an insulating
barrier as shown in Figure 1.1. When the barrier is thin enough cooper pairs can
tunnel from one superconductor to the other. Brian Josephson was the first to predict
the behavior of a Josephson junction and received a Nobel Price in 1973 [4],
3


Josephson junction
Figure 1.1: Geometry of a Josephson junction.
In this section we shall study the Resistively Shunted Junction (RSJ) model (shown in
fig. 1.2) which is the most commonly used equivalent circuit for a superconductor-
isolator-superconductor (SIS) Josephson junction. It has been found that the junction
can be modeled as a differential equation and that this equation is equivalent to that of
a pendulum being rotated in free space [4]. Using the mechanical analog for the
junction we can come up with equations that can help us understand the nonlinear
phenomena that take place in a junction. We will first arrive at some results by
looking at Fig. 1.2(B) and compare these results to the pendulum equation.
4


Vlb
\ ^ lb
X

leap
(A) Josephson Junction
(B) Equivalent Cicuit:
Resistively Shunted Junction (RSJ) Model
Figure 1.2: (A) Josephson junction and (B) its equivalent circuit model.
Inside of the SIS Josephson junction (Fig. 1.2(A)) we have parasitics that are shown
in the equivalent circuit (Fig. 1.2(B)) which is used to model the junction. We can
do a simple nodal analysis to sum up the currents going through the branches. We
know that the total current in the branches will be equal to Ih (bias current).
Therefore,
+10 sin 8
(1.1.2)
5


In equation (1.1.2), V. is the voltage across the junction, 10 is the critical current, and
8 is the phase difference between electron wave functions on either side of the
junction.
The phase difference develops in time according to the voltage-frequency relations
dS_=2q_v
dt h 1
(1.1.3)
h
h = , h represents planks constant, and q is the electron charge magnitude [5].
2k
Substituting (1.1.3) into (1.1.2) for Vj yields
h =
h dS h d28
+ C---+1 sm 8
2qR dt 2q dt'
(1.1.4)
Dividing (1.1.4) by IB
lb _
h dS C h d2S
h 2QRI o dt lo 2<1 dr
+ sin 8
(1.1.5)
We make (1.1.5) dimensionless by introducing a dummy variable 0 = time constant.
i , 2 qRI
The constant will be equal to-----
h
0=t^3^ d0=dt^l^^ dt = _Ji_d0
%
n
2qRi0
(1.1.6)
Next, we plug in the new value for dt into equation (1.1.5) and use the relation for a
h
flux quantum =
2 q
Ib dS 2kR2CI d2S .
=----+-----------^-Fsm 8
10 d0 d02
(1.1.7)
6


In (1.1.7),
ItcRCI t
= jBc and characterizes the capacitance effect of the junctions
and the current-voltage (I-V) characteristics of the SQUID. Finally,
Ib a d28 dS _
= pcT + + sm£
/ d02 d0
(1.1.8)
In understanding equation (1.1.8), it is useful to make the analogy between the current
in a Josephson junction and the torque applied to a pendulum that is free to rotate 360
degrees in a viscous medium. The applied torque (t) on a pendulum can be written:
T = Mr
2 d28
dd
, +k----\-Mgrsin8
dt dt
(1.1.9)
dd .
d28.
where k is the damping torque, Mr is the inertial term, and
dt dt2
Mgr sin 8 is the gravitational restoring torque [6],
Depending on the range of jdc we can get current-voltage (I-V) relations for any
Josephson junction. In theory, if j3c = 0 we get an upper bound on the I-V curve and
when = oo we get a lower bound. For example, in Fig. 1.3(D) an upper and lower
bound on the I-V curve is shown for the CDMS SQUID where the data is measured
from Ib =35 pA to 90 pA The I-V characteristics for the CDMS SQUID are shown
in Fig. 1.3 at different flux values. The vertical axis shows the normalized current
which is applied current Ib divided by critical current I0 and the horizontal axis
shows the normalized voltage. When looking at Fig. 1.3, it should be kept in mind
that the Josephson junction is a symmetric device, so all I-V characteristics have
7


symmetry about the origin. In this report, we will only show the positive portion of
the I-V characteristics. The derivation of these I-V curves will become clearer in
chapter 3 where it is explained under what circumstances the data for the CDMS
SQUID are taken.
(C) (D)
Figure 1.3: I-V characteristics at a given I^ value. (A) I,j, =12.89 pA (B)
Um =5.625 pA (C) I0o =0 (D) The envelope of CDMS SQUID I-V curves from I
=35 pA to 90 pA. Note that /,b is the amount of current required to produce one
flux quantum and is discussed in more detail in section 2.2.
8


From the I-V characteristics a relation for the dynamic resistance can be achieved.
The equation for dynamic resistance is
Slope =
F dV
_1 n =dV
*<, * dl
(1.1.10)
1.3 Physical Layout of Josephson Junction at the Micron Level.
Figure 1.4: Physical layout of the resistively shunted Josephson junction (A)
resistively shunted junction (B) physical layout of junction layers
To better understand the physical realization of the ideal Josephson junction a
physical layout representation is shown in Fig. 1.4. Horizontal dimensions are on the
9


scale of a few microns, vertical dimensions are on the scale of a few hundred
nanometers. Notice that the Capacitance Cs introduced in Fig. 1.2 is built into the
physical layout i.e. it is a parasitic capacitance that is created by the parallel plates.
The capacitance equation for a parallel plate which can be found in any Fields text is
given as
C = y " (1.3.1)
where A is the Area of the junction, s is the dielectric constant of the material that is
being used to represent the junction, and t is the thickness of the junction.
It can be seen in Fig. 1.4 that different materials are symbolized by different shades
and slanted lines and that there are six types of materials used when creating a
junction. The junction material used is aluminum oxide where the oxide is
symbolized by Ox because the oxide has unknown stoichiometry. The insulator is
made of silicon dioxide so the symbol s in equation (1.3.1), the permittivity, will be
calculated according to this insulator. It will be explained in chapter 2 that the type of
insulator that is being used can degrade the performance of the SQUID by amplifying
unwanted resonances. The superconducting wires are made of niobium; niobium is
the most common superconducting material used for SQUIDs. The six layers are
10


deposited on a silicon wafer under specific processes which are beyond the scope of
this work.
The resistor is made of lead-gold alloy (PbAu) with a common sheet resistivity of
(2QJU) where the shunt resistor Rs is fabricated by its length to width ratio, also
called number of squares. The design ofRs is set to make the junction non-hysteretic.
In section 2.2, the area of a Josephson junction is calculated for the Enpuku SQUID
in which a critical current 10 of 15 pA is desired. Therefore for this area the
resistance is calculated accordingly and is given as
P L
d L L
R = p = p-
A. tW tr W
(1.3.2)
where is the sheet resistivity which is constant for film, and is the dimensions
tr W
of each square.
Throughout this discussion we only deal with the fundamental components of the
physical layout. The fundamental components are the non-negligible parasitics in the
junctions such as the overall capacitance, inductance, and resistances.
11


1.4 Two Parallel Josephson Junctions Create the SQUID
A SQUID is a superconducting ring interrupted by two Josephson junctions
connected in parallel. The SQUID can be considered a parallel array of junctions
interconnected by superconducting lines. At the SQUID lab we focus on two junction
SQUEDs that are connected in parallel to create the uncoupled DC SQUID. In
appendix E, we can see a parallel two junction SQUID that is connected in an array of
100; appendix E shows the physical layout of the CDMS SQUID. Fig. 1.5 shows an
uncoupled SQUID that is made up of two resistively shunted Josephson junctions.
Figure 1.5: Model for the DC SQUID made up of two parallel RSJ junctions.
We are interested in the current that is going through the junctions so that we can find
a way to introduce the radio frequency flux noise ) and the internal flux (Oint)
12


that exist in the SQUIDs inductance. The <3?^ noise and Chapter 2 where the SQUID is connected to an input coil. The DC current going
through the superconducting ring in theory is divided equally among the two
branches.
Ib = Il + I2 (1.4.1)
ll = Iex+,I2 = I'ex + (1.4.2)
2 2
Add J1 and 12
11 +12 = lex + l'ex + lb = lb
since Iex = -I'ex = J; where J is the circulating current in the ring
r lb y lb
J = I1-----J = 12---
2 2
J = 11-12-J,2J = 11-12
y = /W2
2
(1.4.3)
(1.4.4)
(1.4.5)
(1.4.6)
J is the circulating current around the ring which creates flux in the SQUID
inductance Ls [5]. This fact will be used in section 2.5 to define the bare SQUID in
the equivalent circuit model which doesnt use any Josephson junctions.
13


