UNIVERSITY OF CINCINNATI September 15 00 _____________ , 20 _____

Jenny Rebecca Holzer I,______________________________________________, hereby submit this as part of the requirements for the degree of:

Doctorate of Philosophy (Ph.D.) ________________________________________________

in: Physics ________________________________________________

It is entitled: Scanning SQUID Microscope Measurements on ________________________________________________ Josephson Junction Arrays ________________________________________________

________________________________________________ ________________________________________________

Approved by: ________________________ Richard S. Newrock ________________________ Leigh Smith ________________________ Brian Meadows ________________________ Steven T. Herbert ________________________

Scanning SQUID Microscope Measurements on Josephson Junction Arrays

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 2000 by

Jenny Rebecca Holzer B.S., Ohio University, 1993 M.S., University of Cincinnati, 1995

Committee Chair : Professor Richard S. Newrock

ABSTRACT

The first part of my dissertation work involved the design, construction, and operation of a scanning microscope that uses a superconducting quantum interference device (SQUID) as a probe. This system can produce two-dimensional (2D) images of the magnetic flux above samples as large as 1cm by 1cm. The microscope has a spatial resolution on the order of 40µm and a flux resolution on the order of a micro flux quantum. Next, I used this scanning SQUID microscope system to study the distribution of currents in 2D arrays of SNS Josephson junctions and attempt to estimate the penetration depth for perpendicular magnetic fields, λ⊥. λ⊥ is an important parameter in that it is a determining factor for the possibility of a Kosterlitz-Thouless phase transition in arrays. Raster scanned images of the flux above an array were produced for various temperatures and currents. The first method I used to determine λ⊥ was transforming the 2D flux images into images of current distribution. Intrinsic problems with the transformation algorithm led to a second approach of fitting a single flux scan to the Biot-Savart law. For the samples studied, λ⊥ was determined to be on the order of the array lattice constant. Finally, we investigated the usefulness of Fisher, Fisher, and Huse dynamical scaling to determine the occurrence of a Kosterlitz-Thouless transition in 2D systems. We simulated current-voltage (IV) curves for a 2D Josephson junction array using appropriate parameters and expressions above and below the Kosterlitz-ThoulessBerezinskii transition temperature (TKT). We also included a contribution arising from

finite-size induced free vortices. The curves were scaled for different voltage cutoffs to simulate the minimum sensitivity of a voltmeter. We found that the value of the dynamical scaling exponent, z, for the best scaling fit and the optimal value of the transition temperature, depended upon the voltage cutoff level chosen; in effect the fit depended upon how much of the finite-sized linear portion of the IV curve that we included.



To My Parents and My Jamesy

Acknowledgements

I am sincerely and extremely grateful to those who have helped me get though the past seven years. I am sure that anything I express here in words will only be understated. First and foremost, I express thanks to my advisor Richard Newrock for his support and guidance. I thank him for the opportunity to do this research and for believing in my ability to complete this work. The many times I was distraught because my work seemed to be going nowhere, he focused on and reminded me of all that I had already accomplished. Thank you to Leigh Smith, Brian Meadows, and Steve Herbert for serving on my committee. I have learned much from Steve Herbert’s involvement in my projects. I especially thank him for his help and advice and time with preparing talks and preparing this dissertation. Richard Gass has been an invaluable source of help and support since day one of my graduate student career. I especially thank him for his help through qualifying exams and with getting this dissertation together. I am grateful to Frank Pinski for looking out for me and making sure I had the time and support to finish my degree. Thank you to our collaborators from the University of Maryland Center for Superconductivity, Chris Lobb, Aaron Nielson, and Brian Straughn. Many thanks to David Will and John Whitaker for solving all my computer crises. Thank you to Bob Schrott for his amazing design and machine work on the SQUID microscope.

.

I am forever indebted to John Markus for absolutely everything! I could have never made it through this without him I am grateful for the friendship of Juliet Evans, Holly Harding, and Jairo Sinova though I have not stayed in close enough contact with any of them due to excuses like qualifying exams or research. I thank Jairo for his help and for giving me a kick in the rear when I needed it to make me work as an undergrad at OU. I am grateful for the love and support of my parents, for their pride in my accomplishments, and for them just wanting me to be happy above all else. Love and gratitude to my boyfriend James Patrick for believing that I could do this and for actually believing that I am intelligent. I thank him for being my friend, confidant, psychiatrist, and for putting up with me through every “hardest test ever”.

Table of Contents page

List of Figures

iv

1. Introduction

1

2. SQUIDs

3

2.1 Introduction to Josephson junctions

3

2.2 SQUID basics

6

2.3 Flux-locked loop

8

3. 2-D Scanning SQUID microscope

11

3.1 Design and construction

11

3.1.1 SQUID Holder

13

3.1.2 Sample Holder

15

3.1.3 Scanning Stage

17

3.1.4 Rotary-to-Linear Motion Feedthroughs

20

3.1.5 Automation

20

3.1.6 Wiring

22

3.1.7 Filters

23

3.1.8 Magnetic shielding

24

3.2 Sample and SQUID preparation

24

3.2.1 SQUID Preparation and Mounting i

24

3.2.2 Sample Preparation and Mounting 3.3 Operation of the Scanning SQUID Microscope

28 31

3.3.1 Leveling the sample

31

3.3.2 Cooling the System

31

3.3.3 Scanning and Data Collection

32

4. SQUID Microscope Images

34

4.1 Flux Images

34

4.2 Resolution

38

4.3 Current Images

39

5. Two-Dimensional Josephson Junction Arrays

43

5.1 Arrays at T=0

43

5.2 Arrays at T>0: The Kosterlitz-Thouless Transition

46

5.3 Experimental Observation of the KT Transition in Arrays

49

5.4 The Importance of λ⊥

53

6. Measurement of Current Distribution and Determination of λ⊥ in Josephson Junction Arrays

54

6.1 Sample Fabrication

54

6.2 First Attempts

55

6.2.1 Sample Characteristics

56

6.2.2 Data

59 ii

6.2.3 Results and Discussion

64

6.3 Second Approach

65

6.3.1 Background Calculations

65

6.3.2 Sample Characteristics

68

6.3.3 Data

71

6.3.4 Results and Discussion

77

7 Finite Size Effects and Dynamical Scaling in Two-Dimensional Josephson Junction Arrays

80

7.1 Introduction

80

7.2 The Two-Dimensional Phase Transition

82

7.3 Scaling

86

7.4 IV Curve Details

91

7.5 Current-Voltage Scaling Results

94

7.6 Discussion and Conclusion

102

Appendix A : SQUID Electronics Schematics

106

Appendix B: Flux-to-Current Algorithm

112

Appendix C: Josephson Junction Array Fabrication Process

116

References

118

iii

List of Figures page Figure 2.1

Schematic diagram for RCSJ model

5

Figure 2.2

SQUID schematic

7

Figure 2.3

Flux-locked loop

9

Figure 2.4

SQUID output

10

Figure 3.1

Schematic of the 2D dc Scanning SQUID microscope system

12

Figure 3.2

SQUID holder

14

Figure 3.3

Sample holder

16

Figure 3.4

Top view of the scanning stage

18

Figure 3.5

End view of scanning stage

19

Figure 3.6

The room temperature flange, SQUID electronics, and stepper motor box

21

Figure 3.7

Filter boxes

23

Figure 3.8

Optical micrograph of a SQUID

25

Figure 3.9

Schematic of SQUID chip mounting technique

27

Figure 3.10 Schematic of sample mounting method

28

Figure 3.11 Side view of scanning stage

30

Figure 4.1

Meander line

36

Figure 4.2

False-color image of the z-component of the magnetic flux above a meander line

37

Flux image of six rows of an SIS Josephson junction array in a small applied magnetic field

38

Typical SQUID noise spectrum

39

Figure 4.3

Figure 4.4

iv

Figure 4.5

Roth et al. algorithm for converting magnetic field data to current distribution data

41

Meander line current distribution calculated from the flux data in Figure 4.2 for different choices of the parameters z and kmax

42

Figure 5.1

Schematic of Josephson junction array

44

Figure 5.2

A single plaquette

44

Figure 5.3

Schematic of uniform phase configuration

47

Figure 5.4

Phase configuration for a single vortex

47

Figure 5.5

An applied current exerts a Lorentz force, FL, on each vortex in a bound pair pushing them apart.

51

Figure 5.6

Typical IV data for a 300x300 array

52

Figure 6.1

SEM picture of an array

55

Figure 6.2

Resistance versus temperature data for 100x100 array

56

Figure 6.3

Current versus voltage characteristics for 100x100 array

57

Figure 6.4

The power law exponent a(T) determined from the slopes of the IV curves in Figure 6.2

58

Image of the magnetic field in the z-direction above the array characterized in Section 6.2.1 at T = 3.25K and I = 100mA

60

Figure 6.6

Current distributions Jx and Jy

62

Figure 6.7

Exponential fit to the Jy data

63

Figure 6.8

λ⊥ calculated from exponential fits to Jy data

64

Figure 6.9

Current through n = 100 wires for Io = 50 µA at different values of λ⊥ 66

Figure 4.6

Figure 6.5

Figure 6.10 The flux through a 10µm square SQUID loop in units of Φo

67

Figure 6.11 Resistance versus temperature for 100x100 array

69

Figure 6.12 Current versus voltage characteristics for 100x100 array

70

v

Figure 6.13 The power law exponent a(T) determined from the slopes of the IV curves in Figure 6.10

71

Figure 6.14 Flux profile at T = 3.97K and I = 25µA

72

Figure 6.15 Flux profile at T = 3.97K and I = 50µA

73

Figure 6.16 Flux profile at T = 3.97K and I = 100µA

74

Figure 6.17 Fit to LHS of data in Figure 6.13 for fixed h = 130µm

76

Figure 6.18 Fit to LHS of data in Figure 6.14 for fixed h = 130µm

76

Figure 6.19 Fit to LHS of data in Figure 6.15 for fixed h = 130µm

77

Figure 6.20 Value of λ⊥ determined from fit to LHS of each data set with h = 139µm

78

Figure 7.1

Simulated current-voltage curves for a Josephson junction array

96

Figure 7.2

Scaled IV curves

98

Figure 7.3

Replot of Figure 7.1(b) showing the voltage cutoffs

99

Figure 7.4

Scaling collapse of finite-size-induced resistive IV data

101

Figure A.1

Bandpass amplifier

106

Figure A.2

Current monitor

107

Figure A.3

Integrator

108

Figure A.4

Phase detector

109

Figure A.5

Preamplifier

110

Figure A.6

Power input filter and regulator

111

vi

1 Introduction Arrays of Josephson junctions are useful model systems for studying a variety of physics. Since they are artificial systems, arrays can be fabricated to display specific desired characteristics. For example, classical two-dimensional (2D) arrays in zero magnetic field are used to study phase transitions such as the Kosterlitz-Thouless (KT) transition [1] where bound pairs of vortices, which are natural thermal excitations of the system, begin to unbind. This work presents two separate studies of issues relevant to understanding the KT transition in Josephson junction arrays. The first is an experimental effort to investigate a required condition for the existence of the transition in an array. The second is a critique of a technique used to determine the occurrence of the transition in 2D systems. To display a Kosterlitz-Thouless transition, certain conditions on the array size must be met: the sample size must be very large (L

’ DQGλ⊥ >> L, where λ⊥ is the

penetration depth for magnetic fields perpendicular to the array plane. However, λ⊥ is not a well-known parameter in these systems. Properties of these Josephson junction arrays such as λ⊥ can be investigated via a Scanning SQUID Microscope system, a system that can produce 2D images of the magnetic flux above a sample. In Chapter 2, I present the basic physics of Josephson junctions and the theory and practical implementation of SQUIDs. Chapter 3 describes the design and operation of my 2D Scanning SQUID Microscope system. Chapter 4 discusses images produced and resolution obtained with this system. Chapter 5 is a discussion of 2D Josephson junction arrays emphasizing the KT transition. Chapter 6 describes the experiment, presents the 1

data, and discusses several approaches to analyzing the data to approximate the value of λ⊥ in these arrays. The dynamic scaling approach of Fisher, Fisher, and Huse [2] has been used to determine the existence of superconducting phase transitions such as the KT transition. If the current-voltage data for a given sample can be made to collapse onto scaling curves, then the KT transition is likely to occur in that sample. However, the scaling collapse alone may not be sufficient evidence to prove the existence of a KT transition. Chapter 7 investigates the role of dynamic scaling in 2D arrays of Josephson junctions.

2

2 SQUIDs

2.1 Josephson Junction Basics A Josephson junction consists of two superconductors connected through a “weak link”. This weak link may be an insulator, (SIS junction), a normal metal (SNS junction), another weaker superconductor (SS’S junction), or a narrow constriction in the superconducting material (a microbridge). Each superconducting region may be described by a complex order parameter [3]

Ψi (r ) = Ψi (r ) e iφ ( r ) , i

(1.1)

where the local density of superconducting particles is represented by the magnitude 2

squared of the order parameter, Ψ (r ) . Josephson predicted that a zero-voltage supercurrent, varying as the sine of the difference in the phases between the two superconductors, could flow between the superconductors [4]

is = ic sin(φ 2 − φ1 ) ,

(1.2)

where ic is the critical current, or the maximum supercurrent the junction can carry. This is the first Josephson equation. When the critical current is exceeded, Josephson further predicted that a voltage would appear across the junction

V=

h d (φ 2 − φ1 ) , 2e dt

and the normal current and supercurrent would be time varying.

3

(1.3)

The energy stored in the junction can be derived by integrating the power ∫ i sVdt to obtain

E=−

hi c cos(φ 2 − φ1 ) = − E J cos(φ 2 − φ1 ) 2e

(1.4)

where E J = hic 2e is called the Josephson coupling energy. In the presence of a magnetic field, the phase difference (φ 2 − φ1 ) must be replaced by a gauge invariant phase difference γ

2π 2 γ = φ 2 − φ1 − ∫ A ⋅ ds , Φo 1 where Φ o =

(1.5)

h = 2.07 x10 −15 T-m2 is the flux quantum. 2e

Equation (1.2) describes the supercurrent through an ideal Josephson junction. In reality, currents can follow other paths across the junction. In the RCSJ (resistively and capacitively shunted junction) model [Figure 2.1] [5,6], the ideal junction is shunted by a resistance R and a capacitance C. R represents the quasiparticle tunneling through the barrier for an SIS junction and the normal metal resistance for an SNS junction. Because there is a voltage across the junction, a current of normal electrons can flow

in =

1 h dγ V = . Ro Ro 2e dt

(1.6)

Since the junction consists of two metal surfaces, there is a capacitance across the junction leading to a displacement current

id = C

h d 2γ dV . =C dt 2e dt 2

4

(1.7)

The sum of the currents from the three branches of the RCSJ model is

i = ic sin γ +

h dγ h d 2γ +C . 2eRo dt 2e dt 2

(1.8)

By making the change of variable τ = ωpt, where 1/ 2

 2ei  ωp =  c   hC 

(1.9)

is the “plasma frequency” of the junction, Equation (1.8) can be written as i d 2γ −1 / 2 dγ = 2 + βc + sin γ . i c dτ dτ

(1.10)

The resulting equation is analogous to a driven harmonic oscillator with damping parameter βc, where

βc

1/ 2

= ω p RC .

ic

(1.11)

Ro C

Figure 2.1. Schematic diagram for RCSJ model. From left to right are the supercurrent, capacitive, and resistive channels.

