ABSTRACT HYSTERETIC DC SQUID. Dissertation directed by: Professor Frederick C. Wellstood. Department of Physics

Transcription

1 ABSTRACT Title of Dissertation: DESIGN OF A LARGE BANDWIDTH SCANNING SQUID MICROSCOPE USING A CRYOCOOLED HYSTERETIC DC SQUID Soun Pil Kwon, Dotor of Philosophy, 006 Dissertation direted by: Professor Frederik C. Wellstood Department of Physis I present the design and analysis of a large bandwidth sanning Superonduting Quantum Interferene Devie (SQUID) mirosope. Currently available SQUID mirosopes are limited to deteting magneti fields with frequenies less than MHz. However, for observing nanoseond time sale phenomena suh as logi operations in today s omputer hips, SQUID mirosopes with GHz bandwidth and larger are required. The major limitation in SQUID mirosope bandwidth is not the SQUID itself but the eletronis and readout tehnique. To inrease bandwidth, the fast transition of a hystereti d SQUID from the zero voltage state to the resistive state an be used as the detetion element in a new SQUID readout tehnique, referred to as pulsed SQUID sampling. The tehnique involves pulsing the bias urrent to the d SQUID while monitoring the

2 voltage aross it. As the pulse length shortens, the SQUID measures the applied external magneti flux with shorter sampling time, whih inreases the bandwidth. Experimental tests of the tehnique have demonstrated the possibility of following signals with frequenies up to GHz using a d SQUID with Nb-AlO x -Nb Josephson juntions at around 4 K. Ringing in the pulse profile permitted the effetive bandwidth of the sampling tehnique to be muh greater than the nominal value suggested by the pulse length setting on the generator. I identify additional means of inreasing bandwidth: redesigning the d SQUID, implementing transmission line wiring, adding high speed superonduting iruits, et. whih should allow bandwidths to reah 40 GHz and higher. Towards reating a large bandwidth SQUID mirosope, I also assembled and tested with ollaborators a fully funtional 4 K sanning SQUID mirosope. With the mirosope, whih used a nonhystereti niobium d SQUID with onventional flux-loked-loop SQUID eletronis, I was able to obtain the magneti field image of a urrent arrying iruit.

3 DESIGN OF A LARGE BANDWIDTH SCANNING SQUID MICROSCOPE USING A CRYOCOOLED HYSTERETIC DC SQUID by Soun Pil Kwon Dissertation submitted to the Faulty of the Graduate Shool of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Dotor of Philosophy 006 Advisory Commitee: Professor Frederik C. Wellstood, Chair Professor Howard Dennis Drew Professor Bryan W. Eihhorn Professor Rihard L. Greene Professor Christopher J. Lobb

4 Copyright by Soun Pil Kwon 006

5 PREFACE I have sometimes wondered whether dotoral dissertations have any signifiane to people other than its author. It seemed to me that most dissertations end up in library or offie shelves just aumulating dust with the passage of time. Only sparingly were these works, books in their own right, taken out of hibernation and read, not over to over but for a single hapter at most. Did the reader find what he or she was looking for? Was the information useful? Does the reader reognize that a lot of time and effort went into the doument s preparation? As I prepared to write my own dissertation, I thought about my own relationship with dissertations prepared by former students and others. A dissertation should foremost represent the author s original sholarly work while working toward a degree. But beyond that, given the effort put into preparing it, should it not also be useful to others? Often while looking up dissertations of former students, I found the best works to be those whih ontained information that diretly helped my own researh. To this end, I deided to make a onsious effort to prepare a dissertation that would be helpful to those interested in onstruting and maintaining a sanning SQUID mirosope as well as ontinuing this line of researh. As a result, this doument is part operation manual, olletion of experimental notes, blueprint, and detailed review of basi theory. It is hoped that those who read this doument find it useful as referene material, frequently looked up to address problems ourring day to day. Soun Pil Kwon ii

6 DEDICATION To my parents who provided me opportunities and to my hildren who are my future iii

7 ACKNOWLEDGEMENTS The work presented in this dissertation overs a period of almost four and a half years. During that time I often was alone, working in the sub-subbasement, literally the lowest plae in the entire physis building, and then walking bak to an off ampus apartment in the wee hours of a hilly dark morning. But if it were not for the help and support of numerous individuals, it may not have been just four and a half years. And, it ould have been muh lonelier. I would like to thank the physis faulty, espeially my advisor Professor Wellstood and onetime o-advisor Professor Lobb. Guidane and diretion are good, but it was the finanial support they provided through the Center for Superondutivity Researh that allowed me to proeed with the work resulting in this dissertation. Considering the delays during the past eighteen months while this dissertation was being prepared, I owe a large debt to Professor Wellstood for his patiene and ontinued finanial support. Without funding, I ould not have sustained myself and my family, and for that I am grateful. I would also like to thank my dissertation ommittee members for their patiene. The delivery of this doument for their review has been long over due. I hope their patiene was not severely tested and that this work is up to their expetations. I would like to thank the tehnial staff of the Physis Department as well as that of the Center for Superondutivity Researh. In partiular, Russell Wood who took over the student shop was vital in providing and setting up equipment needed to iv

8 effiiently onstrut the apparatus for my experiments. His knowledge of tools and materials was muh appreiated, and students are fortunate to have him as a resoure. I hope he feels that he is well appreiated as well. I would like to thank olleagues and ollaborators at Neoera, In., who greatly ontributed to the 4 K SQUID mirosope projet. Most of all, the prototype SQUID mirosope would not have been possible if not for Neso Lettsome, formerly of Neoera, In. At least two people are required to safely assemble and maintain the mirosope. It was Mr. Lettsome who set up the table top infrastruture, designed and redesigned the majority of the omponents, and ordered and assembled nearly all of the mirosope parts. In fat, he performed most of the initial mehanial work on the projet and laid down the foundation on whih work ontinues to date. I hope that he is suessful in whatever endeavor he has now moved onto. I would also like to thank Dr. L. A. Knauss and the people at Neoera for being graious hosts during my time on site working on the mirosope. At the Center for Superondutivity Researh, I would like to thank former students and olleagues who assisted in training me and allowed use of their equipment. Dr. Aaron Nielsen first initiated me to the available equipment and failities at the Center and trained me to prepare SQUID tips, a skill whih was vital to the suess of the projet. I thank Dr. Bin Ming, Dr. Samir Garzon, and Dr. Yuanzhen Chen for help with equipment in Professor Rihard Webb s former laboratory. The departure of Professor Webb has left a gap in available failities that will be missed and will require muh time, money, and effort to reuperate. I also v

9 thank Mr. David Tobias and others who helped me in the use and operation of our wire bonding mahine. Other former students at the Center also helped with important advie and disussion. Dr. Andrew Berkley gave observations and suggested hanges in experimental setup, and along with Dr. Huizhong Xu, Mr. Mark Gubrud, Ms. Hanhee Paik, and Dr. Matthew Sullivan ommented on their experienes with devies and equipment that influened some of my hoies in designing apparatus for my experiments and the prototype SQUID mirosope. I also thank my olleague Mr. Constantine Vlahaos and Dr. John Matthews who now ontinue work on modifying the prototype mirosope and investigating the Pulsed SQUID Sampling tehnique disussed in this dissertation. On a general level, I would like to thank the support staff of the Physis Department and Center for Superondutivity Researh who assisted with preparing and proessing the paperwork required by university poliy. Proessing the dissertation and making sure that I satisfied all the registration requirements was made bountifully easier with the kind assistane of Ms. Jane Hessing. Purhasing of equipment would have been very umbersome if not for Mr. Robert Dahms, Mr. Jesse Anderson, and others. Finanial paperwork for my support and reimbursements for work travel were handled by Ms. Pauline Rirskopa, Ms. Grae Swelall, Mr. Brian Barnaby, and numerous others over the years I have been at the university. I would also like to thank past and present olleagues who helped me and ontributed to the organizing of soial events at the Center for Superondutivity Researh, the Physis Department, and elsewhere. These soial events provided a vi

10 sometimes muh welome diversion from everyday routines and helped lubriate the otherwise grinding gears amongst students, faulty, and staff. It is unfortunate that there has been no one to ontinue this effort as these events were appreiated and enjoyed by all. Along these lines, many thanks go to the staff of the Physis Department, espeially the Offie of Student Servies inluding Ms. Bernadine Kozlowski and Ms. Linda O Hara, who helped with ommuniation and provided spae for these events. Lastly, I thank my family. My parents have provided support in every sense of the word and more. My wife has taken on the majority of the burden in physially raising our young hildren during this time. My parents in-law have provided support for my wife when her burden beame too great. And, my hildren whose smiles relieved some of the stresses assoiated with both parenthood and graduate shool, were a burden that I have gladly arried. They have given me great joy. I hope when they grow up they look bak at this time with fonder memories than I. vii

11 TABLE OF CONTENTS List of Tables... List of Figures... xi xii CHAPTER Introdution.... SQUID Mirosopy... Overview... Motivation Preview of Work Outline of Thesis... 8 CHAPTER Introdution to Josephson Juntions and d SQUIDs Superondutivity and Josephson Juntions Josephson Equations... d Josephson Effet... a Josephson Effet d SQUID... 0 Critial Current Modulation... 0 Cirulating Sreening Current Hysteresis in Josephson Juntions and SQUIDs RCSJ Model Nonhystereti Juntions Hystereti Juntions From Juntions to d SQUIDs CHAPTER 3 SQUID Design Considerations SQUID Parameter Optimization SQUID Loop Indutane and Spatial Resolution SQUID Bandwidth... 5 Magneti Hysteresis and Critial Current Modulation Flux Noise and Optimization... 6 Magneti Indution and Nonlinear Effets Resistively Shunted SQUIDs Niobium SQUID Design and Charateristis Prior SQUID Design Measured SQUID Charateristis SQUID Chip Layout and Leads CHAPTER 4 The 4 K Cryoooled Sanning SQUID Mirosope Overview of Mirosope Cryoooler and Vauum System The Pulse Tube Cryoooler The Vauum Chamber and Pumps... 0 viii

12 Leak Problems Cold Finger and Thermal Anhoring Preparing the SQUID Tip and Cold Finger Motion Isolation Heat Removal... Making Eletrial Contats to the SQUID Sample Sanning Mehanism... 6 Translation System Overview... 6 Hardware... 9 Software and Sanning Parameters SQUID Eletronis and Instrumentation... 3 Flux-Loked-Loop Feedbak Eletronis... 3 Mathing Transformer Ciruit... 9 Signal Proessing and the Integrated Computer System... 3 Temperature and Pressure Monitoring Operation and Maintenane Cool Down Proedure Preparations and Proedures for Sanning Serviing the Mirosope... 4 CHAPTER 5 Magneti Field Image of a Test Ciruit Mirosope and Test Ciruit Preparations Obtained Magneti Field Image Problems with the SQUID Mirosope SQUID Controller Problems Sanning Problems CHAPTER 6 Design of Large Bandwidth SQUID Eletronis Limitations of the FLL Tehnique Inreasing SQUID Mirosope Bandwidth using a Hystereti SQUID with Pulsed Bias Current Basi Priniple and Requirements Pulse vs. Alternative Shemes Feedbak Field Follower New SQUID Control Algorithm Critial Current Detetion Pulse Rate Dependent Signal Following Shemes Implementation into Eletronis Synhronous Alternating Pulse Sheme Pulse Counting and Averaging Superonduting Ciruitry CHAPTER 7 Response of Hystereti Niobium d SQUIDs to Pulsed Bias Current Bakground Large Bandwidth Dip Probe Design and Constrution ix

13 Required Speifiations Final Design External Feedbak Flux Control SQUID and Pulse Signal Charaterization Ciruit Model of Dip Probe and d SQUID Transmission Line Model with SQUID as Voltage Soure 07 Frequeny Domain Analysis of Ciruit and Current Pulse. 0 Simulated Zero Voltage and Resistive State Responses Setting Short Bias Current Pulses Detetion of Mirowave Frequeny Magneti Fields using Pulsed SQUIDs... SQUID Response to Low Frequeny Signals MHz Signal Response... 9 Unexpeted Phenomena in the Voltage Response GHz Measurements and Extensions to Larger Bandwidth CHAPTER 8 Conlusions Summary of Work Large Bandwidth SQUID Mirosopy Limits of the Pulsed SQUID Sampling Tehnique Future Work Needed Mirosope Components and Replaements Upgrading to Large Bandwidth Appendix A: Critial Current of Ideal d SQUID with Asymmetri Juntions Appendix B: Design of a New SQUID Mirosope Cold Finger... 6 Appendix C: Charateristis and Impedane Analysis of Twisted Pair Wire Dip Probe Appendix D: Change in Magneti Flux due to a Rotation Appendix E: Modeling the d SQUID as a Resistor in the Large Bandwidth Dip Probe Referenes... 8 x

14 LIST OF TABLES TABLE I. SQUID parameters for SQUIDs AN and BH xi

15 LIST OF FIGURES FIG... FIG... FIG... (a) A z-squid sanning over a straight wire in the x-y plane at a height z o. (b) Z-omponent of the magneti field due to an infinitely long line urrent in the x-y plane as observed above the plane at a height z o.... B z (z-omponent of magneti field) due to two infinitely long parallel wires arrying urrent in the x-y plane as observed above the plane at a height z o Model of a superondutor-insulator-superondutor Josephson juntion FIG... (a) Ciruit diagram of an ideal d SQUID. (b) Photograph of a 30 µm Nb d SQUID with Nb-AlO x -Nb Josephson juntions, fabriated by Hypres, In.... FIG..3. FIG..4. FIG..5. FIG..6. FIG..7. FIG..8. (a) Diagram showing path of line integral around a superonduting loop deep inside the superondutor. (b) Diagram showing paths of line integrals around d SQUID loop, deep inside the superondutor and through the Josephson juntions.... (a) Graph of the magnitude of the superurrent through an ideal symmetri d SQUID with idential Josephson juntions and zero indutane plotted as a funtion of the total magneti flux through the SQUID hole. (b) Graph of the maximum magnitude of the superurrent through an ideal d SQUID plotted as a funtion of the total magneti flux through the SQUID hole when the ritial urrents of the Josephson juntions are different Equivalent eletroni iruit of a real Josephson juntion desribed by the Resistively and Capaitively Shunted juntion model Series of graphs showing the evolution of the phase spae (γ, v) of the over damped Josephson juntion (β << ) as the bias urrent y = I/I rises from (a) y = 0 to (b) 0 y < to () y = and finally (d) y >, at whih point a voltage appears aross the juntion I-V harateristis of an over damped (β << ) Josephson juntion at zero temperature I-V harateristis of an under damped (β >> ) Josephson juntion at zero temperature when (a) I R < /e and (b) I R > /e xii

16 FIG..9. Approximate I-V harateristis of an under damped (β >> ) Josephson juntion with I R < /e at finite temperature in the RCSJ model FIG..0. I-V harateristis of an under damped (β >> ) Josephson juntion with I R < /e at finite temperature inluding retrapping FIG. 3.. FIG. 3.. FIG FIG FIG FIG FIG FIG FIG (a) Simplified iruit model of a urrent biased Josephson juntion with stati bias urrent where the urrent is just a little greater than the juntion ritial urrent. (b) Simple iruit model of a SQUID loop with an externally applied a magneti field Series of graphs showing the solutions y = y for the total magneti flux through a d SQUID loop of self indutane L and juntion ritial urrent I when there is no externally applied flux, for dereasing values of a = Φ o / LI os γ av Series of graphs showing the disappearane of intersetions between y and y as the line y is translated vertially upward from 0, orresponding to dereasing external magneti flux Φ A through the SQUID hole (a) I-V harateristis of a nonhystereti d SQUID with β at finite temperature. (b) Graph of the SQUID voltage versus applied magneti flux through the SQUID hole when the d SQUID is urrent biased at the urrent I bias shown in (a) (a) Photograph of the resistively shunted niobium d SQUID designed by Cawthorne and Nielsen, whih was used in a LHe ooled sanning SQUID mirosope. (b) Photograph of updated resistively shunted d SQUID I designed with features ompliant with HYPRES design rules I-V harateristis of SQUID AN whih is of the Cawthorne and Nielsen design Series of osillosope pitures showing a omponent of voltage aross SQUID AN at different bias urrents Series of osillosope pitures showing a omponent of voltage aross SQUID AN in LHe, as the amplitude of an externally applied magneti field is inreased I-V harateristis of SQUID BH obtained from a four point measurement with ommon ground xiii

17 FIG I-V harateristis of a hystereti d SQUID showing asymmetry in the positive and negative ritial urrents FIG. 3.. Photographs of niobium d SQUIDs showing their primary ontat pads and leads FIG. 3.. Updated design of HYPRES niobium SQUID hip FIG Diagram showing an x-squid onfiguration with all ontat pads and leads going to one side of the SQUID loop FIG. 4.. FIG. 4.. FIG FIG FIG FIG Shemati diagram of the old region of the prototype 4 K ryoooled Sanning SQUID Mirosope Photograph of prototype Sanning SQUID Mirosope with a 4 K pulse tube ryoooler Photograph of the Cryomeh PT405 4 K pulse tube ryoooler with the old finger attahed to the seond stage heat exhanger Blok diagrams representing the working priniple of a basi single stage pulse tube ryoooler (a) Bottom portion of the ylindrial vauum hamber with the ryoooler assembly in plae. (b) Bottom portion of the assembled vauum hamber Photograph of old finger attahed to the seond stage of the 4 K pulse tube ryoooler FIG Diagram of SQUID tip used in prototype SQUID mirosope FIG FIG Blok diagram of the prototype Sanning SQUID Mirosope eletronis and sanning system Response of a nonhystereti d SQUID with optimally set onstant bias FIG Blok diagram of modulated Flux-Loked-Loop (FLL) SQUID eletronis FIG. 4.. Voltage amplitude versus frequeny of the primary stages of three different SQUID output transformer oils FIG. 4.. (a) Bringing the SQUID mirosope sapphire window lose to the SQUID tip. (b) Positioning an objet lose to the sapphire window prior to sanning it xiv

18 FIG. 5.. FIG. 5.. FIG. 6.. FIG. 6.. FIG FIG (a) Test iruit developed by Neoera, In. for testing sanning SQUID mirosopes. (b) Magneti field image of area indiated by dashed box in (a) obtained using prototype SQUID mirosope Ciruit diagrams of resistor networks used for limiting the urrent from a funtion generator Bakground flux noise spetrum measured by the prototype SQUID mirosope using TRISTAN imag FLL eletronis with a nonhystereti d SQUID at around 5 K Response of a hystereti d SQUID to bias urrent pulses with an external magneti flux signal modulating the ritial urrent Response of an ideal hystereti d SQUID to a bias urrent ramp with an external magneti flux signal Response of an ideal hystereti d SQUID to a sinusoidal bias urrent with period T FIG Osillating feedbak sheme for ritial urrent detetion FIG Alternating pulse sheme for ritial urrent detetion FIG Asynhronous pulsed SQUID sampling tehnique FIG Synhronous pulsed SQUID sampling tehnique FIG Shemati diagram of pulsed SQUID sampling eletronis using a hystereti d SQUID and implementing the synhronous alternating pulse tehnique for large bandwidth magneti field detetion appliations FIG Shemati diagram of eletronis to generate alternating height urrent pulses using two signal generators FIG. 6.. Shemati diagram of superonduting eletronis to generate very short urrent pulses to bias a hystereti d SQUID for large bandwidth pulsed SQUID sampling FIG. 7.. (a) Bottom portion of twisted pair wire LHe dip probe. (b) Chip holder whih attahes to the probe housing on the twisted pair wire dip probe FIG. 7.. Large bandwidth dip probe FIG Bak side view of aluminium hip supporting holder in large bandwidth dip probe xv

19 FIG FIG FIG FIG FIG FIG Configuration of the experimental apparatus for testing pulsed SQUID sampling with a hystereti d SQUID Profile of 0. µs to µs long, 00 µa high urrent pulses from a DG535 pulse generator measured through the large bandwidth dip probe Osillosope trae of the voltage response of a hystereti d SQUID iruit to bias urrent pulses Osillosope trae of the voltage response of a hystereti d SQUID iruit to bias urrent pulses shorter than 0 ns (a) Ciruit model of hystereti d SQUID and large bandwidth dip probe. (b) Simplified iruit of (a) using Thévenin equivalent voltage soure and input impedane. () Ciruit (b) with equivalent output impedane Z(ω) seen by the oaxial able. (d) Equivalent d iruit of (b) with SQUID modeled as a voltage soure V s (a) Plot of alulated spetrum of the output voltage response of the SQUID iruit using the large bandwidth dip probe. (b) Plots of the output voltage response of the SQUID iruit with different parasiti indutane L FIG (a) Plots of the alulated output voltage responses of the SQUID iruit using the large bandwidth dip probe for different SQUID ritial urrents. (b) Plot of the.7 V, 5 ns input voltage pulse to the SQUID iruit that generated the voltage responses alulated in (a).. 8 FIG. 7.. Voltage response of a hystereti d SQUID to pulsed bias urrent and triangular wave flux signal FIG. 7.. Voltage response of SQUID BH to modulating flux with inreasing amplitude FIG Voltage response of SQUID BH showing irregularities to modulating flux with inreasing amplitude FIG Voltage response of SQUID BH triggering to a sinusoidally modulating flux FIG Voltage responses of SQUID BH to 0.9 V, 5ns input voltage pulses and an applied magneti flux signal at different delays, in the large bandwidth dip probe FIG Voltage response of SQUID BH to 0.9 V, 5ns input voltage pulses and an applied magneti flux signal at different delays showing xvi

20 multiple transitions from the zero voltage state to the resistive state, in the large bandwidth dip probe FIG Series of osillosope traes showing the progression of the voltage response of SQUID BH to 0.9 V, 5 ns input voltage pulses and an applied 00 MHz magneti flux signal with varying delay, in the large bandwidth dip probe FIG Osillosope traes showing oupling between the urrent to the one turn oil for the applied magneti flux and the output voltage response signal of the SQUID iruit FIG Two dimensional histogram of SQUID BH triggering events in the presene of a GHz magneti field FIG. B.. Diagram of SQUID mirosope old finger using strips of ultra high purity opper foil to onnet the top and bottom parts FIG. C.. Inverse impedane versus frequeny of able in the twisted pair wire LHe dip probe FIG. C.. Simple model of SQUID iruit using the LHe dip probe with twisted pair wire and oaxial able at the ends FIG. D.. Diagrams showing the orientations of an area n with respet to a uniform magneti field B, as the area goes through a 90 rotation about an axis normal to the diretion of the field FIG. E.. (a) Plots of the alulated output voltage response of the SQUID iruit using the large bandwidth dip probe with L =.9 nh and modeling the d SQUID as a resistor. (b) Plots of the alulated urrent through the d SQUID modeled as a resistor, orresponding to voltage responses in (a) xvii

21 CHAPTER Introdution. SQUID Mirosopy.. Overview Sine its oneption in 964 by R. C. Jaklevi et al., the Superonduting Quantum Interferene Devie or SQUID has firmly established itself as a reliable tool in physis, eletronis, materials researh, and other fields [-7]. In partiular, in the last deade, SQUID mirosopes have made it possible to image extremely weak magneti fields generated by eletri urrents in iruits at mirosopi sales [8-4]. The basi tehnique is to san and image a omponent of the magneti field near the surfae of the soure using a SQUID and then onvert the field image into an image of the soure urrents [4,5]. The mirosope detets the magneti field omponent by diret measurement of the field at a large number of points, i.e. as a near field mirosope. An example of the tehnique is shown in Fig.., where a SQUID is sanned over a surfae ontaining a urrent arrying wire. As the SQUID passes over the wire, the magneti field and onsequently the magneti flux through the SQUID loop hanges with a harateristi signature whih an be used to identify the urrent. By omparing the urrent image to the intended iruit design, the loation of problems or faults in the iruit an be found. Today, fault detetion or diagnostis of miroiruits is the main ommerial appliation of SQUIDs in the eletronis

22 (a) z z o y x x o I (b) B z x x z o o FIG... (a) A z-squid sanning over a straight wire in the x-y plane at a height z o. The wire arries urrent in the y diretion and produes a magneti field around it. If the size of the SQUID loop is suffiiently small, the SQUID will detet the z omponent of the magneti field due to the wire. (b) Z-omponent of the magneti field due to an infinitely long line urrent in the x-y plane as observed above the plane at a height z o. The urrent is along the y diretion and is entered at x = x o. The distane along x is given in units of the height z o above the plane, and the field sale is arbitrary.

23 industry [6]. Full sanning SQUID mirosopes with diagnosti software for planar eletrial iruits are ommerially available, and this appliation is expeted to ontinue as iruits in mirohips and supporting systems beome more omplex [7]. As with any near-field tehnique, the sensor (SQUID) must be brought nearly in ontat with the field generating objet in order to have the field measured at maximum strength and with the best spatial resolution. Although the total magneti field at any point is a sum of all soures, the field at any loation an often be regarded as primarily due to loal soures, as long as the distanes between soures are muh larger than the size of the SQUID and the distane between the SQUID and the objet surfae. As the distane between the objet and SQUID inreases, the relative importane of more distant field generating soures inreases. For example, Fig.. shows the ombined field of two parallel line urrent soures with different separations; they an be regarded as the ombination of two urves shown in Fig..(b) entered at different positions. As the relative separation between the soures dereases, the ombined field takes on a different harateristi [ompare Fig..(a) with.(b)]. If the measured field annot be regarded as only being due to loal soures, a tehnique to derive the soures from the field image is required. Suh a tehnique was first developed by Wikswo s group at Vanderbilt and is now available ommerially [4-7]. Bringing the SQUID as lose as possible to the field soure is also important for other reasons. Maximizing the measured field strength also maximizes the signal to noise ratio (SNR). A larger SNR provides a more preise field measurement with 3

24 (a) B z x z o (b) B z x z o FIG... B z (z-omponent of magneti field) due to two infinitely long parallel wires arrying urrent in the x-y plane as observed above the plane at a height z o. Both urrents flow along the y diretion, but the wire to the left has twie the urrent of the wire to the right. The distane along x is given in units of the height z o above the plane, and the field sale is arbitrary. The two wires are entered at x/z o = ±5 resulting in a separation of 30 in graph (a) and at x/z o = ±.5 with a separation of 5 in graph (b). 4

25 less san time. If the field image is used to derive the soure image, the unertainty in the resulting urrent density will also be smaller for larger SNR. Noise in the SQUID mirosope system ultimately limits the mirosope s apability to measure magneti fields... Motivation Advanes in miroeletronis have made diagnosing some iruit problems more and more diffiult. Very high spatial and temporal resolution is now needed to detet all types of possible failures. In mirohips, submiron line widths are now ommon and iruit lok speeds are in the GHz to 3 GHz range. Deteting and analyzing magneti fields from suh iruits is hallenging even with SQUID mirosopes. Where possible, single iruit leads are ativated individually to help differentiate between losely paked iruit elements. Also, if possible, iruit speeds are slowed down so that the eletronis in a SQUID mirosope an follow the hanging field. Without the derease in speed, present SQUID mirosopes are unable to faithfully monitor the rapidly hanging a omponents of fields. Despite suh triks and various other tehniques to measure high frequeny signals in densely paked omplex iruits, it would be best if the iruits ould be analyzed while working under normal operating onditions. In fat, some dynami problems do not our otherwise. This however is not possible with urrently available SQUID mirosopes. The main limitations lie in the SQUID mirosope s readout eletronis, rather than in the SQUID itself. The most ommon d SQUID eletronis are based on the a modulated Flux-Loked-Loop (FLL) tehnique. The 5

26 eletronis relies on a low noise amplifier, an osillator, a demodulator, an integrator, and negative feedbak to produe a linear output proportional to the flux applied to the SQUID [4,5,8-0]. The range of frequenies the mirosope an measure is speified by its bandwidth. To inrease the bandwidth of the mirosope, the bandwidth of the feedbak system iruitry has to inrease. Unfortunately, the omplexity and diffiulty of building and optimizing large (gigahertz) bandwidth feedbak iruitry has limited the bandwidth of SQUID feedbak eletronis to a maximum range of about MHz to MHz with typial bandwidths between 0 khz and 00 khz [9,0]. Therefore, in order to improve bandwidth to meet diagnosti performane requirements, SQUID readout eletronis based on a different approah is needed.. Preview of Work Researh on hystereti Josephson juntions has shown that the fast transition from the zero voltage state to the resistive state ould be used for high speed measurement. Besides work performed at IBM until 983, the best known measuring instrument using this tehnology was perhaps the HYPRES PSP-000 sampling osillosope whih used Josephson juntions as fast swithes [-4]. Experiments have shown that the voltage transition times of Josephson juntions are dependent on the juntion ritial urrent and an be made muh shorter than a nanoseond [5,6]. If the time sale assoiated with variations in bias urrent or SQUID ritial urrent is muh larger than a nanoseond, the voltage state transition ould be regarded as instantaneous. 6

27 The basi idea of the HYPRES sampling osillosope was to exploit the instantaneous voltage transition of the Josephson juntion as a time resolved preision swith. Timing measurements and pulse generation an be performed relatively easily with high preision using modern eletronis and do not have the same bandwidth issues as does negative feedbak FLL eletronis. If this tehnique an be applied to a d SQUID, instead of a single Josephson juntion, the SQUID ould be used to retrieve magneti field information on very short time sales. In this thesis, I desribe experiments I performed to explore the feasibility of using hystereti d SQUIDs for a large bandwidth mirosope that an follow magneti signals with frequenies on the order of GHz and higher. For this, I designed and prepared hystereti SQUIDs and tested them by observing what happens when they are subjeted to short urrent pulses. In addition, a tabletop ryogeni system for use in a prototype SQUID mirosope was assembled. The ryogenis was based on a low temperature ryoooler with a base temperature below 4 K [7]. One assembled, the prototype mirosope would be a fully funtional Sanning SQUID Mirosope and was tested with a resistively shunted nonhystereti d SQUID [8]. Commerially available FLL eletronis with a 50 khz osillator were used for driving the SQUID and initially operating the mirosope [9]. After initial tests, I was able to produe a magneti field image of a test eletrial iruit, therefore, verifying the feasibility of a 4 K ryoooled sanning SQUID mirosope. The idea is to eventually replae the FLL eletronis and nonhystereti SQUID with new eletronis and a hystereti 7

28 SQUID to turn the prototype mirosope into a fully funtional large bandwidth sanning SQUID mirosope. Although the mirosope ryoooler was available for experiments, I deided to perform experiments on hystereti SQUIDs in liquid helium for onveniene. These experiments mainly onsisted of monitoring the response of the SQUIDs to short bias urrent pulses and mirowave frequeny magneti field signals. Reliable detetion of signals with frequenies up to 00 MHz was observed [30]. My review of the results suggested that with better eletronis, detetion of magneti fields with frequenies GHz and higher ould be ahieved, and subsequent experiments performed shortly thereafter did, in fat, demonstrate GHz detetion [3]. Finally, from my experiments, I devised new SQUID readout eletronis using an approah based on signal sampling with short bias urrent pulses..3 Outline of Thesis The organization of this dissertation is as follows. Chapter provides a general introdution to Josephson juntions and d SQUIDs and ends with a disussion of some speifi harateristis. I disuss both hystereti and nonhystereti Josephson Juntions and the equations desribing them. Chapter 3 desribes d SQUID design riteria, the nonhystereti d SQUIDs that were used in the prototype SQUID mirosope and the hystereti d SQUIDs used in the fast swithing experiments. Chapter 4 desribes the prototype 4 K ryoooled Sanning SQUID Mirosope. I disuss the various subsystem omponents, from the old finger whih holds the SQUID to the translation system whih performs the sanning, 8

29 to the onventional FLL SQUID readout eletronis. The hapter ends with desriptions of and omments on the SQUID mirosope operating proedures. Chapter 5 presents the work performed in produing a magneti field image using the prototype SQUID mirosope. In Chapter 6, I propose my new SQUID readout eletronis, whih makes use of a hystereti d SQUID. The limitations of the a modulated FLL tehnique, and how they are overome by the new tehnique, are disussed in this hapter. Chapter 7 desribes my experiments on urrent pulsed hystereti d SQUIDs and ompares the results with alulations made from a model of the SQUID iruit. This hapter inludes a setion on the large bandwidth dip probe onstruted for the experiments and what were onsidered in its design. Finally, the dissertation onludes with a hapter summarizing the results of my researh with omments and suggestions for future work. 9

30 CHAPTER Introdution to Josephson Juntions and d SQUIDs. Superondutivity and Josephson Juntions The phenomena of superondutivity in the form of omplete loss of eletrial resistane in a material was disovered in 9 by Kamerlingh Onnes [3]. Sine then, muh has been learned about the phenomenon and the study of superondutivity has ontributed to the understanding of many properties of solids. The phenomenon was eventually explained at the mirosopi level by Bardeen, Cooper, and Shrieffer (BCS) in 957 [33,34]. Signifiant questions still remain, however, prinipally the mirosopi mehanism ausing superondutivity in the high transition temperature (T ) uprates suh as YBa Cu 3 O 7-x. Nevertheless, the basi theory of superondutivity in onventional low-t materials first developed in BCS theory is now well established. BCS theory is a quantum mehanial many-body theory in whih eletrons attrat eah other through phonons to form Cooper pairs in a oherent olletive state. This state an also be desribed by a omplex order parameter whih was introdued earlier (950) by Ginzburg and Landau [35]. The order parameter effetively represents the marosopi wave funtion of the superonduting eletrons or Cooper pairs [36,37]. Although the desription of superondutivity by Ginzburg and Landau using the order parameter is only stritly valid near the transition temperature T of the material, it has been suessful in desribing many phenomena outside of this range. 0

31 Sine the disovery and explanation of superondutivity, many appliations have been proposed and exploited. Some of the more interesting appliations have made use of quantum mehanial effets in superondutivity. One suh example is the Josephson juntion whih was proposed by Josephson and fully explained in 963 [38-4]. A Josephson juntion onsists of two superondutors that are separated by a very thin eletrially insulating barrier. Josephson predited that if the barrier is suffiiently thin, Cooper pairs would tunnel through the barrier keeping the quantum mehanial phase information intat and produing no voltage drop aross the barrier. Resistive urrents, though possible, would give rise to a voltage aross the barrier unlike a superurrent and onsequently would not be present under zero voltage bias. Shortly after the predition, experimental verifiation of superurrent tunneling was performed by Anderson and Rowell using a tin-tin oxide-lead juntion [4]. Many other phenomena assoiated with Josephson juntions have also been observed. Perhaps the most important of these onern the a Josephson effet in whih a d voltage drop aross the barrier gives rise to an a superurrent [38,39]. In addition, marosopi quantum effets have been disovered in Josephson juntions inluding marosopi quantum tunneling, quantum energy levels, stimulated tunneling, and quantum oherene [43-5].. Josephson Equations.. d Josephson Effet Equations analogous to the Josephson effets an be derived from a simple one dimensional model [5,53]. Consider the square barrier of length d and height

32 U entered at the origin (see Fig..). The energy assoiated with the marosopi wave funtion is E with the ondition E < U. Outside the barrier, the energy E orresponds to the kineti energy of the arriers of superurrent. E = m * vs (.) where m * is the mass of a harge arrier, and v s is its veloity. This model breaks down if E U as in this ase the barrier no longer ats as an insulator but as a different superondutor. It should be realled that the arriers of superurrent are Cooper pairs, and so the effetive harge and mass of these arriers are e and m e, respetively. The marosopi wave funtion of the superondutor an be expressed by the omplex order parameter ψ, having the following form on eah side of the barrier. ψ = n i e ikx (.) with k = 4m e E h (.3) where n i is onstant and represents the density of Cooper pairs. Assuming zero voltage aross the barrier and idential superondutors on eah side, = n n. n Then, the only differene in the value of ψ on eah side of the barrier is due to the phase. Inside the barrier, the wave funtion has the form ψ = C osh κx + C sinh κx (.4) with

33 Ψ n e = ikx Ψ = + C C osh κx sinh κx Ψ n e = ikx ReΨ, U h k U > E = 4m E -d 0 d x FIG... Model of a superondutor-insulator-superondutor Josephson juntion. Current arrying partiles of energy E and mass me quantum mehanially tunnel through an insulating square potential barrier of thikness d and height U.