1.5 Flux Through the Superconducting Ring
There are four forms of flux in the superconducting ring that are of interest in this
report, flux quantum (), external flux (Oex(), radio frequency flux (Or/ ) noise,
h
and feedback flux (d>/). A flux quantum is calculated to be = 2.07 xlO-15W&
and has become a prominent feature of superconducting science and technology [4].
The Oex( is the external applied quasistatic flux into the ring; it is applied depending
on what is being measured. This flux can be restricted to the range 0 < Ocxt <
without loss of generality since all SQUID responses are periodic in d>exr with period
If we go back to equation (1.4.1), we can say that the fluxes O, and <&2 are
d>. +
proportional to currents II and 12. Defining LI = ^ andL2 = ^ we can
easily show that LI + L2 = L. This is one way to calculate the Oint in the circuit.
The d>int is the flux that is created by the circuit through the Josephson junction
oscillations.
The radio frequency flux noise (3>^) and feedback flux (0 /) describe the resonant
behavior of the coupled SQUID and are generated from the input coil or resonant
circuit. The feedback flux is created when the SQUID is coupled to a multi turn input
14


coil in which a current path is created by the SQUID voltage () that goes through
the input coil inductance (L.) to the feedback capacitance (Cp) and down to ground
(section 2.3). When the current flows through the input coil (L(.) it produces a
magnetic flux in the SQUID inductance (Ls). As a result a feedback loop is formed
for the magnetic flux in the SQUID and is added to the radio frequency flux (^)
noise. In this report when is mentioned we refer to <1total =
The total flux in the presence of the input coil in the superconducting ring is equal to
O ,=0 + +0^=0 + LJ + - 11 5 D
^total ^ ext ^int ~ ^if ^ ext ' ~ ^
where rj < 1 and defines the asymmetries within the SQUID. Equation (1.5.1) will be
used in Chapter 2 to model the Q>total through the coupled SQUID.
A relationship between flux and phase difference is given by Tesche and Clarke [5] as
d\-82=l7^,0,al (1.5.2)
Oo
The phase difference is dependent on the SQUIDs voltage vs. flux (V- ) relation.
The V- d> relation will become more apparent in chapter 2 where aV-O curve will be
attained from the coupled SQUID as shown in Fig. 2.4.
15


2. Characterizing the Enpuku SQUID with WRspice
2.1 Introduction
The discussion in chapter 1 is necessary to understand the concepts that will be dealt
with in chapter 2 where the Enpuku SQUID is introduced. The Enpuku SQUID deals
with all the concepts that have been introduced thus far. It will be shown in section
2.3 that the Enpuku SQUID can be simulated using WRspice and that an equivalent
circuit can be created without Josephson junctions to study the resonance in a
SQUID.
2.2 Parasitics in the Enpuku SQUID
N = 12
Figure 2.1: Single-washer SQUID used in the Enpuku model with a 12-tum input
coil [7]. The cover also shows a single CDMS SQUID that is very similar to this
SQUID except that the junctions are inside the washer.
16


It is believed that the input coil introduces unwanted resonances in a coupled SQUID
[1-2]. The physical layout of a coupled SQUID washer that is used in the Enpuku
model is shown in Fig. 2.1. In his report Robin Cantor gives values for the 12-tum
input coil using a magnification factor of 500 [6], the width wi and spacing s{ of
each turn is equal to 3 pm, the total length of the input coil L = 4680pm, the length of
the slit b = llfjm = N(wt + s;) [7]. These parameters were used to calculate the
input coil and washer fundamental components i.e. inductance, capacitance, and
resistance.
In Fig. 2.1, circles represent Josephson junctions; the size of a Josephson junction
depends on the current density Jc of the material that is being used and on the critical
current of the Josephson junctions at which the device is desired to operate. For
example, to get a critical current 10 = 15pA equation (2.1.1) can be used.
JcA = Ia (2.1.1)
For the Enpuku SQUID, Jc = 500-. Therefore, the area of the junctions using
cm
equation (2.1.1) is given as 3pm2. So, each Josephson junction in figure 2.1 has an
area of 3 pm2, and a critical current 10 =15 pA.
2.3 The Feedback Effect Caused by the Input Coil
It is believed that the resonant radio frequency (O^) flux noise created by the input
coil is the main mechanism that degrades the performance of a coupled DC SQUID.
The input coil resonance creates a feedback loop as shown in Fig. 2.2.
17


input-coil Voltage vs Flux
resonance relation
Figure 2.2: Feedback loop for the magnetic flux caused by the parasitic capacitance
which is introduced by the input-coil [2].
According to Enpuku et al., in the presence of the input coil the rf voltage () in the
SQUID causes current to flow through the input coil inductance and produces a
magnetic flux in the SQUID inductance (Fig. 2.3). This flux produced by V^ acts as
a feedback flux and is added to the initial <5^ noise [1-2]. It is this flux that causes
the degradation in a SQUIDs performance and is introduced into the circuit as a
capacitive feedback C in Fig. 2.3.
2.4 Coupled SQUID using the Enpuku Paper
We have duplicated the results given by Enpuku et al., on the resonant behavior of the
DC SQUID. In his paper, Enpuku explains how the distortion in the V-0 curve is
due to the resonance of the input coil [1-2]. When the input coil has more turns or the
dielectric constant of the insulator is high the degradation of the V-0 curve becomes
more noticeable. Enpuku also explains how to reduce the degradation by putting a
18


damping resistor across the SQUID washer [8] and using a RXCX shunt in the input
coil [9]. Another method that can be used is to increase the shunt resistance of the
SQUID while decreasing the SQUID capacitance [8].
We are interested in Enpukus model which is used to explain how to deal with the
resonant radio frequency flux noise (<5^) of the DC SQUID. The SQUID is
simplified by taking into account only the fundamental components of the input coil
(Fig 2.3).
Ipwl
Figure 2.3: Simple circuit model of the coupled SQUID taking into account the
input coil resonance and the parasitic capacitance. L; C, Rt are the input coil
fundamental components. Ls, C}, Rs are the SQUID inductance, capacitance, and
resistance, respectively. Rp and Cp are the feedback components caused by the input
coil. LI represents the inductance between the input coil and the SQUID and V0 is
the Gaussian noise source.
19


In the circuit, Cp represents the parasitic capacitance between the input coil and the
ground of the SQUID. The resistance Rp represents the radio frequency loss of the
capacitance Cp It is this component that creates the degradation in the SQUID. Cp
and/?p are introduced into the circuit as estimates from actual simulations. In the
actual experiments there was degradation in the SQUID that was being caused by
some unknown element. So, after trial and error calculations it was shown that by
introducing Cp and Rp into the circuit of Fig. 2.3 the output of the circuit match
those of the actual experiment.
The interest in simulating the circuit in Fig. 2.3 is to match the results reported by
Enpuku et al [1-2]. So, the same parameters that were used in that report are used.
Therefore, the temperature of the SQUID will be 77K (K=Kelvin). To get a
temperature of 77K we use a Gaussian noise source which is available in WRspice.
The general form of the Gaussian source is gauss (stddev mean lattice [interp]);
where stddev takes into account the operating temperature of the SQUID, see
Appendix C. The junction critical current IB = 15pA, junction resistance Rs =6Q,
junction capacitance Cj = 0.2pF, Ls = lOOpH, coupling constant a = 0.9 (strong
coupling), L;= lOnH corresponding to a 12-turn input coil, mutual
inductanceMi = a{LtLs)ul = 0.9nH, C(. = 120pF corresponding to a high dielectric
constant of SrTi03, Rt = 900Q, and Ll=500p.
20