5

If C is small so that βc1/2 ic, the voltage across the SQUID varies periodically with the applied field with period Φo [Figure 2.2(c)]. Thus, a SQUID is a flux-to-voltage transducer with transfer coefficient VΦ δV/δΦ, producing an output voltage in response to a small input flux [7].

6

ic

ic

Ib

Figure 2.2 (a) Schematic of the dc SQUID; (b) IV characteristics; (c) V vs. Φ/Φo at constant bias current Ib. From Reference 7.

7

In addition to the externally applied flux, the screening currents circulating around the SQUID loop contribute to the total flux in the loop. The circulating supercurrent producing this screening flux is

I circ =

ic (sin γ 2 − sin γ1 ), 2

(1.14)

assuming the critical current is the same in both junctions. The total flux in the loop is then the sum of the externally applied flux and the self-induced flux

Φ = Φ ext + 12 LI circ = Φ ext + 12 Lic (sin γ 2 − sin γ 1 ).

(1.15)

The phase differences γ1 and γ2 are constrained by [3]

γ 1 − γ 2 = 2πΦ / Φ o (mod 2π).

(1.16)

A special case occurs when Φ = nΦo (see Figure 2.2(b)), for which γ1 = γ2(mod 2π): Icirc= 0 and Φ = Φext. In this case, the flux through the SQUID loop is not affected by the screening. For optimal performance, the SQUID parameters must satisfy certain constrains [7]. Hysteresis in the IV characteristics is avoided by requiring βc ”7RDYRLGPDJQHWLF hysteresis, a screening parameter βm is defined such that βm /,c/Φo ”ZKHUHWKH size and shape of the SQUID determine L, the loop inductance.

2.3 Flux-Locked Loop In practical applications, a SQUID is used as a null detector of magnetic flux. It is usually connected to a “flux-locked loop” feedback circuit, shown schematically in Figure 2.3. This circuit linearizes the response of the dc SQUID to changes in the applied 8

Φ αΦ

Feedback

Figure 2.3 The flux-locked loop.

magnetic field. A feedback coil is used to modulate the flux threading the SQUID. This flux has a peak-to-peak amplitude of Φo/2 and a typical frequency of fm=100kHz. If the quasistatic flux through the SQUID is an integer multiple of Φo the voltage across the SQUID (the “output” voltage) contains only a signal at twice the modulation frequency [Figure 2.4(a)]. If the SQUID output is lock-in detected at the modulation frequency, the output will be zero. If the quasistatic flux is (n+1/4)Φo, the SQUID output voltage is at the modulation frequency [Figure 2.4(b)]. Thus, as the external flux increases from nΦo to (n+1/4)Φo, the output from the lock-in increases; similarly, if the external flux is reduced from nΦo to (n-1/4)Φo, the lock-in’s output will increase in the negative direction [Figure 2.4(c)].

9

nΦo

(a)

(b)

(c)

Figure 2.4 Output voltage across the SQUID for (a) Φ = nΦo and (b) Φ = (n+1/4)Φo. (c) Output from the lock-in detector. From Reference 7.

The SQUID output is coupled via a transformer at liquid helium temperature to a low-noise, room temperature preamplifier. After amplification, the signal is lock-in detected at the frequency fm and integrated. The integrator output drives a resistor. The voltage across this resistor is proportional to the flux change in the SQUID and the current is used to drive the feedback coil. Thus, if the SQUID detects a small flux change δΦ, the feedback circuit generates an opposing flux -δΦ. Using this technique, flux changes as small as a few µΦo and as large as several flux quanta can be measured.

10

3 2D Scanning SQUID Microscope

3.1 Design and Construction Figure 3.1 shows an overall schematic of the 2D scanning SQUID microscope. The design is based a similar system designed and constructed by Professor Fred Wellstood’s group at the University of Maryland Center for Superconductivity Research [8,9]. The basic support structure and scanning stage for our microscope are the same as the previous microscopes, but notable improvements have been made in the SQUID holder, the sample holder, and in the system control and automation. These will be described below.

11

Figure 3.1 Schematic of the 2D dc Scanning SQUID microscope system.

12

3.1.1 SQUID Holder A photograph of the SQUID holder is shown in Figure 3.2. A gold plated copper post holds a sapphire cone [10] onto which the SQUID chip is mounted. The sapphire cone provides an excellent thermal link between the SQUID and the holder while at the same time isolating it electrically. The details of mounting and preparing the SQUID will be discussed in a later section. The post is attached to a U-shaped copper base and brass mounting brackets allow this holder to be attached to the SQUID base plate (see Section 3.1.3) and provide a weak thermal link between the copper and the plastic. The transformer (discussed in section 1.3) is mounted to the post of the SQUID holder. The primary and the secondary of the transformer consists of 4 and 200 turns respectively of 38 AWG copper wire around a ferromagnetic core [11]. A 1Ω resistor is inserted in series with the transformer primary to prevent it from shorting out the SQUID at low frequencies. Resistive wire [12] is wound around the base of the post to form a 50Ω heater. This heater is used to raise the SQUID temperature above its critical temperature to expel any flux trapped in the SQUID. All wires are thermally grounded to the SQUID holder by wrapping them around two small copper spools and securing them with IMI-7031 varnish [13]. Miniature multi-pin connectors [14] are used to connect to the SQUID and transformer wiring.

13

Au-plated Cu

Figure 3.2 The SQUID holder (a) side view of the post and base; (b) top view of the post; (c) side view of assembled SQUID holder.

14

3.1.2 Sample Holder The basic design of the sample holder [Figure 3.3] is similar to that of the SQUID holder. A gold-plated copper post holds a 1cm diameter sapphire rod [10] on which the sample can be mounted. The details of preparing and mounting samples will be discussed later. A U-shaped copper base and brass mounting brackets allow the sample holder to be attached to the sample stage. A three-point leveling system is formed by mounting a spring and a flexible piece of phosphor bronze between the copper base and the brass brackets. A germanium thermometer [15] to measure and control the sample temperature is inserted into a blind hole drilled into the sample holder post. A 50W heater was formed by winding resistive nickel-chromium wire [16] around the bottom of the sample holder post. Thermal grounding of the sample and thermometer wiring is accomplished by using both a pin block on the side of the sample holder and small copper spools mounted to the sample holder post. Miniature multi-pin connectors connect to the sample holder.

15

Figure 3.3 (a) Sample holder post and base; (b) assembled sample holder.

16

3.1.3 Scanning Stage The scanning mechanism [Figure 3.4 and Figure 3.5] consists of a system of sliders which move the sample holder in the horizontal (x) and vertical (y) directions. The sample holder is mounted on a Teflon “internal slider” whose motion is constrained by two Delrin sliders which slide in the vertical direction on Teflon rails. Slider A has a horizontal slot which allows the internal slider to move left to right. Slider B has a diagonal slot, slanted about 14o from the vertical direction. To move the internal slider in the vertical direction, slider A and slider B are moved simultaneously. To move the internal slider in the horizontal direction, slider A is held stationary while slider B is moved. The internal slider moves along the diagonal groove of slider B from left to right but is prevented from moving in the vertical direction by slider A. The internal slider is held firmly against slider B with springs [17] to minimize hysteresis in the horizontal position. A Delrin base plate for mounting the SQUID above the sliders is held under tension by four springs [17]. This base plate pivots on a brass hinge on one end and, on the other end, a Teflon wedge tilts the plate in the z- direction. The scanning stage is constructed from non-metallic and non-magnetic materials to reduce electrical noise and stray magnetic fields that would be detected by the SQUID. The sample is moved in the x- and y- directions while the SQUID remains stationary to minimize SQUID vibrations and pickup from ambient field variations. The scanning mechanism is bracketed to the cold flange and is enclosed in a stainless steel vacuum can to allow for sample temperature variation.

17

Figure 3.4 Top view of the scanning stage. (The SQUID stage has been removed.)

18

Figure 3.5 End view of scanning stage.

19

3.1.4 Rotary-to-Linear Motion Feedthroughs The sliders A and B and the z-wedge are moved using three rotary-to-linear motion vacuum feedthroughs [18]. The feedthroughs are mounted on a stainless steel feedthrough flange with copper gaskets forming the vacuum seal [Figure 3.6]. The feedthrough flange is bolted above the opening of a large 2.75” OD tube leading to the vacuum can. Initially, a viton O-ring was used to seal the feedthrough flange to the room temperature flange, but we suspected this seal to be the source of a slow vacuum leak when the flange got too cold. This O-ring was replaced by an indium gasket. As the feedthrough is rotated, vacuum-sealed bellows are driven on a threaded rod, advancing the shaft. The shafts are attached to 3/8” OD stainless steel tubes which travel through the 2.75” OD tube into the vacuum can and attach to the sliders and wedge. Seizing of the feedthrough drive rods became a constant problem so modifications were made to the feedthroughs. Ball bearing assemblies were constructed and mounted on the top of the feedthroughs to counteract the torque from the drive belts to keep the shaft perpendicular to the rest of the body.

3.1.5 Automation The movement of the z-pusher is controlled manually, and it remains fixed throughout a scan. The movement of the x and y feedthroughs is controlled by computer-driven stepper motors [19] and a system of timing belts and pulleys [20] [Figure 3.6]. I built a motor control box to contain microstep motor drivers for the two stepper motors [21], a power supply [22], and a positioning circuit. This positioning circuit applies a constant voltage to two 10-turn potentiometers geared down from the x and y feedthroughs. The 20

gear

Figure 3.6 The room temperature flange, SQUID electronics, and stepper motor box.

21

outputs of the potentiometers are displayed on the front of the motor control box and can be used to estimate the sample position. A rack-mounted 486 computer contains a motor controller board and connects to the a/d board which is housed in a separate breakout box. A C program written by Fred Cawthorne and updated by Aaron Nielsen and other current students of Chris Lobb controls the scanning motion and the a/d board via both a touch screen interface and a joystick. The program records the SQUID output voltage data as a function of stepper motor steps.

3.1.6 Wiring Five 10-pin Bendix hermetically sealed connectors [23] are mounted on the room temperature flange and connected via Swagelok ½” OD tube fittings [24] to three ½” OD stainless steel tubes. The wiring between room temperature and the cold stage passes through these tubes and each twisted pair of wires is contained in an individual coppernickel tube to provide shielding. The stainless steel and the Cu-Ni tubes end in the liquid helium space above the cold flange. Tinned copper braid is soldered to the end of each Cu-Ni tube to provide a flexible shield. Cryogenic vacuum electrical feedthroughs [25] are mounted on the cold flange to make electrical connections into the vacuum can [Figure 3.4]. Inside the vacuum can, the wires pass through a short piece of the Cu-Ni tubing then miniature multi-pin connectors connect the wires to the sample and SQUID holders [Figure 3.11].

22

The SQUID wiring is isolated from the other wires in its own tube. All of the SQUID connections are made using Duo-Twist [26] wire. The wiring from two of the Bendix connectors passes through the second tube. One connector is used for a liquid helium level sensor and the other connector for the sample thermometer. Wires from the remaining two Bendix connectors pass through the third tube to make connections to the sample. Each connector is wired with 35AWG copper wire for current leads and DuoTwist wires for voltage leads.

3.1.7 Filtering To eliminate high-frequency noise, low pass filters [27] are connected to each sample lead. These filters are enclosed in a cylindrical aluminum box with a 10-pin Bendix connector at each end [Figure 3.7].

Figure 3.7 Filter boxes 23

3.1.8 Magnetic Shielding The SQUID microscope is housed in an rf-shielded room [28] ten feet by twelve feet in area and ten feet in height. Two concentric µ-metal shields are placed around the outside of the dewar. The shields are each approximately 0.03” thick and the outside shield is about 3” higher than the inside shield. Another layer of shielding is possible by inserting a lead liner into the vacuum can, but it is often not used; we suspect that the lead shield traps flux inside the vacuum can.

3.2 Preparing and Mounting the SQUID and Sample 3.2.1 SQUID Preparation and Mounting The low-Tc Nb-NbOx-Nb SQUIDs used in this system [Figure 3.8] are fabricated by Hypres [29]. The SQUID loop is square with 10µm sides. The feedback loop is a single niobium loop fabricated directly onto the SQUID chip. Large gold contact pads for the SQUID and feedback loop wiring form a circle of about 1mm diameter around the SQUID. I mount a SQUID chip to the top of a sapphire cone using Stycast epoxy [30] and allow it to set overnight. Using a rotary tool and a disk of fine sandpaper, I grind the SQUID chip down to fit the tip of the sapphire. Photoresist, which was put on the chip by Hypres prior to shipping to protect it, and the dust from grinding, are cleaned off in a beaker of acetone in an ultrasonic cleaner. I have had frequent trouble with the gold pads peeling off of the chip during the grinding and cleaning process, but this has not

24

caused a major problem; I make contact directly to the SQUID or feedback loop lead if the contact pad is absent.

junctions

feedback loop 10µm shunt resistors

Nb

Figure 3.8 Optical micrograph of a SQUID.

A layer of silver is deposited onto the SQUID and sapphire in a Veeco thermal evaporator system in the Physics Department cleanroom. A small motor was installed in the evaporator chamber to continuously rotate the SQUID so that all sides can be covered during a single deposition.

25

I pattern the silver for etching by hand-painting Shipley 1813 photoresist [31] with a small wire. Gold etch removes the excess silver in a few seconds and the photoresist is removed in acetone leaving strips of silver connecting the wiring from the SQUID chip to the sides of the sapphire cone. The silver does not reliably adhere to the sanded Stycast, so I use silver paint [32] to bridge this gap in the silver film. I put a dab of Apiezon N grease [33] in the top of the SQUID holder and insert the sapphire and tighten the screw to clamp it in place. Two 2-pin miniature connectors are attached to each side of the sample holder and 0.003” gold wires [34] are soldered to each pin then connected to the silver film on the side of the sapphire using silver paint. A schematic of this SQUID mounting configuration is shown in Figure 3.9. Before mounting in the microscope, I test the SQUID in a quick-dip probe. The post part of the SQUID holder can be mounted in this probe. After I am satisfied that I have a working SQUID, I assemble the SQUID holder and bolt the brass brackets to the top of the SQUID base plate and connect the SQUID holder wiring to the probe wiring.

26

SQUID chip

Ag film

Stycast

Ag paint

Au wire

Ag film

Sapphire cone

miniature multi-pin connector

SQUID holder

Figure 3.9 Schematic of SQUID chip mounting technique (not to scale).

27

3.2.2 Sample Preparation and Mounting Since the SQUID must remain as close as possible to the sample for optimum resolution, attaching wires to the samples posed a problem. If wiring were attached above the plane of the sample, the SQUID would run into the wires. A solution to this problem is shown in Figure 3.10. I attach the sample to a slightly larger piece of silicon wafer with Stycast epoxy. Using aluminum foil as a mask, I deposit a silver film in the thermal evaporator to form a connection between the sample contact pads and the wafer below the sample. I then attach #40 copper wires to the silver film on the bottom wafer using silver paint.

Ag paint

Sample chip

Ag film

Cu wire

Stycast epoxy

piece of silicon wafer

Figure 3.10 Schematic of sample mounting method (not to scale).

28

The sample is mounted on the sapphire rod using Apiezon N grease and the wires are soldered to the appropriate posts on the top of the sample holder. To attach the sample holder to the internal slider, I insert the sample holder post through the hole in the internal slider from the front then connect the sample holder base to the post from the back of the slider. The brass mounting brackets are then attached to the back of the internal slider and the sample and thermometer wires are connected to the probe. The SQUID plate must be flipped up out of the way while mounting the sample holder, so it is easier to first mount the sample holder, then put the SQUID plate in place and attach the springs, and then mount the SQUID holder last. Figure 3.11 shows the complete assembly of the sample stage and SQUID holder in the microscope ready for scanning.