34 κ 4m ( U E) e =. (.5) h Mathing boundary onditions at the edge of the barrier, one finds that the values of C and C are given by C C = n = i n os kd osh κd sin kd sinh κd. (.6) From elementary quantum mehanis, the urrent density an be written as J = h e R e im e ψ * ψ = h e m e I m * { ψ ψ} (.7) where J represents the urrent density, and e is the harge of an eletron. Applying Eqs. (.), (.4), and (.6) to Eq. (.7), J = e n hk m e (.8) outside the barrier, and J hκ = e m e I m * { C C } = e n hκ m e os kd sin kd oshκd sinhκd (.9) inside the barrier. As expeted, the urrent density both inside and outside the barrier is onstant with respet to the spae oordinate. Applying trigonometri and hyperboli identities, the magnitude of J inside the barrier an be written as J ( θ θ ) = J sin (.0) where J e n hκ =, (.) m sinh κd e 4

35 θ = kd, and (.) θ = kd. The quantity J is known as the ritial urrent density. It depends on the nature and thikness of the insulating barrier, the superonduting material, and environmental fators like temperature. In this model, J is dependent on the superurrent energy and has a limiting value of e n ћ/m e d as the energy E approahes the barrier height U. However, J will be onsidered a onstant in the rest of this treatment for simpliity. Note also that θ θ is the differene in phase of the wave funtion on either side of the barrier. Given that the alulations remain unhanged if a onstant phase is added to the phase of the wave funtion, only the phase differene aross the barrier is signifiant. On the other hand, if gauge invariane is imposed on the system, an additional term appears in the expression for the phase differene. The gauge invariant phase differene is given by e γ = θ θ + A dl h (.3) where A is the vetor potential and the integration is performed over the region orresponding to the hange of phase from θ to θ, whih is aross the barrier [5]. The additional term omes from a modifiation of the probability urrent due to the presene of an eletromagneti field. Instead of Eq. (.7), the urrent density beomes 5

36 J * h e = e R + e ψ A ψ im e m e h e = e n θ + A m h e (.4) (.5) where in this last equation, ψ outside the barrier is expressed in the more general form of Eq. (.) iθ ψ = n e. (.6) The generalization of Eq. (.0), for the superurrent through the barrier, in the presene of an eletromagneti field is then J = J sinγ (.7) whih is just the d Josephson effet [38-4]. The superurrent is observed to depend on the gauge invariant phase differene γ of the marosopi wave funtion aross the insulating barrier. The dependene of the phase differene on the vetor potential A means that the phase differene an be varied by applying a magneti field to the juntion. This will result in suppression of the d superurrent with respet to its maximum possible value. For example, if magneti flux gets trapped within the juntion, this would effetively result in a derease in the maximum d superurrent [54-57]... a Josephson Effet The dynami behavior of the superurrent an be obtained from the time dependent Shrödinger equation, 6

37 ψ ih = t 4m e h + ea ψ eφψ i (.8) where φ is the eletri potential. Substituting Eq. (.6) for the marosopi wave funtion gives θ h t h = 4m e he m h 4m i he m i e A m ( θ) + A θ + θ + A + eφ e e e e (.9) Separating the real and imaginary parts and solving for θ / t gives θ h = t 4m h = 4m e e ( θ) ea e A θ m m h ea θ + h e e + φ h e e + φ h (.0) and expressing this in terms of the urrent density J gives me = t 4e n θ J h e + φ h (.) Now, applying this equation to Eq. (.3) for the gauge invariant phase γ gives γ θ = t t θ t e + A dl h t (.) me = 4e n h e A ( J J ) + φ + h t dl (.3) Reognizing that the eletri field is given by A E = φ, (.4) t Eq. (.3) redues to 7

38 γ me = t 4e n J h e E dl h (.5) m e e = J + V (.6) 4e n h h where the integration is performed through the insulating barrier from the region orresponding to phase θ to the region with phase θ. Consequently, V is the voltage drop aross the juntion. Here, J is the differene in urrent density squared between one side of the juntion and the other. Assuming urrent onservation and a lumped iruit or l << λ where, l is a harateristi length of the juntion and λ is the wave length of eletrial signals aross the juntion, J = J. So, J = 0 (.7) and thus, γ t = π Φ o V (.8) where Φ o h e T m (.9) is the definition of the flux quantum. Equation (.8) states that the gauge invariant phase aross the insulating barrier evolves in time at a rate proportional to the voltage aross the barrier. Combining this result with Eq. (.7) brings about the onlusion that a d voltage aross the Josephson juntion will give rise to an a urrent through the juntion. This is known as the a Josephson effet [38,39]. 8

39 Using Eqs. (.7) and (.8), a simple expression for the eletrial energy density per unit area within the barrier an be derived [58,59]. Noting that γ is only a funtion of time in this ase, one an write the energy density in the barrier as ε j = = JVdt J Φo sinγ dγ π (.30) = J Φ π o ( osγ ) (.3) where the integration is performed through the barrier from the region with phase θ to the region with phase θ. The total energy in the barrier then sales with E j = I Φ π o (.3) where I is the total ritial urrent through the juntion. From Eq. (.3), note that the phase differene aross the juntion is assoiated with the total energy. If additional energy is supplied from a thermal reservoir, the phase differene will beome unstable. This plaes an upper bound on the temperature for whih the phase differene will be able to assume a well defined value. Therefore, a stable phase would be guaranteed by E j >> k B T (.33) or I >> ek BT h (.34) 9

40 for a temperature T. In pratie, the value of I an be adjusted by hanging the size of the juntion or by reduing the thikness of the barrier whih inreases the ritial urrent density..3 d SQUID.3. Critial Current Modulation The d Superonduting Quantum Interferene Devie or SQUID is a highly sensitive detetor of magneti flux. The working priniple of the SQUID an be explained using the marosopi wave funtion model used to desribe the Josephson juntion [-4,6-9,,,60]. The d SQUID (see Fig..) onsists of a superonduting path whih splits into two branhes that reonnet to form a hole in the middle. Eah branh ontains a Josephson juntion. Only the ideal d SQUID whih ontains no other elements other than the Josephson juntions is onsidered here. What happens when other elements suh as loop indutane are inluded is treated later. The d SQUID is usually urrent biased and an be thought of as an interferometer of the marosopi superurrent wave funtion that splits and reonnets around the hole. The wave funtion must be single valued, or in other words the integrated phase gradient around the hole must be a multiple of π. To appreiate the signifiane of the Josephson juntions and the onsequenes of the single valued wave funtion, first onsider the situation if there were no juntions around the loop. In this ase, the superondutor forms a ontinuous ring [see Fig..3(a)]. If the superondutor is thik enough, most of the superurrent will flow near 0

41 (a) (b) One Turn Feedbak Coil 30 µm Josephson Juntions FIG... (a) Ciruit diagram of an ideal d SQUID. Crosses indiate Josephson juntions. (b) Photograph of a 30 µm Nb d SQUID with Nb-AlO x -Nb Josephson juntions, fabriated by Hypres, In. The SQUID inludes a one turn magneti feedbak oil.

42 (a) (b) C C FIG..3. (a) Diagram showing path of line integral around a superonduting loop deep inside the superondutor. (b) Diagram showing paths of line integrals around d SQUID loop, deep inside the superondutor and through the Josephson juntions.

43 the surfae, within the London penetration depth [6,6]. If so, there will be a ontinuous region deep inside the superondutor where J = 0. A line integral of J around the hole inside this region results in J d l = 0. (.35) From Eq. (.5), this gives θ + e h A dl = 0. (.36) Now, reognize that A d l = A ds = B ds = Φ (.37) where Φ is the total magneti flux through the hole. Thus, Eq. (.36) gives θ dl = πn = e Φ h = π Φ Φ o (.38) where n is any integer value. This result reveals that the total magneti flux through the hole is quantized and is an integer multiple of a flux quantum. Φ = nφ o (.39) The quantization of flux implies that with a onstant applied magneti field, the value of irulating superurrent around the hole is also quantized. Now, onsider the situation where a Josephson juntion is present on eah branh of the SQUID loop. First, the superurrent through a juntion is limited by Eq. (.7). Seond, the line integral of the urrent density J around the loop need not vanish, as J 0 in the juntion barriers. To see how this effets the urrent through both juntions, note that the total quasi stati urrent through the d SQUID is onstant and is given by 3

44 I = I sin γ + I sin γ (.40) ( sin γ sin γ ) = I + (.4) where in the last equation, I assumed that the ritial urrents of the Josephson juntions are equal. Using a trigonometri identity, Eq. (.4) beomes I = I = I os os γ γ sin γ sin γ av γ + γ (.4) where γ γ av = γ = γ γ + γ. (.43) From Eq. (.8), one an write γ t av = π Φ o V (.44) but that γ = 0 t (.45) for the ideal symmetri d SQUID, whih ignores the effets of the loop indutane. When the ontribution of the SQUID loop indutane is taken into aount, Eq. (.45) no longer holds when V 0, due to a time varying irular urrent around the loop [63,64]. Nevertheless, Eqs. (.44) shows that the dynami behavior of γ av is the same as that of the gauge invariant phase differene of a single Josephson juntion. Further analysis shows that γ an be expressed in terms of the total magneti flux through the SQUID hole [60]. Consider the line integral of the urrent density 4

45 5 shown in Fig..3(b). The path of integration goes around the loop inside the superondutor and aross the juntions through the insulating barriers. From Eq. (.5), = C C e C C d e θ m n e d l A l J h h. (.46) The line integral of θ an be expressed as + = + C C d θ d θ d θ d θ l l l l (.47) where paths C and C are deep inside the superondutor exluding the barriers, and paths and are through the insulating barriers. Note that the irular diretion of path around the hole is in the same diretion as paths C and C but that path is in the opposite diretion. The phase hanges aross the juntion barriers are given by Eq. (.3). So, = i i i d e d θ l A l h γ. (.48) With a superurrent flowing through the SQUID loop, the line integral of θ around the loop must be a multiple of π. Thus, + + = + C C d e d e n d θ l A l A l h h γ γ π (.49) Combining Eq. (.49) with the line integral of the vetor potential A, + = + + l A l A d e γ n d e θ C C h h π. (.50) As in the former ase of a ontinuous superonduting ring, if the superondutor is thik enough, the integration an be performed deep inside the

46 superondutor where J = 0. In this ase, the left hand side of Eq. (.50) vanishes resulting in e Φ γ = π n + A dl = πn + π. (.5) h Φ o Substituting this result into Eq. (.4) yields I Φ n Φ = I os πn + π sin γ av = ( ) I os π sin γ av Φ. (.5) o Φ o Equation (.5) shows that the urrent through a d SQUID is similar to that of a single Josephson juntion exept that its ritial urrent is modulated by the total magneti flux through the hole. The magnitude of the modulated ritial urrent is given by I ( Φ ) = I os π Φ Φ o (.53) and is periodi with period Φ o [see Fig..4(a)]. I note that in Eq. (.40), if the ritial urrents of the two juntions are not idential, the superurrent through the d SQUID is given by I = av + Φ Φo ( I ) + 4I I os π sin( γ δ) (.54) where tan δ = I I + I tan π Φ Φ o (.55) and I = I I. Again, this result is only valid for the ideal d SQUID, whih ignores effets due to elements other than the Josephson juntions suh as the loop indutane. Equations (.54) and (.55) are derived by reognizing that the 6

47 (a) I (Φ) I = I I I (b) Φ Φ o I (Φ) I I I + I I I 3-0 Φ Φ o FIG..4. (a) Graph of the magnitude of the superurrent through an ideal symmetri d SQUID with idential Josephson juntions and zero indutane plotted as a funtion of the total magneti flux through the SQUID hole. (b) Graph of the maximum magnitude of the superurrent through an ideal d SQUID plotted as a funtion of the total magneti flux through the SQUID hole when the ritial urrents of the Josephson juntions are different.

48 summation in Eq. (.40) an be expressed as the imaginary part of a sum of two omplex numbers or phasors [60]. It is also seen that the dynami behavior of Eq. (.54) is the same as Eq. (.5), exept that there is a magneti flux dependent phase shift. The maximum magnitude or envelope of the ritial urrent, whih is graphed in Fig..4(b), also depends on the flux and varies between I and I + I. Both the magnitude and phase shift are periodi with period Φ o. The dependene of the total ritial urrent on the magneti flux through the SQUID hole is what makes SQUIDs very sensitive detetors. A small hange in Φ, even if just a fration of a flux quantum Φ o, will signifiantly modulate the ritial urrent. For example, an optimized 4 K d SQUID that is mm on a side is able to distinguish hanges in magneti field that are a fration of a piotesla in one seond..3. Cirulating Sreening Current In many experiments, it is the externally applied magneti flux through the SQUID loop, not the total magneti flux through the hole, that is the quantity of interest. The relation between the total magneti flux Φ and the externally applied flux Φ A is Φ = Φ A + LI s (.56) where L is the self indutane of the SQUID loop, and I s is the irulating urrent or sreening urrent around the hole [60,65]. Applying Eqs. (.43) and (.5) to an ideal d SQUID with idential Josephson juntions, 8

49 I s = I = I = I = I γ sin os γ ( sin γ sin γ ) av Φ Φ n ( ) I sinπ os γav o = I γ γ = I sin Φ sin πn + π os γ Φ o av γ os + γ. (.57) Therefore, the externally applied flux is given by Φ A n Φ = Φ ( ) LI sin π os γav. (.58) Φ o From Eq. (.58), one an see that in order to determine the ritial urrent of the SQUID as a funtion of applied magneti flux, i.e. I (Φ A ), the total magneti flux Φ must be solved by inverting Eq. (.53) and plugging that result into Eq. (.58). In the limiting ase when Φ o >> LI, Eq. (.58) redues to Φ A Φ, (.59) and so I ( Φ ) A Φ A I os π. (.60) Φ o Note that the inversions of both Eqs. (.53) and (.60) are multivalued and an only be solved modulo ½Φ o. In addition, it may not be lear in Eq. (.58) whether the flux due to the irulating urrent should add or subtrat from the total flux. These ambiguities an be resolved by introduing onstraints suh as energy minimization and by keeping trak of the flux, phase differenes, and urrents [60,63]. Equations (.57) and (.58) demonstrate how a SQUID loop differs from a omplete superonduting loop. For a given applied magneti flux, the allowed 9

50 values of the irulating urrent are disrete and onstant in the ase of the ontinuous loop regardless of hanges in the bias urrent. For a SQUID loop, the value of γ av varies ontinuously with the bias urrent through the SQUID. So, the irulating urrent an also hange ontinuously with the bias urrent up to a maximum value within the limits set by Eqs. (.7) and (.58). Correspondingly, the allowed values of the total magneti flux through the SQUID hole are not disrete but have segments in whih they an vary ontinuously..4 Hysteresis in Josephson Juntions and SQUIDs.4. RCSJ Model Equations (.7) and (.8) alone do not fully desribe the harateristis present in a real Josephson juntion. Real juntions show more omplex behavior that require iruit elements in addition to the ideal juntion. The Resistively and Capaitively Shunted Juntion (RCSJ) model (see Fig..5) has been widely used to explain the behavior of real juntions and SQUIDs, both qualitatively and quantitatively [66-7]. The RCSJ model onsists of an ideal Josephson juntion in parallel with a apaitor and a resistor. The apaitor represents the physial fat that a real Josephson juntion has finite size and apaitane between the two superonduting eletrodes that are separated by the insulating barrier. The resistive hannel reflets dissipative losses aross the juntion. These losses an arise from quasipartile tunneling, indutive losses at nonzero frequenies, flux flow of vorties, flux pinning near the surfae, the tunneling of normal eletrons from the breakup of 30

51 I dv I = C dt V I Φo dγ = π dt = I sin γ V I = R FIG..5. Equivalent eletroni iruit of a real Josephson juntion desribed by the Resistively and Capaitively Shunted juntion model. A urrent soure whih provides the bias urrent to the juntion and ground are also shown. 3

52 Cooper pairs or from an extrinsi resistive path deliberately onneted aross the juntion [7-79]. The value of the effetive shunt apaitane is assumed to be independent of the bias voltage, bias urrent, frequeny or temperature, as it is mainly set by the geometri onfiguration of the juntion and the hoie of insulating material. However, the juntion ritial urrent is influened by temperature, the superonduting material, and geometry [40,4,80]. Furthermore, the effetive shunt resistane due to quasipartiles is strongly dependent on temperature and bias voltage [74,79]. For simpliity, I will treat the resistane as pieewise linear. Given this equivalent iruit, the RCSJ model for a Josephson juntion an be analyzed without further need to resort to first priniples. Assuming a lumped iruit, the urrent through the ideal Josephson juntion is given by integrating Eq. (.7) over the juntion area resulting in I = I sin γ. (.6) In Eq. (.6), I is the total ritial urrent and is onsidered onstant for onstant environmental fators. At zero temperature, it is the maximum d superurrent allowed through the juntion. The urrent through the apaitive hannel is given by I = C dv dt (.6) where V is the voltage aross the juntion, and C is the effetive juntion apaitane. The urrent through the resistive hannel is given by Ohm's law, I = V R (.63) 3

53 where R is the effetive shunt resistane and is assumed to be onstant. Thus, the total urrent through the RCSJ model juntion is the sum I dv V = C + + I sin γ. (.64) dt R Reognizing that the gauge invariant phase differene γ is only a funtion of time, Eq. (.8) provides a relation between γ and V: dγ dt π = V. (.65) Φ o Substituting this into Eq. (.64) gives I CΦ o d γ Φ o dγ = + + I sin γ. (.66) π dt πr dt Equation (.66) an be transformed into dimensionless form by using the substitutions and y = I I (.67) x = πi R t t Φ τ o J (.68) where τ J = Φ o /πi R, whih results in RC d γ dγ d γ y = + + sin γ β τ dx dx dx J + dγ dx + sin γ. (.69) where β = τ RC /τ J = πi R C/Φ o and τ RC = RC. From Eq. (.69), τ J is seen to be the time it takes for the gauge invariant phase differene γ of the Josephson juntion to go through a rotation of radian at the bias voltage I R. The onstant τ RC is reognized as the time onstant in an RC iruit. It 33

54 orresponds to the time for a harged apaitor C to disharges through a resistor R to a level that is /e of the original voltage, where e is the base of the natural logarithm. The onstant β is the ratio between τ RC and τ J and is ommonly referred to as the Stewart-MCumber parameter [66,67]. If β >>, the dynamis of γ is dominated by the a Josephson effet. If β <<, the dynamis is dominated by the damping effet or disharging of the RC iruit elements, and Eq. (.69) simplifies to dγ y + sin γ. (.70) dx.4. Nonhystereti Juntions Equation (.69) an be shown to be analogous to the equation for a damped pendulum driven with onstant torque in a gravitational field. The behavior of the damped driven pendulum is nonlinear and is treated in many standard texts on nonlinear dynamis [8-85]. The treatment here will follow the geometri approah of Strogatz. First, it an be seen from Eq. (.69) that analysis an be limited to values of γ between π and π due to periodiity, and that the solutions for γ with y < 0, an be expressed in terms of the solutions with y > 0 due to inversion symmetry. Thus, analysis an be restrited to the range 0 y π. Next, Eq. (.69) an be written in terms of two oupled first order nonlinear differential equations. dγ = dx v (.7) dv dx = ( y v sin γ). (.7) β 34

55 Now, onsider the behavior of Eqs. (.7) and (.7) as β 0, sometimes referred to as the over damped ase following the pendulum analogy. This is the ase dv desribed by Eq. (.70). The term β 0. So, dx y v sin γ = 0 or v = y sin γ. (.73) All points in the phase spae (γ, v) lie on the urve desribed by Eq. (.73) whih is graphed in Fig..6. The evolution of these points with respet to x is determined by Eq. (.7). Consider the ase when y <. For points on the urve where v > 0, the value of γ inreases. For points where v < 0, γ dereases. Therefore, all points on the urve eventually go to the point where v = 0 and y = sin γ (.74) whih gives γ = arsin y (.75) exept for the point given by γ = π arsin y with v = 0 (.76) whih an be shown to be unstable. Equations (.75) and (.76) thus have fixed points that are defined by the ondition dγ /dx = dv/dx = 0. It an be seen that when y = 0, γ = 0, and as y inreases, γ goes through a suession of fixed points until y = when γ = π/. Also, reall that dγ /dt is proportional to the voltage V by Eq. (.65). Then, from v = dγ /dx = τ J dγ /dt, the voltage V = 0 beause v = 0 for the fixed points. By inspetion, this is just the result expeted for the juntion superurrent whih is present when y = I / I <. 35

56 (a) v y = 0 -π 0 π γ (b) v 0 y < -π 0 π γ () v y = -π 0 π γ (d) v y > -π 0 π γ FIG..6. Series of graphs showing the evolution of the phase spae (γ, v) of the over damped Josephson juntion (β << ) as the bias urrent y = I/I rises from (a) y = 0 to (b) 0 y < to () y = and finally (d) y >, at whih point a voltage appears aross the juntion. Filled in dots, outlined dots, and striped dots signify stable fixed points, unstable fixed points, and half stable points, respetively. Arrows signify the diretion of evolution of the points. 36

57 37 For y >, there are no fixed points and the value of γ ontinually inreases. This results in a voltage aross the juntion [68,69]. Also, note that as dγ /dx is not onstant for a given value of y, neither is the voltage. Typially, in experiments only the average voltage is deteted for measurement time sales muh longer than τ J. Using Eqs. (.65) and (.68), the average voltage is given by T Φ dγ T π Φ vdt T πτ Φ v πτ Φ V o π 0 o T 0 J o J o = = = = (.77) where T is the period of v. T an be obtained by solving for γ using Eqs. (.7) and (.73). For y >, the solution is given by = y γ y tan artan y x, (.78) and so + = x y tan y y artan γ. (.79) And thus, x y tan y y x y se y y dx d v + + = = γ. (.80) Both γ and v an be shown to be periodi in time as expeted. Using Eqs. (.67) and (.68), the period is given by o J I I R I Φ y πτ T = =. (.8)

58 Therefore, the average voltage aross the juntion is given by V = Φ T = o I R I I (.8) and is graphed in Fig..7. Equation (.8) shows that a finite voltage aross the juntion gradually appears as I inreases past I and asymptotially approahes Ohmi behavior with inreasing urrent..4.3 Hystereti Juntions As β, the behavior of Eqs. (.7) and (.7) an be treated in the following way. When dv/dx = 0, Eq. (.73) whih was derived for β = 0 is reovered and the same arguments an be made resulting in the onlusion that a superurrent with V = 0 will flow through the juntion up to the point I = I or y =. This result required the existene of fixed points and was only possible for y <. If dv/dx 0, then in the limit β, Eq. (.7) redues to y dv dv = β + v + sin γ β. (.83) dx dx Solving for v gives v = y β x + v o v o (.84) again in the limit β. Thus, γ is given by γ = v o x (.85) from Eq. (.7). 38

59 I I β = 0 0 V I R FIG..7. I-V harateristis of an over damped (β << ) Josephson juntion at zero temperature. Dotted urve shows I = V/R. 39

60 Note that this result implies that a voltage is observed aross the juntion onsistent with the a Josephson effet, and using Eqs. (.65) and (.68), one finds Φ V = Φ o o = v v o. (.86) πτ J πτ J With the rough approximation made in Eq. (.83), the value of v o and onsequently V are independent of y. As will be seen later, this will not be the ase with better approximations. Next, let us assume the quasipartile tunneling through the insulating barrier is Ohmi [69]. Then, the voltage aross the juntion an be given by V = IR. (.87) If no quasipartiles are initially present to give rise to resistive tunneling, the gap energy must be supplied to break up Cooper pairs. At zero temperature with no other urrent path present, resistive urrents will only our after the voltage aross the juntion reahes the gap voltage /e. Therefore, if a superurrent an no longer be sustained through the juntion, a voltage will appear suh that IR for V V = e. (.88) otherwise e The two different ases for I R < /e and I R > /e are presented in Fig..8(a) and.8(b), respetively. I note that Ambegaokar and Baratoff have shown that π ( T ) ( T ) I R = tanh e kt (.89) whih at zero temperature redues to R I (0) = π 4 e (0) 0.79 e (.90) 40

61 (a) I I β >> and I R < e (b) 0 I R e V I I β >> and I R > e 0 e I R V FIG..8. I-V harateristis of an under damped (β >> ) Josephson juntion at zero temperature when (a) I R < /e and (b) I R > /e. Arrows indiate the diretion of jumps during transitions between the zero voltage state and the resistive state. Under ordinary irumstanes, ase (b) I R > /e is not expeted to our. 4

62 using the BCS theory [4,80]. Consequently, as tanh( /kt) <, the latter ase of I R > /e is not expeted to our. Reall that Eq. (.85) is independent of y, but does require that dv/dx 0. In fat, a voltage an be sustained for y < as long as dv/dx 0. In this ase, the voltage ontinues to follow Eq. (.88) until the bias urrent reahes zero or y = 0. When that happens, dv/dx is required to equal zero by Eq. (.83). Thus, the I-V harateristis for the juntion with β >> an be seen to be qualitatively different from the previous ase of β <<. As y inreases from y = 0 and surpasses the point y =, v is seen to jump from v = 0 to v = (πτ J /Φ o )V = V / I R with further inreases in y resulting in V following Eq. (.88). If y is dereased, v does not drop bak to v = 0 at y =, but ontinues to follow v = V / I R where V is given by Eq. (.88) until y = 0. This I-V urve for β, graphed in Fig..8, is known to be valid when the Josephson juntion is at zero temperature. Suh irreversible behavior of v with respet to y, the ontrolling parameter, is an example of hysteresis. Hysteresis ours when β [66,67,8]. At finite temperatures, the I-V harateristis deviate from those shown in Fig..8. Some of these deviations an be explained with the inlusion of an additional term in Eq. (.83). For very large β, γ an still be approximated by Eq. (.85), in whih ase sin γ varies so rapidly that its average effet is zero. Then, instead of Eq. (.83), the governing equation beomes dv y = β + v dx (.9) whih an be solved to give 4

63 v x = y + y e β y (.9) o as x, where y o is the starting value of y in the phase spae of (v, dv/dx). In other words, all points in this phase spae merge exponentially fast toward the point given by v = y. The exeption to this result is when dv/dx = 0 in the original equation of state, Eq. (.7). In this exeption, the result leads to the existene of the superurrent with v = 0. The relation v = y is just Ohm s law stated in normalized units, whih is inorporated in Eq. (.88). However, Eq. (.9) also implies that Ohm s law ontinues to apply for V < /e. An additional fator to onsider is that the effetive resistane of the juntion is different above and below V = /e [69,70,79]. Above /e, the dominant loss mehanism is single eletron tunneling through the break up of Cooper pairs. Below /e, tunneling is by quasipartiles that are exited thermally. In type I superondutors, the ondutane assoiated with quasipartile tunneling at low temperatures and low voltage is very small beause few quasipartiles are present [34,86]. Consequently, the resulting subgap resistane R sg is greater than the normal resistane R above /e. In type II superondutors, however, quasipartile tunneling mehanisms are more easily allowed when magneti flux penetrates into the superondutor, as this auses regions where the gap vanishes. This results in a smooth transition of the resistane from the subgap value to the normal value in fields above H [87]. The effet of a large subgap resistane an be treated through a saling of the normalized units. If the subgap resistane is given by R sg = αr where α >> (.93) 43

64 then, in terms of the normal resistane R, Eqs. (.69) and (.9) transform into d γ dγ y = β + dx α dx + sin γ (.94) and dv v y = β +. (.95) dx α The solution of Eq. (.95) is given by x αβ v = αy + y e αy (.96) o as x giving the expeted result v / y = α >>. Figure.9 shows the resulting I-V harateristis of the hystereti juntion at finite temperature. As y inreases from y = 0, v remains at v = 0 until y = at whih point v jumps to the urve v = V / I R, where here V is given by Eq. (.88), and follows this urve with further inreases in y. If y dereases, v retraes the urve v = y until v = / ei R. Then, despite further dereases in y, v remains at v = / ei R until y = v /α where α = R sg /R. From that point on, v follows the urve v = αy as y dereases to y = 0. Full analysis of Eq. (.69) reveals additional features due to the sin γ term left out in Eqs. (.9) and (.95) [70,8]. As the voltage dereases below /e, Eq. (.85) no longer holds and γ begins to evolve non-uniformly and more slowly. This is similar to the situation with β = 0 at small v. There, the result was a derease in v that was more rapid than v = y. Here, a derease ours that is more rapid than v = αy and implies that the voltage returns to zero before y = 0. This is referred to as retrapping and its onset is determined by the value of β. 44

65 I I β >> and I R < e 0 ei R V I R FIG..9. Approximate I-V harateristis of an under damped (β >> ) Josephson juntion with I R < /e at finite temperature in the RCSJ model. A pieewise linear approximation of the RCSJ resistor is used. 45

66 Analysis by Gukenheimer and Holmes, as ited by Strogatz, shows that the onset of retrapping ours near y r = 4 / π β, as β [8]. An alternate derivation is given in Ref. [70]. Unlike the jump at y =, retrapping is ontinuous. Nevertheless, retrapping is experimentally seen as a sudden jump due to the rapid approah of v to v = 0; v behaves like [ ln y - y r ] - near y = y r. The resulting omplete I-V urve of the Josephson juntion at finite temperature is presented in Fig..0. As expeted, retrapping will not our unless there is hysteresis. Studies of the parameter spae (β, y) as presented in Ref. [8] show that at zero temperature hysteresis will not happen until β is greater than 0.69, i.e. of order unity. Similar results are reported by Stewart and de Waal et al. [64,66]. Levi et al. and other referenes on nonlinear equations, as suggested in Ref. [8], provide a more rigorous analysis. Finally, some of the sharp edges and well-defined limits of I-V urves are rounded off or smeared by noise due to thermal and nonlinear effets. Suh effets have been desribed by Tinkham and others [70,7,88-93]..4.4 From Juntions to d SQUIDs Earlier, I desribed d SQUIDs as being like Josephson juntions with ritial urrents that are modulated by the magneti flux through their holes. Using this idea, it follows that d SQUIDs will also manifest hysteresis and other harateristis desribed by the RCSJ model. All of the results for the RCSJ model an be rudely used to model a d SQUID by substituting I with I (Φ ), where Φ is the magneti flux through the SQUID hole. However in my treatment of the Josephson juntion, I was assumed to be onstant. So, the RCSJ model results will only be valid for d 46

67 I I 4 π 4 π β 0 ei R V I R FIG..0. I-V harateristis of an under damped (β >> ) Josephson juntion with I R < /e at finite temperature inluding retrapping. Arrows on graph indiate diretion of jumps during transitions between the zero voltage state and the resistive state. The vertial referene urrents for retrapping and the gap voltage are the values at zero temperature. 47

68 SQUIDs if the harateristi time of I (Φ) and onsequently Φ is very long ompared to the harateristi time of the juntion. The harateristi time of a Josephson juntion is the greater of τ RC and τ J. The harateristi time of the magneti flux Φ through the SQUID hole is the period of the maximum frequeny Fourier omponent of Φ. If the range of frequenies starts from d, the harateristi time is the inverse bandwidth. Thus, the analysis in this setion should roughly be valid for d SQUIDs given the ondition for for β β >, <, τ τ RC J << << f f (.97) where f is the bandwidth of the externally applied magneti flux signal. Thus, I note that the time onstants τ RC and τ J are of fundamental importane in determining the d SQUID bandwidth for high speed magneti flux sampling. 48

69 CHAPTER 3 SQUID Design Considerations 3. SQUID Parameter Optimization 3.. SQUID Loop Indutane and Spatial Resolution I begin this hapter by disussing pratial onerns in designing d SQUIDs for SQUID mirosopy. The main points of disussion are the spatial, temporal, and magneti flux resolution. The spatial resolution of a SQUID is mainly determined by the dimensions of the SQUID and its distane from the magneti soure. A SQUID is sensitive to the total magneti flux through its hole. Information pertaining to field variations or gradients at sales less than the dimensions of the hole tend to be lost. Consequently, the average magneti field over the area of the hole is what is measured. SQUID dimensions an effet spatial resolution in the following way. For wide SQUID loops, where the widths of the loop are omparable or larger than the diameter of the hole, some of the flux inident outside the hole an be hanneled through the hole by the Meissner effet [4,3,94]. This is referred to as fluxfoussing. Thus, spatial resolution tends to worsen and inrease beyond the hole size for loops with wide line widths. On the other hand, the larger line widths inrease the effetive area, whih in turn inreases field sensitivity by enhaning the field through the hole. An optimal balane an be reahed by first determining the 49

70 maximum spatial resolution required and then adjusting the width and hole size until the desired field sensitivity is reahed. The dimensions of the SQUID loop also determine the self indutane of the loop, whih influenes field sensitivity, as will be seen later. The indutane is primarily dependent on the size and shape of the loop but ould also depend on the position and sizes of the Josephson juntions and any leads onneting them due to their influene on urrent flow [95-97]. Numerial methods have been developed to alulate the indutanes, but some general onlusions on the indutane with respet to dimension an be drawn from basi ideas. First, due to the superonduting nature of the loop, most of the urrent will flow along the inner edge of the hole. This leads to the onlusion that the width of a loop has less influene on the indutane than the size of the hole. In fat, Jayox and Kethen have shown by numerial simulation that the indutane of a square loop is almost independent of the line width of the loop when the line width surpasses the width of the hole [97]. They further show that the limiting value of the self indutane L for a square hole is given approximately by L =.5 µ o d (3.) where d is the inner side length of the square hole. Therefore, for a given effetive SQUID area, the indutane of a SQUID loop an be redued to.5µ o d by inreasing the line width of the loop, whih in turn redues the hole size. I note that Eq. (3.) is qualitatively onsistent with the indutane of a planar loop with irular ross setion, where the self indutane L goes as 50