To calculate the period of a full flux quantum (<&0) we need to know the necessary
current to produce a full period and the mutual inductance between the SQUID
and the input coil. For example,
I* (2.4.1)
It is known thatM, = cc(LjLs)u2 = 0.9nH, and that aOn = 2.07xlO15W&. Plugging
in these values gives 10 = 2.2976 pA. Therefore, to produce two full periods a
o
necessary I(bo = 5 pA is needed. So, we introduce a piecewise linear source into the
circuit of Fig. 2.3 that has a 1^ =2.5 pA (see Appendix D). The Enpuku results are
shown in Fig. 2.4 in which two periods are being considered.
Figure 2.4: The Enpuku results showing two full periods. Notice that the waveform
is degraded due to the input coil resonance. In the actual simulation the V- shown will duplicate at 4.2 pA.
21


In Fig. 2.5, points A and B correspond to points A and B of Fig. 2.3, as shown these
points connect an 8th order Butterworth filter to reduce the intrinsic high-frequency
internal oscillations of the SQUID to a DC average. Using the low pass scaling rules
and the Butterworth low pass filter table we are able to come up with values for the
filter [10].
The values for L0through L3 are calculated using equation (2.4.2)
T , R,L (table)
L {actual) =------------ (2.4.2
CO
and the capacitance values are calculated using equation (2.4.3)
C factual) =C^te)- (2.4.3
coRl
where RL = 1000, co = 2ttf = 2ji{\GHz) and the table values are given in reference
[10].
L3 L2 L1 L0
Figure 2.5: 8th order Butterworth filter that is connected to the circuit of Fig. 2.3 to
reduce the intrinsic high-frequency internal oscillations to a DC average.
22


In Fig. 2.3, if the feedback capacitance is neglected then the V-0 relation will have no
degradation as it is shown in Fig. 2.6.
Fat If*- U
/ r\ \ S' \ \ \
1 1 1 f J \ \ \ 1 \ \ \ / / / \ \ \ \ V \ n
1 l / 1 1 y \ v\ V f fj .a/ \ l 1 \ \ \
r 1
distort. & IS _
529PM 1
Figure 2.6: Voltage vs. Flux relation without the feedback effect. The simulation is
stopped at 4.2 pA where the results start to duplicate.
2.5 Equivalent Circuit Model
Now that it has been shown how to simulate the Enpuku circuit using Josephson
junctions our next step will be to come up with an equivalent circuit that doesnt use
Josephson junctions in an effort to produce an equivalent output to that of the V-0
curve in Fig. 2.4. Enpuku also provides a circuit for studying theOr/ of the circuit in
Fig. 2.3, shown in Fig. 2.7.
23


Figure 2.7: A circuit model for studying the behavior of (t) and (t). The
current In is the noise current whose power spectrum is given by 4 kB T/2 Rs, and
(t) is given by equation (2.4.7). Here Alp represents the coupling constant (a).
In order to understand the circuit in Fig. 2.7 a relationship between the (t) and
the phase difference <81 52> (t) should be derived. The relationship between the
phase difference and voltage is found along the V-0 curve [1-2]. So, the resonant
behavior of the coupled SQUID is determined in terms of the phase difference and the
voltage. First, noise, is defined as > =----(Sl-S2)(t) [1-2, 5], and is
2it
generated from the resonant circuit. This is applied to the SQUID in addition to
the Therefore, the total flux in the SQUID becomes Oext + (t). The
voltage (t) changes along the V-0 relation. If we assume a sinusoidal V-
relation since the V-0 curve is sinusoidal (Fig. 2.4), the average voltage is given by
24


(2.4.4)
< v > (t) = (1 cos( where
51-52 = 271:
O
total

(2.4.5)
and
total =ext+rf(t)
(2.4.6)
Therefore,
< V > (<) = ^(1 cos(^)(4> + (()))
(2.4.7)
It is the above equation that drove us to studying the Enpuku models. This result
gives a starting point in creating a circuit that will be independent of Josephson
junctions. The problem with Josephson junctions is that they are non-linear and
simulations that include Josephson junctions in SPICE programs are time extensive.
Another problem is that current SPICE programs do not allow us to use analysis other
than transient when using Josephson junctions. So far we have introduced a model
that works with Josephson junctions and an equivalent circuit that is independent of
the junctions.
2.6 Results using the Equivalent Circuit Model
The equivalent circuit that is introduced in Fig. 2.7 has been simulated using equation
(2.4.7) and (1.5.1). In equation (2.4.7) when we calculate
into account the internal flux in the circuit. So, on the right hand side of equation
25


(1.5.1)im and we can put ammeters that measure the amount of current that is going through Ls.
These ammeters can then be called by the voltage source (t) of Fig. 2.7 to
account for the current that is going in both directions of Ls. The procedure for
calculating the internal and radio frequency flux is carried out through the use of
ammeters that compensate for the current that is circulating in both directions of the
SQUID inductance. The actual WRspice circuit is shown below in Fig. 2.8.
Figure 2.8: Equivalent circuit of Fig. 2.7 with ammeters across the SQUID
inductance, 8th order Butterworth filter, and Gaussian source representing the
SQUIDs current noise and operating temperature. Notice that this circuit doesnt
have feedback components (Cp andi?p ).
In Fig. 2.8, (t) represents equation (2.4.7) in WRspice and is given as
26


(2.5.1)
2jc 1
< V > (0 = (15//V)(1 -cos()(/(Al)(65p) + i(A2)(65p) + -O0))
In equation (2.5.1) the circulating current is taken into account through the two
ammeters A1 and A2 and multiplied by the SQUID inductance with values of 65 pF
to solve for the internal and radio frequency flux in the circuit. We are able to solve
for the flux in the circuit since the rate of change of flux in the ring yields O = LI
[5].
The input coil fundamental components and Gaussian noise sources used in Fig. 2.8
have identical values to that of Fig. 2.3 which is what this design is based on. Also,
In Fig. 2.8, there is no current path in the input coil so all the current is going into the
SQUID inductance and the V-0 curve should be smooth, as shown in Fig. 2.9(A).
Figure 2.9(A): V-0 curve of equivalent circuit without Josephson junctions and
without the feedback effect. This result agrees to that of Fig. 2.6. The plot duplicates
at 4.2 pA.
27


We can now add the feedback component to the circuit in Fig. 2.8 to create a current
path through the input coil to the feedback capacitance to ground. This will degrade
the V-0 curve as shown in Fig. 2.9(B).
Figure 2.9(B): V-0 curve of equivalent circuit without Josephson junctions and with
the feedback effect. The plot duplicates at 4.2 pA.
The next step will be to get an AC analysis plot for a given flux current. That is for
AC analysis we apply a constant current in the SQUID by replacing the piecewise
linear source by a constant current taken at any point on the V-0 curve of Fig. 2.9(A)
or (B). Then an AC analysis will be done on this circuit so that we can see where the
resonances exist. When doing an AC analysis we are in a way looking at the
impedance versus frequency plot of a SQUID. An AC plot will show us where the
28


resonances occur in the circuit. The Figure below shows an AC plot for the circuit of
Fig. 2.8 without the feedback component.
Figure 3.0(A): AC analysis plot of Fig. 2.8 without the feedback component at
/> =1.5 pA. The resonance seems to occur at about 6.8MHz.
For the same I^ = 1.5 pA with the feedback component, the resonance seems to
occur at the same place. This tells us that for this equivalent circuit the feedback
component doesnt have that much effect on the circuit. This can be one reason why
although there is some change introduced by the feedback component the change is
very minimal and this is why we dont see that much degradation when the feedback
component is introduced.
29


:aagrtl ?Sfl ; 1) - x~ ~{iatwMg*t': v? -- -.s-v.-. *;--
Figure 3.0(B): AC analysis plot of Fig. 2.8 with the feedback component at 1^ =
1.5pA. The resonance seems to occur at about 6.8MHz.
30