29

hinge

z-wedge

Figure 3.11 Side view of scanning stage with sample and SQUID holders in place.

30

3.3 Operation of the Scanning SQUID Microscope 3.3.1 Leveling the Sample A cradle was constructed so that the probe could lie horizontally on a lab bench. The stepper motor box can be mounted to this holder so that the scanning stage can be driven while the probe is in the horizontal position. All leveling is done while the probe is at room temperature and in the horizontal position. After the sample and SQUID holders are in place, I rotate the probe so that I can examine the distance between the SQUID and sample from the side. I set up a microscope above the scanner and measure the SQUID-sample separation at several points around the sample. The leveling screws on the sample holder base are used to adjust the sample level. After leveling the sample, the SQUID is moved several millimeters away from the sample to avoid collisions with the sample while moving the probe. I do not level the SQUID, but a similar three-point leveling system on the SQUID holder base would be a useful future modification.

3.3.2 Cooling the System After all mounting and leveling steps are done, I hang the probe in a vertical position, attach the vacuum can, and pump out the can. If the scanning stage has not been exposed to air for more than a couple of hours, the can evacuates in about an hour. However, if the scanning stage has been exposed to air for a longer time, I usually pump the can out overnight to allow the plastics to outgas. 31

The probe is inserted into a 6” diameter dewar with a liquid nitrogen jacket. I put 500 - 1000 millitorr of He exchange gas into the can and then fill the dewar and the jacket with liquid nitrogen. The sample thermometer reads liquid nitrogen temperature within about 4 hours. To remove the liquid nitrogen, I pressurize the dewar to force the LN2 up the fill tube. The clearance between the cold flange and the dewar wall is very small, so to get liquid into and out of the bottom of the dewar, a small tube was attached to the side of the can. This tube has a funnel to meet the fill tube on top and a piece of latex tubing to reach the middle of the bottom of the dewar. After removing the liquid nitrogen, the remaining N2 gas is purged from the dewar and it is filled with He gas. The first liquid helium transfer takes about an hour and requires about 30 l of LHe. Subsequent transfers require about 10 l of LHe. The microscope stays cold for about 12 hours at 4.2 K, but that time is greatly reduced if the bath is pumped to reach lower temperatures.

3.3.3 Scanning and Data Collection After the system has reached liquid helium temperature, I connect the electronics box to the SQUID connector on the room temperature flange. The SQUID electronics for this system are based on the electronics of Wellstood et al. [35], with changes in some details, including using a 100 kHz modulating frequency. (See Appendix A for SQUID electronics schematics.) The program “SQUIDBOX.EXE” on the rack-mounted 486 computer controls the scanning and the data acquisition. The scan area is set by choosing the beginning and ending points of the scan. The program allows the scan to be performed in either the 32

horizontal or vertical directions, but the vertical direction is the best choice since there will be less motion of the two sliders against one another and thus, less heat generated from friction. The program allows the number of scans, the number of points per scan, the scan speed, and the gain to be chosen. Each line scan is displayed on the screen in real time. The computer stores the SQUID output data as a function of sample position in units of stepper motor steps. I start by performing several rough scans with the SQUID at a large distance from the sample and the scan limits set at the limits of the potentiometers to try and determine the position of the sample. A major problem in this system is the inability to know the height of the SQUID above the sample. The current method I use to try to get the SQUID as close as possible to the sample is to slowly move the SQUID toward the sample until the two touch then move it slightly away. By watching the SQUID output on an oscilloscope, I can determine when the SQUID makes contact with the sample; the SQUID output becomes hectic and unlocks. If the sample is not level, the SQUID and sample may collide during the scan destroying the SQUID and/or the sample.

33

4 SQUID Microscope Images

4.1 Flux Images As discussed in Chapter 2, the feedback electronics holds the total flux in the SQUID, Φtot, constant at an integer multiple of Φo. Φtot may contain contributions from several sources and can be written as

Φ tot = nΦ o = Φ s + Φ a + Φ f + Φ r + Φ b .

(4.1)

The flux due to the sample, Φs, could result from currents applied to the sample or screening currents in superconducting samples. Φa is the flux from an externally applied magnetic field. The flux due to the feedback coil is

Φ f = Vout M f / R f ,

(4.2)

where Vout is the output voltage from the feedback electronics, Rf is the monitoring resistor, and Mf is the mutual inductance between the feedback loop and the SQUID. Φr is the flux from any residual fields trapped in the system. The last term, Φb, is the flux from bias currents in the SQUID and can be neglected if the two arms of the SQUID are geometrically identical and the critical currents of the two junctions are equal [36]. In the absence of an external applied field, the output from the feedback electronics is

Vout =

Rf Mf

(nΦ o − Φ s − Φ r ) .

34

(4.3)

In this system, Mf / Rf was measured as 1.54 V/Φo. The value of n at which the SQUID locks is generally not known, but can be determined using a superconducting sample (see [9], Chapter 5). Without knowing the values of n and the residual field, the absolute value of the flux in the SQUID and hence the actual field above the sample cannot be determined. My solution to this problem is to first, perform a background scan with no current applied to the sample. Next, with the SQUID remaining locked at the same value of n, I scan the same area with the current applied. Then, I subtract the zero-current data from the scan data with the applied current to normalize the data to the background. This eliminates the residual field and the nΦo contributions and the resulting flux should then be only the flux due to the applied current in the sample. I use the software program Transform [37] to produce an image of the magnetic field from the normalized flux data. One of the initial test samples I scanned was a #40 AWG copper wire forming a meander line glued to a glass cover slide with IMI varnish [Figure 4.1]. A magnetic field image of this sample with a 5mA injected current is shown in Figure 4.2. For this image, the data is not normalized to zero-current data and the field units are arbitrary.

35

Figure 4.1 Photograph of a meander line constructed from 40AWG copper wire glued between two glass cover slides with IMI Varnish.

36

mm

40AWG wire, I=5mA 0 1 2 3 4 5 6 7 0.00

2.50

5.00

7.50

10.00 12.50

mm

0 magnetic field in z-direction, arbitrary units

Figure 4.2 False-color image of the z-component of the magnetic field above a meander line made from a #40 wire. The magnetic field increases in the positive direction toward the red end of the spectrum and increases in the negative direction toward the violet end.

37

4.2 Resolution The spatial resolution of the microscope depends on both the size of the SQUID and the separation between the SQUID and sample. Smaller SQUIDs produce better spatial resolution if the SQUID-sample separation is on the order of the size of the SQUID loop. The SQUIDs currently being used have a 10µm square loop. The best spatial resolution I have been able to observe has been on the order of tens of microns. Figure 4.3 is a flux image of the first several rows of an array of SIS Josephson junctions. This array has a lattice spacing of 50µm and the width of the wiring is 10µm.

mm

SIS JJA, ~30mG applied field 0.05 0.10 0.15 0.20 0.25 0.30 0.2

0.4

0.6

0.8

1.0

1.2

1.4

mm

Figure 4.3 Flux image of six rows of an SIS Josephson junction array in a small applied magnetic field. The array has a lattice spacing of 50µm with 10µm wide wiring. The centers of alternating superconducting islands display different intensities of flux due to the construction of the sample.

38

T = 3.50K -10 2 SQUID pickup area = 10 m

-4

-9

1x10

T/ Hz

Φ0/ Hz

-1/2

-1/2

1x10

-5

-10

1x10

1x10

-6

-11

10

10 0

10

1

10

2

10

3

10

4

10

f (Hz)

Figure 4.4 Typical noise spectrum for a Hypres manufactured SQUID with pickup area 10-10m2.

Using a Hewlett-Packard 35665A spectrum analyzer, I measured the root mean square power spectral density of the voltage output of one of the SQUIDs. This was converted to a flux noise spectrum by multiplying by 1.54 volts/Φo. Figure 4.4 shows a typical spectrum from one of the Hypres manufactured SQUIDs. The noise level is on the order of tens of µΦo and for a 10µm square SQUID loop, corresponds to a magnetic field resolution on the order of a hundred pT Hz–1/2.

4.3 Current Images A method of obtaining an image of a two-dimensional current distribution from magnetic field data has been developed by Roth, Sepulveda, and Wikswo [38]. The algorithm is outlined in Figure 4.5. The z-component of the magnetic field is Fourier transformed and then multiplied by an inverse filter function. Since the Fourier transform amplifies high-frequency noise, 39

the magnetic field in frequency space is low-pass filtered by multiplying by a Hanning window. The current density is then obtained by performing an inverse Fourier transform. The cutoff frequency for the low-pass filter, kmax, is somewhat arbitrary. Larger kmax would increase the spatial resolution of the image, but would also increase the background noise; smaller kmax would eliminate much of the noise, but would decrease spatial resolution. Roth et al. found that for a square SQUID with sides L, kmax must be less than 2π/L for the problem to be solved uniquely. Another primary factor contributing to the spatial resolution of the image is the height, z, of the SQUID above the sample. There is currently no method of measuring z in this SQUID microscope system (see section 3.3.3). I have found that changing my estimate for z by as much as tens of microns does not have a significant effect on the spatial resolution of the resulting image. Figure 4.6 shows images of the current distribution obtained from the magnetic field image of the meander wire in Figure 4.2 for different choices of z and kmax. The color spectrum indicates the amplitude of the current density with red again being the most intense. The current appears to be more dense in some regions of the wire than in others. There are several possible explanations for this. First, since this was one of the earliest test samples imaged in the system, I did not attempt move the SQUID very close to the sample to obtain the best possible resolution. Second, the wire is not glued down perfectly flat in the x-y plane, so the SQUID would detect a higher field as it passed over raised portions of the wire. Also, resolution is lost as a result of Fourier transforming data with noise resulting in a widened and rounded current distribution with reduced amplitude.

40

Bz ( x, y ) + ( Noise) ⇓ FFT ⇓ bx (k x , k y ) + (noise) ↓ inverse filter : − i

2 k y kz e µo z k

↓ k  1 cos + π  k max , Hanning window :  2   0, ⇓ j x (k x , k y ) + (noise)

k < k max k > k max

⇓ FFT −1 ⇓ J x ( x, y ) + ( Noise)

Figure 4.5 Roth et al. algorithm for converting magnetic field data to current distribution data. The Transform code used is in Appendix B.

41

0.00

1.25

1.25

2.50

2.50 mm

0.00

3.75

3.75

5.00

5.00

6.25

6.25

7.50 0.0

7.50 0.0

2.0

4.0

6.0

8.0 10.0 12.0

2.0

4.0

6.0

8.0 10.0 12.0

mm

mm

#40 wire, I=5mA, z=50um, kmax=250 1/mm

#40 wire, I=5mA, z=50um, kmax=500 1/mm 0.00

1.25

1.25

2.50

2.50 mm

0.00

3.75

3.75

5.00

5.00

6.25

6.25

7.50 0.0

mm

#40 wire, I=5mA, z=20um, kmax=500 1/mm

2.0

4.0

6.0

7.50 0.0

8.0 10.0 12.0

2.0

4.0

6.0

8.0 10.0 12.0

mm

mm

#40 wire, I=5mA, z=100um, kmax=250 1/mm

#40 wire, I=5mA, z=100um, kmax=500 1/mm

0.00

0.00

1.25

1.25

2.50

2.50 mm

mm

mm

#40 wire, I=5mA, z=20um, kmax=250 1/mm

3.75

3.75

5.00

5.00

6.25

6.25

7.50 0.0

7.50 0.0

2.0

4.0

6.0

8.0 10.0 12.0

mm

2.0

4.0

6.0

8.0 10.0 12.0

mm

Figure 4.6 Meander line current distribution calculated from the flux data in Figure 4.2 for different choices of the parameters z and kmax.

42

5 Two-Dimensional Josephson Junction Arrays Newrock et al.[39] recently published an excellent review article of the physics of two dimensional Josephson junction arrays. In this chapter, I summarize the details relevant to this research. All of the following discussion is for arrays in zero applied magnetic field.

5.1 Arrays at T = 0 The simplest two-dimensional Josephson junction array is a square lattice with M rows and N columns and lattice constant a, shown schematically in Figure 5.1. “Islands” of superconductor are connected to their nearest horizontal and vertical neighbors by Josephson junctions. The unit cell of an array is known as a “plaquette”. Each superconducting island is characterized by a phase γ defined modulo 2π. If we sum the phase differences around any closed path in the array, the total phase difference must change by 2πn, where n is an integer. The contributions from each Josephson junction and each superconducting island around the closed path must be included in the sum. Consider a closed path around a single plaquette as shown by the solid and dotted lines in Figure 5.2. From Equations (1.2) and (1.5), the phase difference across the junction connecting island i to island j is

 iij φ j − φi = sin −1   ic

43

 2e j  + ∫ A ⋅ dr .  h i

(5.1)

a X X superconducting islands

X X

X X

X X

X X

plaquette

X

X

X

X

X

X X X

X X

X X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X X

X

X

X

X

X

X

X

X X

X

X

X

X

X X

Josephson junctions

X X X

Figure 5.1 Schematic of Josephson junction array with lattice constant a. The squares represent the superconducting islands and the X’s are the junctions joining the islands. A single unit cell (“plaquette”) is highlighted in the center.

i

j

Figure 5.2 A single array plaquette. The ith and jth islands are labeled, and φi and φj are the phases at the adjacent edges of the two islands.

44

The phase difference across the superconducting island from point j to a point r along the path through the island is given by r

φ j − φr =

∫ j

 m* J 2e  s  A ⋅ dr . +  2eh ψ 2 h   

(5.2)

Thus, the total phase difference around the path is

∑

 −1  iij  sin   +  ic   junctions

   islands 

∑∫

 2e  m* J  s   ⋅ dr  +  2eh ψ 2   h  

∫

A ⋅ dr = 2πn (5.3)

The first term is the sum of the phase differences across the junctions traversed by the path. The second term is the sum of all of the phase differences across all of the islands in the path. Because of the Meissner effect, a path through the islands with Js=0 can always be found making the second term equal to zero. The third term is the integral of the vector potential around the entire path, which is the total flux enclosed by the path, Φtot. Equation (5.3) can be simplified to

 Φ γ ij = 2π n − tot Φo junctions 

∑

  . 

(5.4)

The equilibrium value of n will be that which minimizes the free energy of the system. For static phase configurations, the energy of the array will be the sum of the individual junction energies; from Equation (1.4),

H = −∑ E J cos γ ij . ij

45

At zero temperature and zero applied magnetic field the energy is minimized when all of the phases are equal. Figure 5.3 shows this phase configuration schematically. The energy will be

E ground state = −2 MNE J .

(5.6)

Now consider the phase configuration shown in Figure (5.4). In going around any closed loop containing the center plaquette in the array, the total phase changes by 2π. These phase differences will induce whirlpools of circulating supercurrents forming a vortex. The energy of this single vortex in an array of width L = Ma = Na in the thermodynamic limit where L→ ∞ can be calculated at T = 0 as

 L E1vortex ≅ πE J ln  − 2 MNE j . a

(5.7)

This energy for a single vortex in an array in zero applied magnetic field is larger than the ground state energy and thus, not energetically favorable.