71 L µ o C 4π ir ln A a rea (3.) where C ir is the irumferene of the loop, A rea is the area enlosed by the loop, and a is the width of the ondutor forming the loop [98]. If C ir = 4d and A rea = d, it an be seen that L d as long as ln(d/a) does not vary very muh. In Eq. (3.), it is assumed that d >> a, so ln(d/a) always omes out positive. 3.. SQUID Bandwidth For a SQUID to have high temporal resolution, its bandwidth must be large. An upper limit on the bandwidth an be inferred from the analysis presented in setion.4. The bandwidth limits are summarized in Eq. (.97): for β > or τ RC > τ J, f < =, (3.3) τ RC RC τ = πi for β < or RC < τ J, f <, (3.4) τ J Φ o R where f is the SQUID bandwidth. In other words, the bandwidth is limited by the longer of the two time onstants τ RC and τ J. If the geometri mean of τ RC and τ J is alulated, it gives f < τ RC τ J = ei hc ω p (3.5) where ω p = πf p is known as the plasma frequeny of the juntion. Although f p is only an upper limit for f, it is a onvenient parameter sine it does not require knowledge of the effetive shunt resistane of the Josephson juntions. Furthermore, the apaitane and ritial urrent are both approximately proportional to the 5

72 juntion area. Thus, the ratio I /C and onsequently some limitations on the bandwidth are determined by the intrinsi juntion harateristis. For the niobium SQUIDs used in this study, I /C is approximately 0 8 A/F resulting in f p 4 GHz as the upper limit of the bandwidth. If the value of β is known, then a better estimate for the SQUID bandwidth is given by multiplying or dividing f p by β depending on whether β is respetively less than or greater than one. Thus, f f p / β for a hystereti SQUID. Equations (3.3) and (3.4) are limitations that arise from the dynamial behavior of the Josephson juntions. It turns out there are other dynamial limitations that an restrit the bandwidth of a d SQUID. For example, onsider the voltage rise time aross the juntion. The analysis in Chapter assumed that the transition from the zero voltage state to the resistive state was instantaneous. In fat, the time needed to make this transition is finite. Although the ultimate origin of the transitions are quantum mehanial, an estimate for the voltage rise time an be derived from the following lassial onsiderations using the RCSJ model [43-5,99]. Assume that the SQUID juntions are initially in the zero voltage state. Then, while the urrent through a juntion is fixed and smaller than the ritial urrent, the voltage aross the juntion remains at zero. However, one the urrent exeeds the ritial urrent, an average voltage appears aross the juntion. This voltage is a onsequene of the harge aumulated aross the insulating barrier and normal urrent tunneling through the break up of Cooper pairs. In determining this average voltage, onsider only the shunt apaitor and resistor in the RCSJ model in parallel with the urrent soure as shown in Fig. 3.(a). In hystereti juntions, I an assume 5

73 (a) I > I with I I C R (b) V = Φω os ωt L FIG. 3.. (a) Simplified iruit model of a urrent biased Josephson juntion with stati bias urrent where the urrent is just a little greater than the juntion ritial urrent. The urrent path through the ideal Josephson juntion in the RCSJ model is ignored for simpliity. (b) Simple iruit model of a SQUID loop with an externally applied a magneti field. The indued emf is modeled by an a voltage supply. The total magneti flux through the loop is Φ, and its frequeny is ω /π. The self indutane of the SQUID loop is L. 53

74 that the a urrent through the ideal Josephson juntion hannel varies too quikly for it to influene the aumulated harge. In nonhystereti juntions, the dynamis are dominated by the RC omponents. The urrent through the shunt apaitor and resistor is then given by dv V I = C + dt R (3.6) where C and R are the shunt apaitane and resistane, respetively. The voltage V aross the juntion appears when the urrent I just exeeds the ritial urrent I. Then, replaing I with I, the solution to Eq. (3.6) is given by t V = I RC R e. (3.7) As the voltage inreases from zero, R orresponds to the subgap value R sg. For small τ RC = RC, as in the nonhystereti ase of β <<, the voltage quikly approahes the limiting value. However, in this limit, Eq. (3.4) yields a more stringent ondition on the bandwidth, and the voltage rise time does not limit the response. For large τ RC, i.e. the hystereti ase of β >>, where R takes on the value of the subgap resistane R sg, the value of I R sg is muh greater than the gap voltage /e, and so the time required to reah /e is relatively small. Approximating the exponential in Eq. (3.7) by its linear expansion for small values of the exponent with V = /e, one finds τ R C = I R sg e I R sg sg = e Thus, the voltage rise time is τ + RsgC I τ. (3.8) C 54

75 τ rise = C ei (3.9) whih has been verified experimentally [5]. Depending on the speifi juntion harateristis, either τ RC or τ rise will determine the maximum bandwidth. For the hystereti niobium d SQUIDs with I /C 0 8 A/F mentioned earlier, Eq. (3.9) gives a rise time of about ps. When pulsing the bias urrent, one should use pulses with durations that are at least twie as long as the rise time. Thus, the maximum bandwidth orresponds to the inverse of τ rise or around 40 GHz. This value is smaller than the SQUID bandwidth limit given by Eq. (3.5) by an order of magnitude. Comparing the two time onstants τ RC and τ rise, τ τ rise RC C = ei R C sg = ei R sg < (3.0) from arguments given earlier. This suggests that Eq. (3.3) still provides a more stringent ondition on the SQUID bandwidth than Eq. (3.9). Nonetheless, Eq. (3.9) an be used as an upper limit for the bandwidth for large β and is partiularly useful when the subgap resistane R sg is not known. So, instead of Eqs. (3.3) and (3.4), Eqs. (3.5) and (3.9) an approximate the bandwidths for nonhystereti and hystereti SQUIDs respetively. It is also interesting to note that for β <<, I R in Eq. (3.7) an be made smaller than /e by a suffiiently small R. In other words, the voltage rise time of a nonhystereti SQUID an be made smaller than a hystereti SQUID with the same I and C. This suggests that nonhystereti SQUIDs have the potential to have larger 55

76 bandwidths than hystereti SQUIDs. In fat, realling the earlier result of Eq. (3.5) with I /C = 0 8 A/F, f p 4 GHz whereas Eq. (3.9) gives 40 GHz. This result points to why development of superonduting iruits have mostly involved nonhystereti Josephson juntions and SQUIDs, as will be disussed in a later hapter. In any ase, I note that these results all predit relatively large bandwidths and imply that the intrinsi bandwidth of a SQUID is unlikely to be the main fator limiting the bandwidth of real SQUID mirosopes Magneti Hysteresis and Critial Current Modulation A SQUID ats as a detetor of magneti field beause the field modulates the SQUID s ritial urrent. In order to maximize the sensitivity to field, the range of ritial urrent values needs to be maximized. From setion.3, this involves inreasing I while reduing L and mathing the two Josephson juntion ritial urrents. It was also observed in setion.3 that if Φ o >> LI, then the frational modulation I / I would be maximal. Maximum modulation an also be ahieved if Φ A >> LI. Neither of these onditions, however, are always attained or satisfied. In general, Eq. (.58) an be used to determine the modulation amplitude. To proeed, Eq. (.58) an be graphially solved by finding the intersetion of the two urves y y Φo Φ ΦA = LI ' Φo LI ' n Φ Φ = ( ) sin π = sin π Φ Φ o o (3.) 56

77 where I = I os γ av and n is an even integer. If n is odd, the onlusions of the analysis do not hange as this only orresponds to a phase shift of the urve y. The line y intersets the horizontal axis when Φ = Φ A. Analysis of Eq. (3.) an be limited to a region where the value of Φ /Φ o ranges from n to n +, or equivalently from to, with the points of intersetion within this domain. For Φ A = 0, the graph of Eq. (3.) is given in Fig. 3. for different values of a given by a Φ = LI ' = o o. (3.) LI Φ osγ av For a < π, it an be seen that there are multiple intersetions between and, and there will be an inreasing number of intersetions as a 0. Now, the value of I ranges from I to 0 as γ av hanges from 0 to π/. Correspondingly, the slope a hanges from Φ o / LI to as the bias urrent goes from 0 to I. For fixed bias urrent, the value of a remains onstant. However, if there is an applied magneti flux, this will ause line y to be vertially translated up or down as the applied flux is dereased or inreased respetively. As y is vertially translated, it an be seen that an intersetion an suddenly disappear if the slope a is less than π. This is represented in the sequene of graphs shown in Fig So, if the system started out with Φ /Φ o = 0 or some multiple of π, then as the applied magneti flux Φ A inreased, Φ /Φ o would have to make a disontinuous jump to another value. If the applied flux then reversed, Φ /Φ o would not neessarily return to its original value. This is another example of hysteresis and is problemati in determining Φ /Φ o from the ritial urrent. 57

78 (a) a > π y y y = a Φ Φ o = sin π Φ Φ o y Φ Φ o (b) y < a < π y y y = a Φ Φ o = sin π Φ Φ o y Φ Φ o y () a < y y y = a Φ Φ o = sin π Φ Φ o y y Φ Φ o FIG. 3.. Series of graphs showing the solutions y = y for the total magneti flux through a d SQUID loop of self indutane L and juntion ritial urrent I when there is no externally applied flux, for dereasing values of a = Φ o / LI os γ av. 58

79 (a) y y = a Φ Φ A Φ o Φ A = Φ y = sin π Φ Φ o y Φ Φ o y Φ Φ o (b) Φ A : Φ Φ Φ > Φ y y y = a Φ Φ A Φ o = sin π Φ Φ o y Φ Φ o y Φ Φ o () Φ A : Φ Φ 3 y y = a Φ Φ A Φ o Φ > Φ 3 y = sin π Φ Φ o y Φ Φ 3 o y Φ Φ o FIG Series of graphs showing the disappearane of intersetions between y and y as the line y is translated vertially upward from 0, orresponding to dereasing external magneti flux Φ A through the SQUID hole. Slope of y is a < π. 59

80 It an also be seen from Fig. 3.3 that as Φ A is ontinually inreased or dereased, a range of values in Φ /Φ o is not attained though sin(πφ /Φ o ) is periodi with Φ A. A disontinuity in Φ /Φ o and onsequently in the ritial urrent is also seen in Fig.3.3. However, the disontinuity in ritial urrent is not seen in numerial simulations when additional iruit elements in the SQUID model are inluded into Eqs. (.58) and (3.) [60,63,00]. Regardless, the result is a derease in the modulation amplitude of the SQUID ritial urrent. In order to avoid the hysteresis, the intersetion between y and y should be unique, whih requires that a satisfy a > π for any value of the bias urrent I < I. This plaes a restrition on the maximum self indutane L in terms of the ritial urrent I of a d SQUID s Josephson juntion, whih is Φ o L πi or (3.3) β LI Φ o π where β is the SQUID sreening or modulation parameter [4,6,3,65]. It an also be seen that if Eq. (3.3) is satisfied, the total flux Φ is ontinuous with Φ A in Eqs. (.58) and (3.), and the relative modulation of the SQUID s ritial urrent I/I is maximized as well. For β, numerial alulations by Teshe et al. and by de Bruyn Ouboter et al. show that the frational modulation I/I diminishes as β with the modulation amplitude I Φ o /L [60,63]. Regardless of the elimination of hysteresis by limiting the value of β, the modulated ritial urrent remains a periodi funtion of the externally applied 60

81 magneti flux Φ A as well as the total magneti flux Φ through the SQUID loop. Therefore, without shielding or additional information, only hanges in magneti flux or field are reliably determined using a SQUID alone. On the other hand, determining the value of Φ/Φ o modulo ½ in Eq. (.53) an be extended to modulo by identifying whether the ritial urrent inreases or dereases with a small inrease in the applied external flux Flux Noise and Optimization Summarizing the above results, one finds that to improve spatial resolution, one needs to minimize the size of the SQUID, whih also dereases its self indutane L. To inrease SQUID bandwidth, one should maximize I /C. To improve flux sensitivity, one needs to minimize β while maximizing I. If the flux modulated ritial urrent I (Φ ) requires a nonzero minimum, one an restrit β so that β > /π, though Teshe and Clarke suggest the ondition β > 0. for nonhystereti d SQUIDs [63]. A similar outome an be obtained by making the ritial urrents of the SQUID Josephson juntions unequal. These results suggest that the SQUID self indutane L and juntion apaitane C be minimized and the juntion ritial urrent I be maximized within fabriation limits. If hysteresis in the I-V harateristis is desired, β should be maximized, meaning that, onsistent with the aforementioned onditions, the juntion shunt resistane R should be maximized. If hysteresis needs to be eliminated, the reverse should be true. These onditions, however, are still not the omplete set of requirements a SQUID must fulfill. In partiular, SQUIDs should have low noise [4-6

82 6,,0,63,64,00]. A signifiant soure of noise is Nyquist noise whih depends on temperature and the shunt resistane. A general limit on thermal energy was given in Eq. (.34). Clarke and Koh suggest the onstraint I > 0ek BT h (3.4) to be suffiient, based on numerial simulations [6]. The shunt resistane in the Josephson juntions produes a urrent noise spetral density S I given by the Johnson noise expression 4k BT S I ( f ) = (3.5) R where S I has the dimensions of urrent squared per unit bandwidth [0]. This noise auses flutuations in the urrent flowing through the juntions and thus also in measurements of the applied flux. Following Tinkham, Eq. (3.5) an be used to dedue an expression for the flux noise density [65]. The transfer funtion between ritial urrent and magneti flux for a symmetri d SQUID with negligible β is given by Eq. (.53), and I Φ max I = π Φ o L (3.6) where in the last inequality, the requirement of β /π is used to ensure no magneti hysteresis. Therefore, the minimum flux noise density beomes S ( f ) k TL I B Φ ( f ) (3.7) min I Rsg Φ S max where the limiting value of /L was substituted for I / Φ max. De Waal et al. suggest that the atual flux noise density is loser to 6

83 S 4k TL B Φ ( f ) (3.8) Rsg whih was obtained from omputer simulations [64]. Other results suggest an even higher proportionality fator in Eq. (3.8) [4,6,63,00]. Equations (3.7) and (3.8) indiate that noise will derease with dereasing temperature and SQUID hole size. Using Eqs. (3.) and (3.8), the magneti flux noise at 5 K over Hz for a SQUID with a 0 µm by 0 µm hole and R sg = 00 Ω is about T m or Φ o. Over a bandwidth of 00 GHz, the flux noise is about T m or Φ o in 0 ps. In omparison, typial high-t SQUID mirosopes using nonhystereti small β SQUIDs have bandwidths of about 00 khz, limited by the readout eletronis, and a flux noise density on the order of 0-5 Φ o / Hz giving a magneti flux resolution of around 0-3 Φ o in 0 µs Magneti Indution and Nonlinear Effets Besides Johnson noise, other effets may limit the performane of a SQUID. For example, onsider the effet of a hanging magneti field. The eletromotive fore (emf) due to the hanging field is given by Maxwell s equations in integral form: d d A. (3.9) t E A = E d l = V = B If the magneti field is B = B o sin ωt, then V = sin ω t d = sin ω t d = Φ A ω os ω t t Bo A t Bo A (3.0) 63

84 where Φ A is the magnitude of the externally applied magneti flux through the SQUID hole. If Φ A Φ o, at the band limited frequeny of 40 GHz for the hystereti SQUIDs in this study, the indued voltage amplitude is on the order of 0-4 V. Although the indued emf may not be enough to overwhelm the dynamis of the SQUID, it ould have a perturbing effet that ould, for example, inrease the voltage noise aross the juntions or ause a large transient. As an example, suppose the SQUID loop is approximated as a simple indutor in series with a Josephson juntion [see Fig. 3.(b)]. The indued urrent around the ring is given by di V = L = Φω os ω t dt di Φ = ω os ω t dt L (3.) where the ontribution of the Josephson juntion impedane has been ignored [0,03]. Solving Eq. (3.) gives I = I o Φ sin ω t L (3.) where I o is the d urrent omponent that is supplied by the stati urrent bias. Inserting Eq. (3.) into Eq. (.66) and transforming into dimensionless form yields d γ dγ β + + sin γ = y + y ext sin ω x dx dx (3.3) where Φ y ext = LI and (3.4) ω = ωτ. J 64

85 Solving for γ in Eq. (3.3) reveals haoti solutions for ertain values of the parameters [88-9,04]. The haoti solutions ause enhaned voltage noise and may limit sensitivity. In addition, the indued emf around the SQUID loop may also affet the juntion in the resistive state, produing enhaned d urrent effets from Shapiro steps and similar phenomena [0,05-09]. I have not fully onsidered the effet of these phenomena. However, as SQUIDs are used to measure high frequeny magneti fields, effets suh as these may well beome apparent. 3. Resistively Shunted SQUIDs As part of developing a large bandwidth SQUID mirosope, I helped build and test a prototype 4 K SQUID mirosope. Testing of the mirosope was performed using d SQUIDs with resistively shunted Josephson juntions and onventional Flux-Loked-Loop (FLL) eletronis [4,5,8-0]. The eletronis require a nonhystereti SQUID (β << ) so that hanges in flux orrespond to hanges in voltage in a reversible manner (see Fig. 3.4). When fabriating SQUIDs, the value of β is ontrolled mainly by adjusting the values of the ritial urrent I, the shunt resistane R, the barrier thikness, and area of the Josephson juntions. High-T superondutor SQUIDs, for example those based on YBCO, have intrinsially small subgap resistane making β small. However, niobium SQUIDs have large subgap resistane and the juntions must be externally shunted with low resistive hannels in order to eliminate hysteresis. The seletion of the shunt resistane R is determined by a few onsiderations. The upper limit is determined by the requirement that β should be less than 0.69, 65

86 (a) I β < 0.69 I β ~ I bias (b) 0 I R V I R V - 0 Φ Φ A o FIG (a) I-V harateristis of a nonhystereti d SQUID with β at finite temperature. The juntion ritial urrent is I, and the subgap resistane is R. The solid urve orresponds to integer multiples of a flux quantum applied through the SQUID hole. The dotted urve represents the I-V harateristis when the applied flux is a half flux quantum more than the solid urve. (b) Graph of the SQUID voltage versus applied magneti flux through the SQUID hole when the d SQUID is urrent biased at the urrent I bias shown in (a). 66

87 where the onset of hysteresis ours [8]. The lower limit to R is mostly determined by noise onsiderations. In FLL eletronis, a room temperature amplifier monitors the voltage signal aross the SQUID instead of the ritial urrent. The magnitude of this voltage is partly dependent on R and sales with I R, where R is assumed to be muh smaller than the intrinsi subgap resistane. For a juntion with I = 00 µa and C = 0.5 pf, the effetive shunt resistane has to be less than.3 Ω to avoid hysteresis. If the shunt resistane R is too small, the SQUID voltage signal will also be small and an be overwhelmed by noise from the amplifier. For example, if R = Ω, the maximum voltage signal will be about 00 µv for a ritial urrent of 00 µa. The SQUID must also operate below a temperature given by Eqs. (3.4) and (3.8) to resolve this level of voltage. For a nonhystereti SQUID with β, the voltage noise spetral density S V is S V ( f ) 6k TR (3.5) B where S V has the dimensions of voltage squared per unit bandwidth [4,0]. Using Eq. (3.5) for a Ω resistor over a bandwidth of 00 GHz, the temperature of the system has to be lower than 4.5 K to derease thermal noise below 0 µv. This alulation ignores noise from other fators, suh as the readout eletronis. So, this estimate is only a lower limit to the atual noise that will be present. In onlusion, the shunt resistor value must be seleted suh that β is less than 0.69 and the SQUID voltage signal is large enough to be measurable but small enough to not affet the bias urrent as the SQUID alternates between the zero voltage state and the resistive state. 67

88 When using nonhystereti d SQUIDs with FLL eletronis, it is ommon to use a transformer to ouple the SQUID output to a low noise preamplifier [4,5,8]. This has a number of advantages. First, standard FLL eletronis uses a flux modulation and detets the SQUID voltage at a partiular frequeny. The transformer naturally filters out the d omponent as well as the very high frequenies. Further, the transformer steps up the voltage signal aording to the transformer turns ratio. Also, without a transformer, the voltage measurement leads would go diretly up to eletronis whih is at room temperature and may send noise signals bak to the d SQUID or even damage it. The transformer helps to isolate the SQUID from the environment. Finally, the transformer an be used to math the output impedane of the SQUID signal to the input impedane of the amplifier and transmission line for high frequeny appliations. This an result in the best signal to noise performane for the SQUID and amplifier ombination. The ideal mathing ondition for lowest noise is obtained when R N = Z n (3.6) where Z n is the noise impedane of the amplifier, N is the turns ratio of the transformer, and R is the dynami resistane of the nonhystereti d SQUID at the operating bias urrent [4,9,]. A small problem with using a transformer is that the primary is in parallel with the d SQUID, and this shunts urrent away from the SQUID bias. If the effetive shunt resistane of the d SQUID is muh larger than the impedane of the transformer primary, the bias urrent to the SQUID will hange onsiderably when the SQUID goes from the zero voltage state to the resistive state and vie versa. This, 68

89 in turn, diminishes the voltage signal and should be prevented. A solution to the transformer problem is to add a small resistane R x in series with the transformer primary to inrease its impedane [9,0]. If the added resistane is too large, the voltage aross the transformer will be too small. So, the resistane is hosen to be somewhat less than the impedane of the transformer primary. When adding shunt resistors to Josephson juntions, the leads of the shunt resistors an ontribute an indutive term to the equation of motion given by Eq. (.69). Speifially, an additional urrent hannel with a resistive and indutive omponent appears aross the juntion. Cawthorne showed that this additional hannel an give rise to ompliated behavior of the SQUID voltage immediately after the bias urrent exeeds the ritial urrent, though the zero voltage state is not affeted [95]. The voltage strutures due to indutive effets were seen to broaden and diminish as the value of β approahed zero. Even if the magneti field deteting properties of a SQUID were not adversely affeted by the presene of some indutane in the shunt resistors, it would be preferable if the SQUID resistive state harateristis were simple and preditable. For ertain values of the parameters, analysis of the juntion with the added indutive element shows that the system an be haoti, as was the ase for Eq. (3.3). Therefore when adding shunt resistors, are should be taken to minimize lead lengths and use intrinsially resistive materials to shorten the shunts so as to minimize the total indutane. The differene in the indutane of the SQUID loop between paths through the shunt resistors and those through the Josephson juntions should be 69

90 minimal as well. Adherene to these riteria will in general have to be verified by numerial simulation in the absene of analytial tehniques [95,96]. 3.3 Niobium SQUID Design and Charateristis 3.3. Prior SQUID Design The SQUIDs I used in my work were made with niobium. The reasons for hoosing niobium were that it has a superonduting transition temperature T 9.3 K with a gap voltage /e of about 3.05 mv at zero temperature [-5]. Seondly, ommerial failities for the fabriation of ustom Nb-AlO x -Nb integrated iruits suh as those operated by HYPRES, In. are readily available. Needless to say, there an be enormous advantages to ordering hips from suh failities ompared to fabriating them in-house. Finally, niobium also has large subgap resistane whih enables hystereti Josephson juntions to be prepared, whih is ritial for my high speed measurement tehnique. The d SQUIDs I tested on the prototype 4 K SQUID mirosope had the same design used on a LHe ooled sanning SQUID mirosope maintained by Cawthorne and Nielsen [,95]. Figure 3.5(a) shows a photograph of one of these SQUIDs. The LHe ooled mirosope had been used for the study of magneti properties of mirosopi superonduting strutures. The Cawthorne and Nielsen mirosope inorporated onventional FLL eletronis, so the SQUID juntions had to be resistively shunted. The SQUID hips were designed using ICED whih is an MS-DOS based integrated iruit editor program [6]. Fabriation of the hips was performed by HYPRES, In. using their niobium hip foundry [7]. The LHe 70

91 (a) One Turn Coil 40 µm (b) Josephson Juntions Shunt Resistor One Turn Coil 3 µm Shunt Resistor Josephson Juntions FIG (a) Photograph of the resistively shunted niobium d SQUID designed by Cawthorne and Nielsen, whih was used in a LHe ooled sanning SQUID mirosope. (b) Photograph of updated resistively shunted d SQUID I designed with features ompliant with HYPRES design rules. 7

92 ooled mirosope required miron sale spatial resolution, so the SQUIDs had outer loop dimensions of 30 µm by 40 µm and a 0 µm by 0 µm square hole. The small SQUIDs have an estimated loop indutane L 6 ph and, in onjuntion with Josephson juntions with ritial urrent I 60 µa, were designed so that β. The Josephson juntions of the d SQUID were designed to be 3 µm by 3 µm squares, the minimum size allowed by HYPRES. However, due to the fabriation proess, the juntions ame out slightly rounded and smaller than designed. HYPRES reported that the loss in area is statistially about 3.0±0.5 µm [8]. This variation was signifiant and had to be fatored into the design, as will be disussed later. Eah SQUID juntion was shunted with a resistor 3 squares long. The resistive material for shunting the juntions was molybdenum with a alibrated resistane of around Ω/square aording to HYPRES speifiations. The atual resistanes were not expliitly determined; interlayer onnetions to the resistors required signifiant area near their ends making the effetive area smaller by around a half square. As there are two shunts per SQUID, one for eah juntion, the ombined parallel resistane is around Ω. The d SQUIDs also have a one turn oil around the inner hole of the SQUID (see Fig. 3.5). The oil is mostly made up of niobium as well. With FLL eletronis, the oil is onneted to the feedbak system and provides a nulling magneti flux through the SQUID loop. In my experiments, it is onneted to a signal generator and sends external high frequeny flux to the SQUID. The small size of the oil allows the field to onentrate the magneti flux through the hole. 7

93 Measurements I made of the I-V urve of a resistively shunted d SQUID (SQUID AN) in LHe showed some problems (see Fig. 3.6). In partiular, the SQUID s shunting resistane was muh greater than Ω. There was also some subtle struture in the resistive region of the I-V urve. This may have been due to indutive effets or a biasing, as disussed earlier [95,08,09]. Nevertheless, the observed large normal resistane is an indiation of a fabriation problem, most likely poor ontat between the resistors and the niobium layer. The problems with SQUID AN, however, were muh less severe than problems with the d SQUIDs left over from the LHe ooled mirosope. Some of those problems were learly visible under a high power optial mirosope. The major problems were with respet to the SQUID leads, their uniformity in partiular, and with the via holes between layers. The leads were sometimes too thin or not properly defined. The result was that they did not make a ontinuous onnetion or failed easily. As it turned out, this was due to lak of adherene to the manufaturer s design rules, i.e. the designed features did not fully adhere to the fabriation riteria [7,8]. Partiularly, some leads were thinner than the minimum requirement, and some via holes did not follow the required size or edge separation. For example, the minimum allowed lead width for a niobium layer was µm to.5 µm depending on the layer. But, this was not followed in some features resulting in exessively narrow setions in the fabriated lead. The one turn oil around the inner hole was a partiular area where the design did not meet the HYPRES design rules; the oil width was designed too thin and would sometimes not ome out as a ontinuous loop. For some devies, exessively narrow 73

94 I (µa) V (mv) FIG I-V harateristis of SQUID AN whih is of the Cawthorne and Nielsen design. Four point measurement made in LHe. Magnet wire leads used to measure signals. SQUID hip attahed to a iruit board with wirebonding. Multiple hanges in the differential resistane an be seen. Beyond the zero voltage state, the differential resistane dv/di is approximately 40 Ω between 5 µa and 50 µa. It then dereases to about 3 Ω near 00 µa, after whih it inreases to around 30 Ω beyond 75 µa. Additional hanges an be seen as the urrent inreases further beyond the urrent values shown here. Modulation of ritial urrent with applied magneti field is observed between 00 µa and 50 µa. A seond urrent modulating region is observed at higher urrent but with smaller amplitude. 74

95 superonduting leads beame resistive under power, produing enough heat to blow out the leads. This was observed to have happened in a few ases, and the damage was visible under an optial mirosope. Regarding via holes, they sometimes laked lear edges. Eletrial onnetions either did not exist or failed in these ases. Other problems with the SQUIDs inluded leads and ontat pads flaking off during hemial leaning. And, there were still problems that ould not be identified visually that were only detetable during testing. After realizing the SQUIDs designed by Cawthorne and Nielsen ontained these problems, I updated the design to bring it into ompliane with the HYPRES design rules [8]. The updated design has resistively shunted SQUID juntions of the same size with similar shunt resistor values ompared with the original design. When all the rules were followed, features in the SQUID were muh more well defined and uniform, as an be seen in Fig. 3.5(b). Unfortunately, the updated SQUIDs arrived too late for them to be installed on the 4 K SQUID mirosope, and I did not have the opportunity to test them after they were made available to me. Notwithstanding, even when the HYPRES design rules are followed, I found deviations in the parameter values of the SQUIDs. Although HYPRES s proess attempts to produe uniformity in the niobium hips, there are variations in the features of the hips, partiularly in the ritial urrent densities of the juntions from hip to hip. These variations an be as muh as 30% from the stated target value and typially differ between 5% and 5%. Another ause of variations in ritial urrent, partiularly for small juntions, is that there is a signifiant differene in the 75

96 areas due to the rounding of the square orners as mentioned earlier. While HYPRES states that these differenes are fairly regular, for the smallest juntion size, the variations an be signifiant enough to ause a mismath between the atual and intended SQUID harateristis. Besides juntion area and ritial urrent density, juntion apaitane and external resistor harateristis also vary. Juntion apaitane depends on the juntion area and thikness of the insulating barrier. The barrier thikness is, in turn, refletive of the ritial urrent density of the juntion. Variations in external resistors, however, were not signifiant as resistor size is larger than the minimum required dimensions. The expeted variation in resistivity is typially less than 0%. Furthermore, for our parameters, the d SQUID s performane is not very sensitive to the atual value of the shunt resistane. Considering that larger areas are less affeted by small variations, inreasing the juntion area by going to a lower ritial urrent density foundry proess would tend to improve the math between designed and atual parameter values without drastially affeting SQUID harateristis Measured SQUID Charateristis Figure 3.6 shows the I-V harateristis in LHe of the resistively shunted d SQUID I used for the prototype 4 K mirosope (SQUID AN). This nonhystereti SQUID shows a zero voltage state with a maximum superurrent of around 5 µa and a resistive region with R d = dv/di hanging pieewise from 40 Ω to 3 Ω then to 30 Ω to within 7% unertainty. This is muh higher than the expeted value of Ω. 76

97 I observed urrent modulation with respet to magneti flux, whih was small in this devie. Figure 3.7 shows the a omponent of the SQUID voltage measured at different bias urrent values while an a magneti field is applied to the SQUID. Maximum modulation ours at a bias urrent around 30 µa resulting in voltage modulation of around 80 µv peak to peak [see Fig. 3.7(e)]. This result shows that peak modulation is atually ourring well inside the resistive region of the I-V urve. This was not expeted and is irregular. Although maximum modulation is ourring near the expeted bias urrent, obtained from alulations of the designed ritial urrent desribed below, the maximum superurrent is muh smaller than expeted. This phenomenon ould be explained by a Josephson juntion with a ritial urrent of 5 µa and a shunt resistane between 0 Ω and 30 Ω in series with the d SQUID. Suh a juntion would be parasiti or aidental and a result of a defet in the fabriation of the SQUID hip. These features in the d SQUID were never fully understood and seemed to be ommon to all SQUID hips in the same bath. Notwithstanding these problems, the d SQUID ould still be used as a magneti field sensor. For example, Fig. 3.8 shows the voltage aross the SQUID as the magneti flux through its hole is linearly inreased then dereased. These measurements show that as the magneti flux ontinuously inreased or dereased, the a SQUID voltage osillated as expeted, similar to Fig. 3.4(b). I found the SQUID parameters for SQUID AN as follows. Using Eq. (3.) with an inner hole length d = 0 µm, the SQUID loop indutane is L 6 ph. The maximum expeted ritial urrent I of eah Josephson juntion is alulated from 77

98 (a) (b) 40 V a (µv) 40 V a (µv) Φ A 40 Φ A () (d) 40 V a (µv) 40 V a (µv) Φ A 40 Φ A (e) (f) 40 V a (µv) 40 V a (µv) Φ A 40 Φ A (g) (h) 40 V a (µv) 40 V a (µv) 0 40 Φ A 0 40 Φ A FIG Series of osillosope pitures showing a omponent of voltage aross SQUID AN at different bias urrents. Bias urrent values are (a) 54 µa, (b) 79 µa, () 96 µa, (d) 6 µa, (e) 30 µa, (f) 48 µa, (g) 80 µa, and (h) 07 µa. Triangular wave represents urrent through one turn oil on SQUID hip, with amplitude of 700±5 µa peak-to-peak. No noise filtering was performed on the signals. Maximum modulation amplitude seen with bias urrent at around 30 µa. 78

99 (a) (b) 40 V a (µv) 40 V a (µv) 0 40 Φ A 0 40 Φ A () (d) 40 V a (µv) 40 V a (µv) 0 0 Φ A Φ A (e) (f) 40 V a (µv) 40 V a (µv) 0 0 Φ A Φ A FIG Series of osillosope pitures showing a omponent of voltage aross SQUID AN in LHe, as the amplitude of an externally applied magneti field is inreased. The SQUID bias urrent was fixed at 30 µa, and the magneti field was applied to the SQUID using the one turn oil on the SQUID hip. The triangular wave represents the urrent through the one turn oil, with peak-to-peak amplitudes of (a) 80 µa, (b) 90 µa, () 389 µa, (d) 499 µa, (e) 579 µa, and (f) 700 µa. No noise filtering was performed on the signals. 79

100 the alibrated value of J Area = 70±6 µa where, the ritial urrent density J of 68.6 A/m was provided by HYPRES, and I took Area = 6.0±0.5 µm. From Eq. (3.3), this yields β.07 whih is greater than /π. Next, from Eqs. (.68) and (.69), β 0.6 for a single 6 µm juntion with a shunt resistane of Ω and a apaitane of 0.30 pf alulated from HYPRES s apaitane formula 0. 0 C = log J 0 (3.6) where C is the apaitane per unit area in ff/µm and the ritial urrent density J is in ka/m [8]. Given the smallness of β, the d SQUID should be nonhystereti in its I-V harateristi, whih is by design and what is observed in Fig With these parameters, the SQUID s maximum ritial urrent should be 40± µa, and the ritial urrent modulation should be about 70 µa orresponding to a voltage modulation of around 70 µv. These expetations are onsistent with the harateristis of the d SQUID, exept for the suppressed ritial urrent and large normal resistane mentioned earlier. From Fig. 3.8, I also find that one period of the voltage orresponding to one flux quantum requires a hange in urrent of 69±4 µa in the one turn oil. Thus, the mutual indutane between the one turn oil and SQUID loop is approximately ph. For omparison, Fig. 3.9 shows the I-V harateristis in LHe of the niobium d SQUID seen in Fig..(b), whih has no shunt resistors (SQUID BH). Figure 3.9 shows two sets of urves, one where the magneti field through the SQUID hole results in maximum SQUID ritial urrent and the other where the ritial urrent is minimum. The I-V urve learly shows hysteresis due to a large β value. 80

101 (a) I (ma) V (mv) 4 V (mv) 0.4 (b) I (ma) FIG I-V harateristis of SQUID BH obtained from a four point measurement with ommon ground. Measurements performed in LHe using miro oaxial able with silver platted opper weld inner ondutor and stainless steel outer ondutor. Connetions to the SQUID performed by wirebonding. Horizontal sale is voltage with mv/div. Vertial sale is urrent with 0. ma/div. Origin entered at middle of piture. An offset voltage of approximately 0.5 mv is seen due to thermal emf. Modulation of ritial urrent between (a) maximum and (b) minimum values is observed. Change in magneti flux through SQUID hole ahieved by rotating d SQUID in ambient magneti field. 8