3. Chapter-Characterizing the CDMS SQUID with WRspice
3.1 Introduction
In this chapter the characterization of a DC SQUID using a Discrete Fourier
Transform (DFT) procedure in SPICE is introduced. The procedure uses a table
function in SPICE in which the Fourier coefficients are stored and multiplied with
sine and cosines to give the desired V-0 curves. The Fourier coefficients used are
taken from a CDMS SQUIDs V-0 relations.
3.2 Characterizing the CDMS SQUID
At the SQUID lab a CDMS SQUID has been probed to acquire the SQUIDs V-0
relation at different bias current settings. The data were measured using room
temperature electronics while the SQUID is submerged in liquid helium. Figure 3.1,
shows the measured V-0 curves for the CDMS SQUID from Ib = 35 pA to 90 pA in
2.73 pA steps.
The amount of current required to produce 1O0 is calculated using a relationship
between mutual inductance, O and U
o
I*0M,= Ofl (3.1.1)
The mutual inductance is equal to Mt = a{LiLs )1/2 (as seen in section 2.3) where a is
the coupling constant.
31


In Figure 3.1, it can be seen that the amount of current required to produce 1 0 is
equal to 22.5 pA, and the value for a flux quantum is equal to 2.07x10~ISWb.
Therefore, equation (3.1.1) can be used to calculate the mutual inductance of this
SQUID, that is, the mutual inductance between the SQUID inductance and the input-
coil inductance. So, using equation (3.1.1) the mutual inductance between the input
coil and the SQUID is Mi = 92pH From this parameter, the coupling constant (a) of
the CDMS SQUID can be calculated.
Iflux Vs. Voltagel
0.004
0.003
0.002
0.001
0
-0.001
-210-5 -1.510"5 -HO* -5 lO*
5104 1 10-5 1.5 10"5
Iflux
Figure 3.1: Simulated V-0 curves for a characterized CDMS SQUID (M=10).
Ib =35 pA to 90 pA in 2.73 pA steps. /* = 22.5 pA .
o
32


3.3 Discrete Fourier Transform
For a constant Ib the V-0 curve can be modeled using a truncated Fourier series, a
finite series of sine and cosine terms with voltage as the dependent variable and flux
as the independent variable.
M
tr/T v-1 f/~, ,2tz&0 2tz&O
v(ib> ) = 2 (ck cos()+Dk sm<^))
*=i
o>
(3.2.1)
Where the summation goes from k=l to 10, M is the number of harmonic terms and
Ck and Dk are the weighting coefficients for each harmonic term. For the CDMS
SQUID M=10 is accurate enough to represent a V-0 curve at a constant current bias.
The data for the CDMS SQUID were taken in N equally spaced data points over 1 Oc
at a fixed Ib so that the following formula can be used
1 AM -y&ttt,
" (3.2.2)
\N 1=0
O
Where the A = is the spacing in flux, V(Ib,lA) is the voltage at the flux state l
for a fixedIb F (Ih, J) is the DFT of V(Ib,lA) with frequency index J [3]. If
equation (3.2.2) is used as the DFT algorithm then
C0=F(Ib,0)N~U2 D0=0 (3.2.3)
(3.2.3) is the value at DC so the imaginary part (D0) is going to be zero
33


Ck =2Real{F(Ib,k)}N~U2 Dk ~-2Rtal{F(Jb,k)}N~xn (3.2.4)
The variable N should be large enough to avoid aliasing. The above procedure can be
accomplished using the Fast Fourier Transform (fft) function in Matlab. When using
the fft function in Matlab the weighting coefficients need to be scaled after Matlabs
results as shown in equations (3.2.3) and (3.2.4).
3.4 Obtaining a Full Period/FIux Quantum
In kaleidograph, a program used to plot data, a linear curve fit function can be used to
truncate the data acquired from probing the CDMS SQUID. This is done to ensure
that only one period or flux quantum (Oe) is being considered. These data can then
be transferred into a Matlab m-file to acquire the Fourier coefficients Ck and Dk. For
example, Fig. 3.2 shows four different original V- SQUID. In order to get one full flux quantum these plots will be truncated by using a
curve fit function that approximates each plot. The goal is to get one full period and
then truncate the average data by discarding everything that is above and below the
full period as seen in Fig. 3.3.
In order to create an accurate curve fit, kaleidograph uses a function that takes into
account the DC value plus three harmonics of each V- doesnt allow for more than three harmonics some curve fits in Fig. 3.2 can be seen to
be inaccurate. Although some of these curve fits are inaccurate, to obtain a full flux
quantum the average of all these V-0 curves is a good estimate (as seen in Fig. 3.3).
34


Iflux Vs. Voltage2
-mo4 -no4 -sio4
2 to4 -mo4 -no4 -5 to4 o 5 to4 no4 15 ic
(A)
6£32Su^
(B)
89^63iJAj
Iflux Vs. VoIlogo2
Bulk Vphl
I Flux
(C) (D)
Figure 3.2: The curve fit curves are the solid lines; the original curves have solid
lines with circles. Notice that the curve fit for I =90 pA (D) is a perfect fit and the
lower Ib goes the harder it is to find a good curve fit to the V-0 curve using DC + 3
harmonics.
35


The calculation that was done is to take the period of each curve and set it equal to
Ik
----; 2k is to compensate for the radians calculation done by kaleidograph. Once
IA
^0
all the curve averages from 35 pA to 90 pA are taken the average period comes out to
be 10.5 pA. So everything above and below 10.5 is deleted from the gathered data.
The results are shown in Fig. 3.3.
| a-5S-3ar3^~| |
IFlux Vs. VoMsgc33
43001 0

r
i.
430U

-1.5104 -119* -JIO4 0 Jiff* IIJ4 13 w4
mux Vs. Vo!lage22
(A)
| e

(B)
HluxVt. Voltago22
Ah
Iflux Vs. Voltagoll
\ / &
N / r
c j?
i \ i i t
i \ i
\ i
% f
uto4 -no4 ^to4 o jio4 no4 I3I44
Ok
(C)
(D)
Figure 3.3: Truncated V-0 curves whose data is ready to be taken into Matlab to
attain the Fourier coefficients of each V-0 curve. This data will later be taken into
WRspice.
36


3.5 Matrix Notation for Matlab
Once the Fourier coefficients are attained for each V-0 curve they are then ready to
be tested using matrices in Matlab. The procedure can be accomplished for each Ib
from 35 pA to 90 pA in 2.73 pA steps to obtain a matrix that is (21X1) + (21X10)
(10X1) + (21X10) (10X1). This matrix will simplify to (21X1) + (21X1) + (21X1)
and output 21 waveforms when executed in Matlab (see Appendix B). This matrix is
composed of Fourier coefficients given by equations (3.2.3) and (3.2.4). Also by
using this procedure it can be verified that the weighting coefficients Ck and Dk are
correct. Once the coefficients are verified to give accurate results they can be
transferred onto WRspice.
This procedure will be illustrated using a CDMS SQUID whose coefficients have
been attained using the Fast Fourier Transform in Matlab. The data for the CDMS
SQUID are taken from Ih = 35 pA to 90 pA in 2.73 pA steps with an amount of
current to produce 1 00 = 22.5 pA (see Fig. 3.1). In Fig. 3.4, the V-0 characteristics
are shown for four Ib values simulated using Matlab.
In Fig. 3.4, it is seen that the coefficients are correct and that they give some type of
V-O curve. The plots represent V-0 relations for Ib = 35 pA, 50 pA, 65 pA, and 90
pA. These V-0 curves are in good agreement with Fig. 3.1.
37


lb=35uA
lb=50uA
x 10'
Figure 3.4: Matlab results for testing the Fourier coefficients of a CDMS SQUID.
3.6 WRspice Results
WRspice produced similar results to that of Matlab as can be seen in Fig. 3.5. The
plots (in WRspice) started by showing pure noise and then a signal started to be
evident at Ib =35 pA where the data is defined. The same thing happened in Matlab
but here each plot depends on the value of Ib. This means that Matlab is just a tool
that is used to verify that the coefficients are correct. In appendix A, the WRspice
38