5.2 Arrays at T > 0: The Kosterlitz-Thouless Transition At temperatures greater than zero, thermal energy will induce vortices in the system. Kosterlitz and Thouless [1] showed that these thermally generated vortices will form in bound pairs of opposite circulation. The energy of this vortex pair bound a distance r apart has been calculated as [40]

r E pair = 2πE J ln  . a

(5.9)

The above equation is valid for a pair of vortices whose separation r is less than λ⊥,

46

Figure 5.3 Schematic of uniform phase configuration (ground state) of array for B=0. The phase of each island is represented by the angle of the arrow on that island. The junctions between the islands have been omitted for simplicity.

Figure 5.4 Phase configuration for a single vortex.

47

where λ⊥ is the penetration depth for magnetic fields perpendicular to the array. [41] It was shown in the previous section that the energy required to add a single free vortex in the system is

 L Evortex = πE J ln  . a

(5.10)

This is valid for very large L and λ⊥ >> L. Since Evortex>>Epair, thermal generation of bound pairs of vortices is energetically favored. At sufficiently low temperatures, no free vortices will be present in the system. At a critical temperature called the Kosterlitz-Thouless temperature, TKT, a phase transition occurs where bound pairs of vortices begin to unbind and free vortices appear in the system. This is the Kosterlitz-Thouless transition. The critical temperature TKT can be estimated by examining the free energy of the system. Introducing a single free vortex in an array causes a change in free energy of

∆F = Evortex − T∆S vortex ,

(5.11)

where ∆Svortex is the positional entropy of the vortex in the array,

 L2  ∆S vortex = k b ln 2  . a 

(5.12)

This gives

 L ∆F = (πE J − 2k bT ) ln  . a When the positional entropy term dominates, the system can create a free vortex to lower its energy. We can estimate the minimum temperature at which free vortices appear in an infinite array as

48

πE J . 2k b

TKT =

(5.14)

The above estimate for TKT is a good approximation, but not exactly correct. The interaction energy of a vortex pair given by Equation (5.9) is correct for only two vortices in the system. Many pairs of thermally generated vortices will be created. Two vortices separated by a large distance may have smaller vortex pairs between them, which weaken the interaction potential and reduce the value of EJ [1]. This is known as renormalization of the vortex-vortex interaction.

5.3 Experimental Observation of the KT Transition in Arrays The density of thermally induced free vortices in an array is given by

T < TKT

 0 n f (T ) =  −2 [ b /(T~ −T~ b1a e 2

1/ 2

KT

)]

T > TKT

,

(5.15)

~

where b1 and b2 are constants of order 1, and T is a dimensionless temperature defined as

~ 2πk BT . T= Φ o ic (T )

(5.16)

If a small external current is applied to the array, the free vortices will move resulting in a voltage proportional to the density of free vortices

V = 2 IRo

L 2 a nf , W

(5.17)

where L and W are the length and width of the array. The resistance above the KosterlitzThouless temperature can be obtained from Equations (5.15) and (5.17) as

49

L V 2 Ro b1 exp[−b2 /(T~ − T~KT )]1 / 2 R(T ) = =  W I  0 

T > TKT . T ≤ TKT

(5.18)

It is only in the limit of zero current that we get a true zero-resistance state below TKT. A nonzero current will unbind vortex pairs resulting in a nonzero free vortex density below the transition temperature. This is illustrated in Figure (5.5). A current density j exerts a Lorentz force on the vortex pair, acting on each member of the pair in an opposite direction. This force causes the pair to unbind more easily than it would due to thermal effects alone. This current-induced unbinding results in a density of free vortices given by 1/ 2

 2πic (T ) Ro   n f (T ) ≈  Φo  

 i     ic (T ) 

πE J

k BT

.

(5.19)

Substituting this into Equation (5.17) gives 1/ 2

V ≈ 2 Ro

3/ 2

 2π   (ic (T )1 / 2−πE La  Φo 

J

/ k BT

(i ) πE

J

/ k BT +1

(5.20)

or

V ∝ I a (T )

(5.21)

where

a (T ) =

πE J (T ) + 1. k BT

(5.22)

Below TKT, the exponent a(T) is greater than 3 and the sample IV curves display power-law behavior described by Equation (5.20). Above TKT and in the zero current limit, a(T) = 1 and the IV curves will be ohmic as in Equation (5.18). At T= TKT, there is 50

a universal jump in a(T) from 1 to 3. Thus, examination of the current-voltage characteristics of an array can determine the presence of the Kosterlitz-Thouless phase transition.

F

j

FL = ± Φ o j × zˆ

F Figure 5.5 An applied current exerts a Lorentz force, FL, on each vortex in a bound pair pushing them apart.

As an example, see Figure 5.6 taken from Herbert et al. [42]. The top graph is a loglog plot of the IV characteristics for a 300x300 array. The value of a(T) for each temperature is equal to the slope of the curve. The two solid lines in the graph have slopes of one and three. The highest temperatures on the left side of the plot are ohmic. The lower temperature curves on the left have power-law behavior. The bottom graph is a plot of the exponent a(T) versus temperature. There is a clear jump in a(T) from one to three at TKT.

51

Figure 5.6 top: Typical IV data for a 300x300 array. The solid lines have slopes of 1 and 3. bottom: a(T) vs. T determined from the slopes of the IV curves. From Reference 42.

52

5.4 The Importance of λ⊥. The preceding discussions assumes that a sample is in the correct regime for a KosterlitzThouless transition to be observed. The following conditions place the sample in this regime: very low currents, very large sample size (L ’ DQGλ⊥ >> L. For very small currents, the number of current-induced free vortices is smaller than the number of thermally-induced free vortices. As the current is increased, the number of current-induced free vortices increases. When the current-induced free vortices dominate, Equation (5.20) can be used to describe the IV behavior both above and below TKT. This results in the IV curves above TKT displaying power law behavior with a(T) ” 3 rather than ohmic behavior. Thus, the sharp jump from a(T) = 3 to a(T) = 1 will not be observed. If the sample is of finite size, meaning the conditions λ⊥ >> L

’DUHQRWPHW

“finite size-induced” free vortices will be present at all temperatures destroying the logarithmic interaction between the vortices described by Equation (5.9).

At low

currents, the presence of free vortices below TKT will cause the slope of the IV curve to deviate from power-law behavior toward ohmic behavior. Again, the sharp jump in the power-law exponent may not be observed. Knowledge of the value of λ⊥ is important in determining the possibility of the existence of a Kosterlitz-Thouless transition; however, this is not a well-known parameter in Josephson junction arrays. λ⊥ is not easily calculated and has not previously been measured in arrays. The next chapter describes an experimental measurement of λ⊥ in Josephson junction arrays and presents the results.

53

6 Measurement of Current Distribution and Determination of λ⊥ in Josephson Junction Arrays

6.1 Sample Fabrication The samples fabricated for this experiment consist of a square lattice of 100 x 100 niobium crosses arranged on a continuous gold film. The cross arms are 1µm wide and 40µm long. A gap of 0.5µm between arms of adjacent crosses forms the Josephson junction. The array dimensions were chosen based on the spatial resolution of the SQUID observed in Figure 4.3. With the 10µm square SQUID loops, features of the usual a = 10µm arrays could not be resolved, so the arrays for this experiment were designed with a = 40µm with the hopes of being able to resolve individual array plaquettes. Brian Straughn from the University of Maryland provided Au-Nb films on 3” silicon wafers. The thickness of each layer was approximately 2000Å. A 50Å layer of Nb was deposited onto the wafer first to aid with the adhesion of the gold. This adhesion layer was made very thin because if the layer becomes superconducting, it will short out the gold layer I fabricated the samples at Cornell Nanofabrication Facility [43]. The Nb crosses and busbars were patterned on the electron beam lithography system and the Nb was etched in the reactive ion etcher. The excess gold was removed by patterning via photolithography and then wet etched. The details of the fabrication process are given in Appendix B. Figure 6.1 shows an electron micrograph of a portion of one of the arrays.

54

Figure 6.1 SEM picture of several crosses of a 100x100 array with lattice constant a = 40µm

6.2 First Attempts A Josephson junction array was scanned in zero applied magnetic field for several current values at various temperatures below the transition temperature. The scan direction was perpendicular to the direction of the current. Each flux image was normalized to the zero current “background” image as described in section 4.1.

55

6.2.1 Sample Characteristics Resistance versus temperature data for the sample analyzed in this section (w13bs2) is shown in Figure 6.1. The large jump in the RT data between 8 and 9 K is a result of current redistribution in the array as the niobium becomes superconducting. The resistance steadily decreases with temperature from this peak due to proximity effects reducing the resistance of the gold between the crosses. A rapid drop in resistance occurs around 3.5 K. The small “knee” in the curve toward the end of this drop is where Kosterlitz-Thouless behavior should occur.

6/23/00

w13bs2 100x100 JJA a=40µm

0.035 0.030

Resistance (Ω)

0.025

I=100µA

0.020 0.015 0.010 0.005 0.000 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

Temperature (K) Figure 6.2 Resistance versus temperature data for 100x100 array.

56

6/23/00

w13bs2 100x100 JJA a=40µm 1E-5

1E-6

voltage (V)

1E-5

3.30K 3.29K 3.28K 3.27K 3.26K 3.25K 3.24K 3.23K 3.22K 3.21K 3.20K 3.19K 3.18K 3.17K 3.16K

1E-6

1E-7

1E-7 slope=1

slope=3

1E-8

1E-8

1E-9

1E-9 1E-5

1E-4

1E-3

total current (A)

Figure 6.3 Current versus voltage characteristics for 100x100 array.

57

0.01

The current versus voltage data is shown in Figure 6.2. As a guide for the eye, solid lines with slopes of 1 and 3 are shown in the plot. The curves display power law behavior with the highest temperature curves toward the left of the graph being nearly ohmic in slope. The lowest temperature curves toward the right are slope 3 or higher over most of the range of current, but begin to bend toward ohmic behavior at the lowest currents displayed for each curve. The values of a(T) determined from the slopes of the IV curves are shown in Figure 6.4. For each curve, the slope was determined in the power law region of the data above the ohmic tails. A sharp jump in a(T) indicative of a Kosterlitz-Thouless transition is not observed in the graph. However, it is possible that KT-like behavior exists in this sample, but is masked by finite size effects.

6

w13bs2 100x100 JJA a=40µm

6/23/00

5

a(T)

4

3

2

1

3.16

3.18

3.20

3.22

3.24

3.26

3.28

3.30

3.32

T

Figure 6.4 The power law exponent a(T) determined from the sloped of the IV curves in Figure 6.2. 58

6.2.2 Data Shown in Figure 6.5(a) is an image of the field data obtained at T = 3.25 K with a 100µA current applied to sample w13bs2. The applied current is in the y- (vertical) direction in the figure. Figure 6.5(b) is a profile of the field along the dotted line across the middle of the sample. In Figure 6.5, the background data has already been subtracted out using the method described in Section 4.1. Since I am only concerned with the distribution of the current and not the magnitude of the B field, I have left Bz in arbitrary units. The magnitude of Bz is related to the color scale at the bottom of Figure 6.5. From Figure 6.5 (a), it can be seen that this sample has a slight rotation in the x-y plane. As a result, the scan direction is not exactly perpendicular to the current direction, but this rotation is slight (a few degrees) , so I will ignore this when calculating the current distribution. I calculated the current distributions in the x- and y- directions using the algorithm outlined in Figure 4.4. The resulting images are displayed in Figure 6.6. The magnitude of the current is related to the color scales at the bottom of the graphs with white being zero current. Since I am only concerned with where the current is and the relative magnitudes of the current but not the actual value of the current, Jx and Jy are shown in arbitrary units. Figure 6.6(a) is the current in the x-direction and Figure 6.6(b) is a profile of the current through the middle of the sample. These images show that the current in the xdirection is essentially zero, as it should be since the current was applied in the ydirection. The very small x component of current near the sample edges are a result of the slight rotation of the sample in the x-y plane.

59

2.0 2.5 3.0

3e-007

(b)

(a)

0e+000

y

Bz

mm

0.0 0.5 1.0 1.5

I

3.5 x 4.0 0.0 1.0 2.0 3.0 4.0 5.0

-3e-007

0.0

1.0

2.0

3.0

4.0

mm

mm

-4e-007 0e+000 4e-007 8e-007

Bz (arbitrary units)

Figure 6.5 (a) Image of the magnetic field in the z-direction above the array w13bs2 characterized in Section 6.2.1 at T = 3.25K and I = 100mA. The background field data has been eliminated in this graph. (b) Profile of the field across the middle of the sample. Bz is in arbitrary units.

60

5.0

Figure 6.6(c) is the image of the current in the y-direction and Figure 6.6(d) is a profile of the current through the middle of the sample. These graphs show that most of the current is confined within a small width along the edge of the array and falls off quickly toward the center of the array. The rotation of the sample in the x-y plane may be a contributing factor to the asymmetry between the right and left sides of the array. It is also quite possible that the sample is tilted in the z-direction so that the SQUID would scan closer on one side of the sample than it would on the other. If the sample has a penetration depth λ⊥, the current should decay toward the center of the sample within a width equal to λ⊥. (For example, jump ahead to Figure 6.9.) To determine λ⊥ from the current distribution data, I fit an exponential e-x/ λ⊥ to the peaks on the edges of the plots of the Jy current profile. As an example, the fit for the peaks on the left hand side of the sample at T = 3.0K are shown in Figure 6.7.

61

0.0 0.5

0.00002

(a)

(b) 0.00001 Jx (Arb.)

mm

1.0 1.5 2.0 2.5

0.00000

3.0 3.5

-0.00001

4.0 0.0 1.0 2.0 3.0 4.0

-0.00002 0.00

5.0

2.50

3.75

5.00

mm

mm

-0.000175

1.25

y

0.000000

x

Jx (Arb.) 0.0 0.5

3.75e-005

(c)

(d) 1.88e-005 Jy (Arb.)

mm

1.0 1.5 2.0 2.5

0.00e+000

3.0 3.5

-1.88e-005

4.0 0.0 1.0 2.0 3.0 4.0

-3.75e-005 0.0

5.0

2.0

3.0

4.0

mm

mm

-6.25e-005

1.0

0.00e+000 Jy (Arb.)

Figure 6.6 (a) Current distribution Jx obtained from the field data in Figure 6.4; (b) Profile of Jx across the middle of the sample; (c) Current distribution Jy obtained from the field data in Figure 6.4; (d) Profile of Jy across the middle of the sample.

62

5.0

7/27/00

w13bs2 T=3.0K 0.00012

-x/λ

J=J0e

0.00011 I=100µA I=500µA I=1mA

0.00010 0.00009

Chi^2 = 8.1257E-13 R^2 = 0.99699 0.12821 ±0.03315 Jo λ 183.222 ±6.341

Jy (Arb)

0.00008 0.00007 0.00006

Chi^2 = 3.5018E-13 R^2 = 0.99563 0.03928 ±0.01156 Jo λ 198.432 ±8.515

0.00005 0.00004 0.00003

Chi^2 R^2 Jo

0.00002 0.00001

λ

= 7.9042E-14 = 0.97156 0.00765 ±0.00691 206.044 ±27.326

0.00000 1000

1200

1400

1600

1800

2000

x (µm)

Figure 6.7 Exponential fit to the Jy data from the left hand side of the sample for three different applied currents at T = 3.0K.

63

6.2.3 Results and Discussion Figure 6.8 shows the estimations of λ⊥ for several different currents and temperatures obtained from exponential fits as in Figure 6.7. This plot shows values for λ⊥ ranging from 110 – 206 µm with no obvious trend.