102 Using the expression for the retrapping urrent given by I r =, (3.7) π 4I β β an be estimated to be approximately 480 with I r = 0±5 µa and I = 7±5 µa [70,8]. However, due to the relatively large unertainty in I r and the effets of noise and temperature in raising the apparent retrapping urrent, β ould be anywhere between 00 and 000. Assuming idential juntions and using Eq. (.68), the subgap resistane of a single SQUID Josephson juntion is expeted to be in the range between 30 Ω and 00 Ω, dependent on the value of β. However, this result onflits with diret observation of the effetive subgap resistane from Fig. 3.9, whih gives a lower limit of 340 Ω aross eah juntion. Suh a resistane would orrespond to a value of β that is greater than 0 4. On the other hand, this onlusion is based on a simple linear model following the analysis in setion.4. A more omplete analysis of the nonlinear behavior of a Josephson juntion based on Eq. (.94) does not fully explain the experimental observations, as disussed by Prober et al. [7]. Consequently, more onfidene is given to the former result of β between 00 and 000 but with prejudie toward the higher limit, not only due to the results of diret observation but also due to thermal effets that inrease the effetive retrapping urrent [70,83]. Some other notable features in the I-V urves of Fig. 3.9 are the gap voltage /e =.7±0. mv and the normal resistane above the gap voltage R = 6.9±0.5 Ω. At the boiling point of LHe, T = 4. K, so T / T = Aording to BCS theory, the gap voltage at 4. K should be very lose to the zero temperature value, differing 8

103 by less than 4% [34,86,5]. However, the observed differene of.5% from the zero temperature value suggests a higher temperature T 0.64T 6.0 K. Although a satisfying explanation for the disrepany has not been determined, there is always the possibility of exess heating due to the urrent in the measurement leads. If one assumes that the higher temperature is orret, the estimated ritial urrent at zero temperature should be I (0) I (T)/0.7 0 µa [4]. Then by Eq. (.90), the gap voltage at zero temperature should be.6 mv. This disrepany ould be a result of ritial urrent suppression or an effetive resistane of R 0 Ω whih lies between the observed normal resistane and the estimated range of the subgap resistane. An alternate explanation for the multiple disrepanies ould be that the superonduting material is not pure niobium. Fortunately, whatever the situation, the magneti field sensing properties and the fast transition from the zero voltage state to the resistive state of the SQUID are not affeted. Using the alibrated ritial urrent density of 60 A/m provided by HYPRES for SQUID BH, the expeted maximum ritial urrent was 39± µa for the two 3 µm 3 µm SQUID juntions. The measured value of 7±5 µa is somewhat larger than this value and is probably due to a slightly larger juntion area than expeted. Compensating for the larger area using the new value of I, the total SQUID apaitane of the two Josephson juntions is approximately 0.75 pf using Eq. (3.6). I an now use Eq. (3.9) to estimate the voltage rise time τ rise of the juntion and find τ rise ps. The maximum SQUID bandwidth is then 44 GHz. Again using Eqs. (3.) and (3.3) with I = 7 µa, the estimated value of β is.3, whih means there ould be magneti hysteresis, though no obvious 83

104 manifestation of this was observed. This value of β also implies that the modulation amplitude of the ritial urrent will be roughly 0.7I 60 µa [60,63]. The observed modulation seen in Fig. 3.9 is 49±7 µa, suggesting that β is slightly larger than.3. To observe the full range of modulation, I rotated the SQUID in the ambient magneti field, whih varied the magneti flux through the SQUID hole. Multiple osillations of the ritial urrent were observed as the SQUID rotated through 90. Examination of Fig. 3.9 also shows that the superurrent ourred at around 0.5 mv instead of zero. I have onluded that this effet was due to the Seebek effet or thermoeletri emf in the leads during I-V measurements. The measurement was performed with miro oaxial able with an inner ondutor made of silverplated opper-lad steel (SPCW) and an outer ondutor of stainless steel. The Seebek oeffiients of opper and iron respetively range from µv/k and 6 µv/k to zero dereasing as T 0 [9]. I measured voltages using the inner ondutor with the outer ondutor onneted to a ommon ground. Given an average differene of µv/k in Seebek oeffiient between the inner and outer ondutors for temperatures between 300 K and 4 K, the resulting thermal emf would be around 0.3 mv. This is the magnitude of the observed voltage offset. I oasionally observed a few other anomalies in the I-V harateristis. Under ertain irumstanes, the zero voltage state was not symmetrial for positive and negative urrents. This an be seen in Fig When this ourred, the modulation of the ritial urrent was suh that one polarity seemed to lag the other. For example, if for positive urrent, the modulated ritial urrent was given by 84

105 I (ma) V (mv) 0. FIG I-V harateristis of a hystereti d SQUID showing asymmetry in the positive and negative ritial urrents. Modulation of ritial urrent in one polarity was found to lag behind modulation of ritial urrent in opposite polarity, as magneti flux through SQUID hole is inreased. Origin entered at middle of piture. An offset voltage of approximately 0.5 mv seen due to thermal emf. 85

106 Φ I ( Φ ) = I os π, (3.8) Φ then for negative urrent, it was behaving like I ( Φ ) o Φ = I sin π. (3.9) Φ o The ause of this behavior was not determined, but the effet sometimes disappeared with thermal yling, suggesting that the phenomenon was assoiated with trapped flux. Finally, in Table I, I summarize the main SQUID parameters for the nonhystereti SQUID AN and the hysteresti SQUID BH SQUID Chip Layout and Leads In addition to the SQUIDs themselves, SQUID hips require leads, ontat pads, and onnetions to external wiring. Several individual SQUIDs an fit onto a single HYPRES hip whih is 5 mm 5 mm square (see Fig. 3.). Although the atual SQUIDs are very small, the ontat pads are onsiderably larger, so most of the hip area is taken up by the ontat pads. These pads are made from titanium, palladium, and gold with gold onstituting most of the pad; this provides a low resistive onnetion of less than 0. Ω/square for the ontat pad leads from external wiring to the SQUIDs [8]. In my design, there are two sets of pads, one for use in the SQUID mirosope and another for testing the hip. Two sets are inluded beause after testing a hip, wire leads onneted to the ontat pads needed to be removed, and during this proess the pads would be damaged. Thus, with two sets, an auxiliary set would be 86

107 TABLE I. SQUID parameters for SQUIDs AN and BH. Values in the Atual olumn are best estimates determined from data or information provided by HYPRES, In. Parameters SQUID AN SQUID BH Designed Atual Designed Atual Size (outer) 30 µm 40 µm 30 µm 30 µm Size (hole) 0 µm 0 µm 0 µm 0 µm L 6 ph 6 ph C 0.30 pf 0.30 pf 0.30 pf 0.38 pf β 0.6 high 00 to 000 β.07.3 I 0 µa 5 µa 0 µa 7 µa I 60 µa 70 µa 60 µa 49 µa V 60 µv 80 µv 3.0 mv.7 mv R Ω 3 to 30 Ω low 6.9 Ω R sg Ω high 30 to 00 Ω 87

108 (a) Primary Contat Pad 940 µm (b) Lead to Seondary Contat Pad 95 µm Primary Contat Pad FIG. 3.. Photographs of niobium d SQUIDs showing their primary ontat pads and leads. (a) Original design by Cawthorne and Nielsen for the LHe ooled sanning SQUID mirosope. (b) Updated design with more irular pads and HYPRES design rule ompliant lead thiknesses. 88

109 used during testing. Then if the SQUID is to be installed into the mirosope, it is died from the hip, and the primary set is used for wiring inside the mirosope. The auxiliary ontat pads used for testing are loated around the rim of the HYPRES hip, and eah pad orresponds to and is eletrially onneted to a primary pad that is loated near the SQUID (see Fig. 3.). The size of a ontat pad is determined by the number of pads needed and the dimensions of the hip. I found the sizes of pads in the original Cawthorne and Nielsen design to be too small. So, I enlarged the pads to allow more onnetions per pad. In the updated design, shown in Fig. 3., the auxiliary ontat pads are about 500 µm 400 µm on average. For the primary ontat pads, I found that inreasing the outer dimension of an individual SQUID hip was not the deiding fator in inreasing the effetive area of the pads. For z-squid onfigurations, seen in Fig. 3., the pads surround the SQUID evenly. However, most of the outer edges of the pads are ground off as a died SQUID must ultimately be plaed on a mm by mm sapphire tip for use in a SQUID mirosope. In order to maximize the remaining pad area, I found that the pads had to fill as muh area around the SQUID as possible, exept near the SQUID to avoid interferene from stray field. This was ahieved by making the pad design more irular than before. Consideration was also given to making ontat pads large enough so that silver paint ould be applied by hand to make eletrial onnetions to the hip. I had attempted making suh onnetions with silver paint for testing purposes, and the pratie is routine for preparing SQUID hips used in high-t SQUID mirosopes 89

110 Original Design Updated Design FIG. 3.. Updated design of HYPRES niobium SQUID hip. Auxiliary ontat pads used for testing are loated around the rim of 5 mm 5 mm hip. Vertial and horizontal stripes indiate diing hannels for extrating individual SQUIDs after testing. The diing hannels divide the hip into nine square areas eah ontaining a single SQUID. There are no auxiliary ontat pads for the enter square whih an ontain a SQUID or some other test iruit, as shown here. The hip shown in this diagram ontains d SQUIDs with both the original design used in the LHe ooled SQUID mirosope and the updated design with HYPRES ompliant features, some with more irular primary ontat pads. 90

111 [7]. However, pad sizes would have to inrease further to make this easy to perform. Furthermore, if the quality of either the silver paint or the ontat surfaes were poor, these onnetions would show a lot of noise and fail in a short time. In the end, I found the best eletrial onnetions were produed by wirebonding. Consequently, the updated ontat pad design was optimized to easily allow multiple wirebonded onnetions, partiularly for the auxiliary pads. In both the original Cawthorne and Nielsen design and in my updated design, four pads are used for eah SQUID. Two are onneted to the d SQUID while the other two are onneted to a one turn oil around the SQUID hole. The SQUID pads are used both to supply the d SQUID with bias urrent and to measure its voltage. Leads to the SQUID are onneted to the SQUID loop in a symmetrial design, as shown in Figs..(b) and 3.5. This prevents any asymmetry in the indutanes of the two branhes of the SQUID loop. Asymmetry may lead to the bias urrent providing a irulating urrent around the loop, whih in turn would lead to a shift in the ritial urrent versus flux relation. Figures.(b), 3.5, and 3. also show how the leads to the one turn oil overlap near the SQUID. This redues the effet of stray magneti field near the SQUID due to the oil leads. The leads were also designed suh that the geometrial shape they formed helped identify and differentiate the SQUID leads from the oil leads. For instane, the SQUID leads formed an L shape, whih is inverted during the hip fabriation proess, and the oil leads formed a T shape (see Fig. 3.). This was espeially helpful in preventing basi wiring mistakes, suh as during wirebonding. Furthermore, in the updated hip, the seondary pads were sequened suh that eah 9

112 pad in a sequene orresponded to the same lead on an individual SQUID regardless of whih SQUID was being tested (see Fig. 3.). This made it more onvenient during testing, as it was easier to identify, verify, and align onnetions. The idea behind x-squids, z-squids, and how they affet ontat pad design is as follows. A SQUID an have two onfigurations depending on whether the plane of the SQUID loop is parallel with or perpendiular to the plane of the surfae it is sanning. In a z-squid onfiguration, the SQUID is oriented to measure the magneti field that is normal to the surfae being sanned, i.e. the SQUID loop is parallel to the sample surfae. On the other hand, the x-squid onfiguration has the SQUID loop normal to the sample surfae, so the SQUID measures the omponent of the magneti field that is parallel with the surfae and normal to the loop. Now, the ability of a SQUID mirosope to spatially resolve individual soures of surfae urrents improves as the distane between the sanned surfae and SQUID dereases [4,5]. So, to maximize spatial resolution, the SQUID must not only be small but also brought as lose as possible to the sanned surfae. In the ase of a z-squid, the whole loop an be brought lose to the surfae, and the ontat pads an be plaed around the SQUID evenly as in Fig. 3.. In ontrast, for an x- SQUID, only one edge of the SQUID loop is brought lose to the surfae. Therefore, all the ontat pads have to be plaed on one side of the SQUID loop. This requires positioning the pads as in Fig Although I did not prepare SQUID hips with primary pads onfigured as x- SQUIDs, the auxiliary ontat pads are in the x-squid onfiguration. In fat, 9

113 FIG Diagram showing an x-squid onfiguration with all ontat pads and leads going to one side of the SQUID loop. The onfiguration inludes leads and pads for a oil to generate a magneti field through the SQUID hole. The oil an be part of a feedbak system or used to test the SQUID. 93

114 Vlahaos et al. have made use of my SQUIDs in the x-squid onfiguration for the prototype SQUID mirosope [0]. 94

115 CHAPTER 4 The 4 K Cryoooled Sanning SQUID Mirosope 4. Overview of Mirosope The main omponents of a sanning SQUID mirosope, besides the SQUID itself, are a ryoooler or ryostat, a vauum hamber, a old finger, a sample translation mehanism, and eletronis for monitoring the SQUID and ontrolling the mirosope. Figure 4. shows a oneptual sketh of the prototype 4 K sanning SQUID mirosope inluding the ryoooler, a radiation shield around the old finger, and a movable sapphire window separating the SQUID from the sample. Figure 4. shows a photograph of the entire system. In the prototype mirosope, the ryoooler takes up the bulk of the volume and is the most expensive single omponent. Until room temperature superondutors are disovered, a ryogeni system will be needed for ooling the SQUID to superonduting temperatures. Several riteria were involved in the design of the SQUID mirosope. Foremost, the mirosope needed to be able to operate with hystereti SQUIDs. This lead to the hoie of niobium SQUIDs whih operate at liquid helium (LHe) temperatures. Also for pratial purposes, it is preferable for the mirosope to be operable and servieable by one person. Samples should be easy to mount and hange. So, to make the system ompat and simple to operate, a large apaity ryoooler is preferred rather than a ryostat whih would take up more spae and require repeated filling of ryogen. Although ryostats are a mature tehnology with 95

116 Cryoooler Cold Finger Cold Heat Exhanger Vauum Chamber Window Positioning Mehanism Radiation Heat Shield Flexible Bellows Sapphire Rod SQUID Chip 5 µm thik Sapphire Window Room Temperature Sample FIG. 4.. Shemati diagram of the old region of the prototype 4 K ryoooled Sanning SQUID Mirosope.

117 Temperature Controller Pulse Tube Cryoooler Translation Stages Helium Compressor m Lok-in Amplifier SQUID Controller Turbo Vauum Pump Pressure Monitor FIG. 4.. Photograph of prototype Sanning SQUID Mirosope with a 4 K pulse tube ryoooler. A diaphragm vauum pump whih supports the turbo pump lies behind the table top hassis and is not seen in the photograph.

118 fewer mehanial vibration issues than ryooolers, it is a major advantage to only need eletrial power and not have to regularly supply LHe to the system. Closed yle ryooolers reahing temperatures below 4 K are urrently available and allow low temperature superonduting material suh as niobium to be used. Furthermore, the lower temperature allows superonduting material with higher T to be used in the future, and the effet of thermal noise is smaller as disussed in setion 3.. It is also advantageous to have samples at room temperature. This is made possible by the moveable sapphire window in the vaum hamber and permits mounting and hanging samples without having to shut down the mirosope [8,,7]. In the following setions, I desribe the main mirosope omponents and some issues and that affet performane. The main soures of noise in the prototype mirosope are mehanial noise from mehanial osillations of the SQUID old finger aused by the pulse tube ryoooler, eletrial noise from the SQUID eletronis, and intrinsi noise from the SQUID itself and from resistive ontats. Some of these noise soures, suh as the mehanial osillations of the old finger, are so signifiant that they will need to be addressed through design hanges. The hapter ends with a setion on the mirosope s operating and serviing proedures, inluding omments on its performane. 4. Cryoooler and Vauum System 4.. The Pulse Tube Cryoooler The ryoooler hosen for the prototype SQUID mirosope is a PT405 pulse tube ryoooler from Cryomeh, In. (see Fig. 4.3) [7,]. The working priniple 98

119 SQUID Tip Radiation Heat Shield Eletroni Wiring Pulse Tubes Vauum Flange 70 m Pressurized He Hose Connetors Cold Head Eletronis Connetors FIG Photograph of the Cryomeh PT405 4 K pulse tube ryoooler with the old finger attahed to the seond stage heat exhanger. The opper radiation shield is attahed to the first stage heat exhanger and overs the old finger and most of the pulse tubes. When installed, the SQUID tip points downward. Hermeti onnetors available on flange for pressure gauge and eletroni wiring. 99

120 behind the pulse tube is similar to that of Stirling ryooolers and is summarized in Fig. 4.4 [-7]. Compressed gas is fored to flow into a tube where it expands and ools the surrounding material. The pulse tube performs this without requiring any moving parts in the old region. This is made possible by substituting the piston, or displaer, in a Stirling ryoooler with a olumn of ompressed gas or gas density wave in the tube. The expanded old gas removes heat from the old region and is then retrieved from the tube. The tube is designed to minimize onvetion urrents whih would mix the old and hot gases in the tube. Before the gas is reompressed and the yle repeated, the gas goes through a regenerator whih is ooled by the old gas. The role of the regenerator is to pre-ool the warm gas passing through before it enters the pulse tube on the following yle. The pulse tube system operates in a losed yle and only needs additional oolant (pure helium) when a suffiient amount is lost due to leaks or when there is signifiant ontamination. The PT405 ryoooler uses a low frequeny (Gifford-MMahon type) pulse tube and operates at a frequeny of around.3 Hz. Compressed helium gas is supplied to the ryoooler from a separate ompressor unit through a flexible metal hose, and the deompressed gas is returned to the ompressor where heat from the gas is removed for reompression. Cold water must be supplied to the ompressor to keep it from over heating. I reommend some type of filtering to the water supply to remove ontaminants whih may ause damage to the ompressor. In order to inrease the ooling effiieny and reah temperatures below 4 K, an intermediary stage exists between the room temperature omponents and the old region seond stage. The first stage ats as the high temperature sink to the seond 00

121 (a) Compressor Reservoir Hot Regenerator Hot Warm Pulse Tube Warm (Cold) Heat Exhanger High Pressure (b) Compressor Reservoir Hot Regenerator Hot Cold Pulse Tube Cold (Cold) Heat Exhanger Low Pressure FIG Blok diagrams representing the working priniple of a basi single stage pulse tube ryoooler. (a) Compression phase: ompressed helium gas passes through the regenerator and old heat exhanger before entering the pulse tube. The old end of the regenerator is thermally anhored to the heat exhanger and is already ooler than the other end due to the previous yle. Gas at the warm end of the pulse tube leaks into a reservoir but out of phase with the hange in pressure inside the tube. (b) Expansion phase: leakage into the reservoir is ut off, and ompressed gas inside the pulse tube expands out through the heat exhanger removing heat in the proess. As the gas returns to the ompressor through the regenerator, the temperature gradient in the regenerator inreases. 0

122 stage. This first stage an reah temperatures down to 3 K with the seond stage reahing below 4 K. The effiieny of the regenerator for the seond stage is maximized for temperatures below that of the first stage. However, overall ooling power of the seond stage is less than that of the first stage. The first stage has a rating of 5 W at 65 K ompared to 0.5 W at 4. K for the seond stage. The first stage is used to ool a radiation shield whih surrounds the seond stage as well as the attahed old finger, reduing radiation heating from the room temperature environment (see Fig. 4.3). Signal wires going to the old finger are also thermally anhored to the first and seond stage heat exhangers to minimize ondutive heating of the old finger and SQUID hip. 4.. The Vauum Chamber and Pumps For good thermal insulation, all the ryogeni omponents are ontained within a vauum hamber. The vauum environment also prevents air and moisture from ondensing on the old parts of the system, thus reduing damage during thermal yling. In the mirosope, a diaphragm pump is used for rough vauum and a turbo pump is used to reah the base pressure of about 0-4 torr before starting the ryoooler [8]. Both pumps are air ooled. One the ryoooler is ativated, the vauum level improves as remaining gas moleules stik to old surfaes inside the hamber. The vauum hamber and hassis are made from nonmagneti stainless steel and aluminium alloy and onsist of three main setions. A large ylindrial setion surrounds the pulse tubes and is bolted to the top of a brae on the table hassis (see 0

123 Fig. 4.). The bottom of this setion, with the ryoooler in plae, is seen in Fig. 4.5(a). The ryoooler is inserted from the top and sealed to the vauum hamber with a rubber o-ring and vauum grease. The ryoooler has to be oriented with the heat exhangers pointing downward for it to operate effiiently. Attahed to the bottom of the ylindrial setion is a metal bellows setion. Finally, a one shaped setion made from fiberglass with a 5 µm thin sapphire window at the tip attahes to the bottom of the bellows. The bellows setion inludes a mehanism to translate the one shaped setion relative to the ylindrial setion. A photograph of the lower portion of the assembled hamber is shown in Fig. 4.5(b). The SQUID must be brought as lose as possible to the sanned objet. The bellows makes this possible by allowing the thin sapphire window to be brought lose to the SQUID hip. The sanned objet on a translation stage is then brought as lose as possible to the sapphire window. I note that this overall design is based on the 77 K ryoooled high-t SQUID mirosope by Fleet et al. for imaging room temperature objets [0,7] Leak Problems Over time, the vauum inside the hamber was found to slowly deteriorate and the temperature of the old finger rose. Small leaks through the fiberglass one are thought to be the main soure as the fiberglass was found to be porous. If the base temperature of the old finger rose too muh, the main remedy was to shut down the ryoooler and allow it to warm up while ontinuously pumping the hamber. Cryoooler operation was resumed when the vauum level returned to desired levels. 03

124 (a) Radiation Heat Shield Vauum Chamber (b) SQUID Tip Position Adjustment Srews 0 m m Fiberglass Cone Metal Bellows FIG (a) Bottom portion of the ylindrial vauum hamber with the ryoooler assembly in plae. The bellows and fiberglass one setions are not installed. The SQUID tip an be seen stiking out of the radiation shield. (b) Bottom portion of the assembled vauum hamber. The thin sapphire window is loated at the tip of the one shaped fiberglass setion of the hamber. 04

125 A permanent fix would involve replaing the fiberglass one with a less porous nononduting and magnetially permeable material suh as erami. An alternate method would be to treat the fiberglass with a vauum seal spray as used by Lee on her high-t SQUID mirosope []. Vauum ontamination and thermal yling also degraded the performane of the SQUID. The degradation appeared to be related to eletrial ontat problems on or near the SQUID hip. Vauum ontaminants ould have seeped into the silver paint used in some eletrial onnetions, and with thermal ontrations and expansions, may have aused onnetions to fail over time. Thus, maintaining a good vauum with onstant temperature was important to the long term performane of the mirosope. 4.3 Cold Finger and Thermal Anhoring 4.3. Preparing the SQUID Tip and Cold Finger For a niobium SQUID to work, it must be ooled below T 9.3 K. However, to san room temperature samples, the SQUID must also be positioned near the sapphire window inside the narrow one setion of the vauum hamber. This requires attahing the SQUID to a old finger (see Fig. 4.6), whih is thermally anhored to the ryoooler s seond stage. Attahing the SQUID to the old finger needs onsiderable preparation. The SQUID is first attahed to the tip of a sapphire rod whih has good thermal onduting properties as well as being a nonmagneti insulator. The width of the tip end where the SQUID hip is mounted is about mm. This allows the SQUID to be 05

126 Sapphire Rod SQUID Chip Position Holder Copper Braid 9 m Wiring Connetor FIG Photograph of old finger attahed to the seond stage of the 4 K pulse tube ryoooler. The wiring is onfigured for FLL eletronis with a nonhystereti d SQUID. 06

127 positioned lose to the sapphire window without touhing any surfaes. The HYPRES SQUID hips, on the other hand, are on 5 mm 5 mm substrates, and a single z-squid with ontat pads and margins takes up about.5 mm.5 mm [7,8]. Thus, the first step is to die out single SQUIDs from a HYPRES hip that will fit on top of the sapphire tip. I died hips by using an automated hip diing mahine or by hand using a diamond sriber. The next step is to hemially lean the hip and epoxy it to the sapphire tip. Cleaning is performed in an ultrasoni bath for a few minutes with aetone followed by a methanol rinse. To perform any neessary degreasing, I did an additional leaning step using TCE at the beginning. To attah the died SQUID to the sapphire tip, I used a thin layer of a two part epoxy, STYCAST 850FT with atalyst 9 from Emerson and Cuming [9]. Although not ideal, the epoxy was the best epoxy readily available for use. I had tested other epoxies, but they either had poor hemial tolerane to solvents like aetone, or their bonds failed under applied fore and thermal stress from repeated thermal reyling. Preparation of the epoxy requires good mixing and an aurate ratio of the two parts, espeially when preparing small quantities. Small air bubbles form during the mixing proess, and these should be avoided to obtain a smooth surfae and void-free finish. The manufaturer reommends removing the bubbles by intermittently plaing the epoxy in a vauum hamber during mixing. During the proess, the epoxy outgases and expands, possibly ontaminating the hamber. Due to a lak of appropriate equipment, the evauation step was skipped. 07

128 For the bond between the SQUID hip and epoxy to reah full strength, the epoxy has to be suffiiently ured. With the layer of epoxy being very thin, the longest uring times were found to produe the strongest bonds. At room temperature, this meant uring the epoxy for 7 hours or more. Higher uring temperatures would shorten the uring time, and a post ure at an elevated temperature is reommended by the manufaturer. However, high temperatures an permanently hange the SQUID harateristis or ause damage. So, temperatures above 50 C were avoided. After uring, I ground the SQUID tip using a turn table grinding mahine to remove epoxy and exess parts of the hip around the SQUID. The top surfae of the SQUID tip must have some protetive layer to prevent damage to the SQUID while grinding. I found that a layer of hardened photoresist on the SQUID hip surfae was suffiient for protetion. I ground the SQUID tip to a roughly mm mm square and then hemially leaned it. The last step remaining in tip fabriation is to make eletrial onnetions, whih is disussed in detail later. One the sapphire rod with the SQUID tip is prepared, it goes into the old finger through a tightly fitting hole. In the prototype mirosope, silver paint and opper sheet shims were used to reate a good strong thermally onduting bond between the sapphire rod and old finger. Thermal ondutane is essential, so the old finger is made of opper. When inserting the sapphire rod into the old finger, the height of the tip needs to be adjusted. The mirosope s radiation shield is onially shaped near the tip and has a hole where the SQUID tip protrudes out as seen in Figs. 4.3 and 4.5(a). 08

129 The height of the SQUID tip determines how muh the tip protrudes outside the hole. If it is too high, the sides of the sapphire rod will touh the radiation shield. If it is too short, the sides of the radiation shield may touh the vauum hamber when the sapphire window is brought lose to the SQUID tip. Both situations should be avoided Motion Isolation For vibration isolation, the old finger omes in two parts. There is a top part whih holds the sapphire rod and a bottom part whih attahes to the ryoooler. The two parts are onneted to eah other with six strips of opper braid for flexibility as seen in Fig A flexible struture is required to provide rude mehanial vibration isolation of the SQUID tip from movement due to ryoooler operation. To hold the SQUID tip fixed inside the prototype mirosope, a fiberglass position holder whih fits over the old finger and onforms to the inside surfae of the radiation shield is used. The fiberglass holder is designed so that the SQUID tip is positioned at the enter of the radiation shield hole. The old finger itself is kept in ontat with the position holder by gravity. Despite the position holder, low frequeny osillations of the SQUID tip were observed in the prototype SQUID mirosope with an amplitude between 5 µm and 30 µm peak-to-peak. Suh osillations are undesirable and need to be suppressed as they ause large imaging noise and lead to aousti pikup as the SQUID moves in the ambient field. The osillations our due to the pulse tube s expansion and 09

130 ontration. This results in the seond stage heat exhanger deforming as the pressure inside the pulse tubes hange [30]. It is still not known whether the problem is only onfined to the seond stage heat exhanger or prevalent throughout the ryoooler. The remedy to the osillation problem will depend on the ause. If the whole ryoooler is affeted, a ompletely different SQUID tip holding sheme is required. If the problem is onfined to the seond stage heat exhanger, the old finger just needs to be rigidly seured to the radiation shield. Another motion problem was high frequeny vibrations that affeted the entire mirosope hassis. These vibrations were found to originate from the helium ompressor. The ompressor generated high frequeny vibrations whih were mehanially oupled not only via the ompressed helium hoses but also through the hard surfae of the floor. In fat, I first notied the signifiane of the high frequeny vibrations while observing the SQUID tip through the sapphire window with an optial mirosope plaed on the floor. The image did not seem to sharply fous and looked fuzzy due to the vibrations. Fortunately, the relative motion was less notieable between omponents on the mirosope table top. Therefore, signal distortions due to the high frequeny vibrations may be less signifiant than the low frequeny osillations. Some vibrational issues had been expeted. Kenyon et al. used a Gifford- MMahon ryoooler to ool a Single Eletron Transistor (SET) to 4 K [3]. There, the use of opper braid and fiberglass position holder to isolate mehanial motion was also found to be insuffiient. Better isolation was ahieved by using strips of 0

131 ultra high purity opper foil (99.999% pure). Suh strips have thermal ondutivity that is orders of magnitude higher than regular opper but are far more flexible than braid, thus dereasing mehanial oupling [3]. The inreased thermal ondutivity of the foil allows less material to be used and makes the onnetion more flexible. Multiple strips an then be added to inrease heat flow as needed. Unfortunately, I did not prepare and implement a opper foil old finger before testing the ryoooler. This was mainly beause the pulse tube ryoooler was expeted to have muh smaller vibrations than the Gifford-MMahon ryoooler. As it turned out, the prototype mirosope would still have signifiantly benefited from the opper foil design Heat Removal The result of heating and insuffiient heat removal by the mirosope old finger is a relatively small temperature differene between the old finger and the tip of the sapphire rod. Suh a temperature differene was observed with the prototype SQUID old finger. The differene was espeially notieable when the ryoooler was turned off and the SQUID allowed to warm up. I observed the I-V harateristis of the d SQUID while the temperature inreased and noted the temperature at whih the superondutor to resistor phase transition ourred. I found that the transition ourred when the temperature of the top part of the old finger was around 8.3 K. As the transition temperature of niobium is approximately 9.3 K, the temperature differene between the SQUID and old finger is approximately K.

132 If the assumption holds that the old finger is at the same temperature as the seond stage heat exhanger, the K temperature differene is likely due to insuffiient thermal ondutane of the sapphire rod. To diminish the temperature differene, I would have to replae the sapphire with a better thermally onduting eletrial insulator, provided I ould obtain suh material. On the other hand, the thermal oupling between the top and bottom parts of the old finger may be a limiting fator. More opper braid would be needed to inrease heat removal, but this would redue vibration isolation. Again, the solution may be to use strips of ultra high purity opper foil instead of braid Making Eletrial Contats to the SQUID The prototype SQUID mirosope used a d SQUID in the z-squid orientation. This orientation does not leave muh spae for eletrial onnetions to be made to the hip. For more spae, eletrial ontats must be extended to the sides of the sapphire rod (see Fig. 4.7). There were two methods I used to make these extensions. One involved a thin layer of silver paint that extended from the gold ontat pads on the surfae of the SQUID hip, over the hip edge and down the sides of the sapphire rod. Although relatively quik and onvenient, the eletrial onnetions made this way were not always reliable and tended to degrade with time due to inonsistent ontat between the paint and gold pads or due to the quality of the silver paint itself. An alternative method, whih is more permanent, involved evaporating a thin layer of gold onto the SQUID hip and sapphire rod, effetively extending the gold

133 Silver Paint or Evaporated Gold Connetion SQUID Chip Epoxy Layer Eletrial Wiring Sapphire Rod FIG Diagram of SQUID tip used in prototype SQUID mirosope. Four eletrial onnetions are present though only one is shown. The thikness of silver paint or evaporated gold is exaggerated to highlight the struture. 3

134 ontat pads to the sides of the rod. This method required masking areas on the surfae of the SQUID hip and on the sides of the sapphire rod, so that the SQUID and leads were proteted and short iruits between leads or ontat pads were prevented. Areas where gold was to be deposited were left exposed while everywhere else was overed by the mask. For the mask, I used photoresist whih I applied by hand following a proedure similar to that of Nielsen []. I lamped a short segment of small diameter wire to tweezers and dipped it in wet photoresist. Small photoresist droplets formed on the wire, whih I applied to the SQUID tip. One the appliation of photoresist was omplete, the SQUID tip was baked at 50 C for about 30 min to harden the resist. Before evaporating gold, I used an argon ion mill on the SQUID tip for about a minute with an aelerating voltage of 79 V to produe a lean surfae. I then evaporated a 5 nm to 0 nm layer of hromium whih helps the gold adhere to the substrate. Finally, I evaporated a 50 nm to 00 nm layer of gold at a rate of 0. nm/s to 0. nm/s onto the SQUID tip. I used an in-house ryopumped high vauum thermal evaporator for all of the evaporations and ion milling. After evaporation, I hemially leaned the SQUID tip to remove the photoresist mask and then examined the tip for defets. With the gold evaporation method, I always found a disontinuity problem. The problem was aused by the epoxy layer that held the SQUID hip to the sapphire rod. After the hip grinding proess, the epoxy surfae was no longer smooth and sometimes deteriorated with time, so that the evaporated gold would not form a 4

135 ontinuous surfae over the epoxy. I resolved the problem by applying a small amount of silver paint to bridge the gap. However, I also investigated another solution in whih I sputtered silion oxide on the epoxy surfae to form a thin insulating layer prior to the evaporation of hromium and gold. The results were mixed, as the insulating layer would only oasionally produe a surfae good enough for the gold to form a ontinuous onnetion. My limited suess may well have been due to problems with the sputtering equipment. In partiular, the vauum system for the ontainment hamber failed to reah reommended pressures before sputtering was initiated. This may have resulted in a poor silion oxide layer being formed. After ompleting the eletrial extensions, I visually hek them for shorts and disontinuities and then measure the resistanes between the leads. For a good SQUID tip, the resistane at room temperature between the d SQUID leads should be smaller than kω. Lower resistane an be expeted between the feedbak oil leads while muh higher resistane should exist between a d SQUID lead and a feedbak oil lead. In LHe, the d SQUID leads look shorted, as do the feedbak oil leads. The onnetion between the d SQUID leads and the feedbak oil should be an open at 4 K. When measuring SQUID hip resistanes at room temperature with a multimeter, the values were not very preise. This was beause I had to set the multimeter to its high resistane range, typially MΩ or greater. This limited the input urrent to less than 00 µa to prevent damage to the SQUID. At these settings, low resistanes all looked like shorts. What was important was that when a high 5

136 resistane was expeted, it would be orders of magnitude higher and therefore easily disernible. I finished the SQUID tip wiring by attahing small diameter opper magnet wire to the eletrial extensions on the sides of the sapphire rod using silver paint (see Fig. 4.7). The opper wires were formed into twisted pairs and soldered to a multipin onnetor attahed to the bottom of the old finger (see Fig. 4.6). This sheme was adequate for the prototype mirosope using a nonhystereti d SQUID with Flux- Loked-Loop (FLL) eletronis. However, hanges would be required for the large bandwidth eletronis desribed in Chapter 6, whih makes use of a hystereti d SQUID. 4.4 Sample Sanning Mehanism 4.4. Translation System Overview The purpose of a Sanning SQUID mirosope is to measure spatial variations in magneti field on the surfae of an objet. In order to aomplish this, a mehanism is required to support and translate the objet with respet to the SQUID. The translation system of the prototype mirosope an be divided into three main omponents. First, there is the human interfae and data proessing unit, whih is an IBM PC ompatible omputer. Seond, ontroller boxes read the position data of the translation stages and onvert instrutions from the omputer into eletrial signals for ontrolling atuators. And third, there are preision x-y translation stages and atuators that support and move the sample. As the stages move the sample, the magneti field deteted by the SQUID is reorded along with the position of the x-y 6

137 stages. This data is then used to reate a magneti field image of the objet. A blok diagram of the sanning system and eletronis used in the prototype SQUID mirosope is shown in Fig Data is taken one line at a time, i.e. as the sample is sanned along one axis whih I will define as the x-axis. Measurements of the magneti field are taken at predetermined time intervals aording to the san speed. The san position is determined by the orresponding time interval. Averaging of measurements is performed by taking multiple suessive measurements, then averaging those values before taking measurements for the next pixel. After a line is sanned, the sample is moved one step along the y-axis. The mirosope is then ready to perform another line san and the proess is repeated until a speified area is overed. If a single step along the y-axis is larger than the SQUID size, some areas between steps in the y diretion will not be sanned. If the harateristi length sale of variations in magneti field is smaller, then the step size should be made smaller to avoid missing any field information. However, for direted measurement, step sizes smaller than or omparable to the SQUID size do not render additional information. I note that in the prototype mirosope, as well as in most other sanning SQUID mirosopes, the sample is moved about the SQUID instead of the SQUID moved over the sample. There are two main reasons for this. First, the SQUID is attahed to a heavy and bulky ryoooler inside a vauum hamber. The mehanial link between the SQUID and ryoooler does not permit signifiant movement of the SQUID without moving the ryoooler and vauum hamber. Preision translation of the ryoooler an be avoided by translating the sample instead. Seond, the SQUID 7

138 Bandpass Filter Lok-in Amplifier DC Data Aquisition Controller AC X imc-303 SQUID Controller IBM PC (ompatible) A to D Converter ESP6000 Motion Controller ifl-30 FLL Eletronis ESP6000 Driver d SQUID (nonhystereti) X-Y Translation Stage & Atuators FIG Blok diagram of the prototype Sanning SQUID Mirosope eletronis and sanning system. Arrows indiate diretion of data flow on oaxial able for onnetions with unidiretional ommuniation. The onnetion between the imc-303 SQUID ontroller and the ifl-30 FLL SQUID eletronis inludes fiber opti able as well as oax. The onnetion between the Data Aquisition Controller and the Analog-to-Digital onverter onsists of a dediated multiwire signal able with bidiretional ommuniation. A similar onnetion exists between the ESP6000 motion ontroller and the driver.