code used is given for a NIST SQUID where the only difference, between the CDMS
and NIST SQUID code, is the value of the coefficients in the look-up table.
(C) (D)
Figure 3.5: WRspice results for Ib = 35 p A to 90 pA in 2.73 pA steps. (A) Ih = 35
pA (B) Ib = 50 pA (C) Ib = 65 pA (D) Ih = 90 pA. Notice that these results are in
good agreement to Matlab Fig 3.4 and to Fig. 3.1.
3.7 Subcircuit Model
Using the device library in WRspice a circuit can be created that calls a subcircuit in
the model library. The code that is used to create the subcircuit is the same code that
is used to create Fig. 3.5 (see Appendix A). Fig. 3.5 is simulated using the editor file
39


in WRspice whereas here the same code will be executed but this time the electrical
layout in WRspice will be used. It is easier to work with an electrical layout in
WRspice because the connections between sub-circuits can be easily implemented.
The goal here is to create an array of 100 SQUIDs in an effort to attempt to put a
value to the parasitics of a SQUID for the in between connections of the array. Fig.
3.6 shows the dummy circuit that will be used.
Sub-circuit SQUID
Figure 3.6: SQUID subcircuit using the text editor file.
Using the NIST SQUID Fourier Coefficients in appendix A, we can create an array of
100 SQUIDs similar to the CDMS SQUID that should give us an amplitude in
40


voltage that is in the milli-volts. The voltage amplitude for a single SQUID such as
the Enpuku and NIST SQUIDs is in the range of micro-volts. So when SQUIDs are
being put in an array the voltage amplitude increases according to the size of the
array. Appendix A has V-0 curves for the NIST SQUID in which it can be seen that
its voltage range is in the micro-volts. This range implies that we are dealing with a
single SQUID.
41


4. Conclusion
The Enpuku SQUID is characterized using Josephson junctions in an attempt to
reproduce the results published by Enpuku et al [1-2]. We successfully accomplished
this goal and analyzed what would happen if the feedback component is taken out of
the circuit. When the feedback component is taken out of the circuit the V-O curve is
smooth and without any distortions. These results allowed us to go to the next step
where we characterized the SQUID without any Josephson junctions. The results
without any Josephson junctions were in good agreement to the results with
Josephson junctions. We proved that the Enpuku results are in fact accurate.
Another method for characterizing a DC SQUID is to use the Discrete Fourier
Transform method. The Discrete Fourier Transform method is implemented by using
the FFT function in Matlab and later tested using equation (3.2.2) in an m-file. This
equation is later built into WRspice as shown in appendix A. We obtained accurate
V-0 curves using this method and built an equivalent electrical model created by
referencing a sub-circuit through the model library in WRspice, Fig. 3.6. This model
can be connected in an array to test the parasitics within these connections. In order
to test an array using this model we need something to compare it to such as the
CDMS SQUID.
42


These methods were both accomplished successfully and a further attempt should be
made to make the Enpuku SQUID dynamic. The word dynamic in reference to the
Enpuku SQUID implies that the output is dependent on the change of bias current that
is introduced into the circuit.
43


Appendix A-WRspice Code
.options acct list node nopage
vntol=lnv reltol=.0001 itll=500
itl2=500 itl4=400
.widthout=80
Vbias 1 2 21u
Rbias 201
Iin 060
Rin 0 6 1MEG
Imod 7 0 pwl(0 0 50ms lu 100ms 0)
Rmod 701
XSQU 607012 101 0 102 0 103 0 104
0 105 0 106 0 107 0 108 0 109 0 110 0
111 0 112 0
+113 0 114 0 115 0 116 0 117 0 118 0
119 0 120 0 121 0 SQUID1
* Iin, IMOD, Vbias
.SUBCKT SQUID1 124567 131 0 132 0
133 0 134 0 135 0 136 0 137 0 138 0 139
0 140 0 141 0 142 0
+143 0 144 0 145 0 146 0 147 0 148 0
149 0 150 0 151 0
.param phio=22.25u
.param bstp=1.786u
.param bstt=5u
RFX 300 0 1MEG
AFX 300 301 V=V(4,5)
VFX 301 0 0
ABSNS 20 21 V=V(6,7)
Aout 21 0 V=V(41)*V(30)
*Eout 1
Eout50 50 0 300 0 function
cos(6.28*x/lu)
R50 50 0 1000k
Routa 51 0 1MEG
Aouta 51 0 v=V(50)*v(132)
Eout52 52 0 300 0 function
sin(6.28*x/lu)
R52 52 0 1000k
Routb 53 0 1MEG
Aoutb 53 0 V=V(52)*v(133)
* Eout2
r54 54 0 1000k
A54 54 0 V= 2*V(300)
Eout55 55 0 54 0 function
cos(6.28*x/lu)
R55 55 0 1000k
A56 56 0 v=V(55)*v(134)
R56 56 0 1000k
Eout57 57 0 54 0 function
sin(6.28*x/lu)
R57 57 0 1000k
A58 58 0 V=V(57)*V(135)
R58 58 0 1000k
*Eout3
r59 59 0 1000k
A59 59 0 V=3*V(300)
Eout60 60 0 59 0 function
cos(6.28*x/lu)
R60 60 0 1000k
A61 61 0 V=V(60) *V(136)
R61 61 0 1000k
Eout62 62 0 59 0 function
sin(6.28*x/lu)
R62 62 0 1000k
A63 63 0 V=V(62)*V(137)
R63 63 0 1000k
*Eout4
r64 64 0 1000k
A64 64 0 V=4*V(300)
Eout65 65 0 64 0 function
cos(6.28*x/lu)
R65 65 0 1000k
A66 66 0 V=V(65)*V(138)
R66 66 0 1000k
Eout67 67 0 64 0 function
sin(6.28*x/lu)
R67 67 0 1000k
A68 68 0 V=V(67)*V(139)
R68 68 0 1000k
*Eout5
r69 69 0 1000k
A69 69 0 V=5*V(300)
Eout70 70 0 69 0 function
cos(6.28*x/lu)
R70 70 0 1000k
A71 71 0 V=V(70)*V(140)
R71 71 0 1000k
44


Appendix A-WRspice Code
Eout72 72 0 69 0 function
sin(6.28*x/lu)
R72 72 0 1000k
A73 73 0 V=V(72)*V(141)
R73 73 0 1000k
*Eout6
R74 74 0 1000k
A74 74 0 V=6*V(300)
Eout75 75 0 74 0 function
cos(6.28*x/lu)
R75 75 0 1000k
A76 76 0 V=V(75)*V(142)
R76 76 0 1000k
Eout90 90 0 89 0 function
cos(6.28*x/lu)
R90 90 0 1000k
A91 91 0 V=V(90)*V(148)
R91 91 0 1000k
Eout92 92 0 89 0 function
sin(6.28*x/lu)
R92 92 0 1000k
A93 93 0 V=V(92)*V(149)
R93 93 0 1000k
*Eoutl0
r94 94 0 1000k
A94 94 0 V=10*V(300)
Eout77 77 0 74 0 function
sin(6.28*x/lu)
R77 77 0 1000k
A78 78 0 V=V(77)*V(143)
R78 78 0 1000k
*Eout7
r79 79 0 1000k
A79 79 0 V=7*V(300)
Eout80 80 0 79
cos(6,28*x/lu)
0 function
Eout95 95 0 94 0 function
cos(6.28*x/lu)
R95 95 0 1000k
A96 96 0 V=V(95)*V(150)
R96 96 0 1000k
Eout97 97 0 94 0 function
sin(6.28*x/lu)
R97 97 0 1000k
A98 98 0 V=V(97)*V(151)
R80 80 0 1000k R98 9E ! 0 1000k
A81 81 0 V=V(80)*V(144)
R81 81 0 1000k
Routl 30 0 : 1000k
Eout82 82 0 79 0 function Aoutl 30 31 v=v(131)+ v(51) + v(53)
sin(6.: 28*x/lu) Aout2 31 32 v=v(56) + v(58)
R82 82 0 1000k Aout3 32 33 v=v(61) + v(63)
A83 83 0 V=V(82)*V(145) Aout4 33 34 VO VO > 11 > + v(68)
R83 83 0 1000k Aout5 34 35 v=v(71) + v(73)
Aout6 35 36 v=v(76) + v(78)
*Eout8
r84 84 0 1000k
A84 84 0 V=8*V(300) Aout7 36 37 v=v(81) + v(83)
Aout8 37 38 v=v(86) + v(88)
Eout85 85 0 84 0 function
cos(6.; 28*x/lu) Aout9 38 39 v=v(91) + v(93)
R85 85 0 1000k AoutlC i 3S l 0 v=v(96) + v(98)
A86 86 0 V=V(85)*V(146)
R86 86 0 1000k
Eout87 87 0 84 0 function
sin(6.28*x/lu)
R87 87 0 1000k
A88 88 0 V=V(87)*V(147)
R88 88 0 1000k
RMAG 41 0 1MEG
AMAG 41 0 V= V(20,21)/ABS(V (20,21) )
*Eout9
r89 89 0 1000k
A89 89 0 V=9*V(300)
RCONV 40 0 1MEG
ACONV 40 0 V={ ( ( (ABS(V(20,21) ) )-
bstt)/bstp)+1)
. ends
45