220

7/27/00

w13bs2 200

50µA 100µA 500µA 1000µA 2000µA

180

λ(µm)

160

140

120

100 2.5

3.0

3.5

Temperature (K)

Figure 6.8 λ⊥ calculated from exponential fits to Jy data.

A major problem with this method becomes apparent upon examination of the images in Figure 6.6 and the peaks in Figure 6.7. The calculated current images show a distribution of current outside the edges of the sample where it could not physically exist. The spatial filtering technique [38] [Figure 4.5] does not incorporate any boundary conditions and, as a result, the current images are rounded and “smeared out”. This smearing out of the current will cause the values of λ⊥ to be calculated as larger than they 64

actually are. This technique functions best as a way to estimate the current pattern for a current source with no actual boundary [44], i.e., current can physically exist anywhere within the scanned area.

6.3 Second Approach Discussion of the problems presented in the previous section with John Wikswo and his group at Vanderbilt (the developers of the spatial filtering technique) led to a different approach to determining λ⊥. Once again, data was collected at various temperatures and various applied currents, but only a single line scan across the midpoint of the sample was needed. A least-squares fit to the Biot-Savart law was performed on the data using λ⊥ and h, the height of the SQUID above the sample, as fitting parameters. To eliminate the background, for an applied current I, scans were taken for both +I and –I. The background then could be given by (Φ + I + Φ − I ) / 2 , where Φ± I is the total flux through the SQUID for the applied current ± I . This background value was then subtracted from the scans at that current I. In practice, instead of only collecting data for a single line scan, I collected data for 4 or 5 scans and used the last scan to give the sample temperature time to stabilize.

6.3.1 Background Calculations Approximating an array as 100 current-carrying wires spaced a = 40µm apart, I calculated the resulting magnetic field. I modeled the current through each of the wires as 65

i n = I o (e − x

n

/ λ⊥

+ e( x −L) / λ ) , ⊥

n

(6.1)

where Io is the total current applied to the sample, xn is the position of the nth wire and L = 100 a = 4mm is the width of the sample. A plot of this current model for different values of λ⊥ is shown in Figure 6.9. At the smallest values of λ⊥, the current is confined to a narrow region along the edges of the array, but as λ⊥ gets larger, more current is allowed to penetrate into the array. Once λ⊥ •L, the current will be evenly distributed across the width of the array.

0.00005

λ⊥=2000µm

0.00004

in 0.00003

λ⊥=1000µm

0.00002

λ⊥=100µm

0.00001

λ⊥=500µm

λ⊥=25µm

0 0

λ⊥.

20

n

40

60

80

Figure 6.9 Current through n = 100 wires for Io = 50 µA at different values of

66

100

h = 20 µm 0.06 0.04 0.02

Φ (Φo)

0

-0.02 -0.04

(a)

-0.06 -0.002

0

x (µm)

0.002

0.004

0.006

0.008

h = 75 µm 0.06 0.04 0.02

Φ (Φo)

0

-0.02 -0.04

(b)

-0.06 -0.002

0

x (µm)

0.002

0.004

0.006

0.008

h = 150 µm 0.06 0.04 0.02

Φ (Φo)

0

-0.02 -0.04

(c)

-0.06 -0.002

0

x (µm)

0.002

0.004

0.006

0.008

Figure 6.10 The flux through a 10µm square SQUID loop in units of Φo from the current distributions pictured in Figure 6.8 at (a) h = 25µm, (b) h = 75µm, and (c) h = 150µm.

67

The magnetic field for the current distributions shown in Figure 6.9 will be a sum of the magnetic field for each of the 100 individual wires. The calculated flux through a 10µm square SQUID loop from the current distributions of Figure 6.9 is shown in Figure 6.10 for different values of h.

6.3.2 Sample Characteristics Resistance versus temperature for the sample analyzed in this section (w8bs2) are shown in Figure 6.11. The large jump in resistance around T ~ 8.5K is due to current redistribution in the array. From this peak, the resistance of the array steadily decreases with temperature due to proximity effects until it sharply drops at T ~ 4.2 K. The small tail at the bottom of this jump is where KT behavior is where KT behavior should occur. The sample IV data is shown in Figure 6.12. The solid lines in the plot have slopes 1 and 3. The curves display power-law behavior with very large linear tails at the lower current end of the curves down to the lowest temperature measured. The linear tails of this sample are much larger than those observed in sample w13bs2 in the last section (Figure 6.3). Figure 6.13 shows the values of a(T) found from the slopes of the power law portions of the curves in Figure 6.12. The plot shows no sharp jump in a(T) indicative of a Kosterlitz-Thouless transition. Since finite size effects are even more prominent in sample w8bs2 than in w13bs2, it is even less likely that KT behavior will be observed in w8bs2.

68

0.024 0.022

w8bs2 100x100JJA a=40µm

1/27/00

0.020 0.018

Resistance (Ω)

0.016

I=100µA

0.014 0.012 0.010 0.008 0.006 0.004 0.002 0.000 -0.002 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.010.5

Temperature (K)

Figure 6.11 Resistance versus temperature for 100x100 array.

69

1/28/00

w8bs2 100x100JJA a=40µm

1E-4

voltage (V)

1E-5

1E-6

4.10K 4.09K 4.08K 4.07K 4.06K 4.05K 4.04K 4.03K 4.02K 4.01K 3.99K 3.98K 3.95K 3.90K 3.70K

slope=1

1E-7

1E-8 slope=3 1E-9 1E-4

1E-3

0.01

total current (A)

Figure 6.12 Current versus voltage characteristics for 100x100 array.

70

4.0

w8bs2 100x100JJA a=40µm

1/29/00

3.5

a (T)

3.0

2.5

2.0

1.5 3.98

4.00

4.02

4.04

4.06

4.08

4.10

Temperature (K)

Figure 6.13 The power law exponent a(T) determined from the sloped of the IV curves in Figure 6.10.

6.3.3 Data Single line scans of the flux profile above the sample at T = 3.97K are plotted in Figures 6.14, 6.15, and 6.16 for I = 25µA, 50µA, and 100µA respectively. These plots show the data after the background data has been subtracted using the method discussed at the beginning on Section 6.3. The circles are the actual data and the solid lines are the leastsquares-fit to the data.

71

Φ

0.007 0.006 0.005 0.004 0.003 0.002 0.001 (Φο) 0.000 -0.001 -0.002 -0.003 -0.004 -0.005 -0.006 -0.007 -0.008 -0.009 -0.010

8/15/00 w8bs2

T = 3.97K, I = 25µA

h = 70µm λperp = 60µm σ = .00053

-0.002

h = 42µm λperp = 194µm σ = .0012

0.000

0.002

0.004

0.006

x (m)

Figure 6.14 Flux profile at T = 3.97K and I = 25µA across the middle of the sample characterized in Section 6.3.2. The circles are the actual data points and the solid lines are a least-squares fit to the data points

72

0.008

Φ

0.009 0.008 T = 3.97K, I 0.007 0.006 0.005 0.004 0.003 0.002 0.001 (Φο) 0.000 -0.001 h = 90µm -0.002 λperp = 50µm -0.003 σ = .00058 -0.004 -0.005 -0.006 -0.007 -0.008 -0.009 -0.004 -0.002 0.000

8/15/00 w8bs2

= 50µA

h = 110µm λperp = 135µm σ = .00138

0.002

0.004

0.006

0.008

x (m)

Figure 6.15 Flux profile at T = 3.97K and I = 50µA across the middle of the sample characterized in Section 6.3.2. The circles are the actual data points and the solid lines are a least-squares fit to the data points

73

0.010

Φ

0.007 0.006 0.005 0.004 0.003 0.002 0.001 (Φ ο) 0.000 -0.001 -0.002 -0.003 -0.004 -0.005 -0.006 -0.007 -0.008 -0.009 -0.010

8/15/00 w8bs2

T = 3.97K, I = 100µA

h = 144µm λperp = 55µm σ = .00069

-0.004

h = 165µm λperp = 105µm σ = .00093

-0.002

0.000

0.002

0.004

0.006

0.008

x (m) Figure 6.16 Flux profile at T = 3.97K and I = 100µA across the middle of the sample characterized in Section 6.3.2. The circles are the actual data points and the solid lines are a least-squares fit to the data points

74

0.01

If the sample was not level, the SQUID height would be different for each side of the sample. Since the flux profile on the left and right sides of the scans was asymmetric for all data obtained, I chose to separately fit the left and right halves of the data to the Biot-Savart law. Each plot shows the results of the fits with h = height of the SQUID above the sample, λperp = λ⊥, and σ = standard deviation of the fit. The data for all other chosen temperatures and currents was analyzed in this manner. The data from the left-hand side of the scans produced better fits and more consistent results than the data from the right side. From Figures 6.13-15, the right-hand side of the data shows some unexpected structure. It is possible that a piece of “dirt” was sitting on that side of the sample and interfered with the SQUID during scanning, or that the sample became scratched during scanning. Whatever the cause, the data from the right-hand side of this sample does not produce a decent fit to the model, so I will continue to analyze the data from only the left-hand-side. After fitting curves to the data for the temperatures T = 3.90K, 3.95K, 3.97K, 4.00K, 4.03K, 4.05K and 4.10K, I calculated the average result for the SQUID height to be h = 139µm. Keeping this parameter fixed at the average value, I again fit the left-hand-

side of each data set data using λ⊥ as the only fitting parameter. The results from the T = 3.95K data of Figures 6.14, 6.15, and 6.16 are shown in Figures 6.17, 6.18, and 6.19.

75

0.005

T = 3.97K, I = 25µA, LHS

8/15/00 w8bs2

0.004 0.003 0.002 0.001

Φ (Φ ο)

0.000 h = 139µm λperp = 75µm σ = .00057

-0.001 -0.002

-0.003

-0.002

-0.001

0.000

x (m)

0.001

0.002

Figure 6.17 Fit to LHS of data in Figure 6.13 for fixed h = 130µm. 0.005

T = 3.97K, I = 50µA, LHS

8/15/00 w8bs2

0.004 0.003 0.002 0.001

Φ (Φ ο)

0.000 -0.001 -0.002

h = 139µm λperp = 60µm σ = .00061

-0.003 -0.003

-0.002

-0.001

0.000

x (m)

0.001

0.002

Figure 6.18 Fit to LHS of data in Figure 6.14 for fixed h = 130µm.

76

0.006

T = 3.97K, I = 100µA, LHS

8/15/00 w8bs2

0.004

0.002

Φ (Φο) 0.000 -0.002

h = 139µm λperp = 55µm σ = .00066

-0.004

-0.003

-0.002

-0.001

0.000

x (m)

0.001

0.002

Figure 6.19 Fit to LHS of data in Figure 6.15 for fixed h = 130µm.

6.3.4 Results and Discussion The value of λ⊥ determined from fitting the left-hand-side of each data set for sample w8bs2 with fixed h = h is displayed in Figure 6.20. The calculated values of λ⊥ ranged between 20 – 75µm. On average, λ⊥ ~ a, much smaller than expected, but not an entirely surprising result for this array. The determined value of the SQUID height above the sample, h, is larger than expected. Since the data gathering on this sample came to an end after the SQUID had crashed into the sample and quit working, I had expected a much smaller value for h. However, it is possible that the SQUID actually crashed into grease, dirt, or something else on the sample.

77

100 90

8/26/00 w8bs2

λ perp determined from fit for havg = 139µm

I = 25µA I = 50µA I = 100µA I = 150µA I = 200µA

80

λperp(µm)

70 60 50 40 30 20 10 3.88

3.90

3.92

3.94

3.96

3.98

4.00

4.02

4.04

4.06

temperature (K) Figure 6.20 Value of λ⊥ determined from fit to LHS of each data set for w8bs2 with h = 139µm.

The IV characteristics for this sample in Figure 6.12 actually support the result of a small λ⊥. The higher current curves toward the right side of the graph show power-law behavior, but very quickly, the curves bend toward linear (ohmic) behavior. This type of behavior in the IV curves is a result of the finite size of the samples as discussed in Chapter 5. Herbert et al. [42] studied finite size effects in a series of Josephson junction arrays with lattice constant 10µm. The arrays were 300 junctions long and varied in width. They found that as the array became more narrow and thus contained more finite sizeinduced free vortices the deviations from power-law behavior became more pronounced. 78

They also found that whenever a significant ohmic tail exists in the IV curve, the powerlaw exponent will be masked. The parameter a for the samples I have studied is much larger than that for the arrays used by Herbert et al. This probably also had significant effects on the results of the experiment. While the overall physical size of my samples was large since the chosen value for a was large, the number of junctions across the width of the sample was apparently too small to place the sample in the thermodynamic limit. While I was able to conclude that λ⊥ was indeed very small for this array, a definite trend in the temperature dependence cannot be determined from Figure 6.20. Since h was determined to be 139µm ~ 3.5 a, the actual value for λ⊥ could vary as much as a few a. Nevertheless, a few a is much smaller than the array width still putting this array geometry out of the thermodynamic limit.

79

7 Finite Size Effects and Dynamical Scaling in TwoDimensional Josephson Junction Arrays

7.1 Introduction The two-dimensional superconducting phase transition in zero magnetic field is of the Kosterlitz-Thouless-Berezinskii (KTB) type. For more than two decades, there has been a great deal of work exploring the details of the KTB transition and whether, in fact, one can truly be observed in physically realizable systems. In the past decade, a new class of superconductors has been added to the mix – high temperature cuprate superconductors. With their layered structures and highly anisotropic coupling strengths, these systems offer the possibility of quasi-two-dimensional (2D) behavior. While the nature of the superconducting phase transition in high temperature superconductors is as yet an unsettled issue [45,46,47,48], several authors [49,50] have published work that purports to show the existence of a KTB-like topological phase transition as part of a larger threedimensional transition mechanism. Others[51] do not observe such a transition and believe that the conditions for it do not exist in these materials. In 1989, Fisher, Fisher, and Huse (FFH) [2,52] offered a general analysis of a superconducting phase transition in D dimensions using a dynamic scaling argument. Their primary focus was on the behavior of superconducting systems in the presence of penetrating magnetic fields, but they pointed out that their scaling law also applied in zero field to the KTB transition for D = 2 and for a dynamical critical scaling exponent z = 2. In the years since, many groups have used this scaling approach as one measure of proof of existence or absence of a KTB transition; in effect, if the properly scaled current-

80

voltage (IV) curves collapse (do not collapse) onto universal scaling curves above and below the transition, then a KTB transition is likely (unlikely) to be present. In most cases, the scaling behavior was offered in support of other more conventional analysis such as the KTB resistive behavior or the existence of a universal jump in the IV exponent. Recently, Pierson et al. [49,50] published a dynamic scaling analysis of IV data taken on ultra-thin (one unit cell thick) high temperature superconducting films [51] as well as on prototypical two-dimensional low-temperature superconducting systems in which it is believed that a KTB transition exists and has been observed [53,54,55]. Based primarily on the results of their scaling, they propose that a re-evaluation of the dynamics of the KTB transition may be in order. In particular, they propose that the dynamical critical scaling exponent z may not be 2, as one would expect for diffusive dynamics in systems which follow the two-dimensional XY model, but may be as high as 5 or 6. In this study, we look to determine the proper role of dynamic scaling in such systems and for a possible source of the very high value of z that Pierson et al. observe in their scaling analysis. In particular, we suggest that it is inappropriate to use evidence of scaling behavior in experimental data as the primary support for the existence of a 2D phase transition. In the case of the KTB transition, the scaling behavior is in fact valid only above the transition temperature where the KT correlation length exists and diverges. Below the transition temperature the correlation length is infinite, and so we should not observe scaling [56]. Nonetheless, an apparent scaling curve is often found below the transition temperature in real IV data. Medvedyeva, Kim, and Minnhagen (MKM) [57] have suggested that the specifics of the scaling behavior below TKT is

81

determined by the finite size of the sample rather than pointing to evidence of some new dynamics, as Pierson et al. suggest. In their analysis, MKM point out that, although for any finite-size sample the resistance only truly vanishes at zero temperature, for a sample of fixed size L and for data within a limited temperature region, the resistance may appear to vanish at some nonzero temperature, and z may be > 2. Thus, one may be lead to believe that a transition to zero resistance may actually occur for values of z > 2, when in fact it does not. MKM have designated such an apparent transition a “ghost” transition. Our results are entirely consistent with those of MKM. We examine this question using the straight-forward approach of generating the IV characteristics of two-dimensional SNS Josephson junction arrays including the finitesize effects. As the IV curves are generated using KTB theory, we would expect that the scaling results would yield parameters consistent with KTB behavior. Instead, we find that the details of the scaling, i.e., the values for z and TKT, depend in a critical way on the effective voltage sensitivity, a purely experimental parameter. We find the mere fact that scaling can be accomplished with values of z other than 2 insufficient evidence of an alternative phase transition.