139 is sensitive to minute hanges in ambient field. If the SQUID moves, it beomes diffiult to differentiate whether measured hanges in field are due to hanges aused by the sample or due to spatial variations in the ambient field. By keeping the position of the SQUID fixed, the only hanges in field should be due to the translation of the sample or due to temporal variations in the field if the objet position is fixed Hardware The translation stages, atuators, motion ontroller, and related omputer interfae ards in the prototype SQUID mirosope were ommerial produts whih were assembled together by Neoera, In. Additional eletronis were also developed by Neoera for the translation system. The translation stages must perform three dimensional positioning of the sanned objet. Sub-miron preision is desired for translations in the horizontal plane. While typial samples are on the order of a few entimeters, the field variations are on the miron sale, for example integrated iruit hips and modules. Preise translation in the vertial diretion is needed to position the sample as lose as possible to the sapphire window of the vauum hamber without rashing into it. Some ommerial SQUID mirosopes an perform vertial translation eletronially and use a feedbak system to keep the objet at a fixed distane from the window [7]. However, the prototype mirosope uses a manual z-stage with µm preision mounted on the x-y stages. The sanning system inorporated two model 850G atuators with linear horizontal translation stages, all made by Newport, In. [33]. One stage was staked 9

140 and bolted on top of the other at a right angle, thus forming the two dimensional x-y horizontal stage. Care was taken to set the angle to 90 as if it were not, the resulting field image would be skewed. The d motor atuators had a minimum step inrement of about µm with an absolute auray of 50 µm over the full range of motion. To obtain faster sanning times, the atuator used for the x diretion was a high speed type 850G-HS. The main motion ontroller and omputer interfae ard was a model ESP6000, also from Newport, In. [34,35]. In onjuntion with a motor driver box, the system was apable of ontrolling up to four atuators, though I only used two for x-y motion. In order to optimize sanning, the driver box was modified to output a dediated signal indiating the position of the translation stages along the x diretion. This signal was sent to the data aquisition ontroller along with signals from the SQUID eletronis. From there, the signals were relayed to the ontrol system omputer where the data was orrelated and proessed into the field image. Translation in both x and y diretions has to be onsistent and reproduible. The model 850G atuators and ESP6000 motion ontroller have some built in features whih support these requirements. However, relying on suh features alone an be insuffiient []. There are some remedies to address position error, whih are disussed in Chapter 8. In general, problems with the translation system an be due to software that ontrol the atuators and other omponents. In other ases, the hardware may ause error signals and halt operation. Without detailed analysis, it is diffiult to determine the ause of suh problems. Indeed, I enountered a very 0

141 serious problem in the translation system of the prototype mirosope, whih is disussed in Chapter Software and Sanning Parameters The software I used for the sanning system was a version of the Magma C ontrol software developed by Neoera speifially for the Newport 850G atuators and ESP6000DCIB motor driver. The software ran under Mirosoft Windows in an IBM PC ompatible omputer equipped with the ESP6000 motion ontroller and an analog-to-digital data aquisition ard for reeiving SQUID data. Besides positioning and translating the sanned objet, the software also reorded, graphed, and normalized the SQUID data as it was being reeived. Separate software utilities were available to analyze magneti field images and perform alibration and diagnostis of the translation hardware. To operate the sanning software, I provide values for ertain parameters inluding san speed S, san area A, number of measurements averaged N, and grid size X Y. Not all parameters are independent, so those parameters of more signifiane determine the others. For example, san speed S is determined by a number of fators, one being the size of the sanned area A and the other the time Τ needed to aquire the image. However, the sanned area is divided into separate pixels aording to a grid size. The grid size determines how many pixels will be stored for the field image. Very often, there is a desired grid size for a given sanned area, so this also partly determines the sanning speed.

142 The sanning software was designed so as to let the operator set ertain parameters, suh as the sanning area and grid size. This determined the spatial resolution R and the x and y step sizes. The number of measurements to average N ould also be speified. But, the san speed S was predetermined from stored profile information set during alibration. Consequently, the san time Τ was mostly determined by the grid size and the measurement time per pixel, where the measurement time per pixel was dependent on the number of measurements to average. The approximate relationship between some of the parameters an be summarized by the following equations. R = S = Τ = L X L Τ X N Τ o (4.) where L is the san length in the x diretion and Τ o is the time for one measurement. The atual time Τ o needed for a single measurement was something that was mostly determined by internal fators and was not readily adjustable. In priniple, the time needed for a single measurement must be long enough to ompensate for any transitory effets in the eletronis. Averaging would then redue the effet of external noise that is present during the time Τ o. Ultimately, the san speed was limited by the hardware. If the speed was too high, the d motors drew a high level of urrent, whih ould damage the motor. Also, if the speed were too high or too low, the positioning would not be reliable. Typially, the speed had to be onstant over a relatively large distane in order for the position to be aurate. Aeleration also had to be ontrolled and well determined

143 for measurements near the starting and ending points. The aeleration profile used in the sanning system for the prototype mirosope was a simple onstant aeleration profile. One of the utilities inluded with the sanning software had the apability to hange and speify the aeleration profile and sanning speed for the atuators. Making hanges using the utility permanently hanged the default settings and ould ause the atuators to reat differently to signals and ignore predetermined limits. Using the utility, I experimented a little in attempts to optimize and determine the ause of some sanning problems. However, adjusting atuator performane was very involved. Some adjustments resulted in dramati hanges while others seemed to have no effet. Consequently, the default settings were kept. To redue san time, it was possible to perform line sans in both diretions along the x-axis, one in the positive x diretion and the next line in the negative x diretion. However, I only performed unidiretional line sanning to avoid alignment problems due to hysteresis in position or speed. For example, there ould be lag in the atuators when the sanning diretion is reversed. For aurate alignment, the sanning software returned the x-y stages to a referene point along the x-axis after eah line san. It also performed a similar ation during system initialization. 4.5 SQUID Eletronis and Instrumentation 4.5. Flux-Loked-Loop Feedbak Eletronis The prototype Sanning SQUID Mirosope uses modulated Flux-Loked- Loop (FLL) eletronis, an imag SQUID system from TRISTAN Tehnologies, In. 3

144 [9]. The system omes in two parts, the ifl-30 FLL eletronis whih diretly onnets to a d SQUID and the imc-303 multihannel ontroller. The ontroller inorporates the user interfae, reads and displays the output from the FLL eletronis, and adjusts parameters suh as the bias urrent, modulation urrent amplitude, and amplifier gain. The FLL eletronis has a 50 khz osillator and a bandwidth of 0 khz under normal operation. The TRISTAN eletronis has fiber opti and oaxial able onnetions between the two separate units. It also features fully automated setup and diagnostis. This is very different from the Berkeley Box FLL eletronis used in earlier SQUID systems suh as those of Blak, Nielsen, and others [36]. The main differenes are that the Berkeley boxes were entirely based on analog eletronis and required manual adjustments with dials and swithes to ontrol various parameters. Nevertheless, despite the differenes in physial implementation, the working priniple is the same in both eletroni systems. The FLL requires a nonhystereti d SQUID whih is urrent biased with a stati d urrent. When the bias urrent level is optimum, the voltage aross the SQUID will be periodi in the magneti flux through the SQUID hole with maximum amplitude [4,5,8-0]. Ideally, the bias urrent I bias I, i.e. I bias should be near the transition between the zero voltage state and the resistive state (see Fig. 3.4). However, this was not always the ase, as for example with SQUID AN whih I used in the prototype mirosope. For SQUID AN, I had to adjust the bias urrent until the maximum signal response, desribed next, was obtained. 4

145 In normal operation, the FLL eletronis supplies an a flux around Φ o /4 to an optimally biased d SQUID at the osillator frequeny f = 50 khz. If there is no external quasi stati flux through the SQUID hole, then the a voltage aross the SQUID will only have omponents with frequenies that are a multiple of f. This omes from the symmetri nonlinear relationship between the SQUID voltage and the external magneti flux. On the other hand, if there is a small externally applied quasi stati flux, then the a voltage aross the SQUID will have a omponent at frequeny f (see Fig. 4.9). The detetion of this signal is the key to the modulated FLL tehnique. To isolate the a SQUID voltage at frequeny f, the voltage from the SQUID is amplified and fed to a phase detetor. The phase detetor mixes the voltage signal with a referene signal from the FLL osillator and produes a new signal that has a pseudo d omponent whih is proportional to the amplitude of the input omponent at frequeny f. Whether the external magneti flux inreased or dereased the total flux through the SQUID an be determined from the relative phase between the osillator and the voltage signals. If the a omponent of the voltage signal at f is in phase with the osillator, the external flux inreased the total flux. If the a omponent is out of phase by 80, the external flux dereased the total flux (see Fig. 4.9). However, it should be noted that there will be phase shifts between the SQUID voltage and the signal deteted by the FLL eletronis. So, instead of looking for the 0 and 80 phase shift in the a omponent, in pratie the FLL eletronis adjusts a phase shift setting that maximizes the signal response at frequeny f during alibration or initialization. 5

146 (a) Φ mod Φ o 4 0 T T t (b) Φ o 4 V 0 < Φ < A Φ 4 o () V 0 T Φ o < Φ < 4 A 0 T t (d) V 0 T Φ A = 0 T t 0 T FIG Response of a nonhystereti d SQUID with optimally set onstant bias. A modulating flux Φ mod ±Φ o /4 is applied on top of a quasi stati external magneti flux Φ A. (a) AC modulation flux of period T versus time. Voltage aross the d SQUID for (b) Φ A > 0, () Φ A < 0, and (d) Φ A = 0. In graphs (b) and (), the solid line is the full voltage and the dotted lines represent the first and seond harmoni Fourier omponents of the voltage. Note the relations between the first harmoni omponents and graph (a). T t 6

147 The output of the phase detetor is then fed bak to the SQUID loop so as to anel out the externally applied flux. When the system is loked, the value of this negative feedbak urrent is a diret measure of the magneti flux applied to the SQUID. The main output of the FLL eletronis is the voltage produed when the feedbak urrent flows through a feedbak resistor R f. However, due to the periodiity of the SQUID voltage versus flux relation, the FLL eletronis does not neessarily lok on to the zero flux state. In fat, it an lok to any state that is an integer multiple of Φ o. As a result, only relative flux hanges are refleted in the output. I note that despite the intrinsi nonlinear and periodi response of the nonhystereti d SQUID to magneti flux, the use of negative feedbak linearizes the response of the eletronis and allows it to follow magneti flux hanges that are greater than Φ o. Figure 4.0 shows a blok diagram of the FLL. In addition to the SQUID, amplifier stage, phase detetor, osillator, and resistor R f, an integrator and a transformer iruit are inluded in the eletronis. The integrator ats as a stabilizing element in the negative feedbak iruit. Due to finite signal propagation speeds through the eletronis, at some frequeny a 80 phase lag develops between the measurement of the signal and the reation to it. The stability riterion states that for the feedbak loop to be stable, the open loop gain of the amplifying iruit must fall faster than /ω at the frequeny where the phase lag is 80 [37]. The integrator ensures that the high frequeny roll off of the amplifier s open loop gain satisfies this riterion. 7

148 Cold Rx Amplifier Stage Phase Detetor Imosωt Rf Integrator Vout FIG Blok diagram of modulated Flux-Loked-Loop (FLL) SQUID eletronis. Elements within the dotted box are at ryogeni temperatures. The modulating urrent from the osillator and the feedbak urrent from the integrator are added to produe the total urrent through the feedbak oil. The magneti flux produed by the feedbak oil is direted through the hole of the d SQUID.

149 4.5. Mathing Transformer Ciruit The purpose of the transformer iruit at the SQUID s output was disussed in setion 3.. I made transformers by wrapping small diameter opper magnet wire wrapped around small toroidal ores. The primary stages had only a few turns while the seondary stages had between 60 and 00 turns. Constrution of a transformer was very time onsuming, and its performane depended on the quality of the wrapped oils as well as on the number of turns. Ideally, the impedane of the transformer primary stage should be greater than the dynami resistane of the nonhystereti d SQUID. In pratie, it is a matter of trial and error to math the impedane of the primary oil to the normal resistane of the SQUID, espeially when the oils are prepared before the SQUID harateristis are known. One ompleted, I tested the transformers using indutane meters and by measuring the frequeny response with a signal generator and osillosope. Measurements using a meter gave signifiantly different values, depending on the speifi instrument used. Consequently, I tended to trust my diret measurements more (see Fig. 4.). Figure 4. shows the voltage amplitude versus frequeny for the primary stages of three different transformers. Dividing the voltage amplitude by the input urrent amplitude provides the impedanes of the primary stages, whih at 00 khz range from.8 Ω to 9.0 Ω. This orresponds to.9 µh and 4 µh in terms of oil indutane. I note that the indutanes do not follow the expeted dependene on the number of turns. In fat, the oils are better modeled by an indutor and apaitor in parallel, where the apaitanes for the 3 and 5 turn oils are on the order of 0. µf 9

150 Voltage (µv) turncoil 3 turn Coil 0 6 turn Coil Frequeny (khz) FIG. 4.. Voltage amplitude versus frequeny of the primary stages of three different SQUID output transformer oils. The voltages aross the primary stages were measured while sending a sinusoidal urrent through them. The output voltage amplitudes are normalized to an input urrent amplitude of 5 µa. The transformer oils were made with small diameter (38 gauge) opper magnet wire with 3, 6, and 5 turns on the primary. The respetive impedanes of the primary stage oils at 00 khz are approximately 5.8 Ω (3 turn oil),.8 Ω (6 turn oil), and 9.0 Ω (5 turn oil). The solid lines represent the best urve fits to an indutor in parallel with a apaitor.

151 and negligible for the 6 turn oil. The 6 turn oil, whih has the smallest indutane, has oils that were not as tightly wound. Furthermore, the transformer was enased in epoxy to hold its form and provide better thermal ondutane. However, the oil had poor noise harateristis ompared with the other oils, so I deided not to enase oils in epoxy again. To enable the d SQUID to be biased with suffiient urrent, I plaed a resistor R x in series with the transformer primary to limit the flow of urrent through the transformer (see Fig. 4.0). As I noted in setion 3., R x should be muh less than the impedane of the primary stage in order for most of the SQUID voltage to appear aross the transformer, whih is satisfied if R x Ω. For the prototype mirosope, I made R x from a m long setion of manganin wire whose alibrated resistivity was 5 Ω/ft. This resulted in R x 0.5 Ω. Unfortunately, the impedanes of all of the transformer oil primary stages were less than the dynami resistane of SQUID AN (see Fig. 3.6). This resulted in a relatively weak signal being deteted by the FLL eletronis and lead to some problems during mirosope operation. These problems are disussed in Chapter Signal Proessing and the Integrated Computer System Most of the instrumentation and ontrol systems for the 4 K prototype SQUID mirosope are installed in a rak next to the ryoooler and vauum hamber hassis (see Fig. 4.). The mirosope s ontrol system an be separated into either integrated or stand alone omponents. The integrated omponents are the translation system, the SQUID data aquisition system, and the magneti field image proessor 3

152 running in an IBM PC ompatible omputer. The stand alone omponents are the vauum system, the ryoooler, the SQUID ontroller, and the temperature monitors. The d SQUID and FLL eletronis were operated by the imc-303 SQUID ontroller whih interfaes to the omputer system through the data aquisition ontroller (see Fig. 4.8). The SQUID ontroller was not integrated into the omputer system but was instead operated manually. Nevertheless, it provided the omputer with ontinuous magneti field data while providing power and instrutions to the ifl-30 FLL eletronis. The SQUID ontroller s output signal was split into a and d signals before it reahed the omputer. The d signal was the ontroller output that went diretly to the data aquisition ontroller. On the other hand, the a signal first went through a bandpass filter to redue low and high frequeny noise and then through a lok-in amplifier where the signal at a partiular frequeny was deteted and sent to the data aquisition ontroller. For eletri iruit diagnosti appliations, the a signal information was more useful as the input urrent to the iruit ould be set to a desired frequeny that was distint from bakground noise Temperature and Pressure Monitoring The SQUID mirosope required a number of diagnosti heks to ensure proper operation. The temperatures of the ryoooler and old finger were monitored to determine whether the ryoooler was operating orretly and whether the SQUID hip had reahed its operating temperature. Furthermore, hanges in temperature affeted the harateristis of the SQUID, suh as the maximum ritial urrent. 3

153 Also, exessive temperature drift was an indiation of aumulating ontamination in the vauum hamber. There were two temperature sensors in the SQUID mirosope. The number of sensors was limited by the wiring. One was plaed inside the top part of the old finger whih holds the sapphire rod and SQUID tip. The other was on the radiation shield whih was thermally anhored to the ryoooler s first stage heat exhanger. Spae was not available for a sensor right next to the SQUID hip. The resistane temperature detetor (RTD) thermometers used in the mirosope have resistanes that hange as a funtion of temperature. Four point resistane measurements were made, requiring two sets of twisted pair 37 gauge opper magnet wire for eah sensor. These wires broke easily, requiring are in design and handling. The wires were also thermally anhored to the ryoooler heat exhangers to prevent heating of the sensors from room temperature due to thermal ondutane in the leads. The ryoooler manufaturer, Cryomeh, reommends in to 4 in of lead be thermally anhored [7]. Thermally onduting vauum grease ould have been used to eliminate gaps within the anhoring points for the wires, but epoxy was used instead making the attahments semi-permanent. The RTD sensors themselves were also anhored to surfaes with either epoxy or silver paint. Temperature measurements were made using a LTC- low temperature ontroller from Neoera [38]. Different RTDs were used for different temperature ranges. For temperatures between 0 K and 00 K, suh as for the mirosope s radiation shield, a platinum RTD was used. For temperatures below 0 K like the old finger, a ruthenium oxide RTD was used. 33

154 The pressure monitor was an integral part of the ontrol system for the turbo vauum pump. A Balzers-Pfeiffer TPR 50 ompat pirani pressure gauge was used to measure the hamber pressure and determine whether the vauum level was low enough for the ryoooler to operate. The gauge was loated at the top of the vauum hamber and an be seen next to the ryoooler old head in Fig. 4.. The operating range of the gauge was limited to values above mbar. As a result, if the vauum was starting to fail, the first indiation was not an inrease in the pressure gauge reading but an inrease in base temperature due to inreased thermal ondutane or onvetion within the hamber. The pump ontroller operated ompletely independently from other systems and inorporated its own internal safety mehanisms to shutdown the turbo pump in ase of vauum failure. If the pressure rose above a set limit, the turbo fans ould not spin at the required frequeny and the pump would shut itself down. An automated vauum valve was available to isolate the hamber from the turbo pump when suh situations ourred, though it was often disabled due to ontrol problems. When a shutdown ourred, the turbo pump had to be manually restarted after a rough vauum was reovered. 4.6 Operation and Maintenane 4.6. Cool Down Proedure To start the prototype SQUID mirosope, I used the following proedure. After assembling the mirosope, I first used the roughing pump to redue the pressure inside the vauum hamber. During normal operation, the roughing pump 34

155 was kept onstantly running, either roughing out the hamber or baking the turbo pump. After a few hours of pumping, the vauum level bottomed out and the turbo pump ould be engaged. The amount of time required to pump out the hamber depended on whether the hamber was left open for an extended period of time and the environmental onditions the hamber was exposed to. When the turbo pump was engaged, a fan fored air around the pump to keep it from over heating due to the amount of urrent that flowed through the turbo pump. In priniple, the system should be heked for vauum leaks at this point, i.e. before starting the ryoooler, but a leak detetor was not available on site. Instead, the only indiators of leaks I had was when the pressure gauge would not drop to prior levels during roughing and when the turbo pump fans were unable to reah their minimum rotation frequeny. The steady state operating frequeny of the pump was 500 Hz, and a safety mehanism isolated the pump from the vauum hamber when the frequeny dropped below 00 Hz. On the other hand, the turbo pump was able to operate at lower frequenies, or start at higher pressures. However, this risked the possibility of exessive heating and damage to the pump if the pressure did not drop. After the hamber pressure dereased to the maximum allowed level for ryoooler operation, the temperature ontroller was turned on, and preparations were made to start the ryoooler. Although the reommended maximum pressure for operation is torr or mbar, on oasion the ryoooler was started with the pressure as high as mbar. Under optimal onditions, the pressure would reah as low as mbar prior to starting the ryoooler. 35

156 Before starting the ryoooler, I heked the eletrial lines, high pressure gas hoses and water ooling lines to the helium ompressor. It was also prudent to ensure that the vauum hamber s thin sapphire window was loated away from the SQUID tip. I then turned on the old water supply to the helium ompressor. The temperature and flow rate must meet the requirements speified by Cryomeh. However, a simple hek was performed; if the temperature of the outflowing water seemed too warm to the touh after about half an hour of ryoooler operation, the flow rate was inreased and the temperature was tested again after several minutes. After starting the water supply, I swithed on the helium ompressor. The ryoooler began to operate in onjuntion with the ompressor whih makes a distintive sound. At first, the temperature readings flutuated but then stabilized and began to steadily derease. I also found that the initial temperature readings of the heat shield and old finger were unphysial; the temperature values beyond the alibrated ranges of the RTDs should not be trusted. Furthermore, with the ryoooler started, the pressure inside the vauum hamber drops suddenly as gas partiles inside the hamber begin to freeze out, partiularly the residual water vapor. The ooling time is typially between one and two hours. In that time, the ryoooler first stage and radiation shield ool below 70 K and gradually bottom out to a limiting temperature of around 3 K. The old finger should reah its minimum temperature of around 3.9 K after about two hours of ryoooler operation. However, I found that this temperature tended to slowly drift upwards, probably due to a slow vauum leak disussed earlier. 36

157 On some oasions, the temperature of the old finger would not reah 4 K. Physial ontat between the SQUID tip and radiation shield or some other omponent at a higher temperature was the likely ause. In suh ases, I suspeted large temperature differenes between the SQUID hip and old finger. To orret the problem, I had to shutdown the ryoooler, warm up the entire system, open the vauum hamber, and adjust the height of the sapphire rod in the old finger Preparations and Proedures for Sanning One the old finger reahes 4 K, the sapphire window is brought lose to the SQUID tip prior to sanning. This tends to inrease the temperature of the SQUID hip by less than K, whih is not very signifiant. To bring the sapphire window lose to the SQUID, I use an optial mirosope and a right angle prism on the translation stage to look through the window and guide it to the SQUID tip [see Fig. 4.(a)]. Positioning srews on the bellows allow fine adjustments of the window position. The idea is to bring the window as lose as possible to the SQUID tip without touhing it. I also used a small fiber opti light soure plaed against the fiberglass one setion to illuminate the SQUID tip inside the vauum hamber. Next, I onneted the able from the SQUID eletronis to the d SQUID before turning on the SQUID ontroller. Caution should be taken to disharge any stati eletriity before onneting the able diretly to the SQUID leads. One the eletronis were onneted and turned on, the imag-303 SQUID ontroller went through a series of diagnosti heks then set the SQUID eletronis parameters and began measuring the ambient field [9]. 37

158 (a) (b) FIG. 4.. (a) Bringing the SQUID mirosope sapphire window lose to the SQUID tip. The operator observes the window through an optial mirosope and prism while positioning the window. (b) Positioning an objet lose to the sapphire window prior to sanning it. 38

159 Adjustments to the automati ontroller settings an be made. The main parameters to adjust are the bias urrent and modulation amplitude. However, I found it diffiult to determine whether modifying the settings atually resulted in better performane. Ideally, the values on the ontroller indiator should be away from the low and high extremes. Whether the parameters are adequate is ultimately determined by operating the mirosope, for example by disturbing the ambient magneti field and observing whether the ontroller reversibly follows the hange. A small magnet or any magneti objet an be used for a quik test. If operating orretly, the SQUID eletronis should hold lok and follow hanges in magneti field. Sudden large hanges ould ause the output to jump lok. Otherwise, the eletronis should be stable and its output signal onstant when the magneti field is onstant. If the output frequently jumped, a phenomenon referred to as losing lok, or if the eletronis did not faithfully follow hanges in magneti field, the SQUID bias and modulation parameters needed adjustment. With the imag-303 SQUID ontroller, the main sign of lost lok was that the output indiator would be at maximum and not follow field hanges. If parameter adjustments did not result in better performane, the SQUID or its onnetions was likely defetive. After starting the SQUID ontroller and heking the SQUID s response, I started the omputer sanning system s hardware and software. To perform sanning, I mounted a sample on the translation stage and brought it lose to the sapphire window. There should be enough spae between the sample and the window over the full x-y sanning range, and the horizontal plane of the sample should be parallel with 39

160 the plane of translation. If not, the distane between the sample and the SQUID will gradually hange giving rise to an artifiial gradient in the measured flux. Furthermore, if the distane is too small, the sample will rash into the window quite possibly ausing damage to both window and sample. To hek the spaing, I used an optial mirosope to look at the gap while moving the translation stages [see Fig. 4.(b)]. Any obstrutions on the surfae of the sample or near the translation stages should be removed. Finally, I enter sanning parameters into the sanning software and the translation stages move the sample to the initial position. Before initiating the san, I hek the SQUID ontroller and verify that the SQUID and its eletronis are operating properly. I also note the SQUID old finger temperature for omparison after sanning. A signifiant temperature drift an hange SQUID harateristis and result in distortion of the magneti field image. While sanning, the field image is displayed and updated in real time on the omputer. If there is any problem with the operation of the eletronis, suh as losing lok, the field image will typially show only low level noise or no features at all. If the SQUID eletronis jumps or loses lok during the middle of a san, there will be a sudden hange in the output, whih persists for the remainder of the san. Image distortion of this kind annot be orreted, so the san must be repeated after the SQUID eletronis is reset. One the san is omplete, I save the data for proessing at a later time. Two separate images, one from the a signal line and the other from the d signal line, are available. 40

161 4.6.3 Serviing the Mirosope To servie the SQUID mirosope, the ryoooler has to be shutdown and the system warmed to room temperature. As a preaution, the sapphire window should first be moved away from the SQUID tip. Warming up to room temperature takes several hours. Monitors an be left on to observe the hange in temperature and pressure. At the same time, vauum pumps should be kept operating to prevent ondensation from forming inside and outside the hamber. Cooling water for the ryoooler ompressor should be left running until the temperature of the outflowing water mathes the inoming water temperature. For safety, the ompressor hoses must be left onneted to the ryoooler, as reommended by Cryomeh, to depressurize helium gas aumulated in the old head [7]. The time to warm up the ryoooler would be less if the vauum hamber were vented or filled with inert gas, but ondensation would also our on the outside of the vauum hamber and possibly damage omponents suh as the sapphire window, pressure gauges, or room temperature eletronis. After the ryoooler reahes room temperature, vauum pumps and other systems an be turned off and eletronis and sensors disonneted from the vauum hamber. Venting the hamber an be performed by slowly removing the pirani pressure gauge. After that, the hamber is unbolted and opened to expose the SQUID tip and radiation shield. In order to aess the old finger and heat exhangers inside the vauum hamber, the radiation shield has to be removed. This in turn requires removing the ryoooler and old finger assembly from the vauum hamber. This provides full aess. Alternatively, part of the shield an be removed while the ryoooler is in 4

162 plae, though this only provides partial aess. In general, with the urrent design, I reommend omplete removal of the ryoooler assembly for any signifiant serviing work. When opening up the vauum hamber, disassembly of the mirosope goes from bottom to top, beginning with the fiberglass nose one setion that has the sapphire window. Reassembly is in the reverse order. This is done as a preaution to avoid damaging the SQUID tip, sapphire window, and radiation shield during ryoooler removal and insertion. It is best to have at least two people working together to remove the ryoooler. The basi old head alone weighs 4 kg and has to be lifted straight up to avoid hitting the hamber walls and onsequently damaging omponents. The most ommon damage to the mirosope during serviing was breaking wire onnetions. Broken wire was a major nuisane requiring frequent maintenane. Some of the wire had more durable insulating oating to help prevent breakage, but it was only available later and not installed everywhere. To help seure the wires and their onnetors, I attahed them to the ryoooler with vauum ompliant adhesive tape. In any ase, the risk of damage was always present and was greatest during ryoooler removal and insertion. Therefore, removal of the ryoooler should be limited and performed as infrequently as possible until the system is redesigned to allow easier aess. 4

163 CHAPTER 5 Magneti Field Image of a Test Ciruit 5. Mirosope and Test Ciruit Preparations To verify the orret operation of the prototype 4 K Sanning SQUID Mirosope, I performed a san of an eletrial test iruit. The iruit was developed by Neoera for alibrating and testing their ommerial sanning SQUID mirosopes. The test iruit onsisted of ondutive leads made of gold on a flat iruit board with preisely determined features, inluding lines with different widths, parallel lines with different spaing, and meshed or webbed leads [see Fig. 5.(a)]. Between two terminals on the iruit board, more urrent flows through the lesser resistive paths with the larger urrents ontributing stronger magneti field omponents over the iruit. The paths of the urrent between the two terminals is refleted in the magneti field image measured by the SQUID mirosope and an be ompared with the expeted result. To drive a urrent through the test iruit, I onneted it to a funtion generator through two terminals on the iruit board. The test iruit ontained no urrent limiting resistors. To adjust the urrent level, I plaed a simple resistor divider network between the funtion generator and test iruit [see Fig. 5.(a)]. Using the resistor network, the funtion generator ould supply up to 00 ma to the iruit, though muh lower urrent levels were used during testing. There was another resistor network I had prepared that provided more linearly varying urrent 43

164 (a) 5 m (b) 4.8 mm.6 mm FIG. 5.. (a) Test iruit developed by Neoera, In. for testing sanning SQUID mirosopes. An a urrent of.6 ma rms at 6 khz was supplied between the two terminals indiated by arrows. (b) Magneti field image of area indiated by dashed box in (a) obtained using prototype SQUID mirosope. The image is 0 69 pixels and shows the normal omponent of the a magneti field (B z ) approximately mm above the board. Relative field values are represented by olor with extreme values indiated by saturated red and saturated blue. The average value is indiated by white. The FLL eletronis lost lok near the top of the image. 44

165 (a) V in 00 Ω Monitor V out kω kω 00 Ω (b) V in 300 Ω Monitor kω 300 Ω 0.03 kω V out FIG. 5.. Ciruit diagrams of resistor networks used for limiting the urrent from a funtion generator. (a) Resistor network that was used with the Neoera test iruit when the prototype SQUID mirosope was tested. (b) Resistor network that was used mostly for d SQUID I-V measurements. 45

166 with hanges in a potentiometer [see Fig. 5.(b)]. It was mostly used for I-V measurements of nonhystereti d SQUIDs. I set the funtion generator to send an a urrent of.6 ma rms to the test iruit at 6 khz. The frequeny was hosen so that the signal did not overlap with any interfering soures within the bandwidth of the FLL SQUID eletronis. I used a SR830 DSP lok-in amplifier from Stanford Researh Systems to detet the 6 khz signal from the SQUID ontroller output [39]. The output of the lok-in amplifier was reorded to reate a magneti field image of the sanned area. The lok-in amplifier took into aount the phase of the a signal so that the field image represented an instantaneous piture of the field. When bringing the test iruit lose to the mirosope s sapphire window, I left a gap suh that the distane between the SQUID and test iruit was approximately mm, though I ould have brought the test iruit loser with more preise adjustment of the translation stages. Furthermore, I did not employ any filtering or shielding other than what was already part of the mirosope s eletronis. Consequently, the magneti field image did not show as muh detail as was possible, and the field measurements inluding the generated field itself were subjet to noise. However, these were not major issues for the test san, as the purpose of the san was to determine whether the prototype SQUID mirosope was operating orretly. 5. Obtained Magneti Field Image The area of the Neoera test iruit that I sanned is indiated in Fig. 5.(a). In that area, there are two terminal lines that are bridged by other lines with different 46