Appendix A-WRspice Code
*****************************************
*****************
global 101 0
global 102 0
global 103 0
global 104 0
global 105 0
global 106 0
global 107 0
global 108 0
global 109 0
global 110 0
global 111 0
global 112 0
global 113 0
global 114 0
global 115 0
global 116 0
global 117 0
global 118 0
global 119 0
global 120 0
global 121 0
global 40 C )
*table part of circuit below
vlOl 101 0 table (tl,v(40))
rlOl 101 0 1MEG
.table tl 0 0 1 0 2 6.766E-08 3 6.863E-
08 4 2.100E-07 5 4.71E-07
+6 1.126E-06 7 2.310E-06 8 4.016E-06 9
5.711E-06 10 7.559E-06
+11 9.572E-06 12 1.105E-05 13 1.295E-05
14 1.504E-05 15 1.704E-05
R102 102 0 1MEG
vl02 102 0 table (t2,v(40))
-table t2 0 0 1 0 2 -2.287E-09 3 -
2.671E-08 4 -2.175E-07 5 -7.055E-07
+ 6 -1.772E-06 7 -3.397E-06 8 -5.261E-
06 9 -6.601E-06 10 -7.061E-06 11 -
6.591E-06
+ 12 -5.360E-06 13 -3.729E-06 14 -
2.261E-06 15 -9.283E-07
R103 103 0 1MEG
vl03 103 0 table (t3, v(40))
-table t3 0 0 1 0 2 2.873E-09 3 2.304E-
08 4 1.278E-07 5 2.755E-07
+ 6 3.914E-07 7 1.337E-07 8 -7.576E-07
9 -2.157E-06 10 -3.696E-06 11 -5.099E-
06
+ 12 -5.984E-06 13 -6.119E-06 14 -
5.726E-06 15 -5.247E-06
R104 104 0 1MEG
vl04 104 0 TABLE (t4, v(40))
-table t4 0 0 1 0 2 1.119E-09 3 7.476E-
09 4 9.865E-08 5 4.112E-07
+ 6 9.846E-07 7 1.553E-06 8 1.750E-06 9
1.510E-06 10 1.183E-06 11 7.186E-07
+ 12 4.211E-07 13 3.532E-07 14 2.418E-
07 15 8.129E-08
R105 105 0 1MEG
vl05 105 0 TABLE (t5,V(40))
-table t5 0 0 1 0 2 -2.641E-09 3 -
3.352E-08 4 -1.821E-07 5 -3.881E-07
+ 6 -5.087E-07 7 -2.679E-07 8 2.394E-07
9 6.016E-07 10 8.017E-07 11 7.076E-07
+ 12 2.899E-07 13 6.378E-08 14 1.24E-07
15 1.948E-07
R106 106 0 1MEG
vl06 106 0 TABLE (t6,V(40))
-table t6 0 0 1 0 2 1.398E-09 3 1.281E-
08 4 9.657E-09 5 -1.315E-07
+ 6 -3.291E-07 7 -4.519E-07 8 -4.668E-
07 9 -4.929E-07 10 -3.173E-07 11 -
8.886E-08
+ 12 1.754E-08 13 1.045E-07 14 1.0135E-
07 15 4.428E-08
R107 107 0 1MEG
vl07 107 0 TABLE (t7,V(40))
-table t7 0 0 1 0 2 -5.626E-10 3
2.425E-08 4 1.526E-07 5 3.311E-07
+ 6 3.815E-07 7 2.591E-07 8 9.030E-08 9
-1.175E-07 10 -2.824E-07 11 -1.195E-07
+ 12 3.632E-08 13 9.829E-08 14 1.255E-
07 15 1.239E-07
R108 108 0 1MEG
vl08 108 0 TABLE (t8,V(40))
-table t8 0 0 1 0 2 -2.699E-09 3 -
2.290E-08 4 -6.189E-08 5 -1.581E-08
+ 6 1.423E-08 7 8.872E-08 8 2.033E-07 9
2.456E-07 10 9.884E-08 11 2.808E-08
+ 12 1.778E-08 13 -6.000E-09 14 -
1.980E-08 15 -6.575E-09
R109 109 0 1MEG
vl09 109 0 TABLE (t9,V(40))
-table t9 0 0 1 0 2 1.361E-09 3 -
6.613E-09 4 -8.064E-08 5 -1.851E-07
+ 6 -1.857E-07 7 -1.576E-07 8 -8.178E-
08 9 1.068E-07 10 1.263E-07 11 2.423E-
08
+ 12 -5.761E-09 13 3.083E-08 14 3.763E-
08 15 1.833E-08
46