7.2 The Two-Dimensional Phase Transition in Superconductors For many years it was believed that many types of phase transitions were not possible in two dimensions. For a superconductor, for example, it was believed that, as the temperature dropped, the resistivity could become exponentially small but would never be zero, and no true phase transition would actually occur. There were theoretical predictions about the impossibility of general long range order in two-dimensional 82

systems. The earliest was by Peierls [58] who argued that the thermal motion of long wavelength phonons would destroy conventional long range order in a two-dimensional crystal. The absence of long range order in two dimensions was rigorously shown by Mermin [59]. The absence of long-range order, however, does not necessarily imply the absence of a phase transition. Such a phase transition would be from a disordered high temperature state to an ordered, but not infinite range, low temperature state. Kosterlitz and Thouless [1], and Berezinskii showed that this was indeed correct by showing that “quasi longrange order”, the algebraic decay of correlations, could occur. They called this topological long range order and applied it to two-dimensional crystals, neutral superfluids, and XY magnets. They did not apply it to two-dimensional superconductors or the isotropic two-dimensional Heisenberg magnet, where, they believed, the proper conditions for observing the transition could not strictly be met. Beasley et al. [60] and Doniach and Huberman [61] demonstrated that Kosterlitz and Thouless’s theory could be extended to superconductors under special conditions. It is these “special conditions” that concern us here. In order to observe a KTB transition – or a KTB-like transition - certainly very stringent conditions must be met, and if they are not, the details of the transition will not be correct and the phase transition will not occur. What are those conditions as applied to the case of a superconductor? In bulk superconductors the energy to create a vortex is proportional to the length of the vortex and as a result is always much greater that the available thermal energy. However, in thin superconducting films where the perpendicular penetrations depth λ⊥ ( = λ2/d) can be made much greater than the sample size, the energy needed to create a bound pair of

83

vortices is 2πns 2/ 2m ln(r/x), where ns is the 2D superfluid density, x is the superconducting coherence length and r is the separations between the two vortices. This can easily be on the order of kBT. (For an array, we consider 2πEJ ln(r/a) where a is the array lattice parameter and EJ is the Josephson coupling energy.) The energy to create a single free vortex on the other hand is πns 2/ 2m ln(L/x), infinite in the thermodynamic limit (L → ’ 7KXVIRUWHPSHUDWXUHVJUHDWHUWKDQ]HUREXWVWLOOVXIILFLHQWO\ORZWKH sample will contain bound pairs of thermally-generated vortices, which cannot be driven by an applied electrical current, and no free vortices. The KTB phase transition occurs when these bound pairs of vortices dissociate; this occurs at the Kosterlitz-Thouless temperature, TKT. These now free vortices may be driven by an applied electrical current, yielding a flux-flow resistance. Thus, below the vortex unbinding temperature the dissipation is zero in the limit of zero current. Above TKT, the resistance is not zero due to the finite density of free vortices and, as is the usual case for flux-flow resistance, the voltage depends linearly on the current, i.e., the system appears ohmic. (Once again, this is strictly correct only in the limit of zero current, as discussed below.) The magnitude of the resistance depends on the density of free vortices, nf, which in turn varies as 1/x+2 where x+, the correlation length, is a measure of the size of the fluctuations (vortices) above the transition temperature. An externally applied current may unbind a pair of vortices via the Lorentz force. Well above the transition temperature, where many pairs of vortices are already unbound, the additional effect of a small current unbinding vortex pairs is not observable. That is, above TKT and at low currents the current voltage characteristics are linear due to the thermally unbound vortices. As the current increases and the additional density of free

84

vortices created by the current begins to be important, the IV characteristics will switch to a power law, V ∝ Ia(T) where 1 < a(T) < 3. Below the transition temperature, where there are no thermally unbound free vortices, current unbinding is always important, and the current-voltage relations are always power law, with a(T) > 3. The result is that for sufficiently small measuring currents, the exponent of the IV characteristics jumps discontinuously from 1 to 3 at TKT. Whether a KTB transition or a KTB-like transition is observable in a particular experimental system depends on the relationships among several length scales: L, the sample size, x+, the correlation length for T •7KT, x−, the characteristic size of a bound vortex pair below TKT, rc, a critical distance between the two members of the bound pair, and λ⊥. The existence of a correlation length is necessary to the scaling we discuss below. x+, the size of fluctuations (vortices) above the transition temperature, is a “true” coherence length – true in the sense of point-to-point correlations of the order parameter. x−, which may be thought of as the mean separation between the two vortices in a bound pair, [62,63,64] is not a true coherence length in that sense. However, the temperature dependence of these two lengths is the same, differing only by a constant. In general, to experimentally observe a KTB phase transition we must be in the thermodynamic limit – L must be large. Second, in a superconductor we also require that λ⊥ >> L so that the vortex-vortex interaction is always logarithmic. Finally, we must be in the low current limit to avoid having too many current-unbound vortices above TKT. These conditions are most often met in high resistance granular low temperature superconductors and in Josephson-junction arrays – i.e., in weakly coupled systems.

85

The first problem that most often arises in an experiment is that two of the limits, L very large and λ⊥ >> L, are violated – either because the sample is too small or because λ⊥ is strongly temperature dependent and crosses over to become smaller than the sample size as the temperature is lowered. In both cases it becomes energetically possible for free vortices to form at all temperatures (either 2πEJ ln(L/a) or 2πEJ ln(λ⊥/a) is no longer much greater that kBT). These additional free vortices, called finite-size-induced free vortices, are most noticeable below the transition temperature at very low currents, where they create a linear or ohmic “tail” on the IV characteristics. As we will see, these finitesize-induced free vortices will have profound effects on dynamic scaling.

7.3 Scaling In a phase transition, sufficiently close to the transition temperature, critical fluctuations are observed. The closer one gets to the transition temperature, the longer these fluctuations will last, and the larger the relevant scale becomes. In a superconductor the relevant length scale is the coherence length x, where

ξ ∝ T − Tc

−ν

.

(7.1)

Without loss of generality we can assume that the lifetime of the fluctuations, τ, varies as

τ ∝ ξz which defines z, the critical exponent. Time “slows down” as T

(7.2) 7c. As we approach

the critical region, all the physics that really matters is in the diverging length and time scales. In the KTB transition we will see that z is expected to be 2. Ammirata et al. [49] have suggested that z is considerably larger in such systems, perhaps as large as 5 or 6. 86

They base this conclusion on a scaling analysis of several experimental systems, some of which [65,66] have heretofore been assumed to display a KTB transition. The scaling analysis is easily understood via dimensional analysis. In a superconductor, the supercurrent Js is related to the gradient of the phase of the order parameter φ,

J s ∝ n s ∇φ ,

(7.3)

where ns is the superconducting electron density. Since the only relevant length is the coherence length,

1 ∇φ ∝ . ξ

(7.4)

Josephson [4] argued that in D dimensions the superfluid density will vary as

ns ∝ ξ 2− D .

(7.5)

If we combine Equations (7.3), (7.4), and (7.5) we obtain

J s ∝ ξ1− D .

(7.6)

Next we note that since the magnetic field B varies as

B ∝ Φ o ξ −2 ,

(7.7)

we obtain a vector potential A varying as

A ∝ Φ o ξ −1

(7.8)

∂A Φ o ξ −1 ∝ ∝ ξ1− Z , E= z ∂t ξ

(7.9)

and an electric field E

87

where Φo is the flux quantum. Combining Equations (7.6) and (7.9) yields, for the resistivity ρ

E ξ −1− z ρ = ∝ 1− D = ξ D −2− z . ξ J

(7.10)

We may extend Equation (7.10) above the critical temperature by assuming

ρ=

E ∝ ξ D −2− z F+ (ξ, E ) , J

(7.11)

where F+ is a function whose argument is a dimensionless combination of x and E. Using Equation (7.9) yields

(7.12)

ρ = ξ D −2− z F+ (ξ1+ z , E ) . For temperatures greater than the critical temperature, as the electric field goes to zero, F+(0) must tend to a constant. At the critical temperature x → ∞ , ρ is measurable and finite, and the current J is not zero. This implies that

E ∝ J 1+ z / D −1

(7.13)

at T = Tc. For z = 2 and D = 2, we recover the KTB result, E = J3. Fisher, Fisher, and Huse [2] obtained the same result from a different route and in a more general form. We write their result in the following way, using the experimentally determined quantities V and I for their E and J:

 Iξ D −1   , V = Iξ D −2− z ρ ±  T  

(7.14)

where ρ± is similar to F+ in Equation (7.11), except that its existence is assumed above and below the transition. Note that using Equation (7.14) implies that correlation lengths exist above and below the transition temperature. In ordinary superconductors this 88

presents no problems. However, in thin films and in Josephson junction arrays this does present a problem since, as discussed above, x- is ’WKLVLVIXUWKHUGLVFXVVHGEHORZ The rest of this chapter will focus on two-dimensional systems and Equation (7.14) will be rewritten as

 Iξ  V = Iξ − z ρ ±   . T 

(7.15)

We can remove a factor of (Ix/T)z from ρ± and rename it P±, yielding 1/ z

II   T V 

 Iξ  = P±   . T 

(7.16)

This is the form often preferred for analysis [48,49,50] since the coherence length, which tends toward infinity as the transition temperature is approached from above (or below in some systems), only appears in the argument of the scaling function. Equation (7.13) tells us that at the critical temperature, the voltage is proportional to Iz+1 for a two-dimensional superconductor. As discussed earlier, this results from the coherence length going to infinity as T approaches Tc while the voltage is finite for nonzero currents. However, this power law behavior is also valid for any temperature and current that makes the argument of ρ± tend toward infinity since this drives Equation (1.14) to the same limiting form. Thus, for high currents V should be a power law function of I at least until other physics enters, e.g., the critical current of the film or junctions is exceeded and the IV curves should once again become ohmic. Above the transition, in the limit Ix/ T goes to zero, if we assume that the power of the exponent is greater than 1, then we can take ρ± ≈ constant and

V = Rlinear ∝ ξ − z . I 89

(7.17)

This is valid for T > Tc and I

DQGLVVLPSO\WKH.RVWHUOLW]-Thouless result just above

the transition due to vortex unbinding. Below the transition, we cannot take the same limit as it leads to the unphysical result of voltages in the superconducting state. At this point, all that is required to do dynamic scaling is the temperature dependence of the correlation length. In KTB theory, the correlation length x+ can be defined above the transition as the size of a fluctuation (i.e., a vortex), or alternatively, as the average distance between two free vortices. It has a temperature dependence given by [4],

 b ± ξ ± ~ exp   T − TKT

1/ 2

  

 . 

(7.18)

Where b± is a constant of order one. The constant of proportionality depends on the system; for 2D Josephson junction arrays it is essentially a, the lattice parameter, while for 2D thin films it is the 2D Ginsberg-Landau coherence length. Below the transition the correlation length is infinite, and we often use x-, the typical separation of a bound vortex pair. This is not a true correlation length in that it does not come from a two-point correlation function. Nevertheless, x- is often used as if it were a correlation length since it has the same temperature dependence as ξ+ to within a constant (b+ and b- differ by a factor of 2π). In practice, we may take the temperature dependence of ξ+ and ξ- to be symmetric without loss of generality. Thus, for the rest of this chapter we will assume b+ = b- = b. The consequences of the correlation length not being well defined below the transition as related to scaling behavior will be discussed below.

90

7.4 IV Curve Details In this section we generate current-voltage characteristics of Josephson junction arrays. We use the standard results from the literature [39] for the power law IV characteristics and for the flux-flow resistance immediately above the transition. Since this system is inherently two-dimensional and thought to display a KTB transition, we should expect any dynamic scaling to yield values consistent with the KTB results, namely z = 2. Next we add the voltages caused by finite-size-induced vortex nucleation above and below the transition [42]. We then use dynamic scaling to study these simulations and to determine the effects of finite samples. Above the transition temperature TKT, thermally generated free vortices add a flux flow resistance of the form [4],

R(T ) t −u

  b Vt −u (T ) L 2 = = 2 Ro b1 exp−  I W   T − TKT

  

1/ 2

 , T > TKT 

where t-u stands for “thermally unbound”, Ro is the normal state resistance, L/W is the length/ width of the array, and b1 and b2 are constants of order one. (Note that b2 is related to b+ in Equation (7.18).) For T ”7KT this thermally unbound flux flow resistance will be zero since there will be no thermally unbound vortices. In addition to the thermally generated voltage, any finite current will unbind vortex pairs, yielding a voltage of the form [4], 1/ 2

Vc−u (T ) = 2 Ro

3/ 2

 2π   La  Φ oα 

[ic (T )]1/ 2−πE (T ) / k T [i]πE (T ) / k T +1 , J

B

J

b

(7.20)

where c-u stands for “current unbound”, a is the lattice array parameter, ic is the critical current per junction, i is the current per junction (roughly I/W), and α is a constant of 91

proportionality. This expression is valid both above and below the transition temperature. Looking at Equation (7.20), we see that we can write it as Vc-u ∝ Ia(T), where

a (T ) =

πE J (T ) + 1, k BT

(7.21)

thus yielding the familiar KTB power law dependence. Above TKT, the IV exponent is 1 ”D7 EXWWKH,9FXUYHVKDYHDORZFXUUHQWIOX[IORZYROWDJHDULVLQJIURPWKH thermally unbound vortices (see Equation (7.20)). Below TKT, a(T) •DQGWKH,9FXUYHV are pure power law. The total voltage signal is closely approximated as the sum of (7.22)

Equations (7.19) and (7.20),

V = Vc−u + Vt −u . Figure 7.1 shows the IV curves generated from Equation (7.22) plotted on a log scale to show the power law behavior. Here we used a square array (L = W = 300a) with a lattice parameter a = 10µm, a normal state resistance Ro = 100µΩ, and TKT = 2.55 K. We determined ic from the universal relation ic(TKT) = (26.706 nA/K) TKT @ 70nA and then calculated EJ  ic/ 2e. For ease of calculation we suppressed the temperature dependence of ic (and hence EJ), a reasonable approximation near the transition and in the weak Josephson coupling limit. We also ignore the renormalization correction, which is assumed to be small. The constant α was set equal to one. Even though the Equations (7.19) and (7.20) contain the array size in their expressions, the data of Figure 7.1(a) assume that we are in the thermodynamic limit (L → ∞). A finite-size sample will contain a population of thermally-generated free vortices

92

both above and below TKT. If we assume that λ⊥ > L, this finite-size-induced free vortex density can be written as [42],

b n f (T ) = 32 e −πE a

J

/ k BT

 L   a

πE J / k BT

,

(7.23)

and the flux flow voltage contributed by these vortices will be of the form

V f −s = a 2WRo n f (T ) i ,

(7.24)

where f-s stands for “finite size,” and b3 in Equation (7.23) is approximately constant for small current. The total voltage for a finite size array will be given by the sum of Equations (7.19), (7.20), and (7.24)

V (T , i ) = c1Vc−u (T , i ) + c2Vt −u (T , i ) + c3V f − s (T , i ) , where we have made the temperature and current dependence explicit and added the constants c1, c2 and c3, all of roughly the same order, to allow us to adjust the IV curves so that they appear in a current-voltage window that is roughly experimentally accessible. We emphasize that these constants do not change the essential character of the IV curves, but rather change where the “bends and wiggles” will appear in current-voltage space. Figure 7.1(b) shows Equation (7.25) plotted on a log scale over an abnormally large voltage scale (the usual range is 10-10 to 0.1 Volts) but over a typical current scale. All other generating parameters were the same for (a) and (b) including the values for c1 and c2.