167 line widths. These bridges are spaed about.5 mm apart and have widths that derease by approximately one half the previous width as one goes up in Fig. 5.(a). Basi resistor network analysis shows that about half of the urrent from one terminal line flows through the onneting bridge that has the largest width, and about half of the remaining urrent flows through the onneting lead with the next largest width, et. Therefore, the vast majority of the urrent flows through the lowest two bridges. The grid size of the.6 mm by 4.8 mm sanned area was 0 69 pixels. This resulted in step sizes that were 80 µm in the x diretion and 359 µm in the y diretion. With a z-squid installed in the prototype mirosope, the magneti field omponent normal to the iruit board (B z ) was measured by the SQUID. Eah flux value in the field image was the average of 5 measurements, and a suessful san took about 80 minutes. Due to problems with the translation system, disussed in the next setion, I only obtained a few suessful magneti field images. Out of many attempts, Fig. 5.(b) shows an example of a omplete field image. In the image, the output from the SQUID ontroller is olor oded with the maximum value of 0.0 mv represented by saturated red and the minimum value of 0.8 mv represented by saturated blue. The mean value of 0.0 mv is represented by white. As the maximum and minimum values are of opposite polarity but of similar magnitude, the average value is a good indiator of the path of the urrent following arguments presented in setion.. Comparing the field image with the sanned area, it an be seen that the path of the urrent is onentrated in a narrow region near the wider bridges, as expeted. The slant in the white average field region between terminal lines has been observed 47

168 in other sans and is also expeted [40]. The distane between adjaent lines bridging the two terminal lines is less than 5 pixels in the y diretion, and the white average field region broadens out between the terminal lines. Consequently, the pixel size and the distane between the SQUID and the iruit surfae were not small enough for one to distinguish adjaent urrent arrying lines diretly from the field image (see setion.). Finally, I note that the field image also shows a jump in the FLL eletronis toward the end of the san. This an be seen at the top of the image where the olored pixels stop following a ontinuous pattern. Jumps were also ommon during sans with the prototype mirosope due to the problems with SQUID AN, as disussed in setion 3.3 and in the following setion. 5.3 Problems with the SQUID Mirosope 5.3. SQUID Controller Problems Although I was suessful in obtaining an image, I had some diffiulties in using the imc-303 SQUID ontroller. The primary diffiulty was in not being able to identify or diretly ontrol various eletronis parameters suh as the SQUID bias urrent and feedbak modulation urrent amplitude. The ontroller was fully automati and did not provide diret values of these parameters. One the ontroller was turned on, it performed diagnostis and automatially set parameter values aording to the results. The settings were provided to the operator in terms of some internally normalized units whih had to be interpreted. Automati realibration of the eletronis and SQUID meant losing the previous settings whih were often very 48

169 different from the new settings. Moreover, it was diffiult to assess whether the new settings would result in better performane, as I disussed in setion 4.6. For example, if the FLL eletronis was not able to provide suffiient bias urrent to the SQUID, there was no indiation of this from the SQUID ontroller. What had to be done was that after determining the harateristis of eah d SQUID, a resistor or resistors in the ifl-30 FLL eletronis was replaed to math the appropriate range in urrent, so that the output voltage avoided the high or low extremes. Knowledge of whih resistor to replae was proprietary information, and so the replaement was performed by a designated tehniian at Neoera. Even after alibration and modifiation of the eletronis, the FLL often did not keep lok. Part of the ause ould be traed bak to problems with the nonhystereti d SQUID (SQUID AN) used. SQUID AN had a large dynami resistane ompared to expetation, and the modulation was ourring well within the resistive range of the I-V urve, as disussed in setion 3.3. These irregularities may not have been antiipated by the SQUID ontroller during automati diagnostis and alibration and may have aused the eletronis to set the parameters inappropriately. Contat resistane and failure of the eletrial onnetion to the SQUID hip would only ompound the problem, resulting in sudden hanges in the SQUID response and loss of lok Sanning Problems The other ommon problem I enountered was that sanning would halt in the middle of a san with the ontrol software immobilized. Speifially, the atuator 49

170 ontrolling the x diretion motion would suddenly stop funtioning. When that happened, the omputer system waited for the atuator to move again, not reognizing a problem ourred until an instrution was entered into the omputer to abort the san, and that would not be reognized until the atuator was manually set to the end of san position. It was unpreditable when or under what irumstanes the san would halt, though the larger and longer the san, the more likely it was that a problem would our. These problems were ompounded by diffiulties with the SQUID eletronis desribed earlier. To take bak ontrol, the mirosope s omputer system had to be shutdown and reinitialized eah time the problem ourred. I had made attempts to remedy the situation by seeking the advie of the translation system developers. One suggestion was that the load on the atuator motors may have been exessive, so that pressure on the translation stages should be eased. Weakening of mehanial pressure on the atuators, however, did not solve the problem. Another suggestion was that a position error heking mehanism was halting the san when the aumulated error exeeded some limit. Suh a problem ould be orreted through modifying the software or hardware. However, when it was determined that it was not pratial to reengineer either the software or hardware, I had to leave the sanning system in an unreliable state. Instead, I modified the operating proedures of the sanning software to irumvent problems as muh as possible. My remedy to the sanning problem was mainly to minimize the san area and san time and reinitialize the omputer just before the san without letting the 50

171 software perform ertain routines. By resetting the omputer, the possibility of aumulated error in position or atuator ontrol would be minimized. On the other hand, it was not known whether the self diagnosing or testing routines that were skipped did not override safety or other mehanisms that should have been in plae. Regardless, this did not eliminate the sanning problem. And with the auses still unidentified, a omplete upgrade of the sanning system is required. It is perhaps worth pointing out that one of the main reasons why I was performing these tests was to run the prototype SQUID mirosope through realisti operations and identify problems that ourred. Disovered problems and related issues would then be addressed before replaing the FLL with large bandwidth eletronis. 5

172 CHAPTER 6 Design of Large Bandwidth SQUID Eletronis 6. Limitations of the FLL Tehnique The fastest SQUID Flux-Loked-Loop (FLL) eletronis have ahieved bandwidths up to around.5 MHz [0]. As disussed in setion 3., the ultimate bandwidth of a hystereti d SQUID is set by Eqs. (3.5) and (3.9), and typially ranges between 0 GHz and 00 GHz. Thus, the bandwidth of SQUID mirosopes that use FLL eletronis is limited by the readout eletronis and not the SQUID. To visualize the bandwidth limit, it is helpful to first look at the harateristi spetrum of the FLL output signal. For FLL eletronis with zero applied flux to the SQUID, the response of the loop is flat from 0 Hz up to a fration (between /0 and /) of the osillator frequeny. At higher frequenies, the response falls off rapidly. The flat regime defines the working frequeny range of the SQUID mirosope, i.e. the bandwidth. In addition to intrinsi limitations set by the bandwidth of the losed loop gain desribed later, the output may also be filtered to eliminate artifats introdued by the eletronis outside the bandwidth. For example, Fig. 6. shows the noise spetrum between 0 Hz and 00 khz produed by the prototype SQUID mirosope using the TRISTAN ifl-30 FLL eletronis with a nonhystereti niobium d SQUID at around 5 K. The spetrum was taken in an unshielded lab at Neoera and shows both noise from the SQUID and external interferene from equipment in the lab. The low frequeny region of the 5

173 Noise ( µ Φ o Hz ) Frequeny (Hz) FIG. 6.. Bakground flux noise spetrum measured by the prototype SQUID mirosope using TRISTAN imag FLL eletronis with a nonhystereti d SQUID at around 5 K. Several peaks are seen inluding 60 Hz noise and its harmonis. Peaks around 50 Hz orrespond to noise generated by the mirosope itself, partiularly the helium ompressor.

174 spetrum with /f noise and the high frequeny region around and above the 50 khz osillator frequeny are not shown. Peaks for a line noise are seen at 60 Hz and its harmonis. Noise peaks due to the ryoooler and ompressor are at around 50 Hz. There are also a number of other peaks from undetermined soures. The broad peak around 40 khz is due to the losed loop gain of the FLL eletronis and along with the rapid derease in response beyond 50 khz is harateristi of eletronis of this type [9,0]. For FLL eletronis to work, the Nyquist sampling riterion requires that the osillator frequeny must be at least twie the bandwidth [4]. For the ifl-30 FLL eletronis, this limits the bandwidth to less than 5 khz, with the atual bandwidth being about 0 khz (see Fig. 6.). In general, the response of the amplifier stages must be good to frequenies up to an order of magnitude higher than the osillator frequeny for the loop to be stable. Thus, to extend the bandwidth to GHz, the FLL osillator would have to operate at frequenies higher than GHz with the amplifier stages good through 0 GHz. As reported by Koh et al., the main diffiulty in inreasing the bandwidth of the FLL tehnique is the diffiulty of using mirowave arrier frequenies and mathing them to eletroni omponents with wide frequeny response and low noise [0]. The tehnial limit is determined by the phase shifts in the amplifiers, whih must operate from d to frequenies an order of magnitude higher than the osillator. In other words, the eletronis still has to be able to reat to slowly varying signals as well as the high frequeny feedbak. For example, the.5 MHz bandwidth system Koh et al. built is the state of the art in FLL eletronis and uses a 6 MHz 54

175 osillator. The system is very sensitive to small hanges in its wiring and needs to be tuned very aurately [4]. In pratie, it has proven very diffiult to build systems with the desired phase response beyond a few megahertz, and requiring amplifier stages to have bandwidths in the tens of gigahertz is beyond urrent tehnology. 6. Inreasing SQUID Mirosope Bandwidth using a Hystereti SQUID with Pulsed Bias Current 6.. Basi Priniple and Requirements In order to go beyond the limitations of the FLL tehnique, an alternate tehnique is required that bypasses the need for onventional large bandwidth FLL iruits. The goal of my researh presented in this thesis is to improve on the fastest FLL eletronis ever built by about three orders of magnitude. The idea is to sample the unknown magneti field at regular time intervals instead of attempting to follow the field ontinuously. The tehnique is analogous to strobosopy in whih the filming of fast moving objets is ahieved by flashing light on the objet [43]. In SQUID signal sampling, instead of flashes of light, short bias urrent pulses are sent through a hystereti d SQUID. During a urrent pulse, the d SQUID will either be in the zero voltage state or in the resistive state depending on the bias urrent pulse and the magneti flux modulated ritial urrent. If the ritial urrent of the SQUID is lower than the pulse height, a voltage will appear aross the SQUID. If the modulated ritial urrent is higher, then no voltage will be observed. This is graphially represented in Fig

176 (a) Φ A ½Φ o (b) t I bias I max I (Φ A ) () t VSQUID e FIG. 6.. Response of a hystereti d SQUID to bias urrent pulses with an external magneti flux signal modulating the ritial urrent. (a) A time varying applied magneti flux Φ A through the SQUID. (b) Solid line shows pulsed bias urrent through the d SQUID. Dotted urve shows SQUID ritial urrent I whih is modulated by Φ A of graph (a). () Voltage aross the d SQUID as Φ A evolves in time. t 56

177 With this sheme, information on the value of the unknown external magneti flux an be determined from whether or not a voltage aross the SQUID is seen during the pulse. If the pulse length is shorter than one half the harateristi time of variations in the magneti field, the SQUID will faithfully sample the field with suessive pulses. Furthermore, if the applied flux is entered at Φ = ±Φ o /4 and varies by no more than ±Φ o /4, then modulation of the ritial urrent will our in a nonlinear but nevertheless well defined one-to-one relation with respet to the total flux. Thus, by following when a urrent pulse triggers a voltage, the ritial urrent of the SQUID and onsequently the external magneti flux through the SQUID hole an be followed. To make the tehnique work for large bandwidth appliations, bias urrent pulses must be very short, on the order of the resolved time or inverse of the bandwidth. This means that for GHz bandwidth, the pulse length should be a fration of a nanoseond. The minimum pulse length is limited only by the restritions imposed by the generating iruit, the wiring, and the SQUID bandwidth. These short urrent pulses must be delivered to the SQUID, and the SQUID must be able to reat quikly with a measurable differene between the triggered and untriggered states. The former ondition requires that the leads delivering the bias urrent pulse have suffiient bandwidth and not be too dispersive up to the desired bandwidth of GHz. The latter ondition an be satisfied by employing a hystereti SQUID instead of a nonhystereti SQUID used in FLL eletronis. The reation time of a hystereti SQUID is given by the SQUID s voltage rise time expressed in Eq. 57

178 (3.9). For the hystereti niobium d SQUID shown in Figs..(b) and 3.9, Eq. (3.9) gives a rise time τ rise ps. To understand why this tehnique works better with a hystereti SQUID, onsider what would happen if a nonhystereti d SQUID were used instead. For a nonhystereti SQUID, there is only a gradual transition from zero voltage to the resistive state, whih near the transition results in a voltage hange of around I R [see Figs. 3.4(a) and 3.6]. For hystereti SQUIDs at the transition, the hange in SQUID voltage not only appears within the short voltage rise time τ rise, the voltage hange of /e is quite large and persists until the bias urrent dereases below the retrapping urrent. Thus, for the same hange in applied magneti flux, the voltage response of hystereti SQUIDs is larger. As an example, for SQUID BH whih is a hystereti niobium d SQUID, /e =.7±0.0 mv at LHe temperatures (see Figs. 3.9 and 3.0). For the nonhystereti SQUID AN, the voltage modulation is around 80 µv. 6.. Pulse vs. Alternative Shemes In addition to the pulse tehnique I outlined above, there were at least two other tehniques that I onsidered to use for fast measurement or readout. One sheme was to use the fast triggering of a hystereti SQUID as a preision swith while the bias urrent is ramped. Figure 6.3 depits the idea; the bias urrent is steadily ramped up from zero as one measures the time from the start of the ramp to the moment the SQUID transitions to the resistive state. In this sheme, the ramped urrent must have a rise time signifiantly longer than that of the SQUID voltage rise time and should preferably inrease linearly. When the urrent reahes the maximum 58

179 (a) Φ A ½Φ o (b) t o T t I bias I max I (Φ A ) () t o T t VSQUID e to T t FIG Response of an ideal hystereti d SQUID to a bias urrent ramp with an external magneti flux signal. (a) A time varying applied magneti flux Φ A through the SQUID. (b) Solid line shows bias urrent through the d SQUID. Dotted urve shows SQUID ritial urrent I modulated by Φ A of graph (a). The bias urrent mathes the modulated ritial urrent at time t o after the beginning of the ramp whih is of duration T. () Voltage aross the d SQUID versus time. 59

180 superurrent, the bias urrent is reset, and the proess is repeated. Essentially, the d SQUID ats as a Shmitt trigger with the applied magneti flux ontrolling the threshold urrent level [44]. When the bias urrent exeeds the ritial urrent, a voltage suddenly appears aross the SQUID due to the transition. If the relation between SQUID magneti flux and ritial urrent is well mapped out, the magneti flux at the moment of transition an be determined from the time it took to trigger the SQUID using the urrent ramp. This sheme relies on the time interval being determined extremely well and that the orresponding bias urrent is onsistent and stable. Noise in the urrent ramp must be extremely low. This sheme is exatly analogous to the tehnique developed in the 980s by Martinis and others for investigating Marosopi Quantum Tunneling (MQT) in Josephson juntions and used by Berkley et al. for Quantum Computing (QC) appliations [44-50]. The tehnique works extraordinarily well for that appliation. However, for large bandwidth appliations, the ramping would have to be very fast. For a GHz bandwidth, the ramp duration would have to be less than a nanoseond, with enough ritial urrent resolution to differentiate small hanges in magneti flux. This is quite fast ompared to the few milliseond to tens of miroseond ramps typially used for MQT and QC experiments. Furthermore, funtion generators apable of produing sawtooth waves or pulses at mirowave frequenies are diffiult to obtain. Moreover, the detailed relation between ritial urrent and flux is nonlinear and different for different SQUIDs. So, eah SQUID has to be alibrated 60

181 individually and the useable dynami range of the ritial urrent ould be small ompared to the full range in bias urrent due to magneti hysteresis or other fators. An alternate readout tehnique involved sending a sinusoidal bias urrent with period T through the d SQUID while observing the voltage response aross it. This sheme was implemented in another experiment, but only at lower frequenies [45]. If the d SQUID behaves ideally as desribed in Chapter, the SQUID voltage will trigger at a time δt determined by the ritial urrent and bias urrent then return to zero when the bias urrent dereases and retrapping ours at rt. On the opposite swing of the bias urrent, the SQUID will respond likewise but with the opposite polarity. Note that δ and r satisfy the relation 0 < δ < r < and that retrapping will always our near T/ for a hystereti SQUID with large β. If the SQUID voltage response is sent through a low pass filter, the lowest harmoni omponents an be seleted. If the Fourier series of the SQUID voltage response is alulated, the omponents will have a relative phase shift ϕ given by δ ϕ = π + r = π δ + r 4 (6.) with respet to the bias urrent (see Fig. 6.4). By measuring this phase shift and knowing when retrapping ours, the trigger time δt an be determined. The ritial urrent and onsequently the applied magneti flux an be dedued from this information as in the urrent ramp sheme. This idea was not implemented for several reasons. First, the dynami range in the phase shift is limited. Triggering ours within the rising edge of the bias urrent, whih is only a quarter period T/4. Seond, the limited dynami range of the 6

182 (a) I bias I max I (Φ A ) I r I (Φ A ) δt T/ δt + T/ T t I max (b) V SQUID e ϕ π T e δt rt T/ T t FIG Response of an ideal hystereti d SQUID to a sinusoidal bias urrent with period T. The d SQUID experienes retrapping from the gap voltage /e to zero when I bias = I r at rt after triggering. The amplitude of the bias urrent is set to a value smaller than the maximum SQUID ritial urrent I max. Graph (a) represents the sinusoidal bias urrent. Graph (b) shows a triggering event whih ours at δt after the start of the bias urrent signal. Dotted urve shows the primary Fourier omponent of the voltage signal. The Fourier omponent lags the bias urrent signal with a phase shift of ϕ. 6

183 phase shift plaes more emphasis on preision measurement while the type of measurement is shifted from time measurements to phase shift measurements. Although the phase shift an be measured with the aid of a lok-in amplifier, suh an amplifier working at mirowave frequenies is not available and would require a repetitive signal demanding that the bias urrent frequeny be even higher than the target bandwidth. Third, irregularities or asymmetries in the harateristis of atual physial SQUIDs ould ause undetermined phase shifts of their own. This is in addition to phase shifts from eletronis and wiring. The shortomings of these two alternate readout tehniques highlight the strengths of the pulsed urrent sheme. The urrent pulse tehnique avoids having to perform a timing measurement. With square pulses, it does not matter when during a pulse the SQUID triggers. It only matters whether or not a pulse triggered the SQUID into the resistive state. At the same time, it produes a large, easily measured, output voltage pulse that requires minimal eletronis for detetion Feedbak Field Follower The pulsed sampling tehnique is made possible in pratie by using a hystereti SQUID. However, to aurately follow the hange in ritial urrent and aurately determine the magneti field from this result, some additional features are required. One suh feature is the need to detet the ritial urrent level or transition and another is to be able to follow it. There are at least two ways of deteting the ritial urrent of a d SQUID with pulsed sampling. One is to adjust the urrent pulse height until a voltage transition is found. The other way is to hold the urrent 63

184 pulse height to a fixed value between I min and I max and apply a feedbak flux that will indue the transition. A feedbak flux ould also be used to anel flux from an externally applied signal. Comparing the two methods, the former method has a number of disadvantages ompared to the latter. First, the ritial urrent versus flux relation is not linear. So, as the pulse height inreases the orresponding flux does not inrease linearly and an even derease. Seond, the former tehnique annot follow flux hanges that are greater than Φ o /. If the externally applied flux gradually hanges by more than this amount, there is ambiguity as to the value of the external flux due to the periodiity between flux and ritial urrent as disussed in setion.3. And third, urrent pulses sent to the d SQUID must be well defined and reproduible. It is far easier to produe suh pulses if their harateristis are kept onstant. From these onsiderations, it should be lear that it would be advantageous to apply feedbak flux to the SQUID rather than adjust the pulse height. Applying feedbak flux also leads to a straightforward way of following the hange in the ritial urrent and external flux. When the externally applied flux varies, the feedbak flux an be adjusted to ompensate for the hange, ideally nulling the total flux through the SQUID. The hange in feedbak flux then follows the hange in the external flux. This is muh like onventional FLL eletronis. To properly ompensate the hanges in external flux, one needs to know first whether the external flux inreased or dereased. The information required to determine this an be obtained during the initial ritial urrent detetion stage. Adjusting the feedbak flux so as to trigger the SQUID will determine whether the 64

185 flux is on an inreasing edge or on a dereasing edge of the ritial urrent versus flux urve (see Fig..4). If inreasing the feedbak flux auses the SQUID to trigger, this means that the ritial urrent is suppressed and the SQUID is on a dereasing edge of the urve. If dereasing the flux auses triggering, the SQUID is on an inreasing edge. With this information, it is simple to determine whether the external flux inreased or dereased by determining whether the ritial urrent inreased or dereased. Given the periodiity of the ritial urrent with respet to flux, it is possible to inrease the feedbak flux in one diretion and loate the ritial urrent on either an inreasing or dereasing edge. This fat ould be used to simplify the eletronis by always loating the SQUID on one type of slope. Then, the hoie of whether to inrease or derease the flux to ompensate for the hange in external flux beomes fixed. The seond piee of information needed for ompensating the hange in external flux is knowing how muh the flux hanged. This, however, is impossible to know without searhing for the new ritial urrent. Assuming that the hange is small ompared to Φ o /4, the searh an be performed by programming the feedbak flux to either inrease or derease in small disrete steps depending on the diretion of hange in the ritial urrent. Eventually, the flux will indue a voltage transition thus revealing the new ritial urrent. The disrete steps should be large enough to quikly ompensate for the hange in external flux but not too large that it over ompensates. This ondition should be satisfied by limiting the step size suh that the hange in flux through the SQUID hole is less than Φ o /4. On the other hand, if 65

186 the steps were too small, the eletronis may not be able to keep up with the hanging external flux. The maximum rate of flux hange that an be handled by the eletronis is its slew rate. As mentioned above, the slew rate is limited by the speed of the eletronis and the step size. Another limit on the slew rate an ome from the need to perform multiple measurements or averaging. Usually, one averages to redue the effets of noise. So, instead of reating immediately to a single pulse measurement, multiple pulse measurements an be performed before adjusting the feedbak flux. This adds to the time needed to reat to the hanging external flux. The external flux then has more time to hange and may inrease faster than the feedbak flux an adjust. It may be possible to address rapidly varying fluxes with adaptive step sizes. However, if one follows this approah, the need for large losed loop bandwidth that limited onventional FLL eletronis reappears. Fortunately, the losed loop feedbak limitation an be irumvented for a ertain lass of magneti field signals, espeially those that an be repeated, suh as from omputer iruits that are yled through the same series of operations. For suh signals, speed limitations set by the slew rate due to multiple measurements are also irumvented. The next setion inludes desriptions of two signal following shemes, one of whih irumvents the losed loop problem. I also present speifi implementations of how the SQUID ritial urrent an be initially determined and how to searh for the new ritial urrent after the applied external flux hanges. 66

187 6.3 New SQUID Control Algorithm 6.3. Critial Current Detetion To implement the basi idea behind the pulsed SQUID sampling tehnique desribed in the previous setion, one needs an effiient way to follow the applied external magneti field. The question then beomes: what is the best way of determining and following hanges in SQUID ritial urrent? Clearly, in the pulsed sampling approah, one needs to make multiple measurements, as a single measurement an only determine whether the ritial urrent is larger or smaller than the urrent pulse. That is, the SQUID only indiates whether its ritial urrent is higher or lower than the bias urrent pulse height. Figure 6.5 shows an example of one sheme. In this sheme, urrent pulses are set to some average height I p I (Φ o /4). The eletronis then adjusts the feedbak field at regular intervals in the same diretion until the voltage aross the SQUID hanges by the gap voltage /e between pulses. After deteting this voltage variation, the feedbak field is adjusted in the opposite diretion and the proess is repeated. After, the initial SQUID ritial urrent level is determined, the feedbak field is swithed bak and forth, modulating the ritial urrent just above and below the average value. When the applied external flux stays onstant, the SQUID ontinuously alternates between the zero voltage and the resistive state with eah bias urrent pulse. As the external flux hanges, the feedbak field will follow the hange in ritial urrent. As long as the hange in the applied external flux, and onsequently the ritial urrent, is not too large, the feedbak field should be able to keep up with the 67

188 (a) Φ A Φ atual Φ Φ estimate = Φ t Φ f (b) t I bias I max I I I (Φ A + Φ f ) () t V SQUID e t FIG Osillating feedbak sheme for ritial urrent detetion. This sheme uses idential bias urrent pulses and a feedbak flux Φ f whih hanges after every pulse by Φ. The diretion of the hange in Φ f is suh that when the SQUID ritial urrent I is lower than the bias urrent pulse height, Φ f inreases I by I. When I is higher than the bias urrent pulse, Φ f dereases I. Consequently, even when the external flux Φ A is onstant, Φ f hanges suh that the d SQUID alternates between the zero voltage state and the resistive state. However, an average flux Φ t Φ o /4 and a orresponding < I > is maintained. (a) Φ A that is initially onstant then inreases. The hange in Φ f follows Φ A and is used to produe Φ estimate. (b) Solid line shows pulsed bias urrent. Dotted urve shows I in response to Φ A + Φ f. () Voltage aross the d SQUID, whih is used to determine whether I is higher or lower than the bias urrent pulse. 68

189 external flux. Under pseudo-d onditions, the external flux through the SQUID hole will orrespond to a level somewhere in between the modulated high and low flux levels. The differene between the high and low magneti flux levels should be large enough that noise does not influene the triggering of the SQUID. The disrete inremental hange in feedbak flux then orresponds to the one shot flux resolution of the eletronis, and the average feedbak field orresponds to the SQUID ritial urrent. Matthews et al. have investigated a variation on this tehnique that instead of searhing for the new ritial urrent, just maps out the triggering frequeny at different feedbak field values [3,0]. One obtains a histogram of triggering events versus flux. The tehnique makes use of noise and the distribution of triggering events to determine the ritial urrent level. In this sheme, the feedbak field is adjusted suh that some of the pulses trigger the SQUID into the resistive state and the rest leave it in the zero voltage state. If the feedbak field exatly orresponds to the ritial urrent, noise in the signal and system will ause only half of the pulses to trigger the SQUID. This sheme has the drawbak of requiring more measurements than the sheme desribed in Fig Figure 6.6 shows an alternative sheme whih needs as few measurements as that of Fig. 6.5 but does not require the feedbak field to be onstantly adjusted. The tehnique involves using bias urrent pulses of two sizes that alternate. First, a small pulse is sent to the SQUID followed by a large pulse. If neither pulse triggers the SQUID into the resistive state, the ritial urrent is larger than both pulse heights. If both pulses ause triggering, the ritial urrent is smaller than both pulses. When 69

190 (a) Φ A Φ estimate = Φ t Φ f Φ Φ atual (b) t I bias I max I I I (Φ A + Φ f ) () t V SQUID e t FIG Alternating pulse sheme for ritial urrent detetion. This sheme uses bias urrent pulses of alternating height and a feedbak flux Φ f whih maintains the SQUID ritial urrent I between the heights of the alternating pulses. (a) External flux Φ A that is initially onstant then inreases. The hange in Φ f follows Φ A and is used to produe Φ estimate. (b) Solid line shows pulsed bias urrent through d SQUID. Dotted urve shows I in response to Φ A + Φ f. If I stays between the two pulse heights, Φ f remains onstant. When Φ A hanges ausing I to drop below the shorter pulse, Φ f hanges by Φ to raise I by I. If Φ A auses I to inrease above the higher pulse, Φ f hanges to derease I. Consequently, an average flux Φ t Φ o /4 and a orresponding < I > is maintained. () Voltage aross the d SQUID, whih is used to determine whether I has moved outside the range between the alternating pulses. 70

191 the SQUID is triggered only by the larger pulse but not by the smaller pulse, the ritial urrent is between the two pulses. This identifies the ritial urrent. When the ritial urrent is outside the upper and lower limits of the two pulses, the eletronis adjusts the feedbak field in the way desribed in setion 6., restoring the total magneti flux through the SQUID hole and bringing the ritial urrent bak between the two pulses. The differene in the two bias urrent pulses orresponds to a SQUID dependent magneti flux differene. Unlike the osillating feedbak sheme of Fig. 6.5, the one shot flux resolution in the alternating pulse sheme of Fig. 6.6 is the larger of this flux differene and the disrete inremental step size of the feedbak flux. Setting the differene between the two pulse heights just a little bit larger than the orresponding differene in feedbak field step size optimizes the step size value. The pulse height differene should be hosen so that its somewhat larger than the rms urrent noise in the SQUID Pulse Rate Dependent Signal Following Shemes Given the means of determining and following the hange in ritial urrent, deteting hanges in the applied external flux on a SQUID is straightforward. However, applying feedbak is more involved. Depending on the relative timing of the measured signal, the bias urrent pulses, and the adjustment of the feedbak field, feedbak an be implemented in pulsed SQUID sampling in two very different ways: synhronous mode or asynhronous mode. Asynhronous mode feedbak requires pulsing the urrent at rates that are large ompared to the inverse of the harateristi time of magneti field variations. It an be used with nonrepetitive 7

192 signals that are suffiiently slow with respet to the pulse rate. Synhronous mode feedbak requires repetitive signals, generally under onditions where time variations in the measured field our muh more rapidly than the maximum possible pulse rate. Synhronous mode is well suited for measurements of hips and miroiruits, sine they an be put through the same set of operations repeatedly. Figure 6.7 summarizes the idea behind asynhronous mode. This mode an be used for both repetitive and nonrepetitive signals and involves pulsing the d SQUID at the maximum reliable onstant rate. Eah pulse in Fig. 6.7 represents a determination of the ritial urrent level and so an orrespond to multiple measurements. For the osillating feedbak sheme of Fig. 6.5, it orresponds to one or more pulses determining whether the SQUID is triggering. In the ase of the alternating pulse sheme of Fig. 6.6, it orresponds to one or more pulse pairs determining whether the ritial urrent is between, above, or below both pulse heights. The voltage aross the d SQUID must be deteted at the pulse rate. After observing how many times the SQUID triggers, the eletronis follows the output signal and adjusts the feedbak field aordingly. In asynhronous mode, the sampled values will represent the entire evolution of the measured signal, provided that during the dead time between measurements, there are no signifiant variation in the signal. In partiular, the variation in applied external flux should be smaller than Φ o /4, as mentioned earlier in this hapter. This is muh like onventional FLL eletronis exept that any signal averaging is performed before adjusting the feedbak field. 7

193 (a) Φ A Φ A Φ Φ f (b) t I bias I max I (Φ A + Φ f ) I () t Φ f t FIG Asynhronous pulsed SQUID sampling tehnique. (a) Solid urve shows the evolution of an external magneti flux Φ A through the d SQUID. The dotted urve shows the evolution of a feedbak flux Φ f responding to and following Φ A suh that Φ A + Φ f <Φ >. (b) Solid line represents urrent pulsing events that determine the SQUID ritial urrent at the indiated times whih are at regular intervals. Dotted urve shows hange in I in response to Φ A + Φ f. After determining I, Φ f is adjusted as neessary, maintaining I < I >. Note that hanges in Φ A are relatively small between pulsing events ompared to its overall evolution. () Feedbak flux Φ f versus time, whih an be used as the output signal. 73

194 The asynhronous pulsed sampling tehnique is limited by the maximum pulse frequeny, minimum pulse length, the time it takes to readout the SQUID voltage, and the time it takes to integrate the feedbak signal for stability purposes. All of these fators ause phase lags just as in onventional FLL eletronis. Consequently, asynhronous mode operation will enounter the same limits that prevent large bandwidth measurement. Even within the bandwidth limit, the pulse period annot be shorter than the pulse length and is usually muh longer, dependent on the time required for the eletronis to register the SQUID voltage, adjust the feedbak field, and reset itself for the next measurement. As an example, given a urrent pulse rise time of ps, the minimum pulse length must be twie that or 4 ps. The maximum possible pulse rate is then around 40 GHz. However, pulse generators with repetition rates that high are not readily available. To follow a GHz signal in asynhronous mode, a repetition rate that is greater and preferably muh greater than GHz is needed. Furthermore, the eletronis must be able to read out the SQUID voltage at this rate as well. These are very serious limitations and suggest asynhronous mode operation is not viable as a large bandwidth tehnique. These limitations are irumvented in synhronous mode operation, desribed below. In synhronous pulsed sampling, the onset of pulses is synhronized to the start of the repetitive signal. Only a single measurement is made per period, thus the pulse rate need not be fast, though multiple periods have to be sampled. In eah subsequent period, an additional measurement an be made with the same synhronization and delay with respet to the start of the signal until the ritial 74

195 urrent level at that speifi delay is determined. After determining the ritial urrent, the eletronis adjusts the feedbak field so as to ompensate for any hanges in the external flux at the speifi delay time. For suessive measurements, a new delay time is hosen, and the signal is sampled repeatedly at the new delay time with respet to the start of the signal until the new ritial urrent level is determined. After adjusting the feedbak field again, the entire proess is repeated at another delay time until the entire signal period is overed. The delay should be small ompared to the period of the signal in order to see the entire signal evolution within a period. On the other hand, if signal variations are small between suessive delays, the offsets do not have to equal the pulse lengths and an be signifiantly longer, so long as the sampled values apture all the features of the signal. As an example, suppose the pulse length is ns and the signal period is µs. At the first measurement, the delay time an be set to zero, so the signal at t = 0 ns is sampled. When measurements are finished at t = 0 ns, the feedbak field is adjusted appropriately, and the next measurements are made with a ns offset from the beginning of the signal period. One the signal is measured at t = ns, the feedbak field is adjusted to ompensate the field at t = ns, and then the system samples the signal at t = ns, and so on. This proess is shown in Fig Eventually, the signal values over the µs period will be sampled, and the measurements an be ombined to reonstrut the entire signal over its period. 75

196 (a) Φ A Φ A Φ f Φ (b) I bias I (Φ A + Φ f ) t I max I () Φ f t t 3 t 4 t t t FIG Synhronous pulsed SQUID sampling tehnique. (a) Solid urve shows the evolution of a fast repeating external magneti flux Φ A. Dotted urve shows the evolution of a feedbak flux Φ f in response to measurements on Φ A. (b) Solid line represents pulsed urrent measurements that determine the SQUID ritial urrent I at the indiated delay times. Dotted urve shows hange in I in response to Φ A + Φ f. After determining I, Φ f is adjusted so that Φ A + Φ f <Φ > and I < I > at the time of the pulsed measurement. The following measurement is delayed by an interval n t with respet to the start of Φ A where n represents an integer sequene. As a result, Φ f follows Φ A but strethed out in time. () Feedbak flux Φ f versus time, whih an be used as the output signal. 76

197 In pratie, many measurements will be needed at eah delay time. Using the osillating feedbak sheme of Fig. 6.5, the SQUID voltage is repeatedly heked at the same delay until the voltage aross the SQUID hanges by the gap voltage /e between pulses. With the alternating pulse sheme of Fig. 6.6, one or more pulse pairs at the same delay, still one pulse per period, are needed to determine whether the ritial urrent is between, above, or below both pulse heights. Averaging to derease the effets of noise will further inrease the number of measurements at the same time offset. But, unlike in asynhronous mode sampling, this does not affet the slew rate; for repetitive or periodi signals that an be generated on ommand, synhronous mode operation allows for an arbitrary amount of time to determine the flux hange. A form of synhronous pulsed sampling of a sinusoidal signal was suessfully used by Matthews et al. in onjuntion with their tehnique of mapping the triggering events of a hystereti d SQUID at different feedbak field values [3,0]. They ahieved the goal of following high frequeny magneti field signals with the tehnique. A sample result is presented in Chapter Implementation into Eletronis 6.4. Synhronous Alternating Pulse Sheme Many variations on the pulsed sampling tehnique with different signal deteting and following shemes are possible. In this setion, I disuss the design of eletronis for the alternating pulse sheme of Fig. 6.6 using synhronous pulsed sampling. 77

198 To summarize the basi idea of the synhronous alternating pulse sheme, the externally applied flux is determined by adjusting the feedbak flux so that the SQUID only triggers when the ritial urrent level is between the two bias urrent pulses of differing height. The differene in pulse height should be larger than the urrent noise or orresponding flux noise to avoid random SQUID triggering. If adjustments of the feedbak field are performed in disrete steps, the step size will limit the flux resolution in addition to the pulse height differene. Adaptive step sizes ould make adjusting the feedbak more effiient. But, in any ase, the smallest inrement in feedbak flux should hange the ritial urrent by no more than the pulse height differene. And finally, a preision master lok must oordinate the relative timings of the pulses with respet to the signal being measured for synhronization. The hoie of the two urrent pulse heights is very important. For fixed step size, the optimum height differene is determined by the flux noise [see Eq. (3.8)]. For SQUID BH used in this study, this orresponds to about 5.5 mφ o in 5 ps at 5 K. In this estimate, I have ignored /f noise in the ritial urrent, whih will have a small effet in omparison [4,6,]. Clearly, the pulse heights must be adjusted to where the slope in the ritial urrent versus external flux urve is maximum, i.e. near Φ = Φ o /4 where I / Φ is large. This maximizes the pulse height differene for the same differene in flux, thus, minimizing effets from bias urrent noise in the pulse height while maximizing flux sensitivity. I note that the absolute maximum useful differene in pulse height is from the bottom of the ritial urrent versus external flux urve to the top, whih orresponds 78

199 to Φ o /. For SQUID BH, this implies that the minimum pulse height differene is about 0 - of the maximum ritial urrent. More experimental details are presented in Chapter Pulse Counting and Averaging Figure 6.9 shows a blok diagram of the eletronis for implementing the synhronous alternating pulse tehnique. In this implementation, instead of diretly determining whether the larger and smaller pulses eah trigger the SQUID, I present an indiret method whih simply ounts the number of pulses and ompares. Other variations are learly possible. If the feedbak field anels the effets of the applied external flux, only the larger pulse will plae the SQUID in the resistive state. So, of the N i pulses sent by the signal generator, only N v = N i / should ause a triggering event. The eletronis an then ompare N i / with N v, and if they are the same, the feedbak field is aneling the external flux. On the other hand, if N i / < N v, then the external flux hanged to suppress the ritial urrent ausing too many triggering events. And, if N i / > N v, then the external flux hanged to augment the ritial urrent allowing too few triggering events. The feedbak field works to restore the equality N v = N i / by ompensating for the hange in applied external flux. Assuming the d SQUID is on a dereasing edge of the ritial urrent versus flux urve, this means dereasing the feedbak field when N i / < N v and inreasing the field when N i / > N v. 79

200 Pulse Generator Counter ½ Flip-Flop Master Clok Delay Not Equal Equal Compare/Reset Reord Ipulse Vf Sample Trigger Counter Shmitt Trigger Amplifier Stage Rf Integrator Φf Vf Sample ΦA FIG Shemati diagram of pulsed SQUID sampling eletronis using a hystereti d SQUID and implementing the synhronous alternating pulse tehnique for large bandwidth magneti field detetion appliations.