Appendix A-WRspice Code
R110 110 0 1MEG
vllO 110 0 TABLE (tl0,V(40))
.table tlO 0 0 1 0 2 -1.643E-10 3
1.319E-08 4 5.976E-08 5 4.871E-08
+ 6 6.077E-08 7 -3.343E-09 8 -1.116E-07
9 -1.207E-07 10 -3.851E-08 11 -9.082E-
09
+ 12 8.046E-10 13 1.124E-08 14 -5.394E-
09 15 -3.084E-09
Rill 111 0 1MEG
Vlll 111 0 TABLE (til,V(40))
.table til 0 0 1 0 2 -1.497E-09 3 -
7.099E-09 4 2.272E-08 5 6.239E-08
+ 6 5.406E-08 7 8.038E-08 8 6.541E-08 9
-3.910E-08 10 -2.918E-08 11 4.285E-08
+ 12 3.240E-08 13 3.673E-08 14 2.761E-
08 15 1.193E-08
R112 112 0 1MEG
Vll2 112 0 TABLE (tl2,V(40))
.table tl2 0 0 1 0 2 -2.676E-09 3 -
4.195E-09 4 -3.541E-08 5 -2.292E-08
+ 6 -4.435E-08 7 -1.933E-08 8 5.723E-08
9 7.901E-08 10 6.299E-09 11 1.778E-08
+ 12 9.404E-10 13 4.105E-09 14 1.680E-
09 15 3.283E-09
R113 113 0 1MEG
vll3 113 0 TABLE (tl3,V(40))
.table tl3 0 0 1 0 2 2.750E-09 3
1.172E-08 4 6.615E-09 5 -6.337E-09
+ 6 3.982E-09 7 -4.282E-08 8 -6.028E-08
9 3.657E-08 10 4.107E-08 11 7.449E-09
+ 12 1.369E-08 13 2.364E-08 14 1.732E-
08 15 1.849E-08
R114 114 0 1MEG
vll4 114 0 TABLE (tl4,V(40))
.table tl4 0 0 1 0 2 2.629E-09 3
9.135E-10 4 1.160E-08 5 -4.202E-09
+ 6 2.184E-08 7 2.629E-08 8 -2.665E-08
9 -4.638E-08 10 4.940E-09 11 -2.716E-09
+ 12 8.558E-09 13 8.917E-09 14 2.895E-
10 15 1.574E-09
R115 115 0 1MEG
vll5 115 0 TABLE (tl5,V(40))
.table tl5 00102 1.795E-09 3 -
6.936E-09 4 -1.509E-08 5 -3.418E-09
+ 6 -2.069E-08 7 2.595E-08 8 5.170E-08
9 -3.255E-08 10 -4.591E-09 11 8.271E-09
+ 12 1.990E-08 13 1.946E-08 14 1.955E-
08 15 1.333E-08
R116 116 0 1MEG
vll6 116 0 TABLE (tl6,V(40))
.table tl6 0 0 1 0 2 1.570E-09 3
4.007E-09 4 -2.043E-09 5 1.162E-08
+ 6 -5.830E-09 7 -2.924E-08 8 1.385E-08
9 2.746E-08 10 1.352E-08 11 -5.683E-09
+ 12 4.159E-09 13 8.676E-10 14 1.994E-
09 15 2.330E-09
R117 117 0 1MEG
vll7 117 0 TABLE (tl7,V(40))
.table tl7 0 0 1 0 2 7.249E-10 3
7.134E-10 4 6.850E-09 5 -6.703E-09
+ 6 1.937E-08 7 -1.416E-08 8 -4.225E-08
9 3.333E-08 10 2.030E-08 11 1.402E-08
+ 12 1.641E-08 13 2.057E-08 14 1.624E-
08 15 1.047E-08
R118 118 0 1MEG
vll8 118 0 TABLE (tl8,V(40))
.table tl8 0 0 1 0 2 -1.370E-09 3 -
3.247E-09 4 -1.307E-09 5 -5.208E-09
+ 6 -3.130E-09 7 3.037E-08 8 -1.332E-08
9 -1.414E-08 10 -1.106E-09 11 3.425E-09
+ 12 2.770E-09 13 3.970E-09 14 1.644E-
09 15 1.009E-09
R119 119 0 1MEG
vll9 119 0 TABLE (tl9,V(40))
.table tl9 0 0 1 0 2 -1.740E-09 3
1.279E-09 4 -4.905E-09 5 1.100E-08
+ 6 -1.442E-08 7 5.298E-09 8 3.061E-08
9 -2.254E-08 10 -5.584E-09 11 1.844E-08
+ 12 1.099E-08 13 1.455E-08 14 1.312E-
08 15 1.316E-08
R120 120 0 1MEG
V120 120 0 TABLE (t20,V(40))
.table t20 0 0 1 0 2 -3.136E-09 3 -
2.699E-10 4 -1.017E-09 5 -2.250E-09
+ 6 4.751E-09 7 -1.827E-08 8 7.246E-09
9 1.218E-08 10 -3.066E-10 11 5.242E-09
+ 12 2.822E-09 13 3.317E-09 14 1.514E-
09 15 -1.482E-09
R121 121 0 1MEG
vl21 121 0 TABLE (t21,V(40))
.table t21 0 0 1 0 2 -1.736E-09 3 -
4.591E-09 4 1.931E-09 5 -1.338E-08
+ 6 8.826E-09 7 -2.540E-09 8 -1.304E-08
9 2.259E-08 10 7.815E-09 11 1.123E-08
+ 12 1.177E-08 13 1.228E-08 14 1.292E-
08 15 8.057E-09
.tran lOOu .1
. end
47


Appendix A-WRspice Code
The code in this appendix is different from the actual code that was
used by GUTT et al [3], to reproduce the V- relations for the NIST
SQUID. The syntax changes for different versions of SPICE. In
their code P-spice version 5.0 was used. Here we are using WRspice
so the code had to be changed accordingly. What was an easy task in
P-spice, here it became very difficult to replicate the results
using WRspice and thus a learning curve has to be considered.
The table function will be demonstrated using an example:
V2 2 0 table (tl, v (2))
R2 2 0 1MEG
.table tl 0 0 1 1 2 3
In the above example tl is the location of the table and v(2) will
be the value at which the table will be referenced. So, if v(2) = 2
then when this line is reached (V2 2 0 ...) the value of the table at
position 2 will be executed. The goal of this code is to calculate
the DFT equation given by equation (3.2.1) where the table
represents the Fourier coefficients. The rest of the code can be
explained using the "WRspice Reference Manual" [11].
The results for the above code using a NIST SQUID are shown below:
(A)
(B)


Appendix A-WRspice Code
(C)
(D)
(G) (H)
49


Appendix A-WRspice Code
(I)
(J)
(M)
(N)
50


Appendix A-WRspice Code
Figure A.l: WRspice results for Ih =8pA to 30 pA in 2 pA steps for a NIST SQUID.
(A-B) Pure noise for Ib =5 pA to 6 pA (C) Noise plus signal Ih =7 pA (D-P) Ih =8
pA to 30 pA (Q) Ib =32 pA (R) Ih =40 pA. Notice that the plots start to degraded
significantly after Ih =30 pA where the coefficients are not defined.
51


Appendix B-Matlab Code
clear;
a0=[0; 1.021E-07; 6.766E-08; 6.863E-08; 2.100E-07; 4.71E-07; 1.126E-06;...
2.310E-06; 4.016E-06; 5.711E-06; 7.559E-06; 9.572E-06; 1.105E-05;...
1.295E-05; 1.504E-05; 1.704E-05;];%(16xl)
x=0:.01:10;
an=[0, 3.630E-09, -2.287E-09, -2.671E-08, -2.175E-07, -7.055E-07, -1.772E-06,...
-3.397E-06, -5.261E-06, -6.601E-06, -7.061E-06, -6.591E-06, -5.360E-06,...
-3.729E-06, -2.261E-06, -9.283E-07;
0, 2.391E-09,1.119E-09, 7.476E-09,9.865E-08,4.112E-07, 9.846E-07 ,...
1.553E-06, 1.750E-06, 1.510E-06,1.183E-06,7.186E-07,4.211E-07 ,...
3.532E-07,2.418E-07,8.129E-08;
0, -4.438E-09,1.398E-09, 1.281E-08, 9.657E-09, -1.315E-07,-3.291E-07 ,...
-4.519E-07, -4.668E-07, -4.929E-07, -3.173E-07, -8.886E-08,1.754E-08 ,...
1.045E-07, 1.0135E-07,4.428E-08;
0,4.945E-10, -2.699E-09, -2.290E-08, -6.189E-08, -1.581E-08,1.423E-08 ....
8.872E-08, 2.033E-07, 2.456E-07, 9.884E-08,2.808E-08,1.778E-08,...
-6.000E-09, -1.980E-08, -6.575E-09;%10xl6
0.1.323E-09, -1.643E-10,1.319E-08, 5.976E-08,4.871E-08, 6.077E-08,...
-3.343E-09, -1.116E-07 .-1.207E-07, -3.851E-08, -9.082E-09, 8.046E-10 ....
1.124E-08, -5.394E-09 ,-3.084E-09;
0 ,5.671E-10, -2.676E-09, -4.195E-09, -3.541E-08 .-2.292E-08 .-4.435E-08 ....
-1.933E-08,5.723E-08,7.901E-08,6.299E-09,1.778E-08,9.404E-10 ,...
4.105E-09, 1.680E-09, 3.283E-09;
0 ,2.245E-09, 2.629E-09,9.135E-10, 1.160E-08, -4.202E-09 ,2.184E-08,...
2.629E-08, -2.665E-08, -4.638E-08,4.940E-09, -2.716E-09, 8.558E-09,...
8.917E-09,2.895E-10, 1.574E-09;
0.2.433E-09,1.570E-09,4.007E-09 ,-2.043E-09,1.162E-08 -5.830E-09,...
-2.924E-08,1.385E-08,2.746E-08,1.352E-08 .-5.683E-09, 4.159E-09 ,...
8.676E-10, 1.994E-09,2.330E-09;
0, -2.304E-09, -1.370E-09, -3.247E-09, -1.307E-09, -5.208E-09, -3.130E-09,...
3.037E-08, -1.332E-08, -1.414E-08, -1.106E-09, 3.425E-09,2.770E-09 ,...
3.970E-09,1.644E-09, 1.009E-09;
0, -1.213E-11, -3.136E-09, -2.699E-10, -1.017E-09, -2.250E-09,4.751E-09,...
-1.827E-08, 7.246E-09,1.218E-08, -3.066E-10.5.242E-09, 2.822E-09 ....
3.317E-09, 1.514E-09, -1.482E-09;];
bn=[cos(x); cos(2*x); cos(3*x); cos(4*x); cos(5*x); cos(6*x); cos(7*x);...
cos(8*x); cos(9*x) ;cos(10*x);]; %10xl
an0=[0 -8.385E-10 2.873E-09 2.304E-08 1.278E-07 2.755E-07 3.914E-07 1.337E-07 ...
-7.576E-07 -2.157E-06 -3.696E-06 -5.099E-06 -5.984E-06 -6.119E-06 -5.726E-06 -5.247E-06;
0 4.790E-10 -2.641E-09 -3.352E-08 -1.821E-07 -3.881E-07-5.087E-07-2.679E-07...
2.394E-07 6.016E-07 8.017E-07 7.076E-07 2.899E-07 6.378E-08 1.24E-07 1.948E-07;
0 3.896E-09 -5.626E-10 2.425E-08 1.526E-07 3.311E-07 3.815E-07 2.591E-07 9.030E-08...
-1.175E-07 -2.824E-07 -1.195E-07 3.632E-08 9.829E-08 1.255E-07 1.239E-07;
0 -2.082E-09 1.361E-09 -6.613E-09 -8.064E-08 -1.851E-07 -1.857E-07 -1.576E-07...
-8.178E-08 1.068E-07 1.263E-07 2.423E-08 -5.761E-09 3.083E-08 3.763E-08 1.833E-08;
0 1.598E-09 -1.497E-09 -7.099E-09 2.272E-08 6.239E-08 5.406E-08 8.038E-08 6.541E-08...
-3.910E-08 -2.918E-08 4.285E-08 3.240E-08 3.673E-08 2.761E-08 1.193E-08;
0 7.250E-10 2.750E-09 1.172E-08 6.615E-09 -6.337E-09 3.982E-09 -4.282E-08 -6.028E-08..
3.657E-08 4.107E-08 7.449E-09 1.369E-08 2.364E-08 1.732E-08 1.849E-08;%10xl6
0 1.215E-10 1.795E-09 -6.936E-09 -1.509E-08 -3.418E-09-2.069E-08 2.595E-08 5.170E-08..
-3.255E-08 -4.591E-09 8.271E-09 1.990E-08 1.946E-08 1.955E-08 1.333E-08;
0 2.623E-09 7.249E-10 7.134E-10 6.850E-09 -6.703E-09 1.937E-08 -1.416E-08 -4.225E-08..
52