93

7.5 Current-Voltage Scaling Results We may now analyze the IV data of Figure 7.1(a) and (b) using scaling, as expressed in Equation (7.16). Our approach is to plot I1+1/z/[TV1/z] as a function of the scaling function variable x = Ix/T and vary the fitting parameters z, TKT, and b to achieve the best collapse onto a scaling curve. The initial guess for TKT was made by determining the highest temperature IV curve which did not display an ohmic tail at low currents. The slope of this IV curve gives an initial estimate for z. A series of scaling plots were then generated with TKT and z varying around these estimated values and with b varying approximately between 1 and 100. The parameters which produced the best scaling were determined by examination of these plots. In practice, we found many values of TKT which gave an acceptable scaling collapse, but there was always a highest value above which no collapse could be achieved. We report below those highest values of TKT, and the corresponding values for z and b, that yield the best collapse. This method closely mirrors the procedure followed by Pierson, et al. For each of the parameters obtained from a scaling collapse, the error in the reported temperature is ±0.01 K and the error in the value of z is ±0.03. The value of b could be varied within a wide range without having a significant effect on the scaling. The fitting parameters TKT and b are contained in the expression for the KT correlation length x shown in Equation (7.18). It is proper to use x in the scaling analysis of the data in Figure 7.1(a) (no finite-size effects) where the thermodynamic limit is assumed, and then only above the transition, since x is not well defined below the transition. For the finite-size effect data [Figure 7.1(b)], it is not proper to use x in the scaling analysis for all temperatures because of the existence of finite-size-induced free 94

vortices presumes that the correlation length is larger than the sample size [67], taking us out of the thermodynamic limit. In this case, we should substitute L for the correlation length, at least for those temperatures for which x > L. Nevertheless, we will proceed by using x for our analysis in order to draw a connection with the work of Pierson, et al. Figure 7.2 shows the scaling behavior of the data of Figure 7.1(a) (no finite-size effects). Here, the best scaling collapse occurs for TKT = 2.55K, in agreement with the value used to generate the data, and z = 2, in agreement with the KT theory. Notice that the data above TKT (lower scaling curve) show excellent scaling behavior in that all of the IV curves collapse onto a single scaling curve with no stray data. This, of course, is not surprising in that the data were generated using the KTB model and evaluated in this regime where the KT correlation length is well defined, so that true scaling behavior is expected. Nevertheless, the lower scaling curve of Figure 7.2 sets the standard by which scaling curves using experimental data should be evaluated. Figure 7.2 inset (b) shows an expanded view of the scaling curve above the transition. The data below the transition (upper scaling curve) does not display nearly as good of a scaling collapse as above. Data very near the transition (right side of scaling curve) are slightly askew and do not seem to lie along the same curve, and data at lower temperatures do not collapse completely on top of each other. This curve is, in fact, strongly reminiscent of many scaling curves using experimental data that are considered good evidence of scaling behavior. Most experimental data, however, have power-law dependence over only one or two orders of magnitude (on rare occasions as high as three or four) with a rollover to ohmic behavior at high and low currents. Thus, the data is often culled to include only the power-law portion – typically a very

95

0

1x10

a) -2

1x10

-4

2.76K

1x10

-6

1x10

-8

v(V)

1x10

-10

1x10

-12

1x10

-14

1x10

-16

1x10 10

-18

10

-20

0.5K

-8

-7

10

-6

10

-5

10

1x10

I(A) 0

1x10

b) -2

1x10

2.76K -4

1x10

-6

1x10

-8

v(V)

1x10 1x10

-10

1x10

-12

1x10

-14

1x10

-16

10

-18

10

-20

0.5K

-8

10

-7

-6

10

10

-5

1x10

I(A)

Figure 7.1 (a) Simulated current-voltage curves for a Josephson junction array in the thermodynamic limit (no finite-size effects) for temperatures varying between 0.5K and 2.76K, with temperature steps of 0.05K. The dark line indicates TKT. (b) Currentvoltage curves including finite-size-induced resistance. Temperatures are shown every 0.1K. 96

short portion of the IV curve – and, as a consequence, the scaling may appear more favorable than it would otherwise. (As an example, in Figure 7.2 inset (a)) we plot the data below the transition, but truncated to include only IV data above 10-9V.) Conversely, here, where the data is as KTB-like as possible, i.e., pure power-law IV curves over 10-15 orders of magnitude with a(T) following the expected temperature dependence, we should expect to see the best scaling collapse possible. That we do not is due to the fact that the KT correlation length is not well defined below the transition and so no true scaling behavior should be expected. In order to show the effects that finite size and experimental limitations have on scaling behavior, we start with the finite-size-induced free vortex data of Figure 7.1(b) and introduce a voltage cutoff. This voltage cutoff plays the role of a minimum voltage sensitivity or experimental voltage noise floor for a measurement system. In Figure 7.3 we replot the data of Figure 7.1(b) with four voltage cutoffs: one at V = 10-7 V, 10-8 V, 10-10 V, and 10-12 V. Notice that as the minimum voltage or noise floor is reduced, the effect is to include progressively more of the finite-size induced linear tail in the IV set. As we shall see, this has a dramatic effect on the parameters of the IV scaling function. Figure 7.4 shows the scaling curves obtained for the four voltage cutoffs V = 10-7 V, 10-8 V, 10-10 V, and 10-12 V. Here, we show the best scaling curve using the highest value of TKT for which a scaling collapse would occur. For the V = 10-7 V cutoff [Figure 7.4(a)], we obtain TKT = 2.29 K and z = 2.23, in contrast to TKT = 2.55 K and z = 2 obtained in Figure 7.2. We note that the 10-7 V cutoff, being the highest of the cutoff voltages, allows for very little of the finite-size-induced linear tail to be included in the IV data set. This is reflected in the scaling parameters being comparatively close to

97

2

10

0

10

TKTB = 2.55 z=2 b = 10

0

T < TKTB -3

10 -7 10

1/z

-2

10

1+1/z

/ [TV ]

10

(a)

1x10

-4

-3

1

10

10

-3

I

10

T > TKTB -6

10

(b) -4

10 -6 10

-8

10

-10

10

-6

10

-2

10

2

10

-1

4

10 6

10

10 10

10

Iξ/ T

Figure 7.2 Scaled IV curves for data of (a) including no finite-size effects. Inset (a) shows a blowup of the data above the transition, but with the IV data truncated below 10-9 V. Inset (b) shows a blowup of the scaling curve below the transition to show the details of the scaling collapse.

98

14

10

-1

10

-2

10

-3

10

-4

1x10

2.76K

-5

1x10

-6

v(V)

10

-7

10

-8

10

-9

10

-10

1x10

-11

10

0.5K

-12

10

-8

10

-7

-6

10

I(A)

10

-5

1x10

Figure 7.3 Replot of Figure 7.1(b) showing the voltage cutoffs. IV curves are shown every 0.05 K. The arrows denote the location of the “ghost transition” for each cutoff value (see text), the point at which the resistive character disappears.

99

those of IV curves without finite-size-induced resistance. As the voltage cutoff is lowered, the value of TKT obtained from the scaling procedure progressively decreases and the value of z increases. For the V = 10-12 V cutoff the values are TKT = 0.84K and z = 5.9. By adjusting a parameter that is determined by the experimental measurement system (i.e., the noise floor) we can vary the fitting parameters of the scaling collapse. Contrast this behavior with non finite-size data where for all voltage cutoffs down to V = 10-20 V (where we stopped), the same values for z and TKT yield the same scaling collapse. Thus, data obtained on the same sample but measured using different measuring systems can yield completely different scaling fits. This simple fact calls into question the practical viability of exploiting the scaling behavior of IV curves to confirm the details of the phase transition in 2D superconductors. We also point out that the value of the voltage cutoff is somewhat arbitrary for the data that we generated. That is, for the same selection of voltage cutoffs, we could have altered the results of the scaling fits by changing the values of c1, c2, and c3 in Equation (7.25) to allow more or less of the resistive portion to appear above (or below) the cutoff. This is akin to the experimental situation where the noise floor of the measuring system is fixed and the coupling strength of the sample determines how much of the resistive tails of the IV curves will be observable. Neither is the quality of the scaling collapse an indication of the reliability of the scaling fit. In Figure 7.4 we have plotted the scaling curves as lines rather than as data points so as to expose any shortcomings in the scaling collapse. We note that the scaling curves look to be quite good, certainly comparable to most experimental data scaling,

100

-4

-4

1x10

1x10

-7

Vcutoff = 10 V

-8

1x10

1/z

10

-7

I

I

-7

TKTB = 1.72K z=3 b = 19

-8

-8

10

10

-9

-9

10

-6

10

1+1/z

1/z

10

/ [TV ]

TKTB = 2.29K z = 2.23 b = 25

-6

1+1/z

/ [TV ]

1x10

10

Vcutoff = 10 V

-5

-5

-8

10

-6

10

-4

1x10

-2

10

0

10

2

10

4

10

6

10

8

10

10

10

10

12

10

Iξ/ T

-4

1x10

-8

10

-6

10

-4

1x10

-2

10

0

10

2

10

4

10

-5

-5

-8

10

10

-7

TKTB = 0.84K z = 5.9 b = 20

I

I

TKTB = 1.37K z = 3.7 b = 20

1/z

/ [T V ]

-7

-6

10

1+1/z

1/z

/ [TV ]

1x10

-6

1+1/z

12

10

-12

1x10

-8

10

-9

10

10

10

Vcutoff = 10 V

Vcutoff = 10 V

10

8

10

Iξ/ T

-4

1x10 -10

10

6

10

-9

-8

10

-6

10

-4

1x10

-2

10

0

10

2

10

4

10

6

10

8

10

Iξ/ T

10

10

12

10

10

-8

10

-6

10

-4

1x10

-2

10

0

10

2

10

4

10

Iξ/ T

Figure 7.4 Scaling collapse of finite-size-induced resistive IV data [] with a voltage cutoff of (a) 10-7 V; (b) 10-8 V; (c) 10-10 V; and (c) 10-12 V.

101

6

10

8

10

10

10

12

10

despite the wide variation of the scaling parameters. In particular, the data above the transition (lower scaling curves) seem to exhibit an especially good collapse in each case. A closer examination, however, reveals a few problems. In Figure 7.4(a) (10-7 V cutoff), for the curves above the transition a slight deviation from KTB behavior at the lowest currents is caused by the addition of finite-size-induced free vortices. This deviation, difficult to discern from the unscaled data [Figure 7.3] yet clearly manifest in the scaled, prevents a total scaling collapse of the data (compare with Figure 7.2). The deviation is also present Figure 7.4(b) through (d). As the voltage cutoff is lowered, TKT is reduced; this causes more of the IV curves to end up on the scaling curve above TKT. The curve becomes “thickened,” making it difficult to distinguish the slight flaws in the collapse. Indeed, if we had used the data points of only moderate size, we might not even notice the effect. In addition, the shape of the scaling curve changes, becoming more rounded (once again, compare with Figure 7.2). For data below the transition (upper scaling curves) the scaling collapse is not good at all. But as the voltage cutoff is lowered and the apparent TKT is reduced, fewer IV curves remain: only those at the lowest temperatures which are now suddenly “near the transition”. Consequently, the scaling collapse may appear better than it really is.

7.6 Discussion and Conclusion We have demonstrated that the interpretation of dynamical scaling of IV curves in 2D systems is subtle. In the thermodynamic limit, while scaling exists and is robust above the transition, it does not exist below the transition (see Figure 7.2). The reason is that the 2D correlation length is well defined above the transition but is infinite at and below 102

the transition. The addition of finite-size effects significantly degrades the scaling behavior. Although a scaling curve can be obtained for finite-size effect data, the scaling parameters are significantly altered, particularly the dynamical critical exponent, z. A change in z would nominally point to a change in the vortex dynamics of the phase transition, indicating other than the diffusive behavior in the KTB picture where z = 2. While it is certainly not unreasonable to believe that the vortex dynamics of a finite-sized system may be different from an infinite system (perhaps even entirely different), we are skeptical that the scaling collapse alone is sufficient evidence for this change. The fact that we may obtain a range of values of z simply by truncating the finite-size data at various voltage cutoffs makes the actual value of z for most experimental data highly suspicious, at least in the absence of corroborating support from other analytical methods. Nevertheless, it is intriguing that Pierson et al. found a value of z ≅ 6 for a variety of 2D systems, both superconducting and superfluid. Rather than pointing to some universality of physics, however, based on our analysis we suspect that this has more to do with the nature of the data collection and the limitation of the instrumentation. In particular we note that the Johnson noise is the universal noise floor for all measurement systems and that many systems are optimized to approach this limit. We end with a discussion of the scaling behavior below the phase transition. As mentioned, IV data below TKT should not scale, and a careful examination of the scaling curves (Figure 7.2 and Figure 7.4) shows that this, indeed, is the case (at least in comparison to the quality of the scaling collapse in the lower curve of Figure 7.2. The data below the transition, however, certainly does show a tendency toward a scaling collapse. MKM have dealt extensively with this question for the 2D-XY model with

103

resistively-shunted-Josephson junction dynamics. They point out that, because the low temperature phase is “quasi-critical” with x = ∞, each temperature is characterized by its own scaling function. For small values of the scaling variable x, however, the scaling function may be taken to be temperature-independent. For a finite-sized system, they assume an approximate form for the correlation length x ∝ R-α, where R is the resistance in the limit of zero current and α will, in general, depend on both L and T. Then they demonstrate that, should α be a constant, the resistance would vanish at a temperature for which z = 1/α. In the KT picture, MKM note that the resistance actually vanishes only at zero temperature (once again, for finite size), but that it could happen that α is approximately constant over a limited temperature regime causing the resistance to appear to vanish at some non-zero temperature. If IV data happen to fall within this limited temperature regime, we would observe an apparent scaling collapse and the apparent vanishing of the resistance at some specific temperature. That is, we may be tempted to conclude that we have evidence for a phase transition. MKM terms this type of transition a “ghost transition”. We may make a connection between this “ghost transition” and our voltage cutoff analysis by noting that the IV exponent a(T) is related to the dynamical critical exponent below the transition [68]: a(T) = z(T) =1. For each of the four voltage cutoffs used, we take the value for z obtained in the scaling process [Figure 7.4] and identify the corresponding temperature for which the IV exponent is z + 1. These IV curves are identified by the arrows above each of the cutoff axes in Figure 7.3. Notice that each of these curves may be identified as the one where evidence of the low current resistive behavior first disappears above the corresponding noise floor. That is, all IV curves at 104

temperatures below this one are pure power-law-like and the ones above show curvature toward an ohmic slope. This observation is illusory. If we look below the cutoff voltage each of the IV curves displays ohmic behavior at lower currents. In the MKM picture, this ghost transition temperature will change depending on the finite sample size. For a fixed voltage cutoff (noise floor), decreasing the sample size will cause more finite-size-induced resistance to appear above the cutoff, and the ghost transition will move to a lower temperature. This is analogous to our picture in which we keep the size of the sample fixed but allow the voltage cutoff to decrease, thereby uncovering more of the resistive character and affecting the ghost transition. In either case, it is obvious that this does not constitute a “true” phase transition and thus, a search for new vortex dynamics is not required.