201 The physial implementation of the above is as follows. A signal generator produes ns or shorter voltage pulses with alternating height. The generator is timed or triggered by a master lok. The voltage signal from the generator is onverted into a urrent pulse and fed to the hystereti d SQUID via a mathed 50 Ω oaxial able. If the SQUID is driven into the resistive state, a voltage will appear aross the SQUID. An amplifier detets this voltage and sends it to a Shmitt trigger whih onverts the analog signal into a digital pulse [44]. This pulse is sent to a ounter. At the same time, pulses from the signal generator are also sent to a ounter through a divide by flip-flop [46]. The results of eah ounter are ompared with one another using digital logi. Depending on the three possible outomes omparing N i / with N v, the output voltage of the feedbak flux line is adjusted. Adjustments in the voltage an be performed in unit steps or adaptively depending on the omparative differene between N i / and N v. The voltage drives an integrator (for feedbak stability) whih in turn drives urrent through the feedbak oil that generates a magneti field to ompensate the hange in externally applied flux through the SQUID hole. After the feedbak stabilizes, the pulse ounters are reset, and the proess is repeated for another measurement. Measurements are repeated at the same delay with respet to the measured signal until the ondition N v = N i / is satisfied. When this ondition is met, the voltage V f aross the feedbak resistor R f is stored as the quantity orresponding to the applied external flux Φ A, just like in onventional FLL, but only at the speified delay. The external flux Φ A an be expressed as 8

202 V f = I f R f = R M f f Φ f = R M f f ( Φ Φ ) t A (6.) where M f is the mutual indutane between the feedbak oil and the SQUID loop, Φ f is the feedbak flux, and Φ t is the total magneti flux that the eletronis attempts to onserve. Finally, a omputer gathering data and ontrolling the system adjusts for the next delay, and the whole proess is repeated. If a single generator apable of produing pulses with alternating heights is unavailable, then two pulse generators synhronized together ould be used in whih one generates regular pulses of onstant height and the other generates positive then negative pulses of small amplitude (see Fig.6.0). If the differene in pulse height is smaller than the retrapping urrent of the hystereti d SQUID, the seond pulse generator ould be replaed by a sine wave soure. The differene in pulse heights would be twie the wave amplitude. If noise in the SQUID bias urrent is signifiant, the d SQUID ould trigger without a urrent pulse being reeived. I observed suh events when I performed initial I-V measurements on SQUID BH. Bias urrent to the d SQUID was ontrolled through a resistor network [see Fig. 5.(b)]. Observations of the resulting I-V urve inluded fast osillatory transitions from the zero voltage state to the resistive state and bak that were unexplainable from basi SQUID harateristis alone. When the resistor network was replaed with a single resistor, the phenomenon disappeared suggesting that it was due to pikup of interferene signals. In priniple, one ould address urrent noise problems by inluding oinidene heking iruitry between the pulse generator and the d SQUID into 8

203 Master Clok Syn Iout Slave Clok FIG Shemati diagram of eletronis to generate alternating height urrent pulses using two signal generators. In this implementation, a small amplitude square wave generator and a onstant height pulse generator are synhronized together. Impedane mathing and proper termination is assumed.

204 the eletronis. A pulse from the signal generator an be ombined with a pulse signal from the d SQUID through the equivalent of a logi AND-gate, whih must also ompensate for the short pulse lengths and timing delays in the two signals. Then, only when both the signal generator sends a pulse and the SQUID triggers is the event reognized and fed to the ounter. If, on the other hand, false triggering is negligible or avoided by other means, the added iruitry should be unneessary Superonduting Ciruitry Other than the SQUID, the key element in the eletronis of Fig. 6.9 that makes pulsed SQUID sampling work is the short pulse signal generator. For GHz bandwidth eletronis, the pulse length has to be 500 ps or shorter. Signal generators apable of produing suh short pulse lengths are available. However, they are expensive, and when pushed to produe even shorter pulse lengths their performane starts to degrade in terms of redued pulse height and distortions in the pulse profile. There is an alternative possibility of produing short pulses and ahieving even larger bandwidth. This involves reating the required short pulses with superonduting eletronis. Based on the work by Faris, Tukerman, and Whiteley et al., superonduting iruits an produe very short pulse signals that have rise times of 0 ps or less [-4,47,48]. Furthermore, superonduting eletronis based on Josephson juntions or Rapid Single Flux Quantum (RSFQ) tehniques have demonstrated the possibility of very fast eletronis that an outperform onventional tehniques [49-5]. In priniple, suh eletronis ould be aommodated on the 84

205 same superonduting hip as the SQUID, so there also would not be muh additional ost one there is a working design. Figure 6. shows superonduting iruitry that uses SQUIDs to produe short pulse signals. The hystereti d SQUID that measures the externally applied magneti flux is SQ 3. Conventional pulse generators are still used but only for synhronizing pulses with the measured signal. In partiular, the onventional generators pulse length is not limited by bandwidth requirements but by the pulse repetition rate. An amplifier does not measure the voltage aross the d SQUID diretly but the voltage signal V out from a Josephson juntion Shmitt trigger that reats to the d SQUID. The iruit of Fig. 6. works as follows. The nonhystereti d SQUID SQ is biased with d urrent suh that maximum voltage modulation ours aross it. The effetive minimum resistane aross SQ should be muh smaller than Ω and the maximum resistane muh larger than Ω. The effetive resistane of SQ is ontrolled by the input signal from a onventional pulse generator. When SQ is in the low resistive state, most of the bias urrent flows through SQ. When a square pulse is reeived through the input signal oil, SQ goes into the high resistive state. The indutane L is large, so initially urrent mostly flows through L. When that urrent reahes a maximum value determined by the Josephson juntion I, the juntion transitions into the resistive state and the urrent then mostly flows through L. This results in a short urrent pulse through L. Faris suggests values of 300 ph and 0 ph for L and L, respetively [47]. When the square pulse through the signal input disappears, urrent flows bak mostly through SQ. 85

206 Pulse Generator Ω Ω R R L3 Ip Ip3 Vout L L I SQ SQ SQ3 I Pulse Generator Feedbak FIG. 6.. Shemati diagram of superonduting eletronis to generate very short urrent pulses to bias a hystereti d SQUID for large bandwidth pulsed SQUID sampling. All elements exept for SQ3 and its feedbak oil reside on top of a superonduting ground plane.

207 The pulse through L an be very short, as short as 6 ps, but omputer simulation by Faris on a similar iruit indiates that it an have noisy features [47]. The iruit ould also behave differently if instead of L, the load was variable like a d SQUID. Therefore, instead of using the urrent pulse through L as the bias urrent pulse to SQ 3, the urrent pulse through L produes a magneti field pulse into SQUID SQ. Like SQ, SQ is nonhystereti and is also urrent biased to maximize voltage modulation. Coupling between L and SQ is set so that the maximum flux Φ max through SQ is Φ max = M I p Φ o (6.3) where M is the mutual indutane between L and SQ, and I p is the height of the urrent pulse through L. SQ is initially in the low resistive state, so most of the urrent flows through SQ rather than through R. However, when the magneti field pulse from L triggers SQ into the high resistive state, more of the urrent is hanneled through R, driving a urrent pulse through SQ 3 with the same duration as the original short pulse through L. Furthermore, an additional nonhystereti d SQUID in series with R an be used to vary the total output impedane to SQ 3, and onsequently modulate the urrent pulse height. This analysis assumes lump iruit behavior. So, for signal rise times of around 0 ps, the analysis would be valid for iruit lengths that are muh less than m/s 0 ps = m = 3 mm. This is aeptable for most of the iruit in Fig. 6., exept near SQ 3. To minimize external field noise, the iruit elements 87

208 should be plaed on top of a superonduting ground plane. However, SQ 3 must be exposed to external field and plaed away from the ground plane. This requires relatively long leads between R, R, and SQ 3. Planar transmission line theory is then required to determine the optimal values of R and R to math SQ 3 and its leads [5]. Finally, the voltage signal aross SQ 3 an be too fast to read out with onventional eletronis. A tehnique adapted from RSFQ eletronis an address this problem with the equivalent of a superonduting Shmitt trigger using a hystereti Josephson juntion I and indutor L 3 [50]. I is urrent biased below its ritial urrent. If SQ 3 does not trigger, the urrent pulse I p3 through L 3 will be small and not enough to trigger I into the resistive state. On the other hand, if SQ 3 does trigger, I p3 will be large enough to trigger I, and a voltage will appear at V out. This voltage will be maintained until the bias urrent to I is reset. Mukhanov et al. suggest that L 3 =.65 ph for I = ma with a bias urrent of 0.63 ma [50]. The bias urrent is set and reset synhronously with the master pulse generator in Fig I have not determined the optimal parameter values for all of the elements in the superonduting iruit desribed above. Even with determined values, deviations during manufaturing may render the iruit inoperable, and therefore a omplete design would need to take suh variations into aount. Implementation of superonduting elements and the onstrution of the large bandwidth SQUID eletronis are left as future work. 88

209 CHAPTER 7 Response of Hystereti Niobium d SQUIDs to Pulsed Bias Current 7. Bakground The pulsed urrent SQUID readout tehnique desribed in Chapter 6 is partly based on a series of experiments I performed on hystereti niobium d SQUIDs. The purpose of the experiments was to test the general idea of pulsed SQUID sampling, speifially using short bias urrent pulses on d SQUIDs to perform flux detetion. The experiments provided insight into the design of the eletronis and heked some aspets of the overall sheme. The experiments involve sending short bias urrent pulses to a hystereti d SQUID while observing its voltage response. One expets that a voltage will be observed aross the d SQUID for the duration of the urrent pulse, depending on the height of the pulse and the magneti flux through the SQUID hole. If the pulse height min max I p and the range in SQUID ritial urrent I is suh that I < I p < I, the voltage signal should appear and disappear as external magneti flux through the SQUID hole varies. Key questions that these experiments needed to answer were: does the pulsed SQUID sampling tehnique work as expeted, and what limitations are there with the available apparatus? Those limitations inluded how short the urrent pulses ould be, how large the signal was ompared to bakground, how easy it was to defeat the 89

210 response, and how fast the pulses ould be repeated. A question that I was not able to answer experimentally was what was the ultimate limit of the tehnique, espeially how short the sampling time or pulse length ould be. This was simply beyond the apability of the pulse generator I used. On the other hand, an integral part of the experiments was determining what type of system or apparatus would be required for the tehnique to work. The first step in my experiments was to assemble hystereti d SQUIDs and prepare an apparatus. As mentioned in setion 3.3, I obtained hystereti SQUIDs from Hypres based on designs arried over from resistively shunted niobium d SQUIDs used in a LHe ooled SQUID mirosope [,95,8]. The harateristis of the main SQUID I used in these experiments (SQUID BH) were summarized in Table I. Although in retrospet I ould have improved on its design, SQUID BH had adequate harateristis and was very durable. Other hystereti SQUIDs used for experiments were damaged or failed during the ourse of my experiments. 7. Large Bandwidth Dip Probe Design and Constrution 7.. Required Speifiations Many prior experiments had been done on resistively shunted niobium d SQUIDs using a LHe dip probe. The probes ould be inserted into LHe dewars whih provide a relatively stable environment for SQUID measurements. Figures 3.6, 3.7, and 3.8 show examples of suh measurements. Noise in Figs. 3.7 and 3.8 is due to noise in the external field as the probe was not magnetially shielded. Compared to onfiguring a ryoooler, it was relatively simple to reonfigure a dip 90

211 probe and repeat measurements. Considering these advantages, I deided it was preferable to ontinue using a dip probe apparatus, rather than the ryoooler, to perform experiments on the pulsed sampling tehnique. However, the existing dip probe was designed for low frequeny measurements and used twisted pair opper magnet wire. This would not be adequate for my purposes sine twisted pair wiring has relatively poor uniformity and low impedane (Z Ω). The pulsed SQUID sampling tehnique required designing apparatus for bandwidths exeeding GHz. Sine there were no ryogeni probes with the right apability available to me, I onstruted a new dip probe. I onsidered two options for replaing the twisted pair able dip probe. One involved generating and deteting short pulse signals loally, espeially near or on the SQUID hip using mirowave integrated iruit designs. In priniple, superonduting eletronis suh as high frequeny Josephson juntion or Rapid Single Flux Quantum (RSFQ) iruits ould be used [-4,47-5]. The required design effort, however, was not ompatible with the given onstraints on time and resoures. A simpler and quiker option was to hange the wiring and onnetors in the dip probe so that they mathed measuring instrumentation. The obvious hoie was to substitute the twisted pair wiring with 50 Ω oaxial able with a bandwidth exeeding GHz. Miro oaxial able with bandwidths exeeding 00 GHz had already been suessfully used in other large bandwidth ryogeni appliations [47,48]. So, despite spae requirements and thermal ondutane onerns installing it later in the ryoooled 4 K SQUID mirosope, I expeted the oaxial wiring to work well in the LHe dip probe. 9

212 After reviewing various hoies, I seleted UT-34-SS, a small diameter 50 Ω oaxial able from Miro-Coax, In. [53]. The outer diameter is in and the inner ondutor diameter is in. The dieletri material is polytetrafluoroethylene (PTFE), and the outer ondutor is made of stainless steel; stainless steel has low thermal ondutivity, and this would limit heat flow. The use of stainless steel for the inner ondutor as well would have limited thermal heating even further. However, attenuation in the signal would have been signifiant, so the inner ondutor was silver-plated opper-lad steel (SPCW). UT-34-SS is a semirigid able, whih means that it ould be deformed to a ertain extent and would hold its shape. The minimum allowed inside bend radius is.3 mm, whih meant that it would be relatively easy to work with. UT-34-SS is a nonstandard able. Some of its speifiations were not readily available from the manufaturer but appeared to be omparable to other small diameter oaxial able. For example, I estimated the apaitane of the able to be about 95. pf/m and the bandwidth approximately 55 GHz from the speifiations of UT-34, whih has an outer ondutor made of opper but is otherwise idential in onstrution [53]. I expeted the attenuation to be less than.9 db/ft at GHz based on the speifiations of UT-0-SS whih is similar to UT-34-SS exept that it has an outer diameter of 0.00 in and an inner ondutor diameter of in. Comparing speifiations, I found that the attenuation is mainly affeted by the hoie of ondutor material, but that there is a marginal inrease in attenuation with smaller ondutor diameter. 9

213 The hoie of UT-34-SS was a ompromise between able diameter, thermal ondutivity, attenuation, and ost. Further, an important fator was that suitable SMA onnetors were available [54]. Mathing onverters were also available to onnet to BNC able and printed iruit board. The onnetors had a muh smaller bandwidth than the able, typially about 8 GHz versus 55 GHz for the able. The SMA-to-BNC onverters had bandwidths of only 4 GHz [54]. Despite the redued bandwidth, there was little impat on my experiments beause the pulse generator I used had a more restrited bandwidth based on the shortest pulse it ould produe. Besides the wiring, making onnetions to the SQUID hip was also a onern. In the dip probe I used for making I-V measurements, I onneted the SQUID to wiring using silver paint [see Figs. 4.7 and 7.(a)]. These silver paint ontats were not very reliable as I disussed in setions 3.3 and 4.3. My I-V measurements of d SQUIDs with silver painted ontats tended to show noise and muh larger ontat resistane ompared to SQUIDs with wire bonded ontats. In order to make the onnetions more robust, I deided to replae the silver paint ontats with wire bonding. This was performed by epoxying the SQUID hip to a hip holder and then wire bonding the SQUID to the holder [see Fig. 7.(b)]. The holder was designed for the twisted pair wire dip probe. For the oaxial able, I had deided to onstrut a new dip probe rather than modify the existing one, so I also designed and onstruted a new hip holder and iruit board (see Fig. 7.). The SQUID hip was epoxied diretly onto the iruit board and then wire bonded to various onnetors whih were attahed to the board as well (see Fig. 7.3). 93

214 (a) Multiturn Coil 3.8 m Sapphire Rod Holder (b) Twisted Pair Wire 3.8 m FIG. 7.. (a) Bottom portion of twisted pair wire LHe dip probe. A multiturn oil for produing a magneti field is seen inside the probe housing. A sapphire rod holder for testing SQUID tips and whih attahes to the probe housing is also seen at bottom right. (b) Chip holder whih attahes to the probe housing on the twisted pair wire dip probe. SQUID hips are glued to the enter of the PC board with either silver paint or photoresist. Eletrial onnetions to the hip are made by wire bonding. 94

215 (a) (b) SMA Adapter Miro Coaxial Cable 5 m.0 m () SMA-PC Board Connetor 4 m Multiturn Coil Twisted Pair Wire FIG. 7.. Large bandwidth dip probe. (a) Full length view of dip probe. (b) Inside view of onnetor box with miro oaxial ables. () Front view of hip holder. Miro oaxial ables and twisted pair wires are attahed to PC board onnetors. A multiturn magnet wire oil an be seen glued on to the middle of the PC board. 95

216 Chip Holder Brass Nut and Ring Wire Bonded Leads Twisted Pair Wire Connetor SQUID Chip Copper Ground Plane SMA-PC Board Connetor FIG Bak side view of aluminium hip supporting holder in large bandwidth dip probe. Square shaped SQUID hip is seen attahed to the PC board with photoresist. Wire bonded leads to and from onnetors, the opper ground plane and SQUID hip are visible. 96

217 7.. Final Design The hip holder was onstruted from a piee of aluminium and a piee of opper plated fiberglass board [see Figs. 7.() and 7.3]. The oaxial ables were insulated from eah other with Teflon tubing and shielded by housing them inside a 0.0 in thik stainless steel tube that also supported the hip holder. Thin stainless steel tubing was hosen to minimize thermal ondutane. Thinner tubing was available but was not strong enough to withstand damage during ordinary handling. The.0 m length of the dip probe s stainless steel tube was determined by the depth of the CMSH-60 LHe dewar (with optional larger nek opening) into whih the dip probe is inserted for ooling [55]. The depth of the hip holder inside the dewar was loked in plae using a quik oupler attahed to a ladish ap around the tube. The aluminium hip holder and stainless steel tube were held together by hard soldering a brass srew ring to the end of the tube and then srewing the holder to the brass ring. I used 4% silver solder and a propane torh for soldering. Aid flux was needed to help the solder flow between the brass and stainless steel. A small brass nut was also made to help seure the holder to the ring. This design permitted the removal and exhange of holders attahed to the end of the dip probe. The iruit board inside the aluminium hip holder had opper plating on one side, whih ated as a ground plane (see Fig. 7.3). I attahed SMA-PC board onnetors to the iruit board by soldering the ground leads to the ground plane [see Figs. 7.() and 7.3]. Inside the holder, there was not enough spae to house many SMA-PC board onnetors. Spae was limited beause the entire holder had to fit 97

218 through the in diameter nek of the LHe dewar, and I had to reserve some spae inside the holder to failitate the onnetion and disonnetion of the oaxial able to the SMA onnetors. Fortunately, I only needed three oaxial ables for my experiments: one for the SQUID bias urrent pulses, another for the output voltage signals, and a third for high frequeny signals to the one turn oil. The outer ondutors of the oaxial ables were all onneted to a ommon ground. Smaller twisted pair wiring was added for low frequeny signals without taking up muh additional spae inside the holder. A onnetor box was attahed to the top of the dip probe by means of a brass flange that was hard soldered to the tube [see Fig. 7.(b)]. The box was not hermetially sealed, so openings around the tube inside the box were overed with some plasti sealant to minimize moisture from going down the tubing. To minimize the possibility of refletions from mismathed impedanes, the oaxial able was kept in one ontinuous straight piee until it onneted with adapters attahed to the box. To provide a ontrolled d magneti field, I prepared a small multiturn oil from opper magnet wire and attahed it to the iruit board on the opposite side of the ground plane. Current to the multiturn oil was supplied by a Hewlett Pakard 330A funtion generator and sent down the probe to the oil through twisted pair wiring in the dip probe [56]. This oil was inadequate for produing high frequeny magneti fields due to the shielding from the ground plane, and I only used it oasionally for applying low frequeny magneti fields to the SQUID [57]. More often, I usually just rotated the dip probe in the ambient field to hange the stati magneti flux through the SQUID. 98

219 To test a SQUID hip, I attahed it to the iruit board using photoresist and then wire bonded gold leads diretly between the SQUID hip and the iruit board SMA onnetors. A ground onnetion was also made between the hip and the opper ground plane (see Fig. 7.3). The onnetions were stable, resilient to repeated thermal yling, and did not show any noise or aging problems. After wiring the SQUID hip and installing the iruit board inside the holder, I wrapped the holder with aluminium foil to provide some rf shielding. I also plaed 50 Ω terminating resistors to the lines on the onnetor box at room temperature to protet the SQUID from possible damage due to eletrostati disharge. Finally, the dip probe is inserted into the CMSH-60 LHe dewar. Insertion of the dip probe was best performed slowly to allow the SQUID and dip probe to reah equilibrium with the temperature inside the dewar, thereby saving LHe and reduing thermal stress and breakage. 7.3 External Feedbak Flux Control As mentioned earlier, one of the three oaxial ables in the dip probe was onneted to the one turn oil around the d SQUID. For most of the experiments, the oil was used to apply mirowave signals to the SQUID, while an adjustable quasi stati flux was provided by rotating the dip probe in the ambient magneti field. Rotations of the dip probe were mehanially stable enough to be performed by hand. On the other hand, if the LHe dewar was disturbed, the hange in flux by rotating the dip probe would not be reversible. So, are was taken to not touh or move the dewar during experiments. 99

220 The probe and LHe dewar did not provide magneti shielding, so the SQUID was tested in the Earth s magneti field of about T. For SQUID BH, whih has outer dimensions of 30 µm 30 µm with a 0 µm 0 µm hole, the effetive area is approximately the geometri mean of the outer area and the area of the hole, i.e. 30 µm 0 µm [4,3,94]. The maximum possible flux hange by rotating the dip probe should then be Φ ± T m = ±7.3Φ o, whih is suffiient to generate multiple osillations of the SQUID ritial urrent, but not too muh as to require overly fine ontrol of the angle. In pratie, I observed between 3 and 4 full osillations of the SQUID ritial urrent when I rotated the SQUID by 90, suggesting that the loal field orientation was not perpendiular to the SQUID. Using this information, I an estimate the hange in flux δφ due to a rotation of the dip probe. The smallest rotation δθ of the dip probe I was able to perform was a fration of a degree. For δθ 0.5, one finds 0.03Φ o < δφ < 0.06Φ o. As δφ orresponded to the minimum step size of an adjustable pseudo stati flux, δφ also orresponded to the one shot flux resolution of a ritial urrent detetion sheme, desribed in setion 6.3, for pulsed SQUID sampling. 7.4 SQUID and Pulse Signal Charaterization After ooling the SQUID, I proeeded by measuring its quasi stati I-V harateristis. These measurements also revealed whether the SQUID was operating orretly, how muh ritial urrent modulation was ourring, what the mutual 00

221 indutane of the one turn oil was, and how muh external magneti flux there was through the SQUID. Next, I onfigured the SQUID and dip probe for measurements with short bias urrent pulses. For this, I used the iruit shown in Fig The iruit is similar to the one I used for I-V measurements, exept that 50 Ω terminators were added to all the oaxial lines at room temperature. Current pulses were produed by a Stanford Researh DG535 digital pulse generator whih was onneted in series with a 0 kω resistor [58]. The resistor, housed in an aluminium box separate from the dip probe, onverted the voltage signal into a urrent signal, before feeding it to the d SQUID through oaxial able. The generator had a speified pulse rise time between ns and 3 ns, limiting the minimum pulse length to around 4 ns. If the pulse length was set to shorter than 4 ns, the pulse height did not reah the generator set value and distortions ourred in its profile. I measured the pulse profile from the generator by observing the output from the pulse generator aross a 50 Ω terminator; this signal traveled about m to the dip probe and bak (see Fig. 7.4). A Tektronix 465B osillosope with 400 MHz bandwidth was used to observe the signal [59]. Figure 7.5 shows the general profile of a 0. µs to µs long pulse. The pulse shows some ringing, a positive peak at the front, a negative peak at the tail, and a rise time that is signifiantly longer than 3 ns. This profile hanges if the measuring iruit hanges. For example, with the addition of 50 Ω terminators, peaks disappear and the rise time dereases signifiantly. This suggests that the effetive pulse height is affeted by the iruit and an be signifiantly higher than the height set on the 0

222 Pulse Generator I p 50 Ω 5 pf 0 kω V out 50 Ω 50 Ω 5 pf I Φ 5 pf kω 50 Ω Signal Generator FIG Configuration of the experimental apparatus for testing pulsed SQUID sampling with a hystereti d SQUID. The d SQUID at bottom left is eletrially onneted to the pulse generator and osillosope through a 50 Ω oaxial able and a 0 kω urrent limiting resistor. The magneti field generating oil is onneted to a signal generator and osillosope through a 50 Ω oaxial able and a kω urrent limiting resistor. The 50 Ω terminators are used to minimize refletions. 0

223 50 ns 40 µa 05 µa 0. µs to µs 40 µa FIG Profile of 0. µs to µs long, 00 µa high urrent pulses from a DG535 pulse generator measured through the large bandwidth dip probe. The middle setion of the pulse is elongated depending on the length of the pulse. 03

224 pulse generator. Consequently, there ould be disrepanies between the optimum pulse height that triggers the d SQUID and the expeted value determined from I-V harateristis. Furthermore, if triggering was aused by the brief peak in the pulse, this ould produe a muh shorter effetive sampling time for the SQUID than the nominal pulse length would indiate. Moreover, due to the limited bandwidth of the osillosope, there ould be additional struture in the pulse profile that is not disernible. Figures 7.6 and 7.7 show the voltage response of a hystereti d SQUID to bias urrent pulses with different pulse lengths. The applied magneti flux is set suh that pulses always trigger the SQUID. The SQUID voltage response suggests that the effetive pulse lengths are the same as the generator set values, despite a possible shorter sampling time due to a peak in the profile. This result ould be understood based on retrapping. A narrow peak at the front of a urrent pulse may have aused the SQUID to trigger, but if the equilibrium pulse height stays above the retrapping urrent, a voltage signal will ontinue to be observed. On the other hand, other results that I present later suggest that the effetive pulse length is dereased under ertain irumstanes. The voltage responses show a rise time on the order of 5 ns. This is about the same as the rise time of the pulse generator, though it should be muh shorter sine it is based on the SQUID voltage rise time τ rise. The bandwidth of the 465B osillosope may be limiting the measurement. Additionally, a relatively flat pulse profile at maximum height is seen for pulses longer than 0 ns. As pulses shorten below 5 ns, the shape of the voltage response hanges notieably. In partiular, the 04

225 (a) V (mv) 0 40 t (µs) 0 00 t (ns) (b) V (mv) 0 40 t (µs) 0 00 t (ns) () V (mv) t (µs) 0 00 t (ns) FIG Osillosope trae of the voltage response of a hystereti d SQUID iruit to bias urrent pulses. The applied magneti flux and pulse height were set so that the SQUID triggered at every pulse. Upper urves show train of pulses 0 µs apart, and bottom urves show a single pulse with expanded time sale. Dotted urves trae out the main pulse. Traes show the responses to (a) 00 ns, (b) 50 ns, and () 0 ns pulses. 05

226 (a) V (mv) t (µs) 0 00 t (ns) (b) V (mv) t (µs) 0 00 t (ns) FIG Osillosope trae of the voltage response of a hystereti d SQUID iruit to bias urrent pulses shorter than 0 ns. To the best of my knowledge, the applied magneti flux and pulse height were set so that the SQUID triggered at every pulse. Upper urves show train of pulses 0 µs apart, and bottom urves show a single pulse with expanded time sale. Dotted urves trae out the main pulse. Traes show the responses to (a) 5 ns and (b).5 ns pulses. The.5 ns pulse response is barely visible. 06

227 pulse shape beomes more triangular and the height dereases signifiantly for lengths shorter than 4 ns. For example, Fig. 7.7(b) shows the SQUID voltage response to a urrent pulse that was set to.5 ns on the generator. In this ase, a roughly triangular output voltage signal with a signifiantly dereased amplitude is barely notieable. Voltage responses to pulses set shorter than ns on the pulse generator were indistinguishable from noise. Given that the voltage response signals are all due to triggering of the SQUID to the gap voltage /e, the output pulse heights should be the same regardless of pulse length. This may naively seem to be in ontradition with the observed results for the 4 ns and shorter pulses. If, on the other hand, the.5 ns pulse is being filtered by the wiring, utoff by the limited rise time, or rolled off by the osillosope s response, the voltage response may not be visible on the osillosope. In addition, there is also the possibility that the bias urrent pulse at the SQUID was not high enough to ause triggering, and some of the voltage response observed at the osillosope was due to diret pikup of the urrent pulse by the voltage leads. 7.5 Ciruit Model of Dip Probe and d SQUID 7.5. Transmission Line Model with SQUID as Voltage Soure In order to better understand the harateristis of the pulse signals, I analyzed a iruit model of the eletrial setup [see Fig. 7.8(a)]. In the model, the dip probe and onneting oaxial able are represented as one ontinuous transmission line. I have also inluded parasiti indutane between the oaxial ables and the d SQUID; the approximately m long wire bonded leads to the SQUID have 07

228 (a) V 0 kω 50 Ω L 50 Ω 50 Ω L R z 50 Ω Z s (b) V 50 Ω 50 Ω L L R z 50 Ω V 0.005V Z s () V 50 Ω 50 Ω Z(ω ) (d) V V 0.005V 50 Ω V 0.005V V s 50 Ω FIG (a) Ciruit model of hystereti d SQUID and large bandwidth dip probe. (b) Simplified iruit of (a) using Thévenin equivalent voltage soure and input impedane. Output signal oaxial able is redued to mathed load R z. () Ciruit (b) with equivalent output impedane Z(ω) seen by the oaxial able. (d) Equivalent d iruit of (b) with SQUID modeled as a voltage soure V s. 08

229 indutane, and the high frequeny Fourier omponents in the pulse signals make the impedane of the leads signifiant. For a m lead, the indutane is L ~ µ o l 3 nh using dimensional analysis. Another order of magnitude estimation omes from the expression for the impedane of a transmission line, Z = L C (7.) where Z and C are, for example, those of a oaxial able with inner ondutor dimensions similar to the wire bonded leads [60]. Using values for UT-34-SS able, L = H/m, so for a m lead at GHz, the impedane Z l = 4.9 Ω. These values are very rough but muh more signifiant than the ontat resistane of the leads, whih an now be safely ignored for high frequenies. The oaxial able in Fig. 7.8(a) was assumed lossless with a uniform impedane of 50 Ω. Despite terminators, there were still some refletions due to impedane mismath in the real iruit. Thus, I expet some disrepanies between the experimentally observed output and the results from the iruit model. The largest impedane mismath is between the oaxial able and the SQUID hip. Sine the d SQUID an go from a virtual short to a high impedane, this is unavoidable. Clearly, better impedane mathing ould further inrease the bandwidth. To model the d SQUID, I treated it as a short when in the zero voltage state and as a voltage soure when in the resistive state. The voltage soure approximation an be used beause when the SQUID is in the resistive state, its voltage is almost onstant at the gap voltage /e (see Figs..0, 3.9, and 3.0). This holds as long as 09