Appendix B-Matlab Code
3.333E-08 2.030E-08 1.402E-08 1.641E-08 2.057E-08 1.624E-08 1.047E-08;
0 -2.704E-09 -1.740E-09 1.279E-09 -4.905E-09 1.100E-08-1.442E-08 5.298E-09 3.061E-08...
-2.254E-08 -5.584E-09 1.844E-08 1.099E-08 1.455E-08 1.312E-08 1.316E-08;
0 2.038E-09 -1.736E-09 -4.591E-09 .931E-09 -1.338E-08 8.826E-09 -2.540E-09 -1.304E-08...
2.259E-08 7.815E-09 1.123E-08 1.177E-08 1.228E-08 1.292E-08 8.057E-09;];
bl=[sin(x); sin(2*x); sin(3*x); sin(4*x); sin(5*x); sin(6*x); sin(7*x);...
sin(8*x) ;sin(9*x); sin(10*x);];%10xl
F= (an.'*bn);
R= (anO.'*bl);
G=(F+R);
plot(x,G) %, axis ([0 20 -2 10])
Tested Coefficients/Voltage VS Flux
Figure B.l Fourier coefficients tested in Matlab without the DC value showing
smooth curves.
The code in appendix B is a basic matrix manipulation in which the goal is to test the
Fourier coefficients obtained from Gutt et al. [3] by using equation (3.2.1). It can be
seen that that the DC values of the Fourier Coefficients are put into matrix a0. The
real parts of the Fourier coefficients are put into matrix an and are multiplied by
53


Appendix B-Matlab Code
matrix bn. In a similar manner the imaginary part of the coefficients an0 is
multiplied by matrix bx. The code can be easily followed to show how the DC value
plus the ten harmonics are being calculated.
Figure B.2 Tested Fourier coefficients with the DC value for a NIST SQUID from
Ib=7pA to 3 Op A in 2pA steps.
In Figure B.2, the DC value plus ten harmonics of a NIST SQUID have been tested.
As seen the waveforms are identical to WRspice results given in appendix A. 1.
54


Appendix C-Gaussian Source
Appendix: Gaussian source
General Form:
Gauss (stddev mean lattice [interp])
Example:
Gauss (43.99pV, 0, lOOp, 1)
In calculating the standard deviation of the Gaussian source the temperature at which
the circuit is being operated is also being taken into account. For Vn (noise voltage),
2& TR
and the interp is given 1. If the interp is 1, it will interpolate if time interval of
simulation does not exactly match the lattice. If it is 0, it will return the value of the
lattice point without adjustment [11].
The standard deviation can be calculated to operate at different values of temperature
using the equation
£72 = 4k B (T)(R)(BW) =
dt
(C.1.1)
Where T = temperature of operation, Bandwidth (BW) is equal to
---where dt is the
2 dt
time interval.
55


Appendix D-Piecewise Linear Function
Figure D.l The piecewise linear function used in the Enpuku SQUID.
The general format of a piecewise linear function on a source statement is
PWL(tl Vl (B.1.1) can be compared to the values used in the Enpuku SQUID for the pwl
function given as
PWL (0 0 200n 0 4200n 5p 8200n 0) (B. 1.2)
In (B.1.2), the period is represented by 5pA and can be seen in Figures 2.4 and 2.6.
56


Appendix E-CDMS NIST SQUID
Figure E.l: CDMS NIST SQUID physical layout series array of 100 SQUIDs (top),
zoom in on array (bottom left), zoom in on single SQUID (bottom right).
57


REFERENCES
[1] Minotani, T., Enpuku, K. & Kuroki, Y. (1997). Effect of Capacitive Feedback on
the Characteristics of Direct Current Superconducting Quantum Interference Device
Coupled to a Multitum Input Coil. Journal of Applied Physics, 82 (1).
[2] Enpuku, K., & Minotani, T. (1998). Distortion of Voltage vs. Flux Relation of
DC SQUID Coupled to Multitum Input Coil Due to Input Coil Resonance Combined
with Capacitive-Feedback Effect. IEEE Trans. Applied Superconductivity, Vol.5
Nos. 7.
[3] G.M. Gutt & N.J. Kasdin (1993). A Method for Simulating a Flux-Locked DC
SQUID. IEEE Tans. On Appl. Supercond., Vol. 3, No.l.
[4] Raymond, S. (1988). Superconductivity Supplement for Physics for Scientist &
Engineers (2nd ed.). New York: Rinehart and Winston.
[5] Tesche, D. & Clarke, J. (1977). DC SQUID: Noise and Optimization. Journal of
Low Temperature Physics, Vol. 29, Nos. 3/4.
[6] Theodore Van Duzer & Charles W. Turner (1999). Principles of Superconductive
Devices and Circuits (2nd ed.). New Jersey: Prentice-Hall.
[7] Cantor, R. (1996). DC SQUIDs: Design, Optimization and Practical
Applications. Kluwer Academic Publishers. Printed in the Netherlands.
[8] Enpuku, K., Cantor, R. & Koch, H. (1993). Resonant Properties of a DC SQUID
Coupled to a Multitum Input Coil. IEEE Trans. On Appl. Supercond., Vol. 3, No.l.
[9] Huber, M.E., Steinbach, A., & Ono, R. (2000). Resonance Damping in Tightly
Coupled DC SQUIDs Via Intra-Coil Resistors. Electromagnetic Technology
Division, National Institute of Standards and Technology.
[10] Zverev, A.I. (1967). Handbook of Filter Synthesis. Wiley US
[11] Whiteley, S. (1998). WRspice Reference Manual. Whiteley Research
Incorporated. Release 2.1.8
58