105

Appendix A : SQUID Electronics Schematics

The electronics for this system were based on the design of Fred Wellstood’s group at Maryland and were constructed by Dexter Clark at U.C.

1

2

3

4

Bandpass D

D

C1 200p C2

J13

J14

15 R4 1K

R5 1K C6 -12

0.01u +12

C1

C3 R3 1K

C

0.1u

J3

J8 1 AD829J 6

2 J1

3

Input From

U1

C7 3 0.01u

5

2

U2

5 AD829J 6

R6 R1 10

C

0.1u

J4

8

R2 1.24 C5

C1

J9 Output to Phase

1

0.47u

8

C4 30p

10

C1 R7 10

J2 Groun

0.1u

0.1u -12

Gain

J10

+12

B

B

J11 J5

L1 1mh

R8 200

+12

+12 Volt Popwer C8 10u

C9 10u

C1 10u

C1 10u

J12 Groun

J6 Groun

J7 -12 Volt Power

L2 1mh

-12

A

A

1

2

3

Figure A.1 Bandpass amplifier

106

4

1

2

4

3

Current Monitor D

D

J4

J6

J5

Current Monitor

External CVurrent Input

J3

C

J1

C

J2

R1

R3

R4

R4

806

1K

1K

1K

+12

J? J7

C1 10uf

C2 0.1uf

R2

D1 6.2v

2K

C1 2.2uf

Current Out J8

C1 2.2uf

Current Out J9 Ground

Current Adjust B

B

A

A

1

2

3

Figure A.2 Current monitor

107

4

1

2

4

3

Integrator R1

R3

R4

10K

10K

J17

+12 806 D

C1 10uf

C2 0.1uf

C3 1uf

R2

D1 6.2v

D

2K J6

J7 S1 J20 AC Modulation Input from Phase Detector

Reset R5 100 J3

R6

C4

100

0.022uf J4

J21 J8

Ground

J5 C5

C

C

-12

J1 Error Input from Phase Detector

R7

R8

10K

10K

8

4 1

0.1uf

S2

2 3

J2

J9

J10 R12

OP-27

In

J11

1K

R10 1Meg

7

R9 20K

Out

6

U1

C6 +12

Ground

Feedback Output

C7

0.1uf

J18

B

B

500pf R11 1Meg J19

C12 -12 J12

Buffered Output

L1 1mh

0.1uf

C9 10uf

C8 10uf

8

+12 4 1

+12 Volt Popwer Supply

2

J13

3

Ground

6

U2 OP-27

C11 10uf

A C13

J14 -12 Volt Power Supply

-12

L2 1mh

J15

49.9

7

C10 10uf

A

R13

J116

+12 0.1uf

1

2

3

Figure A.3 Integrator

108

4

1

2

4

3

1

J6

U3 Input from Bandpass Amplifier 8

Out

+5

D

1 2 11 23 14 S1 5

1 2 3 4

0 1 2 3

n

3 4 5 6 7 8 9 10

S2 5

1 2 3 4

0 1 2 3

n

1

U1

+5

J7 D

U2

S0 S1 ENT ENP CLK

RCO

A B C D E F G H

QA QB QC QD QE QF QG QH

1 2 11 23 14

13

22 21 20 19 18 17 16 15

3 4 5 6 7 8 9 10

S0 S1 ENT ENP CLK

Oscillator Monitor RCO

13

A B C D E F G H

QA QB QC QD QE QF QG QH

22 21 20 19 18 17 16 15

R5 100 R3 10K

J9

C3 .033uf 1

R1 10K

J10

U6 3

X

X0 X1 X2 X3 X4 X5 X6 X7

74LS867DW

74LS867DW

1

J8

C

INH A B C VEE

13 14 15 12 1 5 2 4

7

U4B

5

6 11 10 9

Detector Output

MC4558

6

1

C4 R2 10K

1 100pf

C

J11

J12

AC Modulation Output to Intergator

7 R4

74HCT4051

10K AC Modulation Adjust

R6 25K

+12 B

Oscillator J13

C1

B

8

.1uf J5

C6 .1uf

2

Vout

1

U4A

1

+5

U7 79L05 1

1mh C8 10uf

C7 .1uf

C9 .1uf

Vin

J14

Reference 4

L8

3

GND

1mh

Vin

GND

U5 7805 1

C5 10uf

2 -12

Vout

3

1

J15

C2

C10 .1uf

2

-12 L7

3

1

1

J4 1

J3

1

J2 1

J1

.1uf

A

A

-12

Title: Synchronous Phase Detector Serial: 082019980841HDC 1

2

Organization: By: 3

Figure A.4 Phase detector

109

University of Cincinnati Physics Department Dexter Clark

Date:

Sheet: 1

07-18-99 4

of: 1

1

2

4

3

Preamplifier D

D

C10

J8

47uf R7 5.11K

L1

R5 100K

2.5mh

R11 100

C9 1uf

J8

C5

Output to Bandpass Amplifier

Q6 2N4250

C Adjust R2 and R4 for maximum dynamic range and minimum noise

C

0.01uf C7 Q2 2N5434

R4 20K

R6 715K

Q5 2N3565 10pf

C3 1uf

R8 51.1K

J2 C12 J1

J4

C1 Q1 2N5434

Input from SQUID 330pf

1uf

J3

J5

R1 100Meg B

C6 R2 20K

C2 1uf

R9

C4 0.1uf

J6

549

1uf

B

C8

R3 11

R10 1K

J7

43pf

A

A

1

2

3

Figure A.5 Preamplifier

110

4

1

2

4

3

Power Input Filter

Regulator Circuit Board J2 +12vdc Regulated R3 1.5K

D R1 RES1

R4 1K

R7 3.01K

D

C3 10uf

R6

J1

C1 47uf

J7

10K 1 2 3 4 5 6 7

Ground

Q1 TIP33A

NC I LIM I SEN ININ+ VREF V-

J6

500pf

J4 C

C9 J3 +24vdc Input

U1 LM723CN(14)

NC F COMP V+ VC OUT VZ NC

C11

L1

C

1mh 14 13 12 11 10 9 8

100uf C4 10uf

J8

C5 10uf

J5

J9

-12vdc Regulated

J14

Ground

C12

C13

J12 100pf

1uf

J13

-24vdc Input

R2 RES1

14 13 12 11 10 9 8

C7 10uf

L2

C10 100uf

1mh J11

Q2 TIP34A J16

J15

R9 3.01K

1 2 3 4 5 6 7

U2

B

3.01K

NC F COMP V+ VC OUT VZ NC

C6 10uf J10

NC I LIM I SEN ININ+ VREF V-

B

R8

C2 47uf

LM723CN(14)

A

C8

R10 1.1K

1

2

A

R11 R5 1K

3

Figure A.6 Power input filter and regulator

111

10uf

3.65K

4

Appendix B: Flux-to-Current Algorithm The following code is the Roth et al. algorithm [Figure 4.4] [38] for converting magnetic field data to current distribution data. This is written as a macro for Transform [37] Version 3.3, which is out of date and unsupported by Fortner at this time, but the latest version did not offer any new benefits. This was initially put into Transform Macro code by Aaron Nielsen.

* Notebook macro * Name of macro: flux2current * must enter the following data before executing macro * macro will return x and y components of J in real space

*********************************************************** * Before running this macro, set the following parameters * *********************************************************** * phidata : SQUID output data, assumes that the SQUID * data is in this file. * sample_thickness : must be in millimeters * squid_distance : must be in millimeters * phi_external : must be in the same units at phidata, * namely electronics output * k_max : spatial frequency cutoff, inverse millimeters * initialize *user_interactive = false textformat = "F10.7" PI = 3.1415926 * mu_0 == Tesla * mm / mAmp mu_0 = 4 * PI * 10**(-7)

* t = = thickness of sample (mm) sample_thickness = 2000*10^-7 t = sample_thickness

112

* d == distance of SQUID from sample *(mm) squid_distance = 100/1000 d = squid_distance *L = = length of side of SQUID (mm) L = 10/1000 squid_length = L

* read in data and adjust scales to read mm phidata = ”filename” phi_SQUID = colscales(rowscales(phidata,0,2*rowrange(phidata)/100000),0,colrange(phidata)/10000) call close("phidata")

* subtract applied flux phi_external = 0 phi_internal = phi_SQUID -phi_external * convert flux to h-field measurements hz in Tesla * converting volts out of SQUID electronics to field * averaged over squid area. hz = phi_internal * 1.58 * 10**(-5)

* resample and FFT h field

* the resampling is required because transform can only * do a FFT for 2^n number of data points * all the calculations must be done with complex numbers * so we do the conversion to complex numbers here also hz_r = resamplecols(resamplerows(hz,512),512) hz_rc = complex(hz_r,hz_r*0) hz_kc = fft(hz_rc,1) hz_kc_mag = ampl(hz_kc)

* k vector components associated with *fft data kx=x(real(hz_kc)) ky=y(real(hz_kc)) k = sqrt(kx**2 + ky**2) 113

* convolve field data with SQUID loop geometry squid_filter_kx_temp=2*sin(kx*squid_length/2.0)/squid_length/kx squid_filter_kx=zapnan(squid_filter_kx_temp,1) squid_filter_ky_temp=2*sin(ky*squid_length/2.0)/squid_length/ky squid_filter_ky=zapnan(squid_filter_ky_temp,1) squid_filter = complex(squid_filter_kx*squid_filter_ky,squid_filter_kx*squid_filter_ky) call close("squid_filter_kx_temp") call close("squid_filter_ky_temp") call close("squid_filter_kx") call close("squid_filter_ky")

* low pass FFT data with Hanning window k_max =500 kpass = LTmask(k,k_max) hanning = kpass * 0.5 * (1+cos(PI * k/k_max)) hz_k_pass = complex(hanning,hanning)*hz_kc/squid_filter call close("squid_filter")

* solve for current density in fourier space * g1 and g2 are the Green’s Function g1 = 2/mu_0/t * exp(k * d) / k g2 = complex(g1,g1) hz_k_pass_i = complex(imag(hz_k_pass),-real(hz_k_pass)) jy_k = zapnan(g2 * complex(kx,kx) * hz_k_pass_i,0) jx_k = zapnan(-g2 * complex(ky,ky) * hz_k_pass_i,0)

* inverse FFT to get Jx and Jy in real space * Jx and Jy are the currents we are looking for Jx = real(fft(jx_k,-1)) Jy = real(fft(jy_k,-1))

*Jtot = (Jx**2 + Jy**2)**(1/2)

* close all the datasets generated during macro execution 114

* you may want to look at these to verify that all the steps * in the computation are proceeding correctly call close("phi_internal") call close("phi_SQUID") *call close("hz") call close("hz_kc") call close("kpass") call close("hanning") call close("g1") call close("hz_k_pass") call close("g2") call close("hz_k_pass_i") call close("k") call close("kx") call close("ky") call close("hz_r") call close("hz_rc") call close("jy_k") call close("jx_k") call close("hz_kc_mag")

115

Appendix C : Josephson Junction Array Fabrication Process

The Josephson junction array fabrication was done at the Cornell Nanofabrication Facility [43] with project manager David Spencer.

1. SPIN resist: resist: NEB 31 2:1 PGMEA(TypeP) NOTE: I actually used SAL-603 resist, but the lab has since switched

speed: 1500 RPM spin time: 60 seconds thickness ~ 300nm

2. BAKE resist: vacuum hotplate temperature: 110oC time: 2 min

3. EXPOSE: Leica/Cambridge EBMF 10.5/CS Electron Beam Lithography System beam current: ~2.9nA dose: 9 – 11 µC/cm2 4. POST_EXPOSURE BAKE: hotplate temperature: 95oC time: 4 min

116

5. DEVELOP resist: developer: MF-321

NOTE: I developed the SAL-603 with CD-30 developer

time: ~ 32sec. (10sec per 100nm) rinse: deionized water

6. ETCH: Applied Materials Reactive Ion Etcher [O2 plasma clean chamber 30sccm, 30mTorr, ~10min] [condition chamber at process parameters ~ 3min] gases: CF4, 15sccm SF6, 15sccm pressure: 30mTorr rate: 500-1000 Å/min time: 3:10min

7. PATTERN and ETCH Au: spin resist: Shipley 1813 @ 4000 RPM for 30sec bake: 115oC, 1 min expose: 2sec HTG Deep-UV Contact Aligner develop: CD-26, 1min rinse: deionized water bake: 130oC, 3min etch: gold etch ~ 45-50sec strip resist: acetone

117

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17. Beryllium copper springs from Small Parts Inc., Miami Lakes, FL, (305) 558-1038. 18. Del-Seal linear motion feedthroughs, Models BLM-133-1, BLM-133-4, BLM-133-6, from MDC Vacuum Products Corporation, Hayward, CA, 415-887-6100. 19. ¼” shaft, 8 wire skew pole stepper motors from EADmotors/ Eastern Air Devices, Inc., Dover, NH, eadmotors.com . 20. Timing belts, pulleys, and gear boxes from Stock Drive Products/ Sterling Instrument, New Hyde Park, NY, www.sdp-si.com . 21. MD10A Microstep Motor Drive and PC34-4 4-axis controller from Oregon Micro Systems, Inc., Beaverton, OR, www.omsmotion.com . 22. Model MPS 28-07, 28V, 7A power supply from Cyber Research, Inc., Branford, CT, (203)483-8815. 23. Bendix/ Amphenol connectors supplied by Allied Electronics, Cincinnati, OH (513)771-6990. 24. www.swagelok.com . 25. Cryogenic 10 pin vacuum electrical feedthroughs from Oxford Instruments, Concord, MA. 26. Duo-Twist Cryogenic Wire, DT-36, from LakeShore Cryotronics, Inc. Westerville, OH, www.lakeshore.com . 27. Pi-section filters part no. 9001-100-1010 by Spectrum Control Inc., Fairview, PA, www.spectrumcontrol.com . 28. Ray-Proof Corporation, now Raymond EMC, Ontario, Canada, www.raymondemc.ca . 29. Hypres, Elmsford, NY, www.hypres.com .

30. Stycast 2850FT Black and Catalyst 24LV from Emerson & Cuming Specialty Polymers, Lexington, PA. 31. Shipley Chemical Co., Newton, MA.

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