230 the bias urrent stays above the retrapping urrent but below the region of Ohmi behavior Frequeny Domain Analysis of Ciruit and Current Pulse To analyze the iruit of Fig. 7.8(a) with the SQUID in the zero voltage state, I first redue it to its Thévenin equivalent iruit in the frequeny domain [6,6]. Near the pulse generator, the 50 Ω terminator onneted to the 0 kω resistor results in a generator with a redued output voltage V 0.005V and an output impedane of about 50 Ω. The result is an output impedane that is pratially mathed at the generator end. Near the osillosope, the line is terminated with a mathing 50 Ω resistor R z, whih results in no refletion and the line looking like a 50 Ω resistor [see Fig. 7.8(b)]. This result an then be used to simplify the load impedane presented to the old end of the oaxial able that delivers the urrent pulse. Treating the d SQUID as a short, the load impedane Z is given by Z = jωl + jωl Z Z + jωl (7.) where Z = 50 Ω is the able impedane [see Fig. 7.8()]. The impedane Z seen from the generator end is given by Z = Z Z Z + + jz tan jz tan θ θ = Z Z Z + + jz tan ωlz jz tan ωlz C C (7.3) where C is the apaitane per unit length of the oaxial able, and l is the total length of the able that delivers the urrent pulse to the SQUID [63]. 0

231 The transmitted voltage V t aross Z is given by ( ) = + = = V Z Z Z V TV V t Γ (7.4) where the inident voltage V + is given by ( ) ( )( ) V sin jz os Z Z Z Z Z Z V = + θ θ. (7.5) So, ( )( ) V Z Z Ze V sin jz os Z Z Z ZZ V j t + = + + = θ θ θ. (7.6) Now, the output voltage V out on the osillosope is the voltage aross the 50 Ω resistor R z given by ( )( ) ( ) ( ) V e Z L L j Z L L Z L V e Z L j L j Z L j Z L j Z L j Z Z L j V e Z Z L j Z V Z Z Ze Z L j Z V Z L j V j j j j t out + + = = + = + = = θ θ θ θ ω ω ω ω ω ω ω ω ω ω ω ω. (7.7) Using the approximation L = L L, ( ) + = + L Z Z L j 3 V e V e Z L j LZ 3 LZ V j j out ω ω ω ω ω θ θ. (7.8) The magnitude and phase φ of V out are given by ( ) = = + + = + = L Z Z L 3 tan C lz V Arg L Z Z L 7 V L Z Z L 9 V V out out ω ω ω φ ω ω ω ω. (7.9)

232 This frequeny dependent output an be applied to the Fourier omponents of an input voltage signal. In other words, the output response to a square input pulse is obtained by first alulating the Fourier omponents of an input pulse using the Fourier transform, then applying the omponents to Eq. (7.8), and finally reombining the results in a sum. The Fourier integral an be written as V jωt = e dt π ( ω ) V () t (7.0) where V(t) is the input signal [64]. For a square pulse with a pulse length of τ and pulse height V in entered around t = 0, V(t) is given by V () t = V 0 in τ for - t otherwise τ. (7.) Thus, V ( ω ) = π τ ωτ j ωτ j jωt Vin e e Vin ωτ Vin e dt = = sin. (7.) π jω πω τ To alulate the voltage response V out using V(ω), I make a disrete approximation of V(ω), as in a Fourier series [65]. The even symmetry in V(ω) allows the disrete Fourier omponents to be written as V V πω ωτ jωt jωt in ( t) = V ( ω ) e + V ( ω ) e = V ( ω ) os ωt sin os ωt ω =. (7.3) Combining Eq. (7.3) in omplex number form with Eq. (7.8), the output signal Fourier omponents an be expressed as

233 V out = = jθ e Vin ωτ sin e ωl Z πω 3 + j Z L ω ωτ sin 0.0Vin jω e πω ωl Z 3 + j Z L ω jωt ( t lz C ). (7.4) Reognizing that t = t lz C is just the delay in time of the output voltage signal with respet to the input signal, I sum the results V out (t ) of Eq. (7.4) at different frequenies to determine the total output voltage signal. The sum is given by V out π T N in () t = os ( ωt φ ) n 0.0V πω ωτ sin ωl 7 + Z Z + ωl ω (7.5) where T = πn/ω max, ω = ω max n/n, ω max is the maximum angular frequeny in the sum, N is the number of omponents summed, and φ ω is the frequeny dependent phase shift given by ωl Z φ ω = tan. (7.6) 3 Z ωl Simplifying Eq. (7.5), I find V out 0.0V π N in () t = n nω maxτ nω sin os N N ω max L NZ n n max t φ ω NZ + ω max L (7.7) where T/ < t < T/ due to the disrete bandwidth limited approximation of the Fourier transform. However, the time range of interest is τ < t < τ whih is when the input voltage signal reahes the SQUID. 3

234 The bandwidth of the input pulse Fourier spetrum was hosen by plotting the magnitude of Eq. (7.4) with respet to frequeny and then determining whih frequenies produe the largest signal magnitudes and onsequently the majority of the output response [see Fig. 7.9(a)]. Figure 7.9(a) shows that the output response is onentrated between MHz and GHz for τ = 5 ns and L =.9 nh. Taking into onsideration the number of Fourier omponents to sum and the need for high frequeny detail, I hose 00 disrete omponents between 0 MHz to GHz in intervals of 0 MHz for the analysis of Eq. (7.7). Regarding the d omponent of V out, the iruit of Fig. 7.8(b) redues to that of Fig. 7.8(d) where the SQUID voltage V s is given by V s e = 0 e when SQUID is in resistive state with positive urrent when SQUID is in zero voltage state when SQUID is in resistive state with negative urrent. (7.8) From Eq. (7.8) and Fig. 7.8(d), it an be dedued that while the SQUID is in the zero voltage state, the d omponent of V out = 0. So, the d omponent does not ontribute any signal when the SQUID does not trigger Simulated Zero Voltage and Resistive State Responses Figure 7.9(b) shows the output voltage for t = 6 ns to 6 ns when the SQUID does not trigger for different values of L/L o where L o =.38 nh with τ = 5 ns. Voltages are plotted at 0.5 ns intervals and show a positive peak at the onset of the pulse and a negative peak at its tail. Variations in L/L o from to 9 show hanges in 4

235 (a) V (pv) Frequeny (Hz) 0.6 (b) V (mv) L/L o = 9 L/L o = 7 L/L o = 5 L/L o = 3 L/L o = t (ns).0.5 FIG (a) Plot of alulated spetrum of the output voltage response of the SQUID iruit using the large bandwidth dip probe. The input voltage is a V, 5 ns square pulse. The indutors in the SQUID iruit are L = L = L =.9 nh. (b) Plots of the output voltage response of the SQUID iruit with different parasiti indutane L. The input voltage is a V, 5 ns square pulse. L o =.38 nh. Results are shown for the d SQUID remaining in the zero voltage state. The voltage response appears after a delay t with respet to the input pulse entered around t = 0, where t = t t. 5

236 the pulse shape as well as an inrease in the peaks. Inreases in the pulse length τ only separate the time between the peaks. The peaks are just a bak emf reation of the indutors L and L to the urrent pulse. To model the output voltage response for the ase when the SQUID triggers into the resistive state, I replae the SQUID with a voltage soure and inlude the d omponent of V out in Eq. (7.5). To aomplish this, the d omponent in the frequeny domain must be properly alulated from Eq. (7.): V in in ( ω = 0) = lim sin = = Vin ω 0 V πω ωτ V πω ωτ τ π. (7.9) This omponent is added to the summation inside Eq. (7.5) with the multipliation fator of about to ompensate for the redued output voltage due to the 50 Ω terminator near the pulse generator. The urrent through the SQUID due to this d voltage is added to the SQUID urrent resulting from the a omponents. The latter urrent is I s in () t = sin ( ωt φ ) = π T N n 0.0V πl in 0.0V πω L N n N nω max ωl 7 + Z ω max L NZ ωτ sin n 4 Z + ωl + 7 n nω maxτ nω sin sin N N The urrent due to the d omponent is given by max t φ NZ + ω max L ω ω. (7.0) I s = V ( ω = 0) Z V s = π T τ V π Z in V s = τ V T Z in V s. (7.) 6

237 Whether the SQUID triggers into the resistive state is determined by omparing the urrent through the SQUID with V s = 0 to the SQUID ritial urrent I. If the bias urrent is larger, then the SQUID triggers into the resistive state and the gap voltage V s = /e =.7 mv appears. This gap voltage is maintained as long as the SQUID bias urrent stays above the retrapping urrent I r. In turn, the output voltage response of the iruit is modified by the addition of the SQUID voltage V s to ΣV out of Eq. (7.5) and (7.7). One the SQUID triggers into the resistive state, the d urrent through the SQUID hanges aording to Eq. (7.). The new bias urrent and SQUID voltage state need to be self onsistent. Triggering and the appearane of V s 0 are simulated onditionally in the analysis of the SQUID iruit. This is performed by first alulating the output voltage response and the SQUID bias urrent for both ases, i.e. when the SQUID is in the zero voltage state and when it is in the resistive state, at eah time step in the simulation. Then, with the SQUID initially in the zero voltage state, I ompare the SQUID bias urrent with the ritial urrent and retrapping urrent I r at eah subsequent time step to determine whih state the SQUID is in, taking into aount whih state the SQUID was during the previous time step. Figure 7.0(a) shows the alulated output voltage responses to a.7 V, 5 ns pulse from the generator with L =.9 nh. The different urves show what happens when the SQUID ritial urrent varies from 0 µa to 70 µa, whih is the range of urrents for whih SQUID BH was observed to trigger (see Fig. 3.9 or Table I). For omparison, Fig. 7.0(b) shows the.7 V, 5 ns input voltage pulse used. 7

238 (a) V (mv) 4 3 I = 0 µa I = 40 µa I = 60 µa I = 65 µa I = 68 µa I = 70 µa t (ns) (b) V (V) t (ns) FIG (a) Plots of the alulated output voltage responses of the SQUID iruit using the large bandwidth dip probe for different SQUID ritial urrents. The d SQUID is modeled as a bias urrent dependent voltage soure. The indutors in the SQUID iruit are L = L = L =.9 nh. (b) Plot of the.7 V, 5 ns input voltage pulse to the SQUID iruit that generated the voltage responses alulated in (a). The voltage response appears after a delay t with respet to the input pulse entered around t = 0, where t = t t. 8

239 The alulated voltage response shows that after a delay time of lz C ~ 0 ns, the onset of triggering ours earlier and earlier as the ritial urrent dereases below I = 70 µa. In partiular, for I = 70 µa, the SQUID does not trigger and the response is just that found previously when the SQUID was modeled as a short. The plots also show that the onset of triggering is very sensitive to the ritial urrent near I = 70 µa but beomes less sensitive as I approahes 0 µa. I also found that in order to observe triggering behavior for ritial urrents between I = 0 µa and 70 µa, the pulse amplitude has to be between. V and.7 V, whih averages to.45 V. I note that from Fig. 7.5, the peak of the observed input urrent pulse orresponds to about 40 µa for a pulse set at 00 µa. This suggests that if a pulse struture similar to that of Fig. 7.5 ontinues to persist for pulses as short as 5 ns, the optimal input voltage pulse height setting should be around.04 V instead of.45 V. A signifiant assumption made in my model is that the SQUID voltage and urrent hange instantaneously to the new settings at the onset of triggering. Considering the short voltage rise time aross the SQUID, the approximation an be justified for the voltage. However, the sudden hange in bias urrent ould be in question. As long as the hange in bias urrent is small, the assumption ould still be justified. Yet, this ondition is not assured and is the main flaw in the model. In setion 7.7, I will ompare the results of this model diretly to the measured response. Other models of the d SQUID were also investigated. See for example Appendix E. 9

240 7.6 Setting Short Bias Current Pulses After installing SQUID BH in the large bandwidth dip probe and performing an I-V measurement, I investigated what the optimal input pulse height was for pulsed SQUID sampling with the given iruit. I proeeded to do this by first finding the lowest bias urrent pulse that would still trigger the SQUID and then the highest pulse that would sometimes not. I found the lower limit by sending urrent pulses to the SQUID while rotating the dip probe in the ambient magneti field, whih hanged the ritial urrent of the SQUID. If no voltage signal was observed during the rotation, this meant that the urrent pulse height was less than the minimum SQUID ritial urrent and would not trigger the SQUID into the resistive state. I then inreased the pulse height and again heked for any voltage response as the dip min probe was rotated. The lowest urrent pulse height I p when a voltage signal began to appear while rotating the dip probe identified the lower limit. Similarly, I found max the maximum pulse height I p by identifying the minimum pulse height for whih the urrent pulses always triggered the SQUID despite rotating the dip probe. * The optimal pulse height I p orresponds to the SQUID ritial urrent whih produes the largest sensitivity in ritial urrent per hange in flux. For example, on the ritial urrent versus flux urve in Fig..4(b), it orresponds to the points with the steepest slope on the urve. Assuming a smooth transition of the ritial urrent min max min max from I p to I p, the average between I p and I p should be near the optimal value, * min max i.e. I p (I p + I p )/. min max * For SQUID BH, I p = 6 µa, I p = µa, and thus I p = 9 µa. These urrent values were determined from I p = V in /R g where V in is the voltage setting of the 0

241 pulse generator and R g = 0 kω. In ontrast, from I-V urve measurements, the ritial urrent for SQUID BH ranged between 3 µa and 7 µa. Thus, the atual min max ritial urrent minimum and maximum were larger than I p and I p by 55±5 µa or 40±0%. The likely explanation is that the effetive urrent pulse height at the SQUID is higher than the pseudo-d level inferred from the generator setting. This is onsistent with my observations of the signal pulse height, disussed earlier. In any ase, for flux sensing measurements using SQUID BH, I set pulses to the average * pulse height I p = 9 µa by setting 0.9 V at the pulse generator. I also note that if the pulse profile has peaks similar to those shown in Fig. 7.5, the average pulse height would be around.04 V rather than around.45 V. The experimentally obtained result of 0.9 V is loser to the.04 V predition expeted if ringing is present. This also suggests that there will be fine struture peaks in the pulse profile that are shorter than the pulse length itself. Thus, if there are peaks, they should have durations on the order of ns or less, whih would not be resolved by the 400 MHz Tektronix 465B osillosope. 7.7 Detetion of Mirowave Frequeny Magneti Fields using Pulsed SQUIDs 7.7. SQUID Response to Low Frequeny Signals Finally, to test the pulsed SQUID sampling tehnique, I applied a time varying magneti field on the hystereti d SQUID while ontinually pulsing its bias urrent. If a urrent pulse did not result in a voltage aross the SQUID, this indiated that the modulated ritial urrent was higher than the urrent pulse height I p*. By

242 * augmenting the magneti field, the ritial urrent will eventually drop below I p and trigger a voltage signal aross the d SQUID. I began tests by applying a low frequeny triangular wave field. The signals were produed by a Hewlett Pakard 330A funtion generator in series with a kω urrent limiting resistor whih sent urrent down the dip probe to the one turn oil on the SQUID hip [56]. I hose a triangular wave as it generates pieewise linear variations in magneti flux. The frequeny of the waves was khz to 0 khz. For the urrent pulses, I set the DG535 pulse generator to apply 9 µa, 00 ns bias urrent pulses with a pulse repetition rate between 00 khz and MHz [58]. Figure 7. shows the SQUID response to a khz triangular wave magneti field. The top part of the osillosope piture shows the voltage response from the SQUID iruit. The bottom part represents the magneti field produed by the one turn oil. Eah period of the magneti field produes a lustered region of triggering events in the SQUID output. This is onsistent with the ambient magneti field produing an offset in the applied magneti field. The orrelation between the magneti flux signal and the voltage response signal indiates that the SQUID is triggering on the applied flux. Figures 7. and 7.3 show the SQUID response to a 0 khz triangular wave magneti field with inreasing magnitude. One expets that as the wave magnitude inreases from zero, a voltage response from SQUID triggering would emerge with the same periodiity as the wave. Then with further inreases in amplitude, the lustered triggering regions in the osillosope piture should expand linearly as more pulses per wave period trigger the SQUID. In addition, individual triggering

243 V (mv) Triggering No Triggering t (ms) Ciruit Output Voltage I (µa) t (ms) Applied Magneti Flux FIG. 7.. Voltage response of a hystereti d SQUID to pulsed bias urrent and triangular wave flux signal. Current pulses were 00 ns long with a 00 khz pulse repetition rate. A triangular wave urrent signal with an amplitude of 0 µa and an approximate frequeny of khz was sent through an external oil to produe the applied magneti flux. Clustered regions where the SQUID is triggering in step with the flux wave are visible.

244 (a) (b) () (d) FIG. 7.. Voltage response of SQUID BH to modulating flux with inreasing amplitude. The 9 µa input urrent pulses were 00 ns long and had a pulse repetition rate of MHz. The applied 0 khz triangular wave magneti field was produed by sending urrent through the one turn oil with amplitudes of (a) 400µA, (b) 500µA, () 600µA, and (d) 700µA. 4

245 (a) (b) () FIG Voltage response of SQUID BH showing irregularities to modulating flux with inreasing amplitude. The 9 µa input urrent pulses were 00 ns long and had a pulse repetition rate of MHz. The applied 0 khz triangular wave magneti field was produed by sending urrent through the one turn oil with urrent amplitudes of (a) 800µA, (b) 88µA, and () 850µA. Onset of irregularities seen in (b). 5

246 regions would be symmetri about some point due to the symmetry of the triangular wave. This is seen in Fig. 7. whih shows the symmetri SQUID triggering regions entered around the peaks of the triangular wave, as expeted in the low frequeny limit. It was also expeted that as the flux wave amplitude inreases by more than 3Φ o /4, holes in the lustered SQUID triggering region would appear and grow linearly. That is, the SQUID stops triggering when the ritial urrent exeeds the urrent pulse height, again due to the Φ o periodiity of the response to applied flux. If the wave amplitude inreased further, a small voltage triggering region should appear inside the holes and evolve like the larger lusters. Subsequent features in the SQUID voltage response would be a repeat of this pattern as the flux wave amplitude ontinued to inrease. However, the SQUID voltage response did not exatly follow the expeted behavior. The first hole in the triggering response appeared when the urrent through the one turn oil was 88 µa [see Fig. 7.3(b)]. On the other hand, the position and symmetry of the response suddenly and dramatially hanged at the same time. A sudden hange in the response ourred again at an even higher wave amplitude [see Fig. 7.3()]. For omparison, I substituted the triangular wave flux signal with a sinusoidal signal. Figure 7.4 shows the resulting SQUID response, whih was muh more stable. Abrupt hanges in the response disappeared, and the triggering regions ontinued to be symmetri and entered around the wave peaks. In Fig. 7.4(a), the sinusoidal urrent through the SQUID hip s one turn oil, seen in the bottom half of 6

247 (a) V (mv) 0 I (ma) t (µs) t (µs) (b) V (mv) 0 I (ma) t (µs) t (µs) FIG Voltage response of SQUID BH triggering to a sinusoidally modulating flux. The output voltage responses are seen in the upper portions of the osillosope pitures. The lower portions show the 500 µa amplitude sinusoidal urrent through the one turn oil on the SQUID hip used to generate the magneti flux. The 9 µa input urrent pulses for the bias urrent were 0 ns long and had a pulse repetition rate of MHz. The frequenies of the urrent generating the applied magneti flux were (a) 0 khz and (b) 00 khz. 7

248 the osillosope piture, had an amplitude of 500 µa and a frequeny of 0 khz. The frequeny of the sinusoidal urrent in Fig. 7.4(b) was 00 khz. Bias urrent pulses for these measurements were only 0 ns long with a pulse repetition rate of MHz. The µs intervals between urrent pulses an be seen in the iruit voltage response in the top half of Fig. 7.4(b). Individual pulses are too tightly lustered to be distinguished in Fig. 7.4(a). Although abrupt hanges in the voltage response did not our, neither did I observe growing holes in the SQUID triggering region even with sinusoidal waves. In fat, the voltage response stopped hanging altogether beyond a ertain flux amplitude. There are several possible auses of this nonideal behavior. One possible explanation is that magneti flux is being trapped in the SQUID Josephson juntions at high field, and this is bloking further modulation from ourring. Another possibility ould be due to magneti hysteresis in the SQUID, as the value of its modulation parameter β.3 is larger than /π. With inreasing flux amplitude, the total magneti flux through the SQUID hole ould have gone through hanges that inlude disontinuity in ritial urrent modulation. This last possibility, however, is not supported by other related observations. More likely, the urrent in the one turn oil started affeting the superondutivity of the SQUID when the urrent level beame too high. The lose proximity of the oil with the SQUID loop may have aused part of the loop to go normal, either by heating or by exeeding the ritial field. It is also likely that there was some noise or irregularity in the urrent from the funtion generator. 8

249 Another simple explanation ould be that the expeted holes are too small and are thus masked by noise. The hange in magneti flux through the SQUID hole an be estimated from the mutual indutane of ph, obtained in setion 3.3, between the on hip one turn oil and the SQUID. A hange of 00 µa in the urrent through the one turn oil should result in an approximate hange of 0.6Φ o through the SQUID. A hange of about 3Φ o /4 should have already brought about the appearane of a hole in the lustered triggering regions. However, I did not observe the emergene of holes during a hange of 400 µa in the urrent through the one turn oil (see Fig. 7.). Noise in the pulsing frequeny ombined with insuffiient temporal resolution in the osillosope ould have masked the appearane of holes. Without more investigation and simulation, the ause of why there are no growing holes in the lustered triggering regions and of other irregularities remains undetermined. In any ase, despite the limit to small flux values at low frequenies, the test results learly demonstrate the feasibility of the pulsed SQUID sampling tehnique MHz Signal Response Figure 7.4(b) shows the possibility of deteting signals with frequenies as high as 00 khz using the pulsed SQUID sampling tehnique. This orresponds to a bandwidth that is already higher than what is found in ommerially available SQUID mirosopes. To test the tehnique with even higher frequeny magneti fields, I used a mirowave generator to supply urrent to the one turn oil on the SQUID hip. The mirowave generator was a Hewlett Pakard 8373B signal 9

250 generator that ould produe 0 MHz to 0 GHz mirowaves [66]. For bias urrent pulses, I seleted the shortest reliable pulse length of 5 ns using the DG535 pulse generator. With shorter pulses, the pulse profile was very distorted as disussed in setion 7.4. These urrent pulses were sent to SQUID BH at the maximum pulse rate of MHz. Elementary onsiderations shows that with 5 ns pulses, the highest frequeny signal that an be followed is one with a period of 0 ns, whih orresponds to a frequeny of 00 MHz. Aordingly, I sent mirowave signals with frequenies up to 00 MHz to the one turn oil. In order to observe the onditional triggering of the d SQUID at different delays with respet to the signal, I employed a variation of the synhronous mode signal following sheme of setion 6.3. The idea was to pulse the SQUID at different time delays with respet to the mirowave signal. First, the osillosope was synhronized, i.e. triggered with respet to the bias urrent pulses. The mirowave flux signal was sent to the one turn oil without any synhronization, neither with the urrent pulses nor with the osillosope. When a urrent pulse arrived at the SQUID, it would be at some arbitrary phase with respet to the flux signal. If subsequent pulses arrived at slightly different relative phases, the SQUID's response to the flux signal ould be mapped out. To ensure that pulses arrived at a phase that was shifted with respet to the previous phase between the pulse and flux signal, I inorporated a very small differene in frequeny between the 00 MHz mirowave signal and an integer multiple of the MHz pulse repetition rate. A differene of one part in 0 8 was suffiient. Due to the frequeny differene, the relative phase between the pulses and 30

251 mirowave signal slipped by a small fixed phase after every pulse. Therefore, eah suessive pulse polled the flux signal at a different relative phase with time, whih was the desired result. Figures 7.5, 7.6, and 7.7 show the response of SQUID BH to the 00 MHz flux signal. The top parts of the osillosope pitures show the flux signal. The middle part shows the SQUID iruit s output voltage response. The bottom part shows the input voltage pulse signal. The delay between the input voltage pulse and the output voltage response is partly due to a short ( m) oaxial able between the pulse generator and osillosope and a long (5 m) oaxial able between the dip probe and the osillosope. Figure 7.5(a) shows SQUID BH triggering near the onset of the bias urrent pulse, and Fig. 7.5(b) shows the response when SQUID BH is not triggering. Figure 7.6 shows the intermediary ase when the urrent pulse arrives at the SQUID at a relative phase between the two phases of Fig In Fig. 7.5, simulation results presented in setion 7.5 are superimposed on the osillosope traes of the voltage responses for omparison. The height of the simulated voltage responses are saled to that of a V in =.7 V input voltage pulse, whereas the atual pulse was set to 0.9 V. The simulated pulse length was τ = 5 ns, and the iruit model indutors were set to L = L =.9 nh. The disrepanies between the simulation and experimental results are due to distortions in the input voltage pulse as well as refletions from impedane mismathes. Figure 7.6 shows multiple transitions from the zero voltage state to the resistive state, similar to the simulated results of Fig. 7.0(a). Noise in either the applied flux or the pulse frequeny, in onjuntion with the limited temporal 3

252 (a) I (ma) 0. 0 V (mv) 0 30 t (ns) Magneti Flux 0 V (V) t (ns) t (ns) Voltage Response Input Voltage Signal (b) I (ma) 0. 0 V (mv) 0 30 t (ns) Magneti Flux 0 V (V) t (ns) t (ns) Voltage Response Input Voltage Signal FIG Voltage responses of SQUID BH to 0.9 V, 5ns input voltage pulses and an applied magneti flux signal at different delays, in the large bandwidth dip probe. The applied flux is produed by a 00 µa amplitude, 00 MHz urrent signal through the one turn oil of the SQUID hip. The responses show SQUID BH (a) triggering near the onset of the bias urrent pulse and (b) not triggering. For omparison, simulation results for a.7 V input pulse are superimposed over the voltage response traes, while the input voltage height is saled to 0.9 V. The voltage response is delayed with respet to the input pulse by sending it through a long oaxial able. 3

253 I (ma) 0. 0 V (mv) 0 30 t (ns) Magneti Flux 0 V (V) t (ns) t (ns) Voltage Response Input Voltage Signal FIG Voltage response of SQUID BH to 0.9 V, 5ns input voltage pulses and an applied magneti flux signal at different delays showing multiple transitions from the zero voltage state to the resistive state, in the large bandwidth dip probe. The applied flux is produed by a 00 µa amplitude, 00 MHz urrent signal through the one turn oil of the SQUID hip. The voltage response shows SQUID BH triggering at different relative delays due to noise in either the applied flux or the pulse frequeny. The voltage response is delayed with respet to the input pulse by sending it through a long oaxial able. 33

254 (a) (e) t (ns) t (ns) (b) (f) t (ns) t (ns) () (g) t (ns) t (ns) (d) (h) t (ns) t (ns) FIG Series of osillosope traes showing the progression of the voltage response of SQUID BH to 0.9 V, 5 ns input voltage pulses and an applied 00 MHz magneti flux signal with varying delay, in the large bandwidth dip probe. Vertial markers indiate the onset of the input voltage pulse and the voltage response. The relative delay between the applied flux signal and the onset of the input voltage pulse an be traked by noting the points at whih the flux signal rosses the markers. The relative delays are approximately (a) 0 ns, (b).5 ns, ().5 ns, (d) 3.0 ns, (e) 4.5 ns, (f) 6.5 ns, (g) 8.5 ns, and (h) 9 ns. 34

255 resolution of the osillosope image apturing system, aused triggering at different delay times to be observed in the same osillosope piture. Figure 7.7 shows the progress of the voltage response as the urrent pulse arrives with suessively later delays. A gradual hange in the output voltage response is seen going from Fig. 7.7(a) to Fig. 7.7(f) as the relative phase between the bias urrent pulse and flux signal hanges. The progression of the voltage response resembles the hange in output voltage response obtained from simulations, espeially when the SQUID ritial urrent inreases in Fig. 7.0(a). As the relative phase between the bias urrent pulse and flux signal inreases, the output voltage response progresses from Fig. 7.7(f) to Fig. 7.7(h), eventually returning to the original no-triggering state shown in Fig. 7.7(a). The total shift in the flux signal from Fig. 7.7(a) to Fig. 7.7(h) orresponds to one full period of 0 ns. I an onlude that the progression of the output voltage response in Fig. 7.7 is due to the gradual hange in SQUID ritial urrent aused by the 00 MHz flux signal. Figure 7.7(a) shows SQUID BH not triggering to bias urrent pulses due to high ritial urrent. Figure 7.7(f) shows triggering of SQUID BH near the onset of the bias urrent pulse due to low ritial urrent. Between these points, the intensities of the voltage response urves suggest that the SQUID triggers only part of the time. This behavior is understood as the SQUID being triggered for only a fration of the times it is pulsed and at different delays with respet to the onset of the pulse. Noise and other fators an trigger the SQUID into the resistive state if the differene between urrent pulse height and modulated ritial urrent is suffiiently small. As the differene between the pulse height and ritial urrent inreases above the noise 35

256 level, it beomes more likely that the SQUID will be in one state rather than the other. In order to obtain the results shown in Figs. 7.5, 7.6, and 7.7, I had to adjust the a and d magneti fluxes applied to SQUID BH. To perform this, I first set the mirowave signal generator to a signal amplitude of 00 mv. The pseudo-d urrent amplitude orresponded to 00 µa with a flux amplitude of about 0.6Φ o applied to the SQUID. I alulated the flux amplitude using the mutual indutane of ph, obtained in setion 3.3 with measurements on SQUID AN. However, this did not take into aount possible attenuation of the urrent before it reahed the oil. So, the flux value was only roughly known and ould have been and probably was somewhat smaller. The applied d magneti flux had to be adjusted so that the a flux from the one turn oil would modulate the SQUID ritial urrent suh that the SQUID would trigger about half of the time it was pulsed and not the rest. With an a flux amplitude of 0.6Φ o, this should have automatially been satisfied. However, this was not the ase suggesting that the urrent to the one turn oil was indeed attenuated. I adjusted the d flux by rotating the dip probe, as disussed in setion 7.3. Conditional triggering in response to the a flux was found easily, needing only a small rotation of the dip probe Unexpeted Phenomena in the Voltage Response On a minor point, if I applied large flux signals to the one turn oil, I notied signifiant indutive effets in the system. Figure 7.8 shows the result when instead 36

257 (a) Voltage Response Magneti Flux (b) Magneti Flux Voltage Response Input Voltage Signal () Magneti Flux Voltage Response Input Voltage Signal FIG Osillosope traes showing oupling between the urrent to the one turn oil for the applied magneti flux and the output voltage response signal of the SQUID iruit. The frequeny of the applied flux is 56 MHz for (a) and MHz for (b) and (). Undulations in the voltage response are indued at the same frequeny as the flux signal. As the flux frequeny inreases, there is an inrease in undulation amplitude aompanied by a hange in relative phase between the voltage response and the flux signal. Conditional triggering of SQUID BH by bias urrent pulses and applied flux is seen in (b) and (). The triggering is not affeted by the oupling. 37

258 of 00 mv, the mirowave signal amplitude was set to 500 mv; I observed a small amplitude a voltage signal with the same frequeny as the mirowave signal at the SQUID iruit s output voltage. When the frequeny of the mirowave was inreased from MHz to around 56 MHz, the amplitude of the a voltage signal inreased and was aompanied by a phase shift [see Fig. 7.8(a)]. Sine the SQUID was not triggering for most of the yle, this was a definite sign of indutive pikup between the flux lines or one turn oil and the bias urrent or SQUID voltage lines. Reduing the flux amplitude redued the effet on the output voltage. Although undesirable and like the thermoeletri emf effet desribed in setion 3.3, indutive pikup had minimal impat on magneti flux detetion. In partiular, the response of the d SQUID to bias urrent pulses appeared to be unaffeted. Figures 7.8(b) and 7.8() show the voltage responses to a MHz flux signal induing an a voltage in the output voltage signal while SQUID triggering and no SQUID triggering were being observed, respetively. It is interesting to note that this result is inonsistent with the a bias voltage phenomena mentioned in setion 3., and suggests that nonlinear effets in the SQUID iruit may not be very detrimental. A more interesting phenomenon was observed when urrents with frequenies higher than 00 MHz were sent through the one turn oil. Despite the nominal 5 ns pulse length of the bias urrent, when higher frequeny signals were sent through the oil, SQUID BH ontinued to respond to the flux with onditional triggering. Beyond some limiting frequeny, the SQUID should have been triggering ontinuously as during every urrent pulse, the modulated ritial urrent will at 38

259 some point be below the pulse height. However, this did not our. Instead, the SQUID ontinued to trigger in orrelation with the flux signal well beyond the expeted limiting frequeny of 00 MHz. Due to the limited bandwidth of the Tektronix 465B osillosope, the orrelation between the flux signal and the voltage response ould not be verified beyond 380 MHz, though the voltage response ontinued to show signs of onditional triggering. This unexpeted phenomenon ould be due to an effetive pulse length that is shorter than the generator set value. The ability of the SQUID to learly follow signifiantly higher frequeny signals strongly suggests a shorter pulse length. The result is also onsistent with a pulse profile with peaked strutures produed by the pulse generator, as mentioned in setions 7.4 and 7.6. If I had found a limiting frequeny above 00 MHz where onditional triggering failed, this would have onfirmed the hypothesis. But, I was unable to find suh a limiting frequeny up to the 400 MHz bandwidth of the osillosope. If the hypothesis is orret, this null result suggests that the effetive pulse length is shorter than.3 ns. Further investigation with better diagnosti equipment would be helpful in resolving the issue. 7.8 GHz Measurements and Extensions to Larger Bandwidth Using my apparatus, Matthews and Vlahaos also performed experiments on SQUID BH and deteted magneti fields with frequenies higher than 00 MHz using a variation of my pulsed SQUID sampling tehnique [3]. The tehnique they used is desribed in setion 6.3 and involves finding the distribution of SQUID 39

260 triggering events from multiple measurements. By identifying the transitional point where half of the bias urrent pulses results in triggering and the other half does not, they ould follow magneti flux signals applied to the SQUID. Ultimately, they were suessful in following a GHz mirowave magneti flux signal applied on SQUID BH (see Fig. 7.9). The GHz measurement is remarkable beause Matthews and Vlahaos used 0 ns bias urrent pulses for the measurement. That is, within the duration of a pulse, the flux signal would have aused 0 osillations of the SQUID ritial urrent. Naively, the SQUID should have triggered with every pulse and thus not be able to follow the flux signal. Given that the GHz signal was observed, Matthews et al. also onluded that the effetive bias urrent pulse is muh shorter than the generator set value of 0 ns. A hint to the ause and understanding of the GHz result may lie in the behavior of the noise in the measurement. Examination of Fig. 7.9 shows that dereasing edges of the GHz signal ontain more noise, partiularly a broader distribution of triggering events, than the inreasing edges. Naively, one would expet the noise to be more evenly distributed between both edges if it were due to noise in the mirowave flux signal or in the 0 ps delays used to synhronize the urrent pulses []. On the other hand, if there were signifiant subnanoseond struture to the pulses, the distribution of triggering events would reflet the atual pulse shape. In any ase, these experiments learly demonstrate that the tehnique of pulsed SQUID sampling an be implemented and extended to GHz bandwidth. The 40

261 ΦA (mφo) Counts Time (ns) FIG Two dimensional histogram of SQUID BH triggering events in the presene of a GHz magneti field. There were 0 4 triggering events per data point. The SQUID bias urrent was pulsed with 0 ns pulses. A d feedbak field was applied to the SQUID and inremented after eah data point at the same delay. The vertial axis is the applied flux through the SQUID. The horizontal axis is the time delay in 0 ps steps. The purple urve shows where half of the pulses resulted in triggering. The green urve represents the best sinusoidal fit to the purple urve. (Plot ourtesy of J. Matthews